EFFECT OF IMPERFECT DIRECT SIMPLE SHEAR TEST ...

EFFECT OF IMPERFECT DIRECT SIMPLE SHEAR TEST

BOUNDARY CONDITIONS ON MONOTONIC AND

CYCLIC MEASUREMENTS

by

Man Ho Daniel Wai

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Civil and Mineral Engineering

University of Toronto

© Copyright by Man Ho Daniel Wai 2019

ii

Effect of Imperfect Direct Simple Shear Test Boundary Conditions on

Monotonic and Cyclic Measurements

Man Ho Daniel Wai

Master of Applied Science

Graduate Department of Civil and Mineral Engineering

University of Toronto

2019

Abstract

The direct simple shear (DSS) test is commonly used to assess shear strength of soil, estimate

liquefaction resistance, or calibrate constitutive models. However, constitutive models are

calibrated assuming ideal simple shear stress conditions even though this is not achieved in the

DSS test. Near frictionless vertical boundaries cannot develop the complementary shear stresses

necessary for ideal simple shear conditions. Difficulties in maintaining constant height during

undrained tests violates the constant volume assumption. The top cap holding the top of the

specimen can rock (rotate) during shear also violating the assumed perfect simple shear conditions.

Parametric studies were conducted to investigate the effect of contact friction between the soil and

vertical boundaries, vertical compliance and top cap rocking on the results of undrained DSS tests.

The study simulates an industry-standard SGI type simple shear device using a 3-D finite element

model and an advanced plasticity model to capture soil behaviour.

iii

Acknowledgments

I would like to thank Rocscience Inc. for providing financial and technical support during the

study.

I acknowledge the support of the Natural Sciences and Engineering Research Council of Canada

(NSERC). I would like to thank NSERC for Grant# 401267058 in support of the study.

I would like to thank my supervisor Dr. Mason Ghafghazi for guiding me through my studies.

I would like to thank my fellow graduate students Sartaj Gill, Wyatt Handspiker, Edouardine

Ingabire, Faraz Goodarzi, Wei Liu, Fredric Fong, Mathan Manmatharajan, Mohammad Mozaffari,

Khashayar Nikoonejad, and Emmanuel Sarantonis for their support throughout my studies. Special

acknowledgement to Sartaj Gill, Wei Liu, Khashayar Nikoonejad, and Mathan Manmatharajan for

providing experimental data that is used in this study.

iv

Table of Contents

Acknowledgments.......................................................................................................................... iii

Table of Contents ........................................................................................................................... iv

List of Tables ................................................................................................................................ vii

List of Figures .............................................................................................................................. viii

Chapter 1 Introduction ..............................................................................................................1

1.1 General Remarks ..................................................................................................................1

1.2 Direct simple shear device ...................................................................................................4

1.3 Objective and Scope ............................................................................................................5

1.4 Sign Convention...................................................................................................................6

1.5 Organization .........................................................................................................................6

Chapter 2 Literature Review.....................................................................................................8

2.1 Introduction ..........................................................................................................................8

2.2 Effect of Soil-Ring Friction .................................................................................................9

2.3 Effect of Vertical Compliance on Undrained Tests ...........................................................10

2.4 Effect of Top Cap Rocking ................................................................................................11

2.5 Summary ............................................................................................................................12

Chapter 3 Numerical Analysis of Direct Simple Shear Test Boundary Conditions ...............14

3.1 Introduction ........................................................................................................................14

3.2 Soil Constitutive Model .....................................................................................................14

3.3 Calibration..........................................................................................................................15

3.4 Finite Element Model ........................................................................................................19

3.4.1 Geometry................................................................................................................19

3.4.2 Soil-Ring Interaction ..............................................................................................20

3.4.3 Top Cap ..................................................................................................................20

v

3.4.4 Loading and Boundary Conditions ........................................................................21

3.4.5 Ideal Simple Shear Conditions ..............................................................................21

3.4.6 Mesh sensitivity analysis .......................................................................................22

3.5 Effect of Soil-Ring Friction ...............................................................................................25

3.5.1 Introduction ............................................................................................................25

3.5.2 Results and Discussion ..........................................................................................26

3.6 Effect of Vertical Compliance on Undrained Tests ...........................................................40

3.6.1 Introduction ............................................................................................................40


3.7 Effect of Top Cap Rocking During Shear..........................................................................54

3.7.1 Introduction ............................................................................................................54


3.8 Conclusions ........................................................................................................................62

3.8.1 Effect of Soil-Ring Friction ...................................................................................62

3.8.2 Effect of Vertical Compliance on Undrained Tests ...............................................63

3.8.3 Effect of Top Cap Rocking During Shear..............................................................63

Chapter 4 Summary and Conclusions ....................................................................................65

4.1 Summary and Conclusions ................................................................................................65

4.1.1 Effect of Soil-Ring Friction ...................................................................................65

4.1.2 Effect of Vertical Compliance on Undrained Tests ...............................................66

4.1.3 Effect of Top Cap Rocking During Shear..............................................................66

4.1.4 Recommendations for Future Work.......................................................................66

References ......................................................................................................................................68

Appendix A Dafalias and Manzari (2004) Model Calibration Results ......................................71

A.1 Calibration of Undrained Monotonic Simple Shear Behaviour: experimental data and

Dafalias and Manzari (2004) model simulations ...............................................................72

vi

A.2 Calibration of Undrained Cyclic Simple Shear Behaviour: experimental data and

Dafalias and Manzari (2004) model simulations ...............................................................76

A.3 Calibration of Undrained Cyclic Simple Shear Behaviour: Dafalias and Manzari

(2004) model simulations varying cyclic stress ratios (CSR) to generate (CSR) vs.

number of cycles to liquefaction plot (Figure 3.4) ............................................................81

A.4 Calibration of Undrained Cyclic Simple Shear Behaviour: other experimental data to

generate cyclic stress ratio (CSR) vs. number of cycles to liquefaction plot (Figure

3.4) .....................................................................................................................................85

Appendix B Mesh Sensitivity Analysis Results .........................................................................97

B.1 Frictionless Vertical Boundaries, Constant Height and No Top Cap Rocking..................98

B.2 Effect of Soil-Ring Friction .............................................................................................100

B.3 Effect of Vertical Compliance on Undrained Tests .........................................................101

B.4 Effect of Top cap Rocking During Shear ........................................................................102

Appendix C Simulation Results ...............................................................................................103

C.1 Effect of Soil-Ring Friction .............................................................................................104

C.1.1 Consolidation .......................................................................................................104

C.1.2 Undrained Monotonic Simple Shear ....................................................................106

C.1.3 Undrained Cyclic Simple Shear ...........................................................................110

C.2 Effect of Vertical Compliance on Undrained Tests .........................................................116



C.3 Effect of Top Cap Rocking During Shear........................................................................121



Dafalias and Manzari (2004) Constitutive Model UMAT File ...........................127

D.1 Source Code .....................................................................................................................128

D.2 UMAT File.......................................................................................................................128

vii

List of Tables

Table 3.1. Dafalias and Manzari (2004) model calibrated parameters ......................................... 16

Table 3.2: Total computation time for undrained monotonic and cyclic simulations using

different mesh densities and element types .................................................................................. 25

Table 3.3: Percent change of monotonic test results from higher maximum vertical strain for

loose specimen (Dr = 24%) at peak stress (4% shear strain) ........................................................ 45


dense specimen (Dr = 146%) at 4% shear strain ........................................................................... 49

viii

List of Figures

Figure 1.1: Typical SGI simple shear test boundaries .................................................................... 2

Figure 1.2: Simple shear stress state ............................................................................................... 2

Figure 1.3: SGI type direct simple shear device that is used in this study. .................................... 5

Figure 1.4: Sign convention for key stress and strain components in this study ............................ 6

Figure 3.1: Illustration of the yield, critical state, dilatancy and bounding lines in deviator stress

(q), mean effective stress (p) space (from Dafalias and Manzari, 2004) ...................................... 15

Figure 3.2: Sample calibrated undrained monotonic simple shear test ........................................ 17

Figure 3.3: Sample calibrated undrained cyclic simple shear test ................................................ 18

Figure 3.4: Cyclic stress ratio (CSR) vs. number of cycles to liquefaction for soil relative density

(Dr) of 77% and liquefaction criterion defined by single amplitude shear strain (Liq) ................ 19

Figure 3.5: Finite element model of the DSS test ......................................................................... 20

Figure 3.6: Specimen mesh densities that were analyzed in the mesh sensitivity analysis .......... 23

Figure 3.7: Undrained monotonic test results of loose specimen (Dr = 24%) varying element size

(ES) and element type for frictionless vertical boundaries, constant height and no top cap rocking

....................................................................................................................................................... 24

Figure 3.8: Undrained cyclic test results of loose specimen (Dr = 24%) varying element size (ES)

and element type for frictionless vertical boundaries, constant height and no top cap rocking ... 24

Figure 3.9: Friction on a soil specimen that is encased in a rubber membrane and is confined by

Teflon coated rings ....................................................................................................................... 26

Figure 3.10: End of consolidation stress distribution of loose specimen (Dr = 24%) simulations

for the centre cross section of the specimen (z = 0 mm), varying soil-ring friction angle (soil-ring)

....................................................................................................................................................... 27

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Figure 3.11: End of consolidation vertical effective stress contours of loose specimen (Dr = 24%)

simulation for no slip condition, soil-ring friction angle of 30° ................................................... 27

Figure 3.12: Coefficient of lateral earth pressure (K) during consolidation at the core of loose

specimen (Dr = 24%) simulations for varying soil-ring friction angles (soil-ring)......................... 28

Figure 3.13. Stress contours of loose specimen (Dr = 24%) simulation at 4% shear strain for

frictionless vertical boundaries, soil-ring friction angle of 0° ...................................................... 29

Figure 3.14. Stress distribution of loose specimen (Dr = 24%) simulations at peak shear stress

(4% shear strain) for the centre cross section of the specimen (z = 0 mm), varying soil-ring

friction angle (soil-ring) .................................................................................................................. 29

Figure 3.15: Effective normal stress results for monotonic tests of loose specimen (Dr = 24%) for

varying soil-ring friction angles (soil-ring) ..................................................................................... 30

Figure 3.16. Monotonic test results of loose specimen (Dr = 24%) for varying soil-ring friction

angles (soil-ring) ............................................................................................................................. 31

Figure 3.17. Monotonic test results of dense specimen (Dr = 146%) for varying soil-ring friction

angle (soil-ring) ............................................................................................................................... 31


angles (soil-ring) during shearing phase only ................................................................................. 32

Figure 3.19. Stress contours of dense specimen (Dr = 146%) simulation at the peak amplitude of

the last cycle for frictionless vertical boundaries, soil-ring friction angle of 0° ........................... 34

Figure 3.20. Stress distribution of dense specimen (Dr = 146%) simulations for the centre cross

section of the specimen (z = 0 mm) during the peak amplitude of the (a) first and (b) last cycle

for varying soil-ring friction angles (soil-ring) ............................................................................... 35

Figure 3.21: Normal stress results for cyclic tests of dense specimen (Dr = 146%) for varying

soil-ring friction angles (soil-ring) .................................................................................................. 36

x

Figure 3.22. Cyclic test results of loose specimen (Dr = 24%) for varying soil-ring friction angles

(soil-ring) ......................................................................................................................................... 37

Figure 3.23. Cyclic test results of dense specimen (Dr = 146%) for varying soil-ring friction

angles (soil-ring) ............................................................................................................................. 38


(soil-ring) during shearing phase only ............................................................................................ 40

Figure 3.25: Vertical displacement vs. applied vertical stress for SGI type device that was loaded

without soil specimen ................................................................................................................... 41

Figure 3.26: Stress and strain distribution of loose specimen (Dr = 24%) simulations at peak

shear stress (4% shear strain) for the centre cross section of the specimen (z = 0 mm), varying

maximum vertical strain (y,max) during the shearing phase .......................................................... 43

Figure 3.27: Monotonic test results of loose specimen (Dr = 24%) for varying maximum vertical

strain (y,max) during the shearing phase ........................................................................................ 45

Figure 3.28: Reproduced undrained monotonic test results of Ottawa sand on the effect of

vertical compliance on undrained tests from Zekkos et al. (2018) ............................................... 46

Figure 3.29: Reproduced undrained monotonic test results of pea gravel on the effect of vertical

compliance on undrained tests from Zekkos et al. (2018) ............................................................ 46

Figure 3.30: Vertical displacement vs. applied vertical stress for large-size cyclic simple shear

device developed by Geocomp (2019) without soil specimen ..................................................... 46

Figure 3.31: Monotonic test results of dense specimen (Dr = 146%) for varying maximum

vertical strain (y,max) during the shearing phase ........................................................................... 48

Figure 3.32: Reproduced undrained monotonic test results of dense offshore sand on the effect of

vertical compliance on undrained tests from Dyvik and Suzuki (2018) ....................................... 49

Figure 3.33: Reproduced example of offset in specimen height for Kaolin specimen using an

active height control system from Dyvik and Suzuki (2018) ....................................................... 49

xi

Figure 3.34: Stress-strain response for undrained cyclic simple shear simulations of loose

specimen (Dr = 24%) for varying maximum vertical strain (y,max) during the shearing phase.... 51

Figure 3.35: Stress-strain response for undrained cyclic simple shear simulations of dense

specimen (Dr = 146%) for varying maximum vertical strain (y,max) during the shearing phase .. 52

Figure 3.36: Stress-strain response of cyclic tests for dense specimen (Dr = 146%) using two

approaches to model constant height ............................................................................................ 53

Figure 3.37: Stress and strain distribution of dense specimen (Dr = 146%) simulations for the

centre cross section of the specimen (z = 0 mm) during the peak amplitude of the last cycle using

two approaches to model constant height ..................................................................................... 54

Figure 3.38: Potential top cap rocking that is influenced by loading shaft in SGI type device .... 55

Figure 3.39. Stress and strain distribution of loose specimen (Dr = 24%) simulations at peak

shear stress (4% shear strain) for the centre cross section of the specimen (z = 0 mm), varying

maximum top cap rotation angle (,max) ....................................................................................... 57

Figure 3.40: Monotonic test results of loose specimen (Dr = 24%) for varying maximum top cap

rotation angle (,max) .................................................................................................................... 58

Figure 3.41: Monotonic test results of dense specimen (Dr = 146%) for varying maximum top

cap rotation angle (,max) .............................................................................................................. 58

Figure 3.42: Stress and strain distribution of loose specimen (Dr = 24%) simulations during peak

amplitudes (peak) of the first cycle for the centre cross section of the specimen (z = 0 mm),

varying maximum top cap rotation angle (,max) ......................................................................... 60


specimen (Dr = 24%) for varying maximum top cap rotation angle (,max) ............................... 61


specimen (Dr = 146%) for varying maximum top cap rotation angle (,max) .............................. 62

1

Chapter 1

Introduction

1.1 General Remarks

Numerical modelling is becoming increasingly popular in analyzing geotechnical problems due to

constant improvements in computational ability and the industry’s move towards more

sophisticated and deformation based design methods. Simulations can consider complex site

conditions and soil behaviour using finite element analysis (FEA) software such as RS3

(Rocscience Inc., 2019) or FLAC3D (ITASCA Consulting Group Inc., 2019). These software

packages require advanced constitutive models to capture soil stress-strain response in a realistic

manner. Numerical modelling cannot obtain useful results without accurate calibration of

constitutive models against laboratory tests. It becomes essential that laboratory tests reflect or at

least closely approximate assumed stress conditions when deriving model parameters.

The direct simple shear (DSS) test is commonly used to calibrate constitutive models for soil under

simple shear loading. Specimens are consolidated by a vertical stress before a shear load is applied

to the base pedestal, as shown in Figure 1.1(a). Figure 1.1(b) presents a photo of a typical soil

specimen in an SGI type DSS device, which uses a stack of rings and a membrane to laterally

contain the soil sample, after shearing. The SGI type device is described in detail later. To apply

ideal simple shear conditions, the boundaries must allow for development of the shear stresses

shown in Figure 1.2(a). During shearing, opposite sides must remain parallel. DSS tests

accomplish this by allowing rotation (tilting) of the walls (vertical boundaries) during shear

loading while ensuring no rotation of the top cap or base pedestal. Vertical boundaries must also

enforce constant cross-sectional area. To conduct undrained tests, pore pressure developments are

equated to the change in total vertical stress if constant volume conditions are maintained (Dyvik

et al., 1987). This has the benefit of measuring pore pressures without the need to submerge the

specimen in water. With lateral strains confined by the vertical boundaries, the constant volume

condition is achieved by enforcing no height change during the shearing phase.

2

Figure 1.1: Typical SGI simple shear test boundaries

Figure 1.2: Simple shear stress state

Many DSS testing apparatuses have been developed over the years. As described by

Kjellman (1951), the Royal Swedish Geotechnical Institute (SGI) device was first built in 1936

and applied simple shear loading on a cylindrical soil sample, as shown in Figure 1.1(b). The

sample is contained in a rubber membrane that is laterally confined by a stack of stiff friction-less

rings to ensure no change in diameter. The Norwegian Geotechnical Institute (NGI) device was a

modification of the SGI device that achieved lateral confinement using a wire reinforced rubber

membrane instead of stacked rings (Bjerrum and Landva, 1966). The Cambridge device applied

simple shear loading on a cuboid soil sample that was laterally confined by hinged vertical stiff

plates (Roscoe, 1953). Instead of stiff lateral boundaries that enforced constant cross-sectional

area, the University of Western Australia (UWA)(Mao and Fahey, 2003) and the University of

California at Berkeley (Villet et al., 1985) used devices that applied constant total lateral stress by

conducting the DSS test on a cylindrical specimen within a pressure cell. Multi-directional DSS

testing devices were developed in later years that could apply bi-directional shear loading and

employed similar lateral boundaries as previously mentioned (Ishihara and Yamazaki, 1980;

Boulanger et al., 1993; DeGroot et al., 1993; Duku et al., 2007). For undrained tests, the constant

height condition is achieved either using an active or passive height control system

(Dyvik and Suzuki, 2018). An active control system adjusts the specimen height to maintain a

Teflon coated rings Vertical stress Top cap

Shear stress Base pedestal

(a) Schematic drawing (b) Soil specimen in SGI type device from this study

Teflon coated rings

Base pedestal

Top cap

Rubber

membrane

Before shearing After shearing Before shearing After shearing

(a) Ideal (b) DSS test

y

z

x

3

constant height and measures the induced changes in vertical stress during the shearing phase. A

passive control system mechanically locks the top cap in place using a stiff frame during the

shearing phase. DSS devices that laterally confine soil specimens to ensure no change in cross-

sectional area can use either system as per ASTM D6528-17 standards. As per section 11.3.2 of

ASTM D6528-17 Standard Test Method for Consolidated Undrained Direct Simple Shear Testing

of Fine Grain Soils, the active or passive height control system “must maintain the specimen height

after accounting for apparatus compressibility to within 0.05% of its pre-shear value”.

Current DSS testing apparatuses cannot impose ideal simple shear conditions which is assumed

for the calibration of constitutive models. DSS tests typically impose near frictionless vertical

boundaries due to the inability of the stacked rings and membrane setup to transfer the vertical

friction force. The vertical frictionless boundary is necessary to avoid stress shedding during the

consolidation phase. However, as illustrated in Figure 1.2(b), the near-frictionless vertical

boundaries do not allow development of complementary shear stresses necessary for ideal simple

shear conditions (Roscoe, 1953; Lucks et al., 1972; Saada and Townsend, 1981; Budhu, 1984;

Airey et al., 1985; DeGroot et al., 1994). Lacking complementary frictional forces in the simple

shear test creates a mechanically impossible stress field for a single element meaning that either

the element will have rotational acceleration, or a Mohr circle cannot describe its static

equilibrium. Additionally, active and passive height control systems can minimize but not prevent

vertical strain during undrained tests which violates the constant volume condition and may affect

stress-strain response (Bro et al., 2013; Dyvik and Suzuki, 2018; Zekkos et al., 2018). The amount

of vertical strain during shear is also known as the vertical compliance for undrained tests.

Minimizing top cap rocking (rotation) during shearing is also a concern where the effect on stress-

strain response has not been well studied (Ishihara and Yamazaki, 1980; Boulanger et al., 1993;

Rutherford, 2012; Kwan et al., 2014; Shafiee, 2016). Top cap rocking violates the assumption of

perfect shear deformation and may induce stress non-uniformities near the specimen edges.

4

Experimental and numerical studies have been conducted to study imperfect boundary effects and

estimate the development of stress non-uniformities. Experimental studies suggest that stress non-

uniformities in DSS devices may cause unreliable stress and strain measurements that cannot be

used to approximate real soil response under ideal simple shear conditions (Budhu, 1984;

DeGroot et al., 1994). Numerical studies provided a more in-depth examination of stress and strain

distribution throughout the specimen in a DSS device during shear (Roscoe, 1953;

Lucks et al., 1972; Shen et al., 1978; Saada and Townsend, 1981; Doherty and Fahey, 2011,

Wijewickreme et al., 2013). In numerical studies, the assumed contact behaviour between soil and

vertical boundaries is unclear and the simulations do not appear to consider the effect of vertical

compliance and top cap rocking during the shearing phase. The effect of soil-ring friction on DSS

testing has not been well studied and may impact the calibration of constitutive models. Higher

contact friction allows better development of complementary shear stresses but may induce stress

non-uniformities during the consolidation phase. Bernhardt et al. (2016) showed from a study of

steel spheres in a DSS apparatus that higher steel spheres to ring friction resulted in more strain

hardening and dilative response. Their results suggest that higher soil-ring friction can potentially

overestimate shear strength properties. Numerical studies are not found that studied the effect of

vertical compliance on stress-strain response of undrained tests. Experimental data shows that

vertical strain during shear within the ASTM D6528-17 tolerance of 0.05% may increase or reduce

the measured shear strength (Bro et al., 2013; Dyvik and Suzuki, 2018; Zekkos et al., 2018). The

effect of top cap rocking (rotation) on stress-strain response is not well studied. Ishihara and

Yamazaki (1980) proposed that top cap rocking during experimental tests may have induced

vertical stress concentrations near the edges of the specimen.

1.2 Direct simple shear device

Figure 1.3 presents the SGI type direct simple shear device developed by Geocomp (2018) that is

used in this study. As mentioned previously, the cylindrical soil specimen is encased in a rubber

membrane and is laterally confined by stiff Teflon coated rings. The specimen is vertically

confined by the top cap and base pedestal. Grooved aluminum porous stones are placed at the top

cap and base pedestal to prevent slippage at the top and bottom of the specimen. A loading shaft

connects to the top cap to apply vertical loading. The vertical strain is measured above the loading

shaft by an LVDT.

5

Figure 1.3: SGI type direct simple shear device that is used in this study.

1.3 Objective and Scope

This dissertation summarizes an investigation into the effect of imperfect boundary conditions in

an SGI type device on the calibration of constitutive models. Figure 1.3 presents the SGI type

device that is used in this study. A numerical study is conducted using three-dimensional (3D)

finite element analysis (FEA) of undrained monotonic and cyclic simple shear tests using the

6

software ABAQUS (Dassault Systèmes, 2012). The soil stress-strain behaviour is captured using

the Dafalias and Manzari (2004) constitutive model for its ability to capture sand plasticity during

both monotonic and cyclic shear loading. The simulation models the stacked rings confining a

typical cylindrical soil specimen. The model is calibrated using triaxial and simple shear test data

of a poorly graded medium sand. Parametric studies are performed to investigate the effect of soil-

ring friction, vertical compliance on undrained tests and top cap rocking during shear for various

soil densities. The effect of soil-ring friction is analyzed by simulations using the minimum and

maximum expected soil-ring friction in the laboratory as well as the soil-ring friction necessary to

achieve ideal simple shear conditions. The effect of vertical compliance is analyzed using

simulations with various vertical strain tolerances during the shearing phase to induce unwanted

volume change in undrained tests. The effect of top cap rocking is analyzed using simulations with

various allowed top cap rotations during the shearing phase. Results from the numerical study are

meant to provide insight on the magnitudes of stress and strain non-uniformities due to imperfect

boundary conditions rather than to precisely commute values for design purposes.

1.4 Sign Convention

The study adopts the compression positive sign convention. Positive directions of shear stress and

rotation are shown in Figure 1.4.

Figure 1.4: Sign convention for key stress and strain components in this study

1.5 Organization

The thesis is organized into four chapters as follows.

Chapter 1 presented the introduction that describes the study topic, objectives and scope, sign

convention and chapter organization.

y

x or z (+)

Rotation

Shear

Compression

x (+)

7

Chapter 2 presents the literature review that summarizes the past experimental and numerical

studies on the effect of imperfect boundary conditions in the direct simple shear test on stress-

strain behaviour. The literature review focuses on studies that investigate the effect of soil-ring

friction, vertical compliance on undrained tests and top cap rocking during shear.

Chapter 3 presents the numerical study of an SGI type device on the effect of imperfect boundary

conditions on stress-strain response. Select results are presented in this chapter to avoid redundant

figures. Each section may include simulation results of only loose or dense specimens. However,

all results can be found in the Appendix.

Chapter 4 summarizes the conclusions from this study.

8

Chapter 2

Literature Review

2.1 Introduction

Experimental studies have been conducted on the effect of imperfect boundary conditions on

stress-strain behaviour in the direct simple shear (DSS) test. DeGroot et al. (1994) conducted

undrained simple shear tests on an elastic material and two cohesive soils in a Geonor DSS device.

The results suggest that stress non-uniformities caused by the device decreased vertical stress

which led to decreased measurements of shear resistance. The reduction in vertical stress and shear

resistance worsens with larger shear strain which may have contributed to excessive strain

softening behaviour that does not represent real soil response. Budhu (1984) conducted monotonic

and cyclic strain-controlled tests on dry sand in elaborately instrumented Cambridge and NGI type

devices to investigate stress and strain distributions. Budhu (1984) reported that the flexible

vertical boundaries of the NGI type device produced non-uniform shear deformation during cyclic

loading and did not adequately maintain constant cross-sectional area. Budhu (1984) also reported

that for monotonic loading the Cambridge device could provide reliable results so long as stress

and strain data are taken at the centre of the specimen. During cyclic loading, results from either

device may be unreliable due to stress non-uniformity that increased after each cycle. Experimental

studies confirm the need to consider the soil to apparatus contact behaviour in numerical analysis

of DSS devices to better understand the imperfect boundary effects that exist in real devices.

Numerical solutions have been developed to study boundary effects on stress concentration in

simple shear tests. Roscoe (1953) used a mathematical stress function that qualitatively showed

shear and vertical stress concentrations around the edges of a specimen in a Cambridge device.

The results agreed with a study of boundary effects in the NGI device using 3D linear elastic FEA

by Luck et al. (1972). They reported that approximately 70% of the total area around the specimen

core has a uniform stress distribution. Saada and Townsend (1981) did not agree with the previous

numerical analyses of the Cambridge and NGI device and argued that the assumed boundary

9

conditions, especially at the vertical boundaries, were physically impossible to impose by the

devices. Shen et al. (1978) also conducted 3D linear elastic FEA of the NGI device and provided

recommendations to improve the uniformity of shear strain distribution. Doherty and Fahey (2011)

then modelled the UWA/Berkeley device using 3D non-linear FEA to better capture real soil

response and showed that stress non-uniformities may underestimate shear strength.

Wijewickreme et al. (2013) appears to have modelled the SGI device using Discrete Element

Method (DEM) analysis to access the mobilized friction angle in the soil specimen during shear.

2.2 Effect of Soil-Ring Friction

Numerical analysis is relied on to study the effect of soil-ring friction since there does not currently

exist any tools to measure soil-ring friction in the laboratory. However, there are not many studies

on the effect of soil-ring friction. In many numerical analyses, the assumed contact behaviour

between the soil and vertical boundaries is not clearly defined (Roscoe, 1953; Lucks et al., 1972;

Shen et al., 1978; Saada and Townsend, 1981). The UWA type device that was modelled by

Doherty and Fahey (2011) could not develop significant complementary shear stresses due to its

design. Wijewickreme et al. (2013) appears to have assumed frictionless contact between soil and

vertical boundaries (stiff rings) for the model of the SGI device using Discrete Element Method

(DEM) analysis. Bernhardt et al. (2016) and Chang et al. (2016) considered soil-ring friction in

their study of boundary effects in the DSS device using DEM analyses. The results were interpreted

qualitatively as the simulations were calibrated using DSS tests of steel spheres instead of soil.

Bernhardt et al. (2016) studied the effects of soil-ring friction on stress distribution in the SGI

device using 3D DEM. They showed that higher steel spheres to ring friction resulted in more

strain hardening and dilative response. Their results suggest that higher soil-ring friction can

potentially overestimate shear strength properties. Chang et al. (2016) compared boundary effects

of the SGI and Cambridge type devices using 2D DEM that considered soil to vertical boundary

friction. The results showed no significant differences between SGI and Cambridge type model

results which implies that the effect of soil-ring friction should be similar for both devices.

Bernhardt et al. (2016) and Chang et al. (2016) provided useful insights on the potential effects of

soil-ring friction on stress-strain response, but did not perform parametric studies on the effects of

soil-ring frictions expected in practice or compare them to ideal simple shear conditions. Previous

studies can be improved in the study of contact friction between soil and rings on the stress-strain

response observed.

10

2.3 Effect of Vertical Compliance on Undrained Tests

Experimental research has considered the effect of vertical compliance on stress-strain response

in undrained tests although numerical studies were not found. Many numerical studies appear to

assume constant height during undrained simple shear test simulations (Roscoe, 1953; Lucks et

al., 1972; Shen et al., 1978; Saada and Townsend, 1981; Doherty and Fahey, 2011, Wijewickreme

et al., 2013). Bro et al. (2013) conducted undrained monotonic simple shear tests to study the effect

of strain rate on the strength properties of kaolinite and kaolinite-bentonite clays. The device

employed a passive height control system. While establishing baseline clay strength properties

using a slow strain rate at 50 to 185 kPa vertical consolidation stress, they estimated that vertical

strain of 0.1% to 0.25% measured in 8 out of 10 tests led to 7% increase in shear strength. They

reported that strength increase was related to contraction of the clay, height decrease, as indicated

by the vertical strain direction. Dyvik and Suzuki (2018) conducted undrained monotonic simple

shear tests on soft normally consolidated kaolin clay, stiffer overconsolidated kaolin clay, and

dense offshore sand to study the effect of vertical compliance on undrained tests. An automated

active height control system was employed in an NGI type device to analyze results for vertical

strain tolerance of 0%, 0.05% and 0.1%. Specimens were consolidated to 100 or 200 kPa vertical

stress. The direction of top cap vertical strain during shear followed the contractive or dilative

behaviour of the soil. For an undrained simple shear test, contractive behaviour was identified by

a decrease in effective vertical stress which resulted in height decrease while dilative behaviour

was identified by an increase in effective vertical stress which resulted in height increase. For soft

normally consolidated kaolin clay that exhibited contractive behaviour, decreased specimen height

of 0.05% resulted in increased shear stress by about 2% and increased vertical effective stress by

about 3%. For stiffer overconsolidated clay that exhibited dilative behaviour, increased specimen

height of 0.05% resulted in reduced shear stress by about 2% and reduced vertical effective stress

by about 4%. Dense sand exhibited contractive behaviour at first before transitioning to dilative

behaviour. During dilative behaviour at 5% shear strain, shear stress was reduced by 6% and

vertical effective stress was reduced by 7%. At failure where the soil is transitioning from

contractive to dilative behaviour, the shear stress was increased by more than 20% and the vertical

effective stress was increased by about 30%. Zekkos et al. (2018) considered the effect of vertical

compliance on undrained tests in an SGI type large cyclic direct simple shear device manufactured

by Geocomp (2019) for oversized particles. The performance of an active height control system

was evaluated from undrained monotonic simple shear tests of loose Ottawa sand consolidated to

11

100 kPa vertical stress and loose pea gravel consolidated to 400 kPa vertical stress. Measured

vertical strain varied between 0 and 0.10%. This study did not account for the height changes

allowed by porous stones, which are expected to be larger than the vertical strains that were

measured, so all data likely exceed the ASTM allowed range, and variations were not as significant

as reported. Although the vertical strain during shear measured on tests of Ottawa sand are below

ASTM D6428-17 tolerance of 0.05%, the peak shear stress varied as much as 15%. It was also

reported that the test of pea gravel with measured vertical strain close to ASTM D6428-17

tolerance of 0.05% showed a difference in peak shear stress of 10%. For these tests, higher vertical

strain was shown to increase peak stress and vertical effective stress. Thus, the effect of vertical

compliance on undrained tests cannot be ignored and may have a significant impact on the stress-

strain response. Numerical simulations were not found that studied the effect of vertical

compliance on undrained tests. Numerical simulations can improve the study on the effect of

vertical compliance on undrained tests since they provide more detail than experimental testing on

stress-strain behaviour throughout the specimen.

2.4 Effect of Top Cap Rocking

Numerical studies were not found that investigated the effect of top cap rocking (rotation) on

stress-strain response. Many numerical studies appear to assume no top cap rotation during simple

shear test simulations (Roscoe, 1953; Lucks et al., 1972; Shen et al., 1978; Saada and Townsend,

1981; Doherty and Fahey, 2011; Wijewickreme et al., 2013; Bernhardt et al., 2016; Chang et al.,

2016). However, minimizing top cap rocking is a concern in the design of simple shear devices

(Ishihara and Yamazaki, 1980; Boulanger et al., 1993; Rutherford, 2012; Kwan et al., 2014;

Shafiee, 2016). Ishihara and Yamazaki (1980) used a multi-directional simple shear device to

conduct cyclic tests on saturated sand specimens obtained from the Fuji riverbed. They mentioned

that during cyclic simple shear tests on specimens consolidated to 200 kPa vertical stress, rocking

motions about the horizontal axis may have induced vertical stress differences near the specimen

edge and caused the measured shear strain to be about 2% larger than the actual shear strain applied

to the specimen. However, they do not appear to measure the amount of top cap rotation or the

effect of top cap rocking on stress distribution. Boulanger et al. (1993) conducted undrained cyclic

simple shear tests on saturated sand to evaluate the performance of the University of California at

Berkeley bi-directional simple shear apparatus (UCB-2D). For vertical consolidation stress of 206

kPa and shear stress amplitude of 41.2 kPa, the measured rotation of the specimen based on the

12

vertical strain at the edges was about 0.006°. They noted that the measured rotation was small but

they did not appear to investigate changes in stress-strain response due to the measured rotation.

Kwan et al. (2014) conducted cyclic simple shear tests on uniform Nevada sand using a modified

GCTS cyclic simple shear device. Four loose specimens of about 40% and four dense specimens

of about 70% were tested at a consolidation vertical stress of 100 kPa. It was reported that top cap

rocking caused a difference in vertical displacement of 0.05 mm between the specimen centre and

edge where the max applied shear strain appeared to be about 10%. Based on the reported specimen

dimensions, rocking caused top cap rotation of about 0.03°. They did not appear to report any

changes in stress-strain response due to the measured rotation. Shafiee (2016) conducted undrained

cyclic strain-controlled simple shear tests on dry sand to evaluate the performance of the

University of California at Los Angeles (UCLA) bi-directional broadband simple shear (BB-SS)

device. From the performance data, it appears that for 4% shear strain amplitude and 100 kPa

consolidation vertical stress the maximum rotation angle is about 0.07°. They noted that top cap

rocking caused small vertical strain during shear in comparison to the total average vertical strain

during the tests and did not appear to investigate changes in stress-strain response due to the

measured rotation. These studies provide a range of top cap rotation expected for DSS tests, but

they did not conduct a parametric study on the effect of top cap rocking on stress-strain response.

Higher top cap rotation may induce stress concentrations near the specimen edges. Thus, the effect

of top cap rocking on stress-strain behaviour during the DSS test has not been well studied even

though top cap rocking has been measured during some experimental tests. Numerical studies can

simulate top cap rocking to investigate the effect on stress-strain behaviour for the expected top

cap rotation based on experimental data.

2.5 Summary

Numerical and experimental studies are available in the literature to study the effects of

imperfect boundary conditions on the direct simple shear test. These studies demonstrate that

imperfect boundary conditions develop stress non-uniformities that may cause unreliable stress

and strain measurements. However, the effects of soil-ring friction, vertical compliance on

undrained tests and top cap rocking during shear have not been well studied. Experimental

studies on the effect of vertical compliance on undrained tests suggest that vertical strain induces

unwanted volume change during the shearing phase that can increase shear and vertical stress

during contractive soil behaviour or reduce shear and vertical stress during dilative soil

13

behaviour. Numerical studies are not found that investigate the effect of vertical compliance on

undrained tests. The effect of top cap rocking on stress-strain response has not been studied

although top cap rocking has been measured in some direct simple shear tests. Numerical

analyses that consider imperfect boundary conditions provide information and insights that are

difficult to achieve using experimental studies alone. Thus, the study on the effect of imperfect

boundary conditions in the direct simple shear test can be improved through numerical

simulation of the direct simple shear test.

14

Chapter 3

Numerical Analysis of Direct Simple Shear Test

Boundary Conditions

3.1 Introduction

A numerical study was conducted using three-dimensional (3D) finite element analysis (FEA) to

investigate the effect of imperfect boundary conditions in undrained direct simple shear tests on

stress-strain behaviour. The study investigates the effect of soil-ring friction, vertical compliance

on undrained tests and top cap rocking during shear.

3.2 Soil Constitutive Model

Soil stress-strain behaviour is captured using the Dafalias and Manzari (2004) constitutive model.

The bounding surface model primarily captures sand plasticity due to changes in stress ratio ()

which occur continuously during simple shear tests. Figure 3.1 shows the triaxial space

representation of the formulation. The model uses a thin, open-ended wedge yield surface fixed at

the origin. The model incorporates a rotational hardening parameter () to model sand plasticity

behaviour. The size of the wedge is based on a thickness parameter (m) and defines the region of

elastic behaviour. The thin wedge and rotational hardening parameter are ideal for capturing sand

plasticity during load reversal, such as during cyclic shear loading. The model requires fifteen

parameters that can be categorized according to their use in equations of elasticity, critical state,

yield surface, plastic modulus, dilatancy and fabric-dilatancy. Masin et al. (2018) implemented the

Dafalias and Manzari (2004) model in a user-defined material model (UMAT) file that can be used

in ABAQUS simulations. The UMAT file is publicly available from the database of the

SoilModels project, previously known as the soilmodels.info project, founded by

Gudehus et al. (2008). The UMAT’s numerical stability was improved by adjustments to the

15

convergence criteria and it was verified by comparison to simulation results by Dafalias and

Manzari (2004). The UMAT file is presented in Appendix D.

Figure 3.1: Illustration of the yield, critical state, dilatancy and bounding lines in deviator stress

(q), mean effective stress (p) space (from Dafalias and Manzari, 2004)

3.3 Calibration

Drained triaxial and undrained DSS tests were conducted on poorly graded medium sand using an

SGI type device at University of Toronto. Experimental tests were conducted by other researchers

in the research group. The sand was sourced from the Hutcheson quarry located in Huntsville,

Ontario, Canada. It is referred to as Hutcheson sand (HS). The mineral composition is comprised

of quartz (~30%), potassium feldspar (~30%), amphibole (~15%), plagioclase feldspar (~15%),

biotite (~6%), and other/unknown minerals (~2%) (Gill, 2018). The particle size distribution

median diameter (D50) is 0.46 mm and the uniformity coefficient (Cu) is 2.42. A gradation of 60%

medium sand and 40% fine sand was used in the study. The soil’s maximum and minimum void

ratios are 0.877 and 0.602 respectively. The simulations modelled loose specimens (Dr = 24%)

with a state parameter () of 0.02, and dense specimens (Dr = 146%) with a state parameter () of

0.32. The device uses Teflon coated stacked rings to apply lateral confinement on a 63.5 mm

16

diameter by approximately 20 mm height cylindrical soil sample that is encased in a rubber

membrane. Teflon is a friction reducer that minimizes friction among the rings.

The soil constitutive model was calibrated using experimental data by simulating a single element

model under ideal simple shear conditions which is explained in detail later. Interpretation of

drained triaxial tests established an initial set of parameters using the approach by Taiebat and

Dafalias (2008). Parameters were then adjusted based on undrained monotonic and cyclic simple

shear tests. Initial and adjusted parameters are presented in Table 3.1.

Table 3.1. Dafalias and Manzari (2004) model calibrated parameters

Parameter Type Variable1 Value from

Triaxial Data

Value from Simple

Shear Data

Elasticity Go 75 30

0.2 0.2

Critical state Mtc 1.23 1.23

c 0.715 0.715

c 0.023 0.023

e0 0.81 0.81

0.7 0.7

Yield surface m 0.01 0.01

Plastic modulus h0 8.5 8.5

ch 0.7 0.7

nb 7.0 1.0

Dilatancy A0 0.75 0.34

nd 3.2 3.2

Fabric-dilatancy tensor zmax 4.0 4.0

cz 600 600 1Variables (unit-less) defined by Dafalias and Manzari (2004).

Calibrated undrained monotonic and cyclic simple shear simulation results are presented in

Appendix A from Figure A.1 to Figure A.9. Select results are presented here. Figure 3.2 presents

the (a) shear stress vs. strain and (b) stress path plots for a sample calibrated undrained monotonic

simple shear test. The critical state line (CSL) with the constant volume friction angle ('cv) is also

plotted with the stress path. Figure 3.3 presents plots of (a) shear stress vs. number of cycles, (b)

shear strain vs. number of cycles, (c) vertical stress vs. number of cycles, (d) shear stress vs shear

17

strain and (e) stress path for a sample calibrated undrained cyclic simple shear test. The CSL is

also plotted on the stress path. After 10 cycles, the shear strain sharply increases and stabilizes to

just above 1%, as shown in Figure 3.3(b) and (d). As shown in Figure 3.3(c), the vertical stress

does not continue to decrease after 10 cycles. Instead, the simulation converges to a similar shear

stress to vertical effective stress response, as shown in Figure 3.3(e). Additional cyclic loading

does not generate more shear strain. This locking behaviour after a few cycles is noted by

Boulanger and Ziotopoulou (2017) in their development of a constitutive model based on the

Dafalias and Manzari (2004) formulation. The locking behaviour reduces the single amplitude

maximum shear strain below 3.75% which is a typical criterion for liquefaction during cyclic

loading. As shown in Figure 3.3, the model captures continuous soil plasticity during cyclic

loading such that the general trend in stress-strain response compares well with experimental

results. As such, the locking behaviour does not greatly impact the study on the effect of imperfect

boundary conditions which focuses on stress-strain response prior to liquefaction.

Figure 3.2: Sample calibrated undrained monotonic simple shear test

(a) (b)

18

Figure 3.3: Sample calibrated undrained cyclic simple shear test

The liquefaction criterion was modified to compensate for the locking behaviour. Figure 3.4

presents the plot for cyclic stress ratio (CSR) vs. the number of cycles to liquefaction where the

single amplitude shear strain at which liquefaction occurs was changed from 3.75% to 1%.

Experimental and simulation results to generate this plot are presented in Appendix A from

Figure A.1 to Figure A.25. The cyclic stress ratio (CSR) is the ratio of shear stress to consolidation

vertical stress. As shown in Figure 3.4, experimental results compare well for single amplitude

shear strain liquefaction criteria of 3.75% and 1%. Additionally, the experimental and simulation

results compare well. As such, liquefaction in this study was defined as a single amplitude shear

strain of 1% which corresponds to the approximate maximum shear strain when locking occurs. It

should be noted that the simulations are less sensitive to changes in CSR than is observed from

experimental data, as shown in Figure 3.4. However, the effect of CSR on stress-strain response is

not included in the scope of this study and the locking behaviour described earlier does not impact

(a)

(b)

(c)

(d)

(e)

19

the study as the influence of boundary conditions appears to dominate early cycles as demonstrated

later. As such, the calibrated Dafalias and Manzari (2004) model is sufficient for modelling soil

behaviour to study the effects of imperfect boundary conditions on the stress strain-response.

Figure 3.4: Cyclic stress ratio (CSR) vs. number of cycles to liquefaction for soil relative density

(Dr) of 77% and liquefaction criterion defined by single amplitude shear strain (Liq)

3.4 Finite Element Model

3.4.1 Geometry

The simulations modelled a typical 63.5 mm diameter by 20 mm height soil specimen using the

finite element analysis software (FEA) package ABAQUS (Dassault Systèmes, 2012). The

cylindrical specimen was subjected to unidirectional simple shear loading which resulted in

symmetric stress-strain distribution with respect to a vertical cut at the centre that is parallel to the

shearing direction. The half-cylinder model that is shown in Figure 3.5 took advantage of

symmetry in geometry and loads to improve computation speed by reducing the required number

of elements. 20 rigid rings of 1 mm height around the specimen maintained constant diameter. The

mesh consisted of 5820 8-noded linear isoparametric soil elements, 1600 4-noded rigid plate

elements for the rings, and a mix of 2526 4-noded and 3-noded rigid plate elements for the top

cap.

20

Figure 3.5: Finite element model of the DSS test

3.4.2 Soil-Ring Interaction

Each ring moves due to soil movement as a rigid body. A linear friction model captures the contact

behaviour between the soil and the rings, as shown in Equations 1 and 2. The contact frictional

stress (soil-ring) is based on the normal contact pressure (soil-ring) and the soil-ring friction angle

(soil-ring). The contact pressure is computed by the soil-ring normal stiffness (ksoil-ring,n) and

compression of soil against the ring (dsoil-ring).

soil-ring = soil-ring tan(soil-ring) [1]

soil-ring = ksoil-ring,n dsoil-ring [2]

ABAQUS automatically assigns the soil-ring normal stiffness to be much larger than the adjacent

soil element stiffness and employs the Augmented Lagrange method to ensure each ring acts as a

rigid boundary that displaces horizontally during shear loading (Dassault Systèmes, 2012).

3.4.3 Top Cap

Each plate that defines the top cap moves due to applied loads or soil movement as rigid bodies.

As shown in Figure 3.5, the two plates that define the top cap are connected by springs which can

be adjusted to control the allowed vertical strain, vertical compliance, and rotation, rocking, during

the shearing phase. A linear contact pressure model captures the contact behaviour between the

soil and top cap (bottom plate). It was assumed that the top cap would not slip horizontally along

Top face

Front face

Bottom face

Rings Bottom centre,

(x, y, z) = (0,0,0)

Vertical stress

Shear stress

Top cap &

Springs

21

the soil. The vertical contact pressure of the soil against the top cap (soil-cap) was modelled using

Equation 3.

soil-cap = ksoil-cap,n dsoil-cap [3]

ABAQUS automatically assigns the soil to top cap normal stiffness to be much larger than the

adjacent soil element stiffness and employs the Augmented Lagrange method to ensure the top cap

acts as a rigid boundary that can be fixed in place, displace vertically or rotate during the simulation

(Dassault Systèmes, 2012).

3.4.4 Loading and Boundary Conditions

In all loading phases, the front boundary (Figure 3.5) is restrained from out of plane (z)

displacements. Rings are only in contact with the soil and can displace horizontally in the direction

of applied shear. The top cap is only in contact with the soil and the boundary conditions differ for

each loading phase. The simulation followed two loading phases: In the first phase, the model was

consolidated by a vertical stress where the top cap was restrained from lateral displacement (x and

z) and the bottom face was restrained from lateral and vertical (y) displacement. In the second

phase, shear strain or stress was applied and the bottom face was restrained from vertical and out

of plane displacement. During the shearing phase, the top cap boundary conditions were changed

to study the effect of soil-ring friction, vertical compliance on undrained tests and top cap rocking

during shear. To study soil-ring friction, the top cap and soil on the top face were fixed from

rotation or displacement. To study the effect of vertical compliance on undrained tests, the top cap

can displace vertically and the allowed vertical strain was controlled by the top cap spring stiffness.

To study the effect of top cap rocking, the top cap can rotate at the centre of the top face about the

out of plane (z) axis and the amount of rotation was controlled by the top cap spring stiffness.

3.4.5 Ideal Simple Shear Conditions

As mentioned earlier, to apply ideal simple shear conditions, the boundaries must allow for

development of the shear stresses as shown in Figure 1.2(a). During shearing, opposite sides must

remain parallel. Vertical boundaries must also ensure no change in cross-sectional area. For

undrained tests, no height change may occur to satisfy the constant volume condition. For

undrained monotonic simple shear tests under ideal conditions, the specimens were consolidated

22

by a vertical stress constraining lateral strains. Then shear loading was applied constraining all

strains except shear in the direction of loading.

Ideal simple shear conditions were achieved using two methods: The first method used a single

element model which assumes perfect simple shear conditions. Simulations were conducted using

a single element testing software “Incremental Driver” (Niemunis, 2017). The Incremental

Driver’s simplicity allows it to be more computationally efficient than ABAQUS for single

element models under ideal simple shear conditions. The second method works by applying

different soil-ring friction angles during each loading phase of the multi-element SGI type stacked

ring simulation in ABAQUS that is shown in Figure 3.5. Frictionless boundaries were set during

the consolidation phase to ensure uniformly distributed stresses. Conversely, no slip conditions

were set during the shearing phase to develop the necessary complementary shear stresses for ideal

simple shear conditions. The two models produced practically identical results, so the single

element model was used to produce ideal simple shear condition due to its efficiency. The single

element model was also used for model calibration.

3.4.6 Mesh sensitivity analysis

Mesh sensitivity was analyzed by computing simulations using various element sizes (ES) and

meshing pattern. The ES is an approximate length of each element. As shown in Figure 3.6, the

analysis considered uniform meshes of 8-noded isoparametric linear soil elements with ES of (a)

8 mm, (b) 4 mm, and (e) 1 mm, as well as graded meshes with (c) 8-noded linear, and (d) 20-noded

quadratic isoparametric soil elements. The graded meshes begin at an ES of 2 mm at the vertical

boundary and gradually increases to 5 mm at the specimen centre. Smaller elements capture the

expected stress concentrations near the vertical boundaries due to the lack of complementary shear

stresses, while larger elements are sufficient in capturing more uniform stress distribution that is

expected near the centre. The height of elements is restricted to the height of a ring which is 1 mm.

In general, 20-noded quadratic elements perform better than 8-noded linear elements in capturing

highly non-linear stress-strain distribution but require more computation time. In many cases, 8-

noded linear elements are sufficient for capturing the average stress-strain response and

understanding the regions of stress or strain concentrations.

23

Figure 3.6: Specimen mesh densities that were analyzed in the mesh sensitivity analysis

The average stress-strain response and the stress distribution were analyzed to evaluate the quality

of the mesh for this study. Figure 3.7 and Figure 3.8 presents the plots of (a) stress path and (b)

shear stress distribution for undrained monotonic and cyclic simulations of loose specimens (Dr =

24%). The monotonic test was sheared until the peak shear stress was reached which was

approximately 4% shear strain. The simulation with graded mesh of 20-noded elements did not

converge after 3.5% shear strain due to high stress concentrations. The cyclic test was sheared for

one cycle as subsequent cycles had similar stress-strain response. The shear stress distribution plot

is based on the average stress of elements along the top of the specimen at the centre cross section.

The critical state line (CSL) with the constant volume friction angle ('cv) is also plotted with the

stress paths when applicable. The simulations assumed frictionless vertical boundaries, constant

height and no top cap rocking. Figure 3.7(a) and Figure 3.8(a) shows that the stress path is not

sensitive to the tested mesh densities. However, Figure 3.7(b) and Figure 3.8(b) show that the

graded mesh (ES = 2 to 5 mm) or uniform mesh of 1 mm ES performs better in capturing the stress

concentrations near the vertical boundaries. The simulation with graded mesh of 20-noded

elements shows much higher stress concentrations than other simulations near the vertical

boundaries which may have contributed to non-convergence before the target 4% shear strain.

Similar findings were observed in dense specimen (Dr = 146%) simulations which are presented

in Appendix B Figure B.2 and Figure B.4. As shown in Table 3.2, for loose (Dr = 24%) and dense

(Dr = 146%) specimen simulations, the computation time sharply increases when using the graded

mesh (ES = 2 to 5 mm) with 20-noded elements or the uniform mesh of 1 mm ES with 8-noded

elements. To ensure the chosen mesh is sufficient for all analysis in this study, the mesh sensitivity

analysis was also carried out for high soil-ring friction, high vertical strain during undrained tests

(e) ES = 8, Uniform 8-noded linear elements

(c) ES = 2 to 5 mm, Graded 8-noded linear elements

(d) ES = 2 to 5 mm, Graded

20-noded quadratic elements

(a) ES = 8 mm, Uniform

8-noded linear elements

(b) ES = 4 mm, Uniform 8-noded linear elements

24

and high top cap rotation during shear. The stress path plots are presented in Appendix B from

Figure B.5 to Figure B.10. The stress path is not sensitive to the tested mesh densities. As such,

the graded mesh (ES = 2 to 5 mm) with 8-noded elements was selected for the study since the

stress non-uniformities and average stress response were reasonably captured with less

computation time when compared to 20-noded elements or a denser mesh. The chosen graded

mesh is presented in Figure 3.6(c) and the results from using this mesh are shown in green in the

Figure 3.7 and Figure 3.8.

Figure 3.7: Undrained monotonic test results of loose specimen (Dr = 24%) varying element size


Figure 3.8: Undrained cyclic test results of loose specimen (Dr = 24%) varying element size (ES)

and element type for frictionless vertical boundaries, constant height and no top cap rocking

(a) Stress path

(b) Top (y = 19.5 mm) along

centre cross section (z = 0 mm)

at peak shear stress

(a) Stress path



at peak of load reversal

25

Table 3.2: Total computation time for undrained monotonic and cyclic simulations using

different mesh densities and element types

Uniform (U) or graded

(G) mesh of different

element size (ES) and

type (8-noded or 20-

noded)

Total computation time of all CPU’s (hours)1

Monotonic loading Cyclic loading

Loose specimen

(4% shear strain)

Dense specimen

(4% shear strain)

Loose

specimen

(1 cycle)

Dense

specimen

(1 cycle)

ES = 8 mm (U), 8n 0.5 0.6 0.5 0.5

ES = 4 mm (U), 8n 0.8 0.8 0.8 0.8

ES = 2 to 5 mm (G), 8n 2.3 2.5 2.7 2.6

ES = 2 to 5 mm (G), 20n 25.5 26.5 43.5 42.2

ES = 1 mm (U), 8n 22.0 25.6 56.5 49.2 1Computer Processing Unit (CPU) information: Intel® Core™ i7-7800X CPU @ 3.50GHz

3.5 Effect of Soil-Ring Friction

3.5.1 Introduction

A parametric study was conducted by varying the soil-ring friction between 0, 5 and 30 degrees in

undrained monotonic and cyclic test simulations. The soil-ring friction angle was assumed to be

constant to study the effect of friction throughout the DSS test. The friction angles represent a

combination of Teflon coated rings and rubber membrane sliding against soil, as shown in

Figure 3.9. Tatsuoka and Haibara (1985) studied the interface friction angle between a Teflon sheet

and air-pluviated Toyoura sand and found that for a Teflon sheet that is fixed on both sides, the

friction angle is around 7°. However, the DSS device encases soil within a rubber membrane which

compresses during consolidation from vertical stress and flexes during shearing. The flexibility of

rubber does not allow significant development of complementary shear stresses. Furthermore, the

Teflon coated rings are not fixed at either end and their light weight would not be able to resist

much upward complementary shear stress. Under these conditions, the study assumed the range of

achievable friction angle in the laboratory is between 0 and 5 degrees. The lower range of 0 degrees

represents typical assumptions of frictionless boundaries. The upper range of 5 degrees represents

maximum achievable friction angle from current laboratory conditions. The ultimate friction angle

of 30° represents a no slip condition between rings and soil that is necessary for ideal simple shear

conditions in the multi-element simulation using ABAQUS. The results are compared to ideal

simple shear conditions.

26

Figure 3.9: Friction on a soil specimen that is encased in a rubber membrane and is confined by

Teflon coated rings

3.5.2 Results and Discussion

3.5.2.1 Consolidation

The soil specimen was consolidated to 100 kPa vertical stress (’vc). Only the loose specimen

(Dr = 24%) results are presented here and all results, including those of dense specimens

(Dr = 146%) are presented in Appendix C from Figure C.1 to Figure C.4.

Soil-ring friction affects stress non-uniformities that develop during the consolidation phase.

Figure 3.10 presents the shear and vertical stress distribution of loose specimen simulations based

on the average stress of the elements along the top and sides of the specimen at the centre cross

section. As shown in Figure 3.10(a) and (b), higher soil-ring friction increases shear stresses near

the boundaries. As a result, vertical stress concentrations develop at the top and bottom edges, as

shown in Figure 3.10(c) and Figure 3.11, to maintain uniform axial displacement that is imposed

by the stiff top cap and bottom pedestal. Near the middle, shear stresses are zero and vertical

stresses are uniformly distributed as shown in Figure 3.10(a) and (c). Since the average vertical

stress over the top area must equal the 100 kPa stress that is applied, higher vertical stress

concentrations at the top edges decreases the vertical stress of the top middle area as shown in

Figure 3.10(c). For frictionless vertical boundaries (0°), vertical stress is 100 kPa throughout the

specimen. The same observations on stress non-uniformities are also found in dense specimen

simulations as shown in Appendix C.

Rubber membrane

Teflon coated rings

Soil specimen Membrane to ring

friction

Soil to membrane

friction

27

Figure 3.10: End of consolidation stress distribution of loose specimen (Dr = 24%) simulations for

the centre cross section of the specimen (z = 0 mm), varying soil-ring friction angle (soil-ring)

Figure 3.11: End of consolidation vertical effective stress contours of loose specimen (Dr = 24%)

simulation for no slip condition, soil-ring friction angle of 30°

The coefficient of lateral earth pressure (K), ratio of lateral to vertical stress, at the core (centre) of

the multi-element simulations are compared with the ideal simple shear simulation. Figure 3.12

presents the coefficient of lateral earth pressure at the core (centre) of loose specimens during

consolidation. The coefficient of lateral earth pressure at the core compares well with ideal simple

shear conditions for all soil-ring friction. This is shown in Figure 3.12 for loose specimen

simulations, but is also the case for dense specimen simulations as shown in Appendix C. As such,

stress conditions near the specimen core approximate ideal simple shear conditions despite the

stress non-uniformities near the vertical boundaries due to higher soil-ring friction.

(a) Top (y = 19.5 mm) (b) Left (x = -30.75 mm)

Right (x = 30.75 mm)

(c) Top (y = 19.5 mm)

Max = 143 kPa, Min = 36 kPa

Average over top face = 100 kPa

28

Figure 3.12: Coefficient of lateral earth pressure (K) during consolidation at the core of loose

specimen (Dr = 24%) simulations for varying soil-ring friction angles (soil-ring)

3.5.2.2 Undrained Monotonic Simple Shear

Loose (Dr = 24%) and dense (Dr = 146%) specimen simulations were consolidated to 100 kPa

vertical stress (’vc). They were then sheared until the simulations reached non-convergence or

20% shear strain. The multi-element simulations of the loose and dense specimens reached non-

convergence between 4% and 6% shear strain, respectively. Select results are presented here. All

results are presented in Appendix C from Figure C.5 to Figure C.12.

Soil-ring friction affects stress non-uniformities that develop during the shearing phase.

Figure 3.13 presents the shear and vertical stress contours of the loose specimen simulation at 4%

shear strain, which approximates the peak shear stress for frictionless vertical boundaries. As

shown in Figure 3.13(a), the shear stress contours for the simulation with frictionless vertical

boundaries shows near zero shear stresses, blue colour, near the sides. As a result, frictionless

vertical boundaries cannot develop significant complementary shear stresses. Maximum shear

stress and vertical stress are concentrated at the top and bottom edges of the specimen as shown in

Figure 3.13(a) and Figure 3.13(b). Stresses near the centre are uniformly distributed. As shown in

Figure 3.13(a), shear stresses adjacent to the frictionless vertical boundaries quickly go from zero

to the applied shear stress level. Figure 3.14 presents the shear and vertical stress distribution based

on the average stress of elements along the top and sides of the specimen at the centre cross section.

Figure 3.14(b) shows that complementary shear stresses develop immediately adjacent to the

frictionless vertical boundaries which results in a sharp shear stress gradient that transitions to

core

29

uniformly distributed stresses near the centre, as also shown in Figure 3.13(a). Higher soil-ring

friction increases complementary shear stresses near the vertical boundaries that improve stress

uniformity as shown in the shear stress distribution, Figure 3.14(a) and (b), and in the vertical

stress distribution, Figure 3.14(c). The same observations on stress non-uniformities are also found

in dense specimen simulations as shown in Appendix C. Thus, higher soil-ring friction improves

stress uniformity in the specimen due to the development of complementary shear stresses.

However, stress concentrations exist for all soil-ring friction angles as shown in Figure 3.14 and

in Figure C.8 from Appendix C. Convergence criteria becomes more difficult to satisfy with large

stress concentrations at high shear strain which may have contributed to non-convergence of the

simulations before the target 20% shear strain.

Figure 3.13. Stress contours of loose specimen (Dr = 24%) simulation at 4% shear strain for

frictionless vertical boundaries, soil-ring friction angle of 0°

Figure 3.14. Stress distribution of loose specimen (Dr = 24%) simulations at peak shear stress (4%

shear strain) for the centre cross section of the specimen (z = 0 mm), varying soil-ring friction

angle (soil-ring)

(b) Vertical effective stress Max = 233 kPa, Min = 5 kPa


(a) Shear Stress Max = 46 kPa, Min = 3 kPa


(a) Top (y = 19.5 mm)

(b) Left (x = -30.75 mm)


(c) Top (y = 19.5 mm)

30

The vertical and lateral stresses at the core (centre) of the multi-element simulations are compared

with the ideal simple shear simulation. Figure 3.15 presents the effective normal stress and lateral

earth pressure coefficient (K) vs. shear strain plots for loose specimen (Dr = 24%) simulations. As

shown in Figure 3.15(a) and (b), under ideal simple shear conditions, vertical and in-plane (x)

effective stresses progress to the same value or a coefficient of lateral earth pressure of 1.0. The

out-of-plane (z) coefficient of lateral earth pressure progresses to about 0.6 for this calibration of

the constitutive model. As shown in Figure 3.15(b), the in-plane and out-of-plane coefficient of

lateral earth pressure at the specimen core compares well with ideal simple shear conditions for

all soil-ring friction. The same trends are found in dense specimen simulations as shown in

Appendix C. Thus, the results show that the vertical and lateral stress behaviour near the core

approximates ideal simple shear conditions despite stress non-uniformities near the boundaries.

The changes in the lateral stress and their differences in in-plane and out-of-plane directions is an

important flaw for University of Western Australia (UWA)(Mao and Fahey, 2003) and the

University of California at Berkeley (Villet et al., 1985) apparatus type that maintains equal and

constant lateral stress during the test.

Figure 3.15: Effective normal stress results for monotonic tests of loose specimen (Dr = 24%) for

varying soil-ring friction angles (soil-ring)

Soil-ring friction affects the average stress-strain response during the shearing phase. Figure 3.16

and Figure 3.17 presents the (a) shear stress vs. shear strain and (b) stress path plots for loose and

dense specimen simulations. The critical state line (CSL) with the constant volume friction angle

core

(b) Coefficient of lateral earth pressure (𝐾)

at core of specimen

(a) Effective normal stress results for

ideal simple shear condition

31

('cv) is also plotted with the stress paths. For the multi-element stacked ring simulations, the plots

present the average stress and strain values over the top area. As shown in Figure 3.16(a) and (b),

higher soil-ring friction reduces the average shear stress for loose specimen simulations when

compared to the single element ideal simple shear simulation. For the expected achievable range

of soil-ring friction between 0 to 5°, loose specimen simulation results compare well with the ideal

simple shear simulation. Additionally, as shown in Figure 3.17(a) and (b), dense specimen

simulations compare well with the ideal simple shear simulation for all tested soil-ring friction

angles. Thus, the effect of soil-ring friction (or lack thereof) is negligible for typical DSS testing

conditions. Higher soil-ring friction does not improve the average stress-strain response in

simulating ideal simple shear conditions. Practically, near frictionless vertical boundaries that are

currently used in direct simple shear tests achieve the best comparison with ideal simple shear

conditions.


angles (soil-ring)

Figure 3.17. Monotonic test results of dense specimen (Dr = 146%) for varying soil-ring friction

angle (soil-ring)

(a) (b)

(a) (b)

32

Stress non-uniformities that develop during the consolidation phase due to soil-ring friction,

especially for soil-ring = 30°, affect the average stress-strain response during the shearing phase.

As mentioned previously, Figure 3.10 shows that higher soil-ring friction increases shear stress

and vertical stress concentrations around the specimen edges during the consolidation phase. To

study the effect of soil-ring friction on the shearing phase only, frictionless vertical boundaries

were set during the consolidation phase (soil-ring,c = 0°) to ensure uniform stress distribution. The

soil-ring friction angle was varied for the shearing phase only. Figure 3.18 presents the (a) shear

stress vs. shear strain and (b) stress path plots for simulations of frictionless (soil-ring,s = 0°) and

no slip soil-ring,s = 30°) vertical boundaries during the shearing phase. The critical state line (CSL)

with the constant volume friction angle ('cv) is also plotted with the stress paths like before. As

mentioned previously, applying frictionless vertical boundaries during the consolidation phase and

no slip vertical boundaries (soil-ring,s = 30°) during the shearing phase simulates ideal simple shear

conditions. The shear stress vs. strain plot in Figure 3.18(a), and the stress path plot in

Figure 3.18(b) show that varying the soil-ring friction during the shearing phase does not affect

the average stress-strain response. Both soil-ring friction conditions compare well with the one

element ideal simple shear simulation. Hence, the average stress-strain response during the

shearing phase is primarily affected by the development of stress concentrations during the

consolidation phase. Higher soil-ring friction increases stress concentrations during the

consolidation phase that reduces the shear stress during the shearing phase, as previously shown

in Figure 3.16.


angles (soil-ring) during shearing phase only

(b) (a)

CSL 'cv

= 30°

33

3.5.2.3 Undrained Cyclic Simple Shear

Loose (Dr = 24%) and dense (Dr = 146%) simulated specimens were consolidated to 100 kPa

vertical stress (’vc) before applying cyclic shear loading of 12 kPa amplitude. The applied cyclic

stress ratio (CSR), ratio of cyclic shear loading to vertical consolidation stress, was 0.12. Loose

and dense specimens were sheared until the simulations reached liquefaction or the stress-strain

response did not vary significantly with more cycles due to locking behaviour. The liquefaction

criterion and locking behaviour were explained in more detail in the section 3.3. The loose

specimen simulations were ended after 4 cycles when liquefaction occured. The dense specimen

simulations were ended after 40 cycles due to locking behaviour and did not reach liquefaction.

Select results are presented here. All results are presented in Appendix C from Figure C.13 to

Figure C.20.

Soil-ring friction affects stress non-uniformities that develop during the shearing phase. Only

dense specimen simulation results are presented here as loose specimen simulation results are

similar. Figure 3.19 presents the vertical and shear stress contours of the dense specimen

simulation at the peak amplitude of the last cycle for frictionless boundaries (soil-ring = 0°). As

observable by the bands of low shear stress contour near the vertical boundaries in Figure 3.19(a),

frictionless vertical boundaries cannot develop significant complementary shear stresses. As a

result, maximum shear stress and vertical stress are concentrated near the top and bottom edges of

the specimen as shown in Figure 3.19(a) and Figure 3.19(b). Figure 3.20 presents the shear and

vertical stress distribution for the first and last cycle of the dense specimen (Dr = 146%)

simulations based on the average stress of elements along the top and sides of the specimen at the

centre cross section. During consolidation, higher soil-ring friction increases shear stresses in the

upwards direction near the boundaries which in turn increases the vertical stress near the top edge

as shown previously in Figure 3.10. During the first cycle of shearing, higher soil-ring friction also

increases shear stress in the upward direction near the boundaries as shown in Figure 3.20(a-i) and

(a-ii), and increases vertical stress near top edge as seen in Figure 3.20(a-iii). Consequently, the

increase in shear and vertical stress during consolidation due to higher soil-ring friction propagates

into the shearing phase as shown by the stress distribution of the first cycle in Figure 3.20(a-i), (a-

ii) and (a-iii). The increase in shear and vertical stress during consolidation due to higher soil-ring

friction does not appear to influence the stress distribution in the last (40th) cycle as shown in

Figure 3.20(b-i), (b-ii) and (b-iii). During the last cycle, Figure 3.20(b-ii) shows that higher soil-

34

ring friction increases complementary shear stresses near the vertical boundaries that improve

stress uniformity as shown in the shear stress distribution, Figure 3.20(b-i) and Figure 3.20(b-ii),

and in the vertical stress distribution, Figure 3.20(b-iii). Thus, higher soil-ring friction develops

complementary shear stresses that improve stress non-uniformity but are not noticeable until the

stress concentrations that develop during consolidation dissipate with more cycles. Higher soil-

ring friction increases shear and vertical stress concentrations during consolidation that transfer to

the shearing phase. After stress concentrations that develop during consolidation dissipate with

more cycles, the results show that higher soil-ring friction improves stress uniformity in the

specimen due to the development of complementary shear stresses. The same observations are

found in loose specimen simulations, as shown in Figure C.13 from Appendix C. However, for

loose specimen similations, the stress concentrations that develop during consolidation do not fully

dissipate before the simulations end after about 4 cycles due to liquefaction.

Figure 3.19. Stress contours of dense specimen (Dr = 146%) simulation at the peak amplitude of

the last cycle for frictionless vertical boundaries, soil-ring friction angle of 0°

(b) Vertical effective stress Max = 128 kPa, Min = 28 kPa


(a) Shear Stress Max = 19 kPa, Min = -4 kPa


35

Figure 3.20. Stress distribution of dense specimen (Dr = 146%) simulations for the centre cross

section of the specimen (z = 0 mm) during the peak amplitude of the (a) first and (b) last cycle for


The vertical and lateral stresses at the core (centre) of the multi-element simulations are compared

with the ideal simple shear simulation. Figure 3.21 presents the plots of (a) effective normal stress

and (b) lateral earth pressure coefficient (K) vs. number of cycles for dense specimen (Dr = 146%)

simulations. As shown in Figure 3.21(a) and (b), under ideal simple shear conditions vertical and

in-plane (x) stresses progress to the same value or a coefficient of lateral earth pressure of 1.0 .

The out-of-plane (z) coefficient of lateral earth pressure does not vary greatly and progresses to

about 0.5. As shown in Figure 3.21(b), the in-plane and out-of-plane coefficients of lateral earth

pressure at the specimen core (centre) compare well with ideal simple shear conditions for all soil-

ring frictions. The same observations are found in loose specimen simulations as shown in

Appendix C. Thus, despite stress non-uniformities that are caused by higher soil-ring friction,

especially in the first few cycles, the stress behaviour near the core approximates ideal simple

(a-i) Top (y = 20 mm)

(a-ii) Left (x = 0 mm)

Right (x = 63.5 mm)

(a-iii) Top (y = 20 mm)

(b-ii) Left (x = 0 mm)

Right (x = 63.5 mm)

(b-i) Top (y = 20 mm)

(b-iii) Top (y = 20 mm)

(a) Peak amplitude of first cycle (b) Peak amplitude of last cycle

36

shear conditions throughout the simulation. The changes in the lateral stress and their differences

in in-plane and out-of-plane directions is an important flaw for University of Western Australia

(UWA)(Mao and Fahey, 2003) and the University of California at Berkeley (Villet et al., 1985)

apparatus type that maintains equal and constant lateral stress during the test.

Figure 3.21: Normal stress results for cyclic tests of dense specimen (Dr = 146%) for varying soil-

ring friction angles (soil-ring)

Soil-ring friction affects the average stress-strain response during the shearing phase. Figure 3.22

and Figure 3.23 present plots of (a) shear stress vs. number of cycles, (b) shear strain vs. number

of cycles, (c) vertical stress vs. number of cycles, (d) shear stress vs shear strain and (e) stress path

for loose (Dr = 24%) and dense (Dr = 146%) specimen simulations. The critical state line (CSL)

with the constant volume friction angle ('cv) is also plotted with the stress paths when applicable.

For the multi-element stacked ring simulations, the plots present the average stress and strain

values over the top area. As shown in Figure 3.22(c) and (e), and Figure 3.23(c) and (e), higher

soil-ring friction reduces the vertical stress when compared to the single element ideal simple shear

simulation. The reduction in vertical stress for higher soil-ring friction leads to an earlier prediction

of liquefaction, as shown in Figure 3.22(b). In other words, higher soil-ring friction reduces the

liquefaction resistance in cyclic loading. Similarly, higher soil-ring friction reduces the measured

shear stress (strength) in monotonic loading as observed in Figure 3.16. Higher soil-ring friction

primarily reduces the vertical stress for the first few cycles, as shown in Figure 3.22(e) and


at core of specimen

(a) Normal stress results for


core

37

Figure 3.23(e). After a few cycles, the decrease in vertical stress per cycle becomes similar for all

soil-ring friction angles. For dense specimen simulations, the shear strain per cycle compares well

for all soil-ring friction, as shown in Figure 3.23(b). For the expected achievable range of soil-ring

friction between 0 to 5°, loose and dense specimen simulations compare well with the ideal simple

shear simulation. Thus, the effect of soil-ring friction (or lack thereof) is negligible for typical DSS

testing conditions. Higher soil-ring friction does not improve the average stress-strain response in

simulating ideal simple shear conditions. Practically, near frictionless vertical boundaries that are

currently used in direct simple shear tests achieve the best comparison with ideal simple shear

conditions.


(soil-ring)

(a)

(b)

(c)

(d)

(e)

38

Figure 3.23. Cyclic test results of dense specimen (Dr = 146%) for varying soil-ring friction angles

(soil-ring)

Stress non-uniformities that develop during the consolidation phase due to soil-ring friction,

especially for soil-ring = 30°, affect the average stress-strain response during the shearing phase.

As mentioned previously, Figure 3.10 shows that higher soil-ring friction increases shear stress

and vertical stress concentrations around the specimen edges during the consolidation phase. To

study the effect of soil-ring friction on the shearing phase only, frictionless vertical boundaries

were set during the consolidation phase (soil-ring,c = 0°) to ensure uniform stress distribution. The

soil-ring friction angle was varied for the shearing phase only. Figure 3.24 presents the (a) shear

stress vs. shear strain and (b) stress path plots for simulations of frictionless (soil-ring,s = 0°) and

no slip (soil-ring,s = 30°) vertical boundaries during the shearing phase. The critical state line (CSL)

with the constant volume friction angle ('cv) is also plotted with the stress paths like before. As

(a)

(b)

(c)

(d)

(e)

39

mentioned previously, applying frictionless vertical boundaries during the consolidation phase and

no slip vertical boundaries (soil-ring,s = 30°) during the shearing phase simulate ideal simple shear

conditions. The shear strain vs number of cycles plot in Figure 3.24(a), and the stress path plot in

Figure 3.24(b) show that simulation results for both soil-ring friction conditions compare well with

the single element ideal simple shear simulation. The no slip condition shows a slightly smaller

decrease in vertical stress per cycle than the frictionless condition which is due to an increase in

frictional resistance of the vertical boundaries against the contractive behaviour of the soil as

vertical stress decreases. Essentially, transfer of vertical stress from the soil to the top cap is

reduced near the sides due to the increased friction of the vertical boundaries. As shown in

Figure 3.24(b), the no slip vertical boundary condition does not show the large reduction in vertical

stress during the initial cycles as seen in Figure 3.22 when no slip vertical boundaries are used

during both consolidation and shearing phase. Hence, the average stress-strain response during the

shearing phase is primarily affected by the development of stress concentrations during the

consolidation phase. This is also the case with dense specimen simulations as shown in Appendix

C. Higher soil-ring friction increases stress concentrations during the consolidation phase that

reduce the vertical stress during the initial cycles of the shearing phase, as previously shown in

Figure 3.22 and Figure 3.23.

40


(soil-ring) during shearing phase only

3.6 Effect of Vertical Compliance on Undrained Tests

3.6.1 Introduction

Undrained monotonic and cyclic simple shear simulations were conducted on loose (Dr = 24%)

and dense (Dr = 146%) specimens to study the effect of vertical compliance on undrained tests.

The maximum allowed vertical strain was between 0 and 0.4%. A vertical strain of 0% represents

a fixed top cap that cannot displace during the shearing phase. Simulations were also analyzed for

ASTM D6528-17 tolerance of 0.05% height change (vertical strain) during shear. The maximum

allowed vertical strain also considers the apparatus deformation. As shown in Figure 1.3, changes

in vertical stress induce vertical strain in the soil specimen, top cap, base pedestal, loading shaft,

and two aluminum porous stones which are all detected by the LVDT sensor. The LVDT sensor

(a)

(b)

(c)

(d)

(e)

41

measures vertical strain that is used in an active height control system to adjust the sample height

as to maintain a constant height during undrained tests. Figure 3.25 presents the (a) vertical

displacement vs. applied vertical stress plot for the (b) SGI type DSS device that was loaded

without a soil specimen. As shown in Figure 3.2 in section 3.3, experimental data from this study

shows that vertical stress increases from about 100 to 350 kPa during monotonic tests. From the

loading curve in Figure 3.25(a) that reaches a maximum vertical stress of 400 kPa, a vertical stress

increase from 100 to 350 kPa results in a change in vertical displacement of 0.048 mm. As a result,

the apparatus deformation can increase vertical strain in the specimen by about 0.24% for a

specimen height of 20 mm. As shown in Figure 3.25(a), an increase in vertical stress causes

compression of the apparatus. An active height control system corrects vertical displacement that

is detected by an LVDT sensor. Depending on the positioning of the LVDT, the compliance of

various components of the apparatus may act as a cushion to the height changes of the specimen,

limiting the effectiveness of active control. The same components, as well as the rest of the

apparatus would contribute to specimen height changes in a passive device. It is physically

difficult, if not impossible, to exclude the top cap, bottom pedestal, the water bath and its bearings,

and the two porous stones and filter papers from the sandwich that acts as a cushion to the LVDT

measurements in an active device. Hence, the analysis considered a range of maximum vertical

strains during shearing between 0% and 0.4% which covers the range of compliance expected

around the ASTM standard, plus the “sandwich compliance” that is commonly missed in reporting

compliance (e.g. Zekkos et al. (2018)). The simulations assumed frictionless vertical boundaries

(soil-ring friction angle of 0°) which closely represent expected laboratory conditions.

Figure 3.25: Vertical displacement vs. applied vertical stress for SGI type device that was loaded

without soil specimen

(a) Results (b) Test setup without soil specimen

Top cap

Base pedestal

Aluminum grooved

porous stones

Loading shaft

42


3.6.2.1 Undrained Monotonic Simple Shear


vertical stress (’vc). They were then sheared until the simulations reached non-convergence or

20% shear strain. The multi-element simulations of the loose and dense specimens reached non-

convergence between 4% and 13% shear strain. Select results are presented here. All results are

presented in Appendix C from Figure C.21 to Figure C.24.

Vertical compliance of undrained tests does not affect stress non-uniformities. Figure 3.26 presents

the shear and vertical stress distribution at 4% shear strain which approximates the peak shear

stress for loose specimen simulations. The vertical strain distribution along the top nodes of the

specimen is also presented, Figure 3.26(a), and follows a compression positive sign convention.

Stress distributions are based on the average stress of elements along the top of the specimen at

the centre cross section. Figure 3.26(b) and (c) show shear and vertical stress concentrations near

the specimen edges due to frictionless vertical boundaries as previously discussed in section 3.5

on the effect of soil-ring friction. These stress concentrations are also shown in desnse specimen

simulations in Figure C.23 from Appendix C. Convergence criteria becomes more difficult to

satisfy with large stress concentrations at high shear strain which may have contributed to non-

convergence of the simulations before the target 20% shear strain. Higher vertical compliance does

not appear to affect stress concentrations, but does increase the average shear and vertical stress

near the top of the specimen for the loose specimen simulations, as shown in Figure 3.26(b) and

(c), and decrease the average shear and vertical stress near the top of the speciment for dense

specimen simulations, as shown in Figure C.23(b) and (c) in Appendix C. Higher vertical

compliance has negligible effect on shear stresses along the sides, as shown in Figure 3.26(d) and

Figure C.23(d).

43

Figure 3.26: Stress and strain distribution of loose specimen (Dr = 24%) simulations at peak shear

stress (4% shear strain) for the centre cross section of the specimen (z = 0 mm), varying maximum

vertical strain (y,max) during the shearing phase

Vertical compliance of undrained tests affects the average stress-strain response of loose specimen

(Dr = 24%) simulations. Figure 3.27 presents the average stress and strain response over the top

area for loose specimen simulations which includes plots for (a) vertical strain during shear vs.

shear strain, (b) effective vertical stress vs. shear strain, (c) shear stress vs. strain, and (d) stress

path. The critical state line (CSL) with the constant volume friction angle ('cv) is also plotted with

the stress paths. By convention, positive vertical strain represents compression. The effective

vertical stress decreases during the shearing phase which indicates contractive behaviour as the

specimen height decreases, as shown in Figure 3.27(a) and (b). Higher vertical compliance

increases vertical stress as shown in Figure 3.27(b), and increases shear stress as shown in

Figure 3.27(c). This is also shown in the stress path from Figure 3.27(d). Table 3.3 presents the

percent difference of shear and vertical stress at the peak shear stress (4% shear strain) for higher

maximum allowed vertical strain. For maximum vertical strain close to or above the ASTM

D6528-17 tolerance of 0.05%, shear and vertical stress increase significantly. Shear and vertical

stress increase by 6% and 7% at 0.05% maximum vertical strain and increase by 33% and 41% at

(c) Top (y = 19.5 mm)

(b) Top (y = 19.5 mm)

(d) Left (x = -30.75 mm) Right (x = 30.75 mm)

(a) Top (y = 20 mm)

44

0.28% maximum vertical strain. Similar trends were reported by Zekkos et al. (2018) for the effect

of vertical compliance on undrained tests in a large-size cyclic simple shear device on Ottawa sand

with relative density of 43 to 46% and pea gravel with relative density of 35 to 40%. Figure 3.28

and Figure 3.29 presents the reproduced plots of (a) shear stress vs shear strain, (b) stress path, and

(c) vertical strain during shear vs shear strain for Ottawa sand and pea gravel. Their results show

that higher vertical compliance causes similar increases in shear and vertical stress for soils

exhibiting contractive behaviour. Although the vertical strain that was measured on tests of Ottawa

sand were below the ASTM D6428-17 tolerance of 0.05%, the peak shear stress varied as much

as 15%. It was also reported that the test of pea gravel with maximum measured vertical strain

close to the ASTM D6428-17 tolerance of 0.05% showed a difference in peak shear stress of 10%.

As shown in Figure 3.30, a machine identical to the large-size cyclic simple shear device that was

used by Zekkos et al. (2018) was loaded and unloaded without a soil specimen to measure the

“sandwich compliance” due to changes in vertical stress. From the unloading curve that begins at

100 kPa, the reduction in vertical stress from 100 to 20 kPa in the monotonic test results of Ottawa

sand (Figure 3.28) may have resulted in vertical displacement of 0.34 mm due to apparatus

deformation. For a typical specimen height of 110 mm in the large-size cyclic simple shear device,

apparatus deformation may have resulted in additional vertical strain of 0.31% that does not appear

to be accounted for in the study by Zekkos et al. (2018). Therefore, the actual vertical strain may

be much higher than was reported by Zekkos et al. (2018), and closer to the upper range of strains

considered in this study. The effect of vertical compliance on undrained tests of loose specimens

is similar to the effect of partial drainage conditions. Higher vertical compliance increases shear

and vertical stress which is the same behavior as allowing more drainage for tests of loose

specimens. A drained test is the same as a very high vertical compliance where the specimen is

free to change height without restraint.

45

Figure 3.27: Monotonic test results of loose specimen (Dr = 24%) for varying maximum vertical

strain (y,max) during the shearing phase


loose specimen (Dr = 24%) at peak stress (4% shear strain)

Maximum Vertical Strain

During Shear (%)

Percent Change from Higher Maximum Vertical Strain1 (%)

Shear Stress Vertical Stress

0.00 0.0% 0.0%

0.05 6.0% 7.1%

0.07 9.4% 11.2%

0.19 24.7% 30.0%

0.28 33.4% 41.1%

1positive indicates increase in value from no vertical strain results

(a)

(c)

(b)

(d)

46

Figure 3.28: Reproduced undrained monotonic test results of Ottawa sand on the effect of vertical

compliance on undrained tests from Zekkos et al. (2018)

Figure 3.29: Reproduced undrained monotonic test results of pea gravel on the effect of vertical

compliance on undrained tests from Zekkos et al. (2018)

Figure 3.30: Vertical displacement vs. applied vertical stress for large-size cyclic simple shear

device developed by Geocomp (2019) without soil specimen

Vertical compliance of undrained tests affects the average stress-strain response of dense specimen

(Dr = 146%) simulations. Figure 3.31 presents the average stress and strain response over the top

area for dense specimen simulations which includes plots for (a) vertical strain during shear vs.

(a) (b) (c)

(b) (a) (c)

47

shear strain, (b) effective vertical stress vs. shear strain, (c) shear stress vs. strain, and (d) stress

path. The critical state line (CSL) with the constant volume friction angle ('cv) is also plotted with

the stress paths. The results show primarily dilative behaviour due to an increase in specimen

height, as shown in Figure 3.31(a). From 0 to 1% shear strain, the simulation exhibits contractive

behaviour. However, higher vertical compliance has negligible effect on stress-strain response

during this period. After 1% shear strain, the simulation exhibits dilative behaviour. During

dilative behaviour, higher vertical compliance reduces vertical stress, as shown in Figure 3.31(b),

and reduces shear stress, as shown in Figure 3.31(c). This is also shown in the stress path from

Figure 3.31(d). Table 3.4 presents the percent difference of shear and vertical stress at 4% shear

strain for higher maximum allowed vertical strain. For maximum vertical strain above the ASTM

D6528-17 tolerance of 0.05%, shear and vertical stress are reduced. Shear and vertical stress

decrease by 1% and 1% at 0.05% maximum vertical strain and decrease by 8% and 7% at 0.37%

maximum vertical strain. The results compare well with the work by Dyvik and Suzuki (2018)

who studied the effects of vertical compliance on undrained tests in dense offshore sand with a

relative density of 95%, as shown by the reproduced results in Figure 3.32. Dyvik and Suzuki

(2018) varied the specimen height between zero and an offset constant vertical strain using an

active height control system to study the effect on stress-strain response during the same test. As

shown in Figure 3.33, Dyvik and Suzuki (2018) only presented the vertical strain vs shear strain

plot for Kaolin, but mentioned that a similar pattern was used to offset specimen height for dense

offshore sand. For dense sand, higher vertical compliance increased vertical and shear stress during

contractive behaviour between 0% and about 2% shear strain. Then higher vertical compliance

reduced vertical and shear stress during dilative behaviour from 2% shear strain onwards. During

dilative behaviour at 5% shear strain, it was reported that shear stress was reduced by 6% and

vertical effective stress was reduced by 7%. At failure where the soil is transitioning from

contractive to dilative behaviour, the shear stress was increased by more than 20% and the vertical

effective stress was increased by about 30%. Dyvik and Suzuki (2018) does not appear to consider

the “sandwich compliance” in the reported vertical strain of the soil specimen either. Apparatus

deformation can significantly increase vertical strain as previously shown in Figure 3.25 and

Figure 3.30. Therefore, the actual vertical strain (compliance) of the soil specimen may be greater

than was reported.

48

As discussed in the Chapter 2, the effect of higher vertical compliance on undrained tests is dictated

by contractive or dilative behaviour of the soil (Dyvik and Suzuki, 2018). As shown by the loose

specimen (Dr = 24%) simulation results in Figure 3.27 and reproduced results of Zekkos et al.

(2018) in Figure 3.28 and Figure 3.29, higher vertical compliance increases shear and vertical

stress during contractive behaviour. As shown by the dense specimen (Dr = 146%) simulations

results in Figure 3.31 and reproduced results of Dyvik and Suzuki (2018) in Figure 3.32, higher

vertical compliance reduces shear and vertical stress during dilative behaviour. Changes in shear

and vertical stress are significant close to or above the ASTM D6528-17 vertical strain tolerance

of 0.05%.

Figure 3.31: Monotonic test results of dense specimen (Dr = 146%) for varying maximum vertical


(a)

(c)

(b)

(d)

49


dense specimen (Dr = 146%) at 4% shear strain

Maximum Vertical

Strain During Shear (%)

Percent Change from Higher Maximum Vertical Strain1 (%)

Shear Stress Vertical Stress

0.00 0.0% 0.0%

0.05 -1.4% -1.1%

0.17 -4.5% -3.6%

0.37 -8.1% -6.5%

1positive indicates increase in value from no vertical strain results

Figure 3.32: Reproduced undrained monotonic test results of dense offshore sand on the effect of

vertical compliance on undrained tests from Dyvik and Suzuki (2018)

Figure 3.33: Reproduced example of offset in specimen height for Kaolin specimen using an active

height control system from Dyvik and Suzuki (2018)

3.6.2.2 Undrained Cyclic Simple Shear


vertical stress (’vc) before applying cyclic shear loading of 12 kPa in amplitude (CSR = 0.12).

(b) (a)

50

Loose and dense specimens were sheared until the simulations reached liquefaction or the stress-

strain response did not vary significantly with more cycles due to locking behaviour. The

liquefaction criterion and locking behaviour were explained in more detail in the section 3.3. The

loose specimen simulations were ended after 4 cycles when liquefaction occured. The dense

specimen simulations were ended after 40 cycles due to locking behaviour and did not reach

liquefaction. All results are presented here and in Appendix C as described later.

Vertical compliance of undrained tests affects the stress-strain response of undrained cyclic simple

shear tests. As shown previously in Figure 3.26 and Figure C.23, higher vertical compliance does

not increase stress concentrations, but does affect the average stress-strain response. Figure 3.34

and Figure 3.35 present plots of (a) vertical strain during shear vs. number of cycles, (b) shear

strain vs. number of cycles, (c) vertical stress vs. number of cycles, (d) shear stress vs shear strain

and (e) stress path for loose and dense specimen simulations. The critical state line (CSL) with the

constant volume friction angle ('cv) is also plotted with the stress paths where applicable. Results

from Figure 3.34 and Figure 3.35 are also presented in Appendix C as shown in Figure C.25 and

Figure C.26. Figure 3.34(a) and Figure 3.35(a) shows that loose and dense simulated specimens

simulations exhibit contractive behaviour while specimen height decreases throughout the

simulations due to vertical compliance. As shown in Figure 3.34(c) and Figure 3.35(c), higher

vertical compliance increases the vertical stress for each cycle as compared to the constant height

simulation. As such, higher vertical compliance increases the number of cycles to liquefaction, as

shown in the shear strain vs. number of cycles plots from Figure 3.34(a) and Figure 3.35(a). Higher

vertical compliance has negligible effect on the magnitude of shear strain. The trends are similar

to loose specimen monotonic test simulations as shown in Figure 3.27, which show that higher

vertical compliance during contractive behaviour increases the vertical stress. An increase in shear

stress, as observed during contractive behaviour in monotonic tests, is not seen in cyclic tests since

the analysis is load-controlled by the applied shear stress amplitude, as shown in Figure 3.34(d)

and (e), and Figure 3.35(d) and (e).

51

Figure 3.34: Stress-strain response for undrained cyclic simple shear simulations of loose specimen

(Dr = 24%) for varying maximum vertical strain (y,max) during the shearing phase

(a)

(b)

(c)

(d)

(e)

52


specimen (Dr = 146%) for varying maximum vertical strain (y,max) during the shearing phase

As shown in Figure 3.35(b), higher vertical compliance appears to cause a bias in the direction of

shear strain development during dense specimen simulations which may be due to the boundary

conditions at the top face. Figure 3.36 presents the (a) shear strain vs number of cycles and (b)

stress path plots for two modelling approaches to maintain constant height in an undrained test

simulation. In the first approach, the soil on the top face was directly fixed from displacement and

rotation. The approach was used for studying the effects of soil-ring friction in the previous section.

In the second approach, the top cap was restricted from displacement or rotation by applying highly

stiff vertical springs on the edges, as shown in Figure 3.5. The soil could only displace horizontally

and was restricted from vertical displacement by the top cap contact. The second approach

modeled the top cap to soil contact behaviour to study the effects of vertical compliance on

undrained tests and top cap rocking during shear. Figure 3.37 presents the vertical strain and shear

stress distribution during the peak amplitude of the last cycle using both approaches. Vertical strain

(a)

(b)

(c)

(d)

(e)

53

distribution is based on the nodal displacements along the top of the specimen at the centre cross

section and follows a compression positive sign convention. Shear stress distribution is based on

the average stress of elements along the top of the specimen at the centre cross section. As shown

in Figure 3.37(a), the first approach guarantees uniform soil displacement at the top face while the

second approach cannot guarantee perfectly uniform vertical displacements due to complex soil to

top cap interaction. However, the maximum vertical strain during the shearing phase in the second

approach is about 0.0004% which is sufficient for approximating constant height conditions. In

the second approach, the opposing vertical strain directions near the edges of top face develops a

slight rotation of the specimen. As shown in Figure 3.36(b) and Figure 3.37(b), the stress path and

shear stress distribution are not affected by the modelling approach. However, Figure 3.36(a)

shows that non-uniform top face displacements and a small top face rotation may develop a bias

in the direction of shear strain development. The bias in shear strain direction can also be seen in

dense specimen simulations when comparing plots of shear strain vs. number of cycles from the

study of soil-ring friction (Figure 3.23) and the study of vertical compliance on undrained tests

(Figure 3.35). The bias in shear strain direction is noticeable but not significant as the magnitude

of shear strain is much less than the liquefaction criterion of 1% single amplitude shear strain.

Current active and passive height control systems cannot guarantee zero vertical strain during the

shearing phase or uniform soil displacement at the top cap. Even for a perfect height control

system, soil may lose contact with the top cap especially near the specimen edges. In reality, it is

possible that existing contacts that cannot guarantee uniform soil displacement at the top face may

contribute to a bias in the direction of shear strain development.

Figure 3.36: Stress-strain response of cyclic tests for dense specimen (Dr = 146%) using two

approaches to model constant height

(a) (b)

54

Figure 3.37: Stress and strain distribution of dense specimen (Dr = 146%) simulations for the

centre cross section of the specimen (z = 0 mm) during the peak amplitude of the last cycle using

two approaches to model constant height

3.7 Effect of Top Cap Rocking During Shear

3.7.1 Introduction

Undrained monotonic and cyclic simple shear simulations of loose (Dr = 24%) and dense (Dr =

146%) specimens were conducted for maximum top cap rotation angles ranging from 0° to 0.27°.

As mentioned previously, studies on the effect of top cap rocking during the shearing phase on the

stress-strain response could not be found in the literature review. However, top cap rocking

measurements have been reported when evaluating the performance of simple shear devices. Based

on previous work covered in Chapter 2, the measured top cap rocking in direct simple shear devices

ranged between 0.006° to 0.07° (Boulanger et al., 1993; Kwan et al., 2014; Shafiee, 2016). Thus,

the rotation angle of 0° represents a fixed top cap with no rocking. The rotation angle around 0.1°

represents the maximum expected rocking of the top cap for current DSS devices (see Figure 3.38

for example). Higher rotation angles were considered to analyze potential problems in the top cap

setup that can increase rocking, such as the connection of the loading shaft and bushing, as shown

in Figure 3.38. The bushing is meant to stop the bending of the loading shaft to reduce rocking.

For cyclic simulations where the top cap was allowed to rotate without restraint, the maximum

rotation angle was found to be 0.14° for loose specimen simulations and 0.035° for dense specimen

simulations. As such, cyclic simulations analyzed top cap rocking between 0° and the maximum

rotation angles, 0.14° or 0.035°. Monotonic simulations can undergo much higher top cap rotation

angles and were restrained to a maximum rotation angle of about 0.27°. The simulations assumed

(a) Top (y = 20 mm) (b) Top (y = 19.5 mm)

55

frictionless vertical boundaries (soil-ring friction angle of 0°) which closely represent expected

laboratory conditions.

Figure 3.38: Potential top cap rocking that is influenced by loading shaft in SGI type device


3.7.2.1 Undrained monotonic simple shear


vertical stress (’vc). They were then sheared until the simulation reached non-convergence or 20%

shear strain. The multi-element simulations of the loose and dense specimens reached non-

convergence between 5% and 10% shear strain. Select results are presented here. All results are


Top cap rocking affects stress and strain distribution during the shearing phase. Figure 3.39

presents the shear and vertical stress distribution for loose specimen simulations at 4% shear strain

which approximates the peak shear stress. The vertical strain distribution along the top nodes of

the specimen is also presented and follows a compression positive sign convention. Stress

distribution values are based on the average stress of elements along the top and sides of the

specimen at the centre cross section. Dense specimen stress and strain distribution plots are not

presented here as the results are similar to loose specimen results. For loose specimen simulations,

the sharp increase in shear and vertical stresses on the left side and sharp decrease in shear and

vertical stresses on the right side, as shown in Figure 3.39(b) and (c), are due to frictionless vertical

Loading shaft

Top cap

Bushing

56

boundaries which were discussed in section 3.5 on the effect of soil-ring friction. These stress

concentrations are due to the lack of complementary shear stresses, as shown in Figure 3.39(d).

Similar stress concentrations due to frictionless vertical boundaries are also observed in dense

specimen simulations, as shown in Figure C.29 in Appendix C. Convergence criteria becomes

more difficult to satisfy with large stress concentrations at high shear strain which may have

contributed to non-convergence of the simulations before the target 20% shear strain. The sharp

decrease in shear and vertical stress on the right side causes soil near the right edge to lose contact

with the top cap, as shown by the sudden increase in vertical strain in Figure 3.39(a). The shear

and vertical stress concentrations near the sides are not due to the effect of top cap rocking. The

effect of top cap rocking is analyzed by comparing stress results of no rotation with increased

rotation. As shown in Figure 3.39(a), higher top cap rocking induces height increase (extension)

on the left half of the specimen and induces height decrease (compression) on the right half. As

shown in Figure 3.39(b) and (c), on the right half, where rocking induces compression, shear and

vertical stress are increased and on the left side, where rocking induces extension, shear and

vertical stress are reduced. Higher top cap rocking has negligible effect on the shear stress near

the sides as shown in Figure 3.39(d). Similar results are found in dense specimen simulations as

shown in Figure C.29 in Appendix C. The results agree with Figure 3.27 and Figure 3.31 on the

effect of vertical strain during undrained tests which shows that height decrease will increase shear

and vertical stress and height increase will reduce shear and vertical stress.

57

Figure 3.39. Stress and strain distribution of loose specimen (Dr = 24%) simulations at peak shear


top cap rotation angle (,max)

Since top cap rocking induces compression on half the specimen and induces extension on the

other half, higher top cap rocking does not seem to affect the average stress-strain response.

Figure 3.40 and Figure 3.41 presents plots of (a) top cap rotation vs. shear strain, (b) effective

vertical stress vs. shear strain, (c) shear stress vs. shear strain, and (d) stress path for loose and

dense specimen simulations. The critical state line (CSL) with the constant volume friction angle

('cv) is also plotted with the stress paths. The plots present the average stress values over the top

area. As shown in Figure 3.40(a) and Figure 3.41(a), the maximum top cap rotation ranges between

0 to 0.27°. As shown in Figure 3.40(d) and Figure 3.41(d), higher top cap rotation has negligible

effect on the stress path. This is also shown in the plots of vertical stress vs. shear strain and shear

stress vs. shear strain that are presented in Figure 3.40(b) and Figure 3.41(b), and Figure 3.40(c)

and Figure 3.41(c). As such, higher top cap rocking has negligible effect on the average stress-

strain response.

(d) Left (x = -30.75 mm)


(b) Top (y = 19.5 mm)

(c) Top (y = 19.5 mm)

(a) Top (y = 20 mm)

58

Figure 3.40: Monotonic test results of loose specimen (Dr = 24%) for varying maximum top cap

rotation angle (,max)

Figure 3.41: Monotonic test results of dense specimen (Dr = 146%) for varying maximum top cap


(a)

(b)

(a)

(c)

(b)

(d)

CSL 'cv = 30°

(a)

(c)

(b)

(d)

59

3.7.2.2 Undrained cyclic simple shear


vertical stress (’vc) before applying cyclic shear loading of 12 kPa (CSR = 0.12). Loose and dense

specimens were sheared until the simulations reached liquefaction or the stress-strain response did

not vary significantly with more cycles due to locking behaviour. The liquefaction criterion and

locking behaviour were explained in more detail in section 3.3. The loose specimen simulations

were ended after 4 cycles when liquefaction occured. The dense specimen simulations were ended

after 40 cycles due to locking behaviour and did not reach liquefaction. Simulations allowed the

maximum top cap rotation possible which was found by allowing the top cap to rotate freely

without restraint. The maximum top cap rotation was about 0.14° for loose specimen simulations

and 0.035° for dense specimen simulations. Select results are presented here. All results are


Top cap rocking affects stress non-uniformities during the shearing phase. Figure 3.42 presents the

shear and vertical stress distribution of the loose specimen simulations at the peak amplitudes of

the first cycle. The vertical strain distribution along the top nodes of the specimen at the centre

cross section are also presented and follows a compression positive sign convention. Stress

distribution values are based on the average stress of elements along the top and sides of the

specimen at the centre cross section. Stress and strain distribution plots for the dense specimen

simulations are not presented here as the results are similar to loose specimen simulation results.

Figure 3.42(c) shows vertical stress concentrations near the specimen edges due to frictionless

vertical boundaries as previously discussed in section 3.5 on the effect of soil-ring friction. Vertical

stress concentrations are due to the lack of complementary shear stresses from frictionless vertical

boundaries, as shown in Figure 3.42(d). Similar vertical stress concentrations due to frictionless

vertical boundaries are found in dense specimen simulations as shown in Figure C.33 in

Appendix C. The vertical stress concentrations near the specimen edges are not due to the effect

of top cap rocking. The effect of top cap rocking is analyzed by comparing stress results of no

rotation with increased rotation. As shown in Figure 3.42(a), higher top cap rocking induces height

decrease (compression) on half the specimen and height increase (extension) for the other half. As

shown in Figure C.33(c), half the specimen that is compressed from rocking increases in vertical

stress while the other half that is extended reduces in vertical stress. Load reversal flips the

direction of rotation such that half the specimen that is compressed from rocking becomes

60

extended and vice-versa. Higher top cap rocking has negligible effect on the shear stress as shown

in Figure 3.42(b) and (d). Similar shear and vertical stress behaviour is found in dense specimen

simulations as shown in Figure C.33 in Appendix C. Higher top cap rocking affects stress

distribution in the same manner as monotonic tests as shown previously in Figure 3.39 and

Figure C.29.

Figure 3.42: Stress and strain distribution of loose specimen (Dr = 24%) simulations during peak

amplitudes (peak) of the first cycle for the centre cross section of the specimen (z = 0 mm), varying

maximum top cap rotation angle (,max)

Top cap rocking does not affect the average stress-strain response. Figure 3.43 and Figure 3.44

presents plots of (a) top cap rotation vs. number of cycles (b) shear strain vs. number of cycles, (c)

effective vertical stress vs. number of cycles, (d) shear stress vs. shear strain, and (e) stress path

for loose and dense specimen simulations. The critical state line (CSL) with the constant volume

friction angle ('cv) is also plotted with the stress paths when applicable. Higher top cap rotation

has negligible effect on the stress-strain response as shown in plots of shear strain vs. number of

cycles, vertical stress vs. number of cycles, shear stress vs shear strain, and stress path from

Figure 3.43(b) to (e) and Figure 3.44(b) to (e). However, as shown in Figure 3.44(b), there is a

bias in the direction of shear strain development for dense specimen simulation results with and

without top cap rocking. As previously shown in Figure 3.36 of section 3.6.2.2 for studying the

(d) Left (x = -30.75 mm)


(b) Top (y = 19.5 mm)

(c) Top (y = 19.5 mm)

(a) Top (y = 20 mm)

61

effect of vertical strain in cyclic loading, modelling the soil to top cap contact behaviour that allows

non-uniform vertical strain to develop at the top face may contribute to a bias in the direction of

shear strain development. A small top face rotation may be the cause of the bias in shear strain

direction. However, the bias in shear strain direction is not significant as the magnitude of shear

strain is much smaller than the liquefaction criterion of 1% single amplitude shear strain.


specimen (Dr = 24%) for varying maximum top cap rotation angle (,max)

(a)

(b)

(c)

(d)

(e)

62



3.8 Conclusions

3.8.1 Effect of Soil-Ring Friction

Simulations of the SGI type device using stacked rings suggest that typical high quality DSS

equipment adequately capture the ideal simple shear conditions. Increasing soil-ring friction

increases the magnitude of stress non-uniformities that develop adjacent to the vertical boundaries

during the consolidation phase. As a result, shear stresses are reduced during monotonic shear

loading and the vertical stress is reduced for the first few cycles during cyclic shear loading. Higher

soil-ring friction did improve stress uniformity during the shearing phase, but the effect on

consolidation stresses overwrites any advantages of having frictional vertical boundaries.

Practically, near frictionless vertical boundaries achieve the best comparison with ideal simple

shear conditions.

(a)

(b)

(c)

(d)

(e)

63

3.8.2 Effect of Vertical Compliance on Undrained Tests

Simulations of the SGI type device using stacked rings suggest that higher vertical compliance of

undrained tests can significantly affect the stress-strain response. Undrained monotonic

simulations of loose specimens (Dr = 24%) exhibited contractive behaviour (height decrease)

which increased the peak shear stress and corresponding vertical stress by 6% and 7% for ASTM

D6528-17 vertical strain tolerance of 0.05%. Undrained monotonic simulations of dense

specimens (Dr = 146%) exhibited dilative behaviour (height increase) which reduced each of the

shear and vertical stress at 4% shear strain by 1% for ASTM D6528-17 vertical strain tolerance of

0.05%. Vertical strains and their effects on DSS test measurements are likely much higher than

reported in the literature due to unmeasured apparatus deformations (i.e. top cap, base pedestal,

loading shaft, and two aluminum porous stones), which were shown to increase the vertical strain

in a typical direct simple shear apparatus by about 0.24%. This is based on an increase in vertical

stress from about 100 to 350 kPa during experimental monotonic tests. Accounting for ASTM

vertical strain tolerance of 0.05%, the vertical compliance may be as high as 0.29%. For loose

specimen simulations, vertical compliance of 0.28% resulted in contractive behaviour that

increased the peak shear stress and corresponding vertical stress by about 33% and 41%,

respectively. For dense specimen simulations, vertical compliance of 0.37% resulted in dilative

behaviour that reduced the shear and vertical stress at 4% shear strain by 8% and 7%, respectively.

Undrained cyclic simulations exhibited contractive behaviour. As such, higher vertical compliance

increases the vertical stress for each cycle in turn increasing the number of cycles to liquefaction.

Thus, vertical compliance of undrained tests needs to be minimized after accounting for apparatus

deformation to better approximate ideal simple shear conditions during undrained tests.

3.8.3 Effect of Top Cap Rocking During Shear

Simulations of the SGI type device using stacked rings shows that top cap rocking affects stress

non-uniformities in DSS specimens. At the top of the specimen, simulations showed that top cap

rotation induced height decrease (compression) for half the specimen that increased shear and

vertical stress, and induced height increase (extension) for half the specimen that reduced shear

and vertical stress. However, the change in stress distribution is negligible for the expected range

of top cap rotation. Higher top cap rocking within the expected range has negligible effect on the

64

average stress-strain response and is not a problem for approximating ideal simple shear conditions

in direct simple shear tests.

65

Chapter 4

Summary and Conclusions

4.1 Summary and Conclusions

This study investigated the effects of imperfect boundary conditions in the direct simple shear test

on undrained monotonic and cyclic stress-strain response. Imperfect boundary conditions do not

allow the direct simple shear test to achieve ideal simple shear conditions that are assumed for

model calibration. Imperfect boundary conditions were studied through numerical analysis of the

nearly ubiquitous SGI type direct simple shear (DSS) device that uses stacked rings for lateral

confinement. Results from the numerical study are meant to provide insight on the influence of

stress and strain non-uniformities due to imperfect boundary conditions on test measurements

rather than to precisely compute values for design purposes. Where experimental data were

available, simulation results were compared to them and shown to be in general agreement with

them.

4.1.1 Effect of Soil-Ring Friction

A major problem with DSS tests is its near frictionless vertical boundaries that cannot develop

complementary shear stresses necessary for ideal simple shear conditions. Simulations of the SGI

type device demonstrate that the results compares well with ideal simple shear conditions for the

expected range of soil-ring friction. Thus, near frictionless vertical boundaries in current testing

apparatuses are sufficient for approximating ideal simple shear conditions for the calibration of

constitutive models. Higher soil-ring friction did improve stress uniformity during the shearing

phase, but the effect on consolidation stresses overwrite any advantages of having frictional

vertical boundaries. Practically, near frictionless vertical boundaries that are currently used in

direct simple shear tests adequately capture ideal simple shear conditions.

66

4.1.2 Effect of Vertical Compliance on Undrained Tests

Active and passive height control systems can minimize but not eliminate vertical strain during

the shearing phase to approximate undrained (constant volume) conditions. Apparatus

deformation, such as those from the loading shaft, top cap, base pedestal, and porous stones, can

significantly increase the vertical strain during the shearing phase above ASTM D6528-17

tolerance of 0.05% and may not be corrected by the active height control system. These

unmeasured, or very difficult to measure deformations were shown to increase the vertical

compliance in a typical direct simple shear apparatus by about 0.24%. This is based on an increase

in vertical stress from about 100 to 350 kPa during experimental monotonic tests. Accounting for

ASTM vertical strain tolerance of 0.05%, the vertical compliance may be as high as 0.29%. For

loose specimen simulations, vertical compliance of 0.28% resulted in contractive behaviour that

increased the peak shear stress and corresponding vertical stress by about 33% and 41%,

respectively. For dense specimen simulations, vertical compliance of 0.37% resulted in dilative

behaviour that reduced the shear and vertical stress at 4% shear strain by 8% and 7%, respectively.

Simulations of the SGI type device demonstrate that high vertical compliance can significantly

affect the stress-strain response. Undrained cyclic simulations exhibited contractive behaviour. As

such, higher vertical compliance increased the vertical stress for each cycle which increased the

number of cycles to liquefaction.

4.1.3 Effect of Top Cap Rocking During Shear

A problem in DSS tests is top cap rocking that violates the assumption that top and bottom faces

remain parallel in ideal simple shear conditions. Simulations of the SGI type device using stacked

rings demonstrate that the top cap rotation affects the stress non-uniformities, but the change in

stress distribution is negligible for the expected range of top cap rotation. Additionally, higher top

cap rotation within the expected range has negligible effect on the measured stress-strain response.

Thus, top cap rocking for the expected amount of rotation appears not to be a problem in direct

simple shear tests.

4.1.4 Recommendations for Future Work

This study focuses on numerical analysis of imperfect boundary conditions in the direct simple

shear device and can be greatly enhanced by further validation against experimental studies.

67

Experimental studies can validate numerical results that show the effect of higher soil-ring friction

on consolidation stresses overwrite any advantages of having frictional vertical boundaries.

However, there are many challenges in the laboratory to control and measure friction between the

soil and vertical boundaries that needs to be overcome to study the effect of soil-ring friction. As

presented earlier, experimental studies exist on the effect of vertical compliance on undrained tests.

However, more attention is needed on the actual vertical strain of the soil specimen which may be

much higher than is reported after considering apparatus deformation. Numerical results suggest

that the effect of top cap rocking is negligible for the expected range of top cap rotation. Future

experimental studies can verify the expected range of top cap rotation for different apparatus

configurations and conduct a parametric study on the effect of top cap rotation on stress-strain

response. Finally, the current study can be expanded to include a parametric study on the effect of

friction between the soil and porous stones on stress-strain response. A primary function of the

porous stones at the top cap and base pedestal is to prevent soil slippage during shear. Numerical

and experimental studies can study the effect of porous stone material and design on stress-strain

response under monotonic and cyclic loading.

68

References

Airey, D.W., Budhu, M., and Wood, D.M. 1985. Some Aspects of the Behaviour of Soils in Simple

Shear. In Developments in Soil Mechanics and Foundation Engineering: Vol. 2: Stress-

Strain Modelling of Soils. Edited by P. Banerjee, K. and R. Butterfield. Elsevier Applied

Science Publishers. pp. 185-213.

ASTM D6528-17. 2017. Standard Test Method for Consolidated Undrained Direct Simple Shear

Testing of Fine Grain Soils. ASTM International, West Conshohocken, PA, USA.

Available from www.astm.org [accessed 29 May 2018].

Bernhardt, M.L., Biscontin, G., and O'Sullivan, C. 2016. Experimental validation study of 3D

direct simple shear DEM simulations. Soils and Foundations, 56(3): 336-347.

Bjerrum, L., and Landva, A. 1966. Direct Simple-Shear Tests on a Norwegian Quick Clay.

Géotechnique, 16(1): 1-20.

Boulanger, R.W., and Ziotopoulou, K. 2017. PM4Sand (Version 3.1): A sand plasticity model for

earthquake engineering applications - Report No. UCD/CGM-17/01. Center for

Geotechnical Modeling, Department of Civil and Environmental Engineering, University

of California, Davis, CA.

Boulanger, R.W., Chan, C.K., Seed, H.B., Seed, R.B., and Sousa, J.B. 1993. A Low-Compliance

Bi-directional Cyclic Simple Shear Apparatus. Geotechnical Testing Journal, 16(1): 36-45.

Bro, A.D., Stewart, J.P., and Pradel, D.E. 2013. Estimating Undrained Strength of Clays from

Direct Shear Testing at Fast Displacement Rates. In Geocongress 2013: Stability and

Performance of Slopes and Embankments III. Edited by C.L. Meehan and D. Pradel and

M.A. Pando and J.F. Labuz. ASCE, San Diego, California. pp. 106-119.

Budhu, M. 1984. Nonuniformities imposed by simple shear apparatus. Canadian Geotechnical

Journal, 21(1): 125-137.

Chang, W.J., Phantachang, T., and Ieong, W.M. 2016. Evaluation of size and boundary effects in

simple shear tests with distinct element modeling. Journal of GeoEngineering, 11(3): 133-

142.

Dafalias, Y.F., and Manzari, M.T. 2004. Simple plasticity sand model accounting for fabric change

effects. Journal of Engineering Mechanics, 130(6): 622-634.

Dassault Systèmes. 2012. ABAQUS Version 6.12 Unified Finite Element Analysis. Vélizy-

Villacoublay, France.

DeGroot, D.J., Germaine, J.T., and Ladd, C.C. 1993. The Multidirectional Direct Simple Shear

Apparatus. Geotechnical Testing Journal, 16(3): 283-295.

DeGroot, D.J., Germaine, J.T., and Ladd, C.C. 1994. Effect of Nonuniform Stresses on Measured

DSS Stress-Strain Behaviour. Journal of Geotechnical Engineering, 120(5): 892-911.

Doherty, J., and Fahey, M. 2011. Three-dimensional finite element analysis of the direct simple

shear test. Computers and Geotechnics, 38: 917-924.

Duku, P.M., Stewart, J.P., Whang, D.H., and Venugopal, R. 2007. Digitially controlled simple

shear apparatus for dynamic soil testing. Geotechnical Testing Journal, 30(5): 368-377.

http://www.astm.org/

69

Dyvik, R., and Suzuki, Y. 2018. Effect of Volume Change in Undrained Direct Simple Shear

Tests. Geotechnical Testing Journal, 42(4): 1075-1082.

doi:https://doi.org/10.1520/GTJ20170287.

Dyvik, R., Berre, T., Lacasse, S., and Raadim, B. 1987. Comparison of truly undrained and

constant volume direct simple shear tests. Géotechnique, 37(1): 3-10.

Geocomp. 2018. Below the Surface - New Addition to our Soil Testing System [online]. Available

from https://www.geocomp.com/files/newsletters/Q2-2018-Geocomp-Newsletter.pdf

[cited 17 April 2019].

Geocomp. 2019. Large Cyclic Direct Simple Shear (Cyclic DSS) Load Frame / Test System

[online]. Available from https://www.geocomp.com/Products/LabSystems/TT-Cyclic-

DSS-Large [cited 20 August 2019].

Gill, S.S. 2018. Minerology of a Sand Sample Using Scanning Electron Microscopy [Unpublished

manuscript]. University of Toronto, Department of Civil and Mineral Engineering,

Toronto, Canada.

Gudehus, G., Amorosi, A., Gens, A., Herle, I., Kolymbas, D., Masin, D., Wood, D.M., Niemunis,

A., Nova, R., Pastor, M., Tamagnini, C., and Viggiani, G. 2008. The soilmodels.info

project. International Journal of Numerical and Analytical Methods in Geomechanics,

32(12): 1571-1572.

Ishihara, K., and Yamazaki, F. 1980. Cyclic simple shear tests on saturated sand in multi-

directional loading. Soils and Foundations, 20(1): 45-59.

ITASCA Consulting Group Inc. 2019. FLAC3D Version 7.0 - Explicit continuum modeling of

non-linear material behaviour in 3D. Minneapolis, USA.

Kjellman, W. 1951. Testing the Shear Strength of Clay in Sweden. Géotechnique, 2(3): 225-232.

Kwan, W.S., Sideras, S., El Mohtar, C.S., and Kramer, S. 2014. Pore pressure generation under

different transient loading histories. In Proceedings of the 10th U.S. National Conference

on Earthquake Engineering: Frontiers of Earthquake Engineering. Earthquake Engineering

Research Institute, Anchorage, Alaska, USA.

Lucks, A.S., Christian, J.T., Brandow, G.E., and Höeg, K. 1972. Stress conditions in NGI simple

shear test. Journal of the Soil Mechanics and Foundations Division, 98(1): 155-160.

Mao, X., and Fahey, M. 2003. Behaviour of calcareous soils in undrained cyclic simple shear.

Géotechnique, 53(8): 715-727.

Masin, D., Martinelli, M., Miriano, C., and Tamagnini, C. 2018. SoilModels project - SANISAND

Abaqus UMAT and Plaxis implementations [online]. Available from

https://soilmodels.com/download/plaxis-umat-sanisand/ [cited 29 May 2018].

Niemunis, A. 2017. SoilModels project - Incremental Driver [online]. Available from

https://soilmodels.com/idriver/ [accessed 29 May 2018].

Rocscience Inc. 2019. RS3 - 3D finite element analysis for advanced analysis. Toronto, Canada.

Roscoe, K.H. 1953. An apparatus for the application of simple shear to soil samples. In

Proceedings of the 3rd International Conference on Soil Mechanics and Foundation

Engineering, Zurich, Switzerland. pp. 186-191.

70

Rutherford, C.J. 2012. Development of a Multi-Directional Direct Simple Shear Testing Device

for Characterization of the Cyclic Shear Response of Marine Clays. Ph.D. thesis,

Department of Civil Engineering, Texas A&M University, Texas, USA.

Saada, A.S., and Townsend, F.C. 1981. State of the Art: Labratory Strength Testing of Soils. In

STP740-EB Laboratory Shear Strength of Soil. Edited by R.N. Yong and F.C. Townsend.

ASTM, West Conshohocken, PA, USA. pp. 7-77.

Shafiee, A. 2016. Cyclic and Post-Cyclic Behavior of Sherman Island Peat. Ph.D. thesis,

Department of Civil Engineering, University of California, California, USA.

Shen, C.K., Sadigh, K., and Herrmann, L.R. 1978. An Analysis of NGI Simple Shear Apparatus

for Cyclic Soil Testing. In STP654-EB Dynamic Geotechnical Testing. Edited by M. Silver

and D. Tiedemann. ASTM, West Conshohocken, PA, USA. pp. 148-162.

Taiebat, M., and Dafalias, Y.F. 2008. SANISAND: Simple anisotropic sand plasticity model.

International Journal for Numerical and Analytical Methods in Geomechanics, 32(8): 915-

948.

Tatsuoka, F., and Haibara, O. 1985. Shear resistance between sand and smooth or lubricated

surfaces. Soils and Foundations, 25(1): 89-98.

Villet, W.C.B., Sitar, N., and Johnson, K.A. 1985. Simple shear tests on highly overconsolidated

offshore silts - Paper No OTC 4918. In Proceedings of the 17th annual Offshore

Technology Conference, Houston, USA. pp. 207-218.

Wijewickreme, D., Dabeet, A., and Byrne, P. 2013. Some Observations on the State of Stress in

the Direct Simple Shear Test Using 3D Discrete Element Analysis. Geotechnical Testing

Journal, 36(2): 292-299.

Zekkos, D., Athanasopoulos-Zekkos, A., Hubler, J., Fei, X., Zehtab, K.H., and Marr, W.A. 2018.

Development of a Large-Size Cyclic Direct Simple Shear Device for Characterization of

Ground Materials with Oversized Particles. Geotechnical Testing Journal, 41(2): 263-279.

71

Appendix A

Dafalias and Manzari (2004) Model Calibration

Results

72

A.1 Calibration of Undrained Monotonic Simple Shear Behaviour:

experimental data and Dafalias and Manzari (2004) model simulations

Figure A.1: Test HVS001M calibrated undrained monotonic simple shear results

73


74


75


76

A.2 Calibration of Undrained Cyclic Simple Shear Behaviour:

experimental data and Dafalias and Manzari (2004) model simulations

Figure A.5: Test HVS001 calibrated undrained cyclic simple shear results

77


78


79


80


81

A.3 Calibration of Undrained Cyclic Simple Shear Behaviour: Dafalias

and Manzari (2004) model simulations varying cyclic stress ratios (CSR)

to generate (CSR) vs. number of cycles to liquefaction plot (Figure 3.4)

Figure A.10: Test HVS005 undrained cyclic simple shear test simulation at CSR = 0.085

82


83


84


85

A.4 Calibration of Undrained Cyclic Simple Shear Behaviour: other

experimental data to generate cyclic stress ratio (CSR) vs. number of

cycles to liquefaction plot (Figure 3.4)

Figure A.14: Test HVS022 undrained cyclic simple shear experimental results at CSR = 0.12

86


87


88


89


90


91


92


93


94


95


96


97

Appendix B

Mesh Sensitivity Analysis Results

98

B.1 Frictionless Vertical Boundaries, Constant Height and No Top Cap

Rocking

Figure B.1: Undrained monotonic test results of loose specimen (Dr = 24%) varying element size


Figure B.2: Undrained monotonic test results of dense specimen (Dr = 146%) varying element size


(a) Stress path



at peak shear stress

(a) Stress path



at 4% shear strain

99

Figure B.3: Undrained cyclic test results of loose specimen (Dr = 24%) varying element size (ES)


Figure B.4: Undrained cyclic test results of dense specimen (Dr = 146%) varying element size (ES)


(a) Stress path




(a) Stress path




100

B.2 Effect of Soil-Ring Friction

Figure B.5: Stress path for (a) loose (Dr = 24%) and (b) dense (Dr = 146%) specimen monotonic

simulations for no slip condition (soil-ring friction angle of 30°), varying element size (ES)

Figure B.6: Stress path for (a) loose (Dr = 24%) and (b) dense (Dr = 146%) specimen cyclic

simulations for no slip condition (soil-ring friction angle of 30°), varying element size (ES)

(a) Loose specimen (b) Dense specimen

(a) Loose specimen (b) Dense specimen

101

B.3 Effect of Vertical Compliance on Undrained Tests


simulations for highest allowed vertical strain during the shearing phase, varying element size (ES)


simulations for highest allowed vertical strain during the shearing phase, varying element size (ES)

(a) Loose specimen (y,max = 0.28%) (b) Dense specimen (y,max = 0.37%)

(a) Loose specimen (y,max = 0.36%) (b) Dense specimen (y,max = 0.21%)

102

B.4 Effect of Top cap Rocking During Shear


simulations for highest top cap rotation (,max) in the analysis, varying element size (ES)


simulations for highest top cap rotation (,max) in the analysis, varying element size (ES)

(a) Loose specimen (,max = 0.24%) (b) Dense specimen (,max = 0.27%)

(a) Loose specimen (,max = 0.14%) (b) Dense specimen (,max = 0.035%)

103

Appendix C

Simulation Results

104

C.1 Effect of Soil-Ring Friction

C.1.1 Consolidation

Figure C.1: End of consolidation stress distribution of loose specimen (Dr = 24%) simulations for


Figure C.2: Coefficient of lateral earth pressure (K) during consolidation at the core of loose

specimen (Dr = 24%) simulations for varying soil-ring friction angle (soil-ring)



(c) Top (y = 19.5 mm)

core

105

Figure C.3: End of consolidation stress distribution of dense specimen (Dr = 146%) simulations

for the centre cross section of the specimen (z = 0 mm), varying soil-ring friction angle (soil-ring)

Figure C.4: Coefficient of lateral earth pressure (K) during consolidation at the core of dense

specimen (Dr = 146%) simulations for varying soil-ring friction angle (soil-ring)



(c) Top (y = 19.5 mm)

core

106

C.1.2 Undrained Monotonic Simple Shear

Figure C.5: Stress distribution of loose specimen (Dr = 24%) simulations at peak shear stress (4%

shear strain) for the centre cross section of the specimen (z = 0 mm), varying soil-ring friction

angle (soil-ring)

Figure C.6: Effective normal stress results for monotonic tests of loose specimen (Dr = 24%) for


(a) Top (y = 19.5 mm)

(b) Left (x = -30.75 mm)


(c) Top (y = 19.5 mm)

core


at core of specimen



107

Figure C.7: Monotonic test results of loose specimen (Dr = 24%) for varying soil-ring friction

angle (soil-ring)

Figure C.8: Stress distribution of dense specimen (Dr = 146%) simulations at 4% shear strain for


(a) (b)

(c) (d)

(a) Top (y = 19.5 mm)

(b) Left (x = -30.75 mm)


(c) Top (y = 19.5 mm)

108

Figure C.9: Effective normal stress results for monotonic tests of dense specimen (Dr = 146%) for


Figure C.10: Monotonic test results of dense specimen (Dr = 146%) for varying soil-ring friction

angle (soil-ring)

core


at core of specimen



(c) (d)

(a) (b)

109

Figure C.11: Monotonic test results of loose specimen (Dr = 24%) varying soil-ring friction angle

(soil-ring) during the shearing phase only

Figure C.12: Monotonic test results of dense specimen (Dr = 146%) varying soil-ring friction angle


(b) (a)

(c) (d)

(b) (a)

(c) (d)

110

C.1.3 Undrained Cyclic Simple Shear

Figure C.13: Stress distribution of loose specimen (Dr = 24%) simulations for the centre cross

section of the specimen (z = 0 mm) during the peak amplitude of the (a) first and (b) second cycle

for varying soil-ring friction angles (soil-ring)

(a-i) Top (y = 20 mm)


Right (x = 63.5 mm)



Right (x = 63.5 mm)

(b-i) Top (y = 20 mm)


(a) Peak amplitude of first cycle (b) Peak amplitude of second cycle

111

Figure C.14: Normal stress results for cyclic tests of loose specimen (Dr = 24%) for varying soil-


Figure C.15: Cyclic test results of loose specimen (Dr = 24%) for varying soil-ring friction angle

(soil-ring)


at core of specimen



core

(a)

(b)

(c)

(d)

(e)

112

Figure C.16: Stress distribution of dense specimen (Dr = 146%) simulations for the centre cross

section of the specimen (z = 0 mm) during the peak amplitude of the (a) first and (b) last cycle for


Figure C.17: Normal stress results for cyclic tests of dense specimen (Dr = 146%) for varying soil-


(a-i) Top (y = 20 mm)


Right (x = 63.5 mm)



Right (x = 63.5 mm)

(b-i) Top (y = 20 mm)


(a) Peak amplitude of first cycle (b) Peak amplitude of last cycle


at core of specimen



core

113

Figure C.18: Cyclic test results of dense specimen (Dr = 146%) for varying soil-ring friction angle

(soil-ring)

(a)

(b)

(c)

(d)

(e)

114

Figure C.19: Cyclic test results of loose specimen (Dr = 24%) for varying soil-ring friction angle


(a)

(b)

(c)

(d)

(e)

115

Figure C.20: Cyclic test results of dense specimen (Dr = 146%) for varying soil-ring friction angle


(a)

(b)

(c)

(d)

(e)

116

C.2 Effect of Vertical Compliance on Undrained Tests


Figure C.21: Stress and strain distribution of loose specimen (Dr = 24%) simulations at peak shear


vertical strain (y,max) during the shearing phase

(c) Top (y = 19.5 mm)

(b) Top (y = 19.5 mm)


(a) Top (y = 20 mm)

117

Figure C.22: Monotonic test results of loose specimen (Dr = 24%) for varying maximum vertical


Figure C.23: Stress and strain distribution of dense specimen (Dr = 146%) simulations at 4% shear

strain for the centre cross section of the specimen (z = 0 mm), varying maximum vertical strain

(y,max) during the shearing phase

(a)

(c)

(b)

(d)

(c) Top (y = 19.5 mm)

(b) Top (y = 19.5 mm)


(a) Top (y = 20 mm)

118

Figure C.24: Monotonic test results of dense specimen (Dr = 146%) for varying maximum vertical


(a)

(c)

(b)

(d)

119


Figure C.25: Stress-strain response for undrained cyclic simple shear simulations of loose

specimens (Dr = 24%) for varying maximum vertical strain (y,max) during the shearing phase

(a)

(b)

(c)

(d)

(e)

120

Figure C.26: Stress-strain response for undrained cyclic simple shear simulations of dense

specimen (Dr = 146%) for varying maximum vertical strain (y,max) during the shearing phase

(a)

(b)

(c)

(d)

(e)

121

C.3 Effect of Top Cap Rocking During Shear


Figure C.27: Stress and strain distribution of loose specimen (Dr = 24%) simulations at peak shear


top cap rotation angle (,max)

Figure C.28: Monotonic test results of loose specimen (Dr = 24%) for varying maximum top cap


(d) Left (x = -30.75 mm)


(b) Top (y = 19.5 mm)

(c) Top (y = 19.5 mm)

(a) Top (y = 20 mm)

(a)

(b) CSL 'cv = 30°

(a)

(c)

(b)

(d)

122

Figure C.29: Stress and strain distribution of dense specimen (Dr = 146%) simulations at 4% shear

strain for the centre cross section of the specimen (z = 0 mm), varying maximum top cap rotation

angle (,max)

Figure C.30: Monotonic test results of dense specimen (Dr = 146%) for varying maximum top cap


(d) Left (x = -30.75 mm)


(b) Top (y = 19.5 mm)

(c) Top (y = 19.5 mm)

(a) Top (y = 20 mm)

(a)

(b)

(a)

(c)

(b)

(d)

123


Figure C.31: Stress and strain distribution of loose specimen (Dr = 24%) simulations during peak



(d) Left (x = -30.75 mm)


(b) Top (y = 19.5 mm)

(c) Top (y = 19.5 mm)

(a) Top (y = 20 mm)

124

Figure C.32: Stress-strain response for undrained cyclic simple shear simulations of loose


(a)

(b)

(c)

(d)

(e)

125

Figure C.33: Stress and strain distribution of dense specimen (Dr = 146%) simulations during peak



(d) Left (x = -30.75 mm)


(b) Top (y = 19.5 mm)

(c) Top (y = 19.5 mm)

(a) Top (y = 20 mm)

126

Figure C.34: Stress-strain response for undrained cyclic simple shear simulations of dense


(a)

(b)

(c)

(d)

(e)

127

Dafalias and Manzari (2004) Constitutive Model

UMAT File

128

D.1 Source Code

Masin et al. (2018) implemented the Dafalias and Manzari (2004) model in a user-defined material

model (UMAT) file that can be used in ABAQUS simulations. The UMAT file is publicly

available from the database of the SoilModels project, previously known as the soilmodels.info

project, founded by Gudehus et al. (2008). The UMAT’s numerical stability was improved by

adjustments to the convergence criteria and it was verified by comparison to simulation results by

Dafalias and Manzari (2004).

D.2 UMAT File

c------------------------------------------------------------------------------

c complete file suite for Dafalias & Manzari (2004) SANISAND model for sand

c------------------------------------------------------------------------------

subroutine umat(stress,statev,ddsdde,sse,spd,scd,

& rpl,ddsddt,drplde,drpldt,

& stran,dstran,time,dtime,temp,dtemp,predef,dpred,cmname,

& ndi,nshr,ntens,nstatv,props,nprops,coords,drot,pnewdt,

& celent,dfgrd0,dfgrd1,noel,npt,layer,kspt,kstep,kinc)

c------------------------------------------------------------------------------

c user subroutine for Abaqus 6.3

c------------------------------------------------------------------------------

c

c Implemented constitutive law:

c -----------------------------

c Dafalias & Manzari(2004) SANISAND model for sand

c

c Ref: Dafalias, Y. F. and Manzari, M. T.

c Simple plasticity sand model accounting for fabric change effects

c J. Engng. Mechanics, ASCE (2004), 130(6):622-634.

c

c ----------------------------------------------------------------------------

c The string for the material name may contain 9 characters.

c ----------------------------------------------------------------------------

c Material constants:

c

c ---------------------------------------------------------------------

c props(j)

c ---------------------------------------------------------------------

c 1 p_a Atmospheric pressure

c 2 e0 Void ratio on CSL at p = 0

c 3 lambda CSL parameter (e:p plane)

c 4 xi CSL parameter (e:p plane)

c 5 M_c Slope of CSL in q:p plane, TX compression

c 6 M_e Slope of CSL in q:p plane, TX extension

c 7 mm opening of yield surface cone

c 8 G0 Shear modulus constant

c 9 nu Poisson's ratio

c 10 h0 Plastic modulus constant

c 11 c_h Plastic modulus constant

c 12 n_b Plastic modulus constant

129

c 13 A0 Dilatancy constant

c 14 n_d Dilatancy constant

c 15 z_max Fabric index constant

c 16 c_z Fabric index constant

c 17 bulk_w Pore water bulk modulus (undrained conditions)

c 18 p_tmult shift of mean stress pt=p_tmult*p_a

c 19 initial value of void ratio

c ----------------------------------------------------------------------

c

c Solution dependent state variables (statev):

c definition via sdvini

c

c group 1: internal variables (14 variables)

c

c 1 ... alpha_11 back stress, orientation of yield surface cone

c 2 ... alpha_22

c 3 ... alpha_33

c 4 ... alpha_12

c 5 ... alpha_13

c 6 ... alpha_23

c

c 7 ... void void ratio

c

c 8 ... Fab_11 fabric tensor z

c 9 ... Fab_22

c 10 ... Fab_33

c 11 ... Fab_12

c 12 ... Fab_13

c 13 ... Fab_23

c

c 14 ... not used

c

c group 2: memory variables for shear reversal (SR) and other purposes

c

c 15 ... alpha_sr_11 alpha value at stress reversal points (discrete update)

c 16 ... alpha_sr_22





c

c 21 ... not used

c 22 ... not used

c 23 ... not used

c 24 ... not used

c 25 ... not used

c 26 ... not used

c 27 ... not used

c

c 28 ... not used

c

c group 3: variables saved for post processing or other purposes

c

c 29 ... pore excess pore pressure (undrained case)

c 30 ... p mean effective stress

c 31 ... q deviator stress

c 32 ... z Lode parameter (cos(3theta))

130

c 33 ... dtsub suggested size of first time substep

c 34 ... nfev number of function evaluation

c 35 ... not used

c 36 ... not used

c

c Authors:

c C. Tamagnini ([email protected])

c Dipartimento di Ingegneria Civile e Ambientale

c Università degli Studi di Perugia, Italy

c

c M. Martinelli

c Dipartimento di Ingegneria Strutturale e Geotecnica

c Università di Roma "La Sapienza", Italy

c

c C. Miriano

c Dipartimento di Ingegneria Strutturale e Geotecnica

c Università di Roma "La Sapienza", Italy

c

c Modifications: D. Masin, 2015

c

c NOTES:

c - sign convention for stress and strain: tension and extension positive

c - stress and strain sign convention changed upon entering the SP algorithm

c - tangent stiffness operator evaluated according to two alternative options

c selected setting the logical flag "cons_lin":

c cons_lin.eq.0 -> numerical linearization via

c direct perturbation of dstran

c cons_lin.eq.1 -> continuum tangent stiffness

c (not optimal for full N-R iterative solver)

c cons_lin.eq.2 -> elastic tangent stiffness

c (even less optimal for full N-R iterative solver, but sometimes more stable)

c

c

c Last change: 4/2013

c

c----------------------------------------------------------------------------

c

implicit none

c

character*80 cmname

c

integer ntens, ndi, nshr, nstatv, nprops, noel, npt,

& layer, kspt, kstep, kinc

c

double precision stress(ntens), statev(nstatv),

& ddsdde(ntens,ntens), ddsddt(ntens), drplde(ntens),

& stran(ntens), dstran(ntens), time(2), predef(1), dpred(1),

& props(nprops), coords(3), drot(3,3), dfgrd0(3,3), dfgrd1(3,3)

double precision sse, spd, scd, rpl, drpldt, dtime, temp,

& dtemp, pnewdt, celent

c

c ... 1. nasvdim = maximum number of additional state variables

c 2. tolintT = prescribed error tolerance for the adaptive

c substepping scheme

c 3. maxnint = maximum number of time substeps allowed.

c If the limit is exceeded abaqus is forced to reduce

c the overall time step size (cut-back)

131

c 4. DTmin = minimum substeps size allowed.

c If the limit is exceeded abaqus is forced to reduce

c the overall time step size (cut-back)

c 5. perturb = perturbation par. for computation of Jacobian matrices

c 6. nfasv = number of first additional state variable in statev field

c 7. prsw = switch for printing information

c

c ... declaration of local variables

c

integer prsw,elprsw,cons_lin,abaqus,chiara,check_ff,drcor

c

integer i,error,maxnint,nfev,mario_DT_test,inittension

integer nparms,nasvdim,nfasv,nydim,nzdim

integer nasvy,nasvz,nyact,nzact,plastic,testing

c

double precision dot_vect

c

double precision parms(nprops),theta,tolintT,dtsub,DTmin,perturb

double precision sig_n(6),sig_np1(6),DDtan(6,6),pore,PI

double precision deps_np1(6),depsv_np1,norm_D2,norm_D,tolintTtest

double precision eps_n(6),epsv_n,alphayield(6)

double precision norm_deps2,norm_deps,pp,qq,cos3t,ddum

double precision zero,tol_f,fact_thres,p_thres,stran_lim,eps_debug

double precision p_atm,ptshift,phimob,tol_f_test,youngel,nuel

double precision avoid,apsi,aec,yf_DM,fyield,psi_void_DM,Mb

double precision dummy,sdev(6),I1,alpha(6),cM,tau(6),gth,etanorm

double precision sinphinorm

c

parameter (nasvdim = 36)

parameter (nydim = 6+14)

parameter (nzdim = 14)

parameter (tolintT = 1.00d-3)

parameter (tolintTtest = 1.0d-2)

c

parameter (maxnint = 50000)

parameter (DTmin = 1.0d-18)

parameter (perturb = 1.0d-4)

parameter (nfasv = 1)

parameter (prsw = 0)

parameter (cons_lin = 1)

parameter (abaqus = 0)

c chiara

parameter (eps_debug = 0.9d-3)

c

parameter (zero = 0.0d0)

parameter (PI = 3.14159265358979323846264338327950288)

parameter (fact_thres=0.000000001d0)

c

c ... additional state variables

c

double precision asv1(nydim-6),asv2(nzdim)

c

c ... solution vector (stresses, additional state variables)

c

double precision y(nydim),y_n(nydim),z(nzdim),z_n(nzdim)

c

c common /z_nct_errcode/error

132

c common /z_tolerance/tol_f

c common /z_check_yeld/check_ff

c common /z_drift_correction/drcor

c common /z_threshold_pressure/p_thres

c

tol_f=1.0d-6

tol_f_test=1.0d-6

check_ff=0

drcor=1

plastic=0

phimob=0.0d0

ptshift=0.0d0

c

c

c ... Error Management:

c ----------------

c error = 0 ... no problem in time integration

c error = 1 ... problems in evaluation of the time rate, (e.g. undefined

c stress state), reduce time integration substeps

c error = 3 ... problems in time integration, reduce abaqus load increment

c (cut-back)

c error=10 ... severe error, terminate calculation

c

error=0

c

c ... check problem dimensions

c

if (ndi.ne.3) then

c

write(6,*) 'ERROR: this UMAT can be used only for elements'

write(6,*) ' with 3 direct stress/strain components'

write(6,*) 'noel = ',noel

error=10

c

endif

c open(unit=6,position='Append',file=

c . 'C:/users/david/data/Zhejiang/plaxis-sani/UMATdebug.txt')

c

c ... check material parameters and move them to array parms

c

nparms=nprops

call check_parms_DM(props,parms,nparms)

c

c ... print informations about time integration, useful when problems occur

c

p_atm=parms(1)

p_thres=fact_thres*p_atm

c

elprsw = 0

if (prsw .ne. 0) then

c

c ... print only in some defined elements

c

c if ((noel.eq.101).and.(npt.eq.1)) elprsw = 1

c

endif

133

c

c ... define number of additional state variables

c

call define(nasvy,nasvz)

nyact = 6 + nasvy

nzact = nasvz

if (nyact.gt.nydim) then

write(6,*) 'ERROR:nydim too small; program terminated'

error=10

elseif (nzact.gt.nzdim) then

write(6,*) 'ERROR:nzdim too small; program terminated'

error=10

endif

c

c ... suggested time substep size, and initial excess pore pressure

c

pore = statev(29)

c

c ... changes sign conventions for stresses and strains, compute

c current effective stress tensor and volumetric strain

c

ptshift=parms(18)*parms(1)

do i=1,3

stress(i) = stress(i)-ptshift

enddo

call move_sig(stress,ntens,-1*ptshift,sig_n)

call move_sig(stress,ntens,pore,sig_n)

call move_eps(dstran,ntens,deps_np1,depsv_np1)

call move_eps(stran,ntens,eps_n,epsv_n)

norm_D2=dot_vect(2,deps_np1,deps_np1,6)

norm_D=sqrt(norm_D2)

c chiara

if (eps_n(1).gt.eps_debug) then

chiara=1

end if

c

c ... initialise void ratio and yield surface inclination

c

if (statev(7) .lt. 0.001) then

do i=1,6

alphayield(i)=zero

end do

call deviator(sig_n,alphayield,ddum,pp)

avoid=0

if(parms(19) .le. 5.0) then

avoid=parms(19)

else if(parms(19) .gt. 5.0) then

apsi=parms(19)-10.0d0

aec=parms(2)-parms(3)*(pp/parms(1))**parms(4)

avoid=aec+apsi

endif

statev(7)=avoid

do i=1,6

statev(i)=alphayield(i)/pp

134

statev(i+14)=alphayield(i)/pp

end do

end if

c

c ... move vector of additional state variables into asv1(nydim) and asv2(nzdim)

c

do i=1,nasvy

asv1(i) = statev(i-1+nfasv)

enddo

c

do i=1,nasvz

asv2(i) = statev(i-1+nfasv+nasvy)

enddo

c

c --------------------

c ... Time integration

c --------------------

c

call iniyz(y,nydim,z,nzdim,asv1,nasvy,asv2,nasvz,sig_n,ntens)

call push(y,y_n,nydim)

call push(z,z_n,nzdim)

c

if (elprsw.ne.0) then

write(6,*) '================================================='

write(6,*) ' Call of UMAT - DM SANISAND model: '

write(6,*) '================================================='

call wrista(3,y,nydim,deps_np1,dtime,coords,statev,nstatv,

& parms,nparms,noel,npt,ndi,nshr,kstep,kinc)

endif

c

c ... local integration using adaptive RKF-23 method with error control

c

if((dtsub.le.zero).or.(dtsub.gt.dtime)) then

dtsub = dtime

end if

testing=0

c For use in PLAXIS, activate the following line

c if(kstep.eq.1 .AND. kinc.eq.1) testing=1

c For use in ABAQUS, the line above should be inactive

if(norm_D.eq.0) testing=2

c FEM asking for ddsdde only

nfev = 0 ! initialisation

if(testing.eq.1) then

call rkf23_upd_DM(y,z,nyact,nasvy,nasvz,tolintTtest,maxnint,

& DTmin,deps_np1,parms,nparms,nfev,elprsw,

& mario_DT_test,

& error,tol_f_test,check_ff,drcor,p_thres,plastic)

c ... give original state if the model fails without substepping

if(error.ne.0) then

do i=1,nyact

y(i)=y_n(i)

end do

error=0

135

end if

else if(testing.eq.2) then

do i=1,nyact

y(i)=y_n(i)

end do

c ... Normal RKF23 integration

else !testing.eq.0

call rkf23_upd_DM(y,z,nyact,nasvy,nasvz,tolintT,maxnint,

& DTmin,deps_np1,parms,nparms,nfev,elprsw,

& mario_DT_test,

& error,tol_f,check_ff,drcor,p_thres,plastic)

end if

c

c ... error conditions (if any)

c

if(mario_DT_test.eq.1) then



endif

c

if(error.eq.3) then

c

c ... reduce abaqus load increment

c



write(6,*) 'subroutine UMAT: reduce step size in ABAQUS'

write(6,*) 'error 3 activated'

if(abaqus.ne.0) then

pnewdt = 0.25d0

else

c ... write a message and return the original state

do i=1,nyact

y(i)=y_n(i)

end do

endif

c

c if(abaqus.eq.0) then

c write(6,*) 'analysis ended because number of time '

c write(6,*) 'substeps exceeded maximum number allowed'

c write(6,*) ' (maxnint)'

c call xit_DM

c endif

return

c

elseif(error.eq.10) then

write(6,*) 'error 10 activated'

c



call xit_DM

c

endif

c

c ... update dtsub and nfev

136

c

if(dtsub.le.0.0d0) then

dtsub = 0

else if(dtsub.ge.dtime) then

dtsub = dtime

end if

statev(33)=dtsub

statev(34)=dfloat(nfev)

c

c ... computation of Jacobian matrix

c

error=0

if(cons_lin.eq.0) then

c

c ... parameter of the numerical differentiation

c double precision

c

norm_deps2=dot_vect(2,deps_np1,deps_np1,ntens)

norm_deps=dsqrt(norm_deps2)

theta=perturb*max(norm_deps,1.0d-6)

c

c ... compute consistent tangent via numerical perturbation

c

call pert_DM(y_n,y,z,nyact,nasvy,nasvz,tolintT,maxnint,DTmin,

& deps_np1,parms,nparms,nfev,elprsw,theta,ntens,DDtan,


c

else

c

call tang_stiff(y,z,nyact,nasvy,nasvz,parms,nparms,

& DDtan,cons_lin,


c

endif

c

if(error.ne.0) then

write(6,*) 'some error in pert or tang_stiff, value=',error

c

c call wrista(2,y,nydim,deps_np1,dtime,coords,statev,nstatv,

c & parms,nparms,noel,npt,ndi,nshr,kstep,kinc)

c call xit_DM

c

endif

c

c ... convert solution (stress + cons. tangent) to abaqus format

c update pore pressure and compute total stresses

inittension=0

c just checks for NAN

call check_RKF_DM(inittension,y,nyact,nasvy,parms,nparms)

if (inittension.ne.0) then

do i=1,nyact

y(i)=y_n(i)

end do

end if

c

137

call solout(stress,ntens,asv1,nasvy,asv2,nasvz,ddsdde,

+ y,nydim,z,pore,depsv_np1,parms,nparms,DDtan)

c

c ... updated vector of additional state variables to abaqus statev vector

c

do i=1,nasvy

statev(i-1+nfasv) = asv1(i)

end do

c

do i=1,nasvz

statev(i-1+nfasv+nasvy) = asv2(i)

enddo

c

c ... transfer additional information to statev vector

c

do i=1,6

sig_np1(i)=y(i)

end do

call inv_sig(sig_np1,pp,qq,cos3t)

c

statev(29) = pore

statev(30) = pp

statev(31) = qq

statev(32) = cos3t

cM=parms(6)/parms(5)

alpha(1)=y(7)

alpha(2)=y(8)

alpha(3)=y(9)

alpha(4)=y(10)

alpha(5)=y(11)

alpha(6)=y(12)

call deviator(sig_np1,sdev,I1,pp)

do i=1,6

tau(i)=sdev(i)-pp*alpha(i)

end do

call lode_DM(tau,cM,cos3t,gth,dummy)

etanorm=gth*qq/pp

sinphinorm=3*etanorm/(6+etanorm)

statev(33) = asin(sinphinorm)*180/PI

statev(34) = nfev

c check that bounding surtface is not violated

c if (noel.eq.324 .and. npt.eq.3) then

c fyield=yf_DM(y,nyact,parms,nparms)

c apsi=psi_void_DM(statev(7),pp,parms,nparms)

c Mb=parms(5)*dexp(-parms(12)*apsi)

c write(6,*) 'fyield=',fyield

c write(6,*) 'psi=',apsi

c write(6,*) 'Mb=',Mb

c write(6,*) 'qq/pp*gth=',qq/pp*gth

c write(6,*) '---------------------'

c end if

do i=1,3

stress(i) = stress(i)+ptshift

enddo

138

c close(6)

c

c -----------------------

c End of time integration

c -----------------------

c

return

end

c

c

cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

c

c

c-----------------------------------------------------------------------------

subroutine alpha_th_DM(flag,n,gth,psi,parms,nparms,alpha)

c-----------------------------------------------------------------------------

c calculate tensors:

c

c alpha_c => flag = 1

c alpha_b => flag = 2

c alpha_d => flag = 3

c

c Dafalias & Manzari (2004) SANISAND model for sands

c

c variables allocated in parms

c


















c

c written 10/2008 (Tamagnini)

c-----------------------------------------------------------------------------

implicit none

c

integer flag,nparms,i

c

double precision n(6),gth,psi,parms(nparms),alpha(6)

double precision M_c,mm,n_b,n_d

c

double precision M,alpha_th

double precision two,three,sqrt23

c

data two,three/2.0d0,3.0d0/

139

c

sqrt23=dsqrt(two/three)

c

c ... recover material parameters

c

M_c=parms(5)

mm=parms(7)

n_b=parms(12)

n_d=parms(14)

c

c ... select which alpha tensor to evaluate

c

if(flag.eq.1) then

c

c ... critical state cone

c

M=M_c

c

elseif(flag.eq.2) then

c

c ... bounding surface cone

c

M=M_c*dexp(-n_b*psi)

c

else

c

c ... dilatancy cone

c

M=M_c*dexp(n_d*psi)

c

endif

c

c ... tensor alpha_ij

c

alpha_th=M*gth-mm

c

do i=1,6

alpha(i)=sqrt23*alpha_th*n(i)

end do

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine check_crossing(y,y_tr,n,parms,nparms,prod)

c-----------------------------------------------------------------------------

c

c computes

c

c prod := dsig_tr*grad(f)_k

c

c useful for checking if crossing of yield locus occurs whenever

c f_k = 0 and f_tr > 0

140

c

c variables allocated in vector y(n):

c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c

c-----------------------------------------------------------------------------

implicit none

c

integer i,n,nparms

c


c

double precision y(n),y_tr(n),parms(nparms)

double precision P(6),P1(6),dsig_tr(6)

double precision prod

c

c ... gradient of yield surface at state y_k

c

call grad_f_DM(y,n,parms,nparms,P,P1)

c

do i=1,6

dsig_tr(i)=y_tr(i)-y(i)

end do ! i

c

prod=dot_vect(1,P,dsig_tr,6)

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine check_parms_DM(props,parms,nprops)

c-----------------------------------------------------------------------------

c checks input material parameters for Dafalias & Manzari (2004)

c SANISAND model for sand

141

c

c Material constants:

c

c ---------------------------------------------------------------------

c props(j)

c ---------------------------------------------------------------------


















c ---------------------------------------------------------------------

c

c Solution dependent state variables (statev):

c definition via sdvini

c

c group 1: internal variables (14 variables)

c


c 2 ... alpha_22

c 3 ... alpha_33

c 4 ... alpha_12

c 5 ... alpha_13

c 6 ... alpha_23

c


c

c 8 ... Fab_11 fabric tensor z

c 9 ... Fab_22

c 10 ... Fab_33

c 11 ... Fab_12

c 12 ... Fab_13

c 13 ... Fab_23

c

c 14 ... not used

c

c group 2: memory variables for shear reversal (SR) and other purposes

c







c

142

c 21 ... not used

c 22 ... not used

c 23 ... not used

c 24 ... not used

c 25 ... not used

c 26 ... not used

c 27 ... not used

c

c 28 ... not used

c

c group 3: variables saved for post processing or other purposes

c

c 29 ... pore excess pore pressure (undrained case)

c 30 ... p mean effective stress

c 31 ... q deviator stress

c 32 ... z Lode parameter (cos(3theta))

c 33 ... dtsub suggested size of first time substep

c 34 ... nfev number of function evaluation

c 35 ... not used

c 36 ... not used

c

c-----------------------------------------------------------------------------

implicit none

c

integer nprops

c

double precision props(nprops),parms(nprops)

double precision p_a,e0,lambda,xi,M_c,M_e,mm

double precision G0,nu,h0,c_h,n_b,A0,n_d,z_max

double precision c_z,bulk_w,sinphi,PI,sinphiext

c

double precision zero

c

parameter(zero=0.0d0)

parameter(PI=3.14159265358979323846264338327950288)

c

c ... recover material parameters and initial state info

c

p_a=props(1)

e0=props(2)

lambda=props(3)

xi=props(4)

M_c=props(5)

M_e=props(6)

mm=props(7)

G0=props(8)

nu=props(9)

h0=props(10)

c_h=props(11)

n_b=props(12)

A0=props(13)

n_d=props(14)

z_max=props(15)

c_z=props(16)

bulk_w=props(17)

c

143

c ... move vector props into local vector parms

c

call push(props,parms,nprops)

if(parms(5) .gt. 5) then

sinphi=sin(parms(5)/180*PI)

parms(5)=6*sinphi/(3-sinphi)

else

sinphi=3*parms(5)/(6+parms(5))

end if

if(parms(6) .gt. 5) then

sinphiext=sin(parms(6)/180*PI)

parms(6)=6*sinphiext/(3+sinphiext)

else if ((parms(6) .le. 5) .and. (parms(6) .gt. 0.01)) then

sinphiext=3*parms(6)/(6-parms(6))

else

parms(6)=parms(5)*(3-sinphi)/(3+sinphi)

end if

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine define(nasvy,nasvz)

c-----------------------------------------------------------------------------

implicit none

integer nasvy,nasvz

c

c number of additional state variables stored in vectors y and z

c must 14 each (otherwise change nasvdim in umat)

c

c components of ASV(i) stored in y (14 variables)

c


c 2 ... alpha_22

c 3 ... alpha_33

c 4 ... alpha_12

c 5 ... alpha_13

c 6 ... alpha_23


c 8 ... Fab_11 fabric tensor

c 9 ... Fab_22

c 10 ... Fab_33

c 11 ... Fab_12

c 12 ... Fab_13

c 13 ... Fab_23

c 14 ... not used

c

c components of ASV(i) stored in z (14 variables)

c




144




c 21 ... not used

c 22 ... not used

c 23 ... not used

c 24 ... not used

c 25 ... not used

c 26 ... not used

c 27 ... not used

c 28 ... not used

c

nasvy = 14

nasvz = 14

c

return

end

c

c


c

c

c------------------------------------------------------------------------------

subroutine deviator(t,s,trace,mean)

c------------------------------------------------------------------------------

c calculate deviator and trace of 2nd order tensor t(6)

c

c NOTE: Voigt notation is used with the following index conversion

c

c 11 -> 1

c 22 -> 2

c 33 -> 3

c 12 -> 4

c 13 -> 5

c 23 -> 6

c

c------------------------------------------------------------------------------

c

implicit none

c

double precision t(6),s(6),trace,mean

double precision one,three,onethird

c

data one,three/1.0d0,3.0d0/

c

c ... some constants

c

onethird=one/three

c

c ... trace and mean value

c

trace=t(1)+t(2)+t(3)

mean=onethird*trace

c

c ... deviator stress

c

s(1)=t(1)-mean

145

s(2)=t(2)-mean

s(3)=t(3)-mean

s(4)=t(4)

s(5)=t(5)

s(6)=t(6)

c

return

end

c

c


c

c

c------------------------------------------------------------------------------

double precision function distance(alpha_k,alpha,n)

c------------------------------------------------------------------------------

c computes distance function

c

c d = (alpha^k_{ij}-alpha_{ij})n_{ij} (k=sr,b,d)

c


c

c------------------------------------------------------------------------------

implicit none

c

integer i

c


c

double precision alpha_k(6),alpha(6),n(6),delta(6)

c

do i=1,6

delta(i)=alpha_k(i)-alpha(i)

end do

c

distance=dot_vect(1,delta,n,6)

c

return

end

c

c


c

c

c------------------------------------------------------------------------------

double precision function dot_vect(flag,a,b,n)

c------------------------------------------------------------------------------

c dot product of a 2nd order tensor, stored in Voigt notation

c

c flag = 1 -> vectors are stresses in Voigt notation

c flag = 2 -> vectors are strains in Voigt notation

c flag = 3 -> ordinary dot product between R^n vectors

c------------------------------------------------------------------------------

implicit none

integer i,n,flag

double precision a(n),b(n)

double precision zero,half,one,two,coeff

146

c

parameter(zero=0.0d0,half=0.5d0,one=1.0d0,two=2.0d0)

c

if(flag.eq.1) then

c

c ... stress tensor (or the like)

c

coeff=two

c

elseif(flag.eq.2) then

c

c ... strain tensor (or the like)

c

coeff=half

c

else

c

c ... standard vectors

c

coeff=one

c

end if

c

dot_vect=zero

c

do i=1,n

if(i.le.3) then

dot_vect = dot_vect+a(i)*b(i)

else

dot_vect = dot_vect+coeff*a(i)*b(i)

end if

end do

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine drift_corr_DM(y,n,z,nasvz,parms,nparms,tol,switch2,

& mario_DT_test,


c-----------------------------------------------------------------------------

c performs consistent drift correction (see Sloan et al. 2001)


c


c-----------------------------------------------------------------------------

implicit none

c


c

integer switch2,mario_DT_test

c

external matmul

147

double precision yf_DM

c

integer n,nasvz,nparms,i,n_drift,max_ndrift,switch

integer iter, itermax

integer error,check_ff,drcor,plastic

double precision tol_f,p_thres

c

double precision y(n),y0(n),y1(n), z(nasvz),parms(nparms)

double precision gradf(6),gradf1(6),gradg(6),gradg1(6)

double precision DDe(6,6),UU(6),VV(6),h_alpha(6),Kpm1,p1,pp1

double precision f0,tol,zero,one,denom,fnm1,p,three,onethird,f0_p

double precision factor,f1,p_atm

c

parameter(zero=0.0d0,one=1.0d0,three=3.0d0)

parameter(max_ndrift=10000, itermax=1000)

c


c

c ... initialize constants and vectors

c

call push(y,y0,n)

c

c ... check if current state is inside the elastic nucleus

c

c Chiara Miriano 15 maggio 2009

c onethird=one/three

c p=(y0(1)+y0(2)+y0(3))*onethird

c if(p.lt.zero) then

c do i=1,3

c y0(i)=y0(i)-p

c end do

c p=(y0(1)+y0(2)+y0(3))*onethird

c end if

c Chiara Miriano 15 maggio 2009

c

f0=yf_DM(y0,n,parms,nparms)

onethird=one/three

p=(y0(1)+y0(2)+y0(3))*onethird

n_drift=0

switch=0

f0_p=f0/p

c DWAI 10 October 2018

c Check error tol to be consistent with rest of UMAT,

c In the future may need to create function to be used by all UMAT to handle error tol calc

if(p.gt.one) f0_p=f0

c END DWAI

c p_atm=parms(1)

c if(p.lt.(p_atm/100)) f0_p=f0_p/1000000

c f0_p=f0

switch2=0

c

do while(f0_p.gt.tol)

cccc do while(f0.gt.tol)

148

c

fnm1=f0

c

c ... current state outside yield surface, correct it until f0 < ftol

c

n_drift=n_drift+1

c

c ... elastic stiffness and gradients of f and g

c

call el_stiff_DM(y0,n,parms,nparms,DDe,


call grad_f_DM(y0,n,parms,nparms,gradf,gradf1)

call grad_g_DM(y0,n,parms,nparms,gradg,gradg1)

c

c ... vectors UU=DDe*gradg and VV=DDe*gradf

c

call matmul(DDe,gradg1,UU,6,6,1)

call matmul(DDe,gradf1,VV,6,6,1)

c

c ... hardening function h_alpha and plastic modulus (1/Kp)

c

call plast_mod_DM(y0,n,z,nasvz,parms,nparms,h_alpha,Kpm1,

& switch2,mario_DT_test,


if (switch2.gt.zero) return

c

if(one/Kpm1.le.zero) then

c write(6,*) 'ERROR: subroutine DRIFT_CORR:'

write(6,*) 'subcritical softening condition'

c write(6,*) 'Kp = ',one/Kpm1

c error=10

error=3

c switch2=1

return

cc switch=1

end if

c

c ... correction for stress (y(1):y(6))

c

if(switch.eq.0) then

do i=1,6

y1(i)=y0(i)-Kpm1*f0*UU(i)

end do

c

c ... correction for hardening variable alpha (y(7):y(12))

c

do i=1,6

y1(6+i)=y0(6+i)+Kpm1*f0*h_alpha(i)

end do

do i=13,n

y1(i)=y0(i)

end do

c

c ... recompute drift at the new state

c


if(f0.gt.fnm1) then

149

switch=1

c switch2=1

p1=(y1(1)+y1(2)+y1(3))*onethird

c write(*,*)'switch2=',switch2

c write(*,*)'p=',p

c write(*,*)'p1=',p1

c return

else

call push(y1,y0,n)

end if

else

c

c ... normal correction in place of consistent correction

c

c call push(y,y0,n)

call push(y0,y1,n)


c f1=yf_DM(y1,n,parms,nparms)

denom=dot_vect(1,gradf,gradf,6)

factor=one

f1=f0

ccc iter=0

ccc do while(f1.ge.f0)

ccc iter=iter+1

ccc if (iter.gt.itermax) then

ccc error=10

ccc endif

do i=1,6

y1(i)=y0(i)-f0*gradf(i)/denom/factor

end do

do i=13,n

y1(i)=y0(i)

end do

pp1=(y1(1)+y1(2)+y1(3))*onethird

ccc factor=factor*2

if(pp1.lt.zero)then

c write(*,*)'pp1<0=',pp1

switch2=1

return

endif


ccc enddo

c write(*,*) 'drift_corr: normal correction'

call push(y1,y0,n)

c pause

end if

c

c ... recompute drift at the new state

c


p=(y0(1)+y0(2)+y0(3))*onethird

f0_p=f0/p

c DWAI 10 October 2018

150

c Check error tol to be consistent with rest of UMAT,

c In the future may need to create function to be used by all UMAT to handle error tol calc

if(p.gt.one) f0_p=f0

c END DWAI

c if(p.lt.(p_atm/100)) f0_p=f0_p/1000000

c f0_p=f0

c

if(n_drift.gt.max_ndrift) then

write(6,*) 'ERROR: subroutine DRIFT_CORR:'

write(6,*) 'too many iterations, increase tolerance'

write(6,*) 'n_drift = ',n_drift

write(6,*) 'drift = ',f0_p

c error=10

c error=3

c return

f0_p=0

end if

c ... bottom of while loop

c

c write(*,*) 'drift_corr: f0 = ',f0

c write(*,*) 'drift_corr: n_drift = ',n_drift

end do

c

c ... return corrected stress and q into vector y

c

call push(y0,y,n)

c

return

end

c

c-----------------------------------------------------------------------------

subroutine el_stiff_DM(y,n,parms,nparms,DDe,


c------------------------------------------------------------------------------

c subroutine to compute elastic stiffness

c Dafalias &Manzari SANISAND model (2004)

c

c material parameters

c

















151


c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c

c NOTE: soil mechanics convention (compression positive)

c all stress and strain vectors are 6-dimensional

c------------------------------------------------------------------------------

implicit none

c

integer i,j,n,nparms

c

double precision y(n),parms(nparms)

double precision p_a,G0,nu,ratio

double precision sig1,sig2,sig3,p,void

double precision coeff1,coeff2

double precision Kt,Gt,fe

double precision Id(6,6),IxI(6,6),DDe(6,6)

double precision zero,half,one,two,three

double precision pp,p_thres_E,tenm3

c



parameter(zero=1.0d0,half=0.5d0)

parameter(one=1.0d0,two=2.0d0,three=3.0d0)

parameter(p_thres_E=0.001d0)

cc parameter(tenm3=0.001d0)

c


c

c ... initialize matrices

c

call pzero(Id,36)

call pzero(IxI,36)

call pzero(DDe,36)

c

152

Id(1,1)=one

Id(2,2)=one

Id(3,3)=one

Id(4,4)=half

Id(5,5)=half

Id(6,6)=half

c

IxI(1,1)=one

IxI(2,1)=one

IxI(3,1)=one

IxI(1,2)=one

IxI(2,2)=one

IxI(3,2)=one

IxI(1,3)=one

IxI(2,3)=one

IxI(3,3)=one

c


c

p_a=parms(1)

G0=parms(8)

nu=parms(9)

c

c ... recover state variables

c

sig1=y(1)

sig2=y(2)

sig3=y(3)

c

void=y(13)

c

c ... mean stress

c

p=(sig1+sig2+sig3)/three

c

pp=p

ccc if(p.lt.p_thres_E)then

ccc pp=p_thres_E

ccc end if

if(p.lt.p_thres)then

pp=p_thres

end if

c

c ... max. shear modulus, tangent shear modulus Gt, tangent bulk modulus Kt

c

ratio=three*(one-two*nu)/(two*(one+nu))

fe=(2.97d0-void)*(2.97d0-void)/(one+void)

Gt=G0*p_a*fe*dsqrt(pp/p_a)

Kt=Gt/ratio

c

c ... elastic stiffness, stored in matrix DDe(6,6)

c

coeff1=Kt-two*Gt/three

coeff2=two*Gt

c

do i=1,6

do j=1,6

153

DDe(i,j)=coeff1*IxI(i,j)+coeff2*Id(i,j)

end do

end do

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine f_hypoelas_DM(y,n,parms,nparms,deps,F,


c-----------------------------------------------------------------------------

c

c ... computes the function F(y) for (hypo)elastic processes

c Dafalias & Manzari SANISAND Model (2004)

c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c

c variables allocated in vector z(nasvz):

c

c z(1) = alpha_sr(1)






c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

c z(12) = not used

c z(13) = not used

154

c z(14) = not used

c

c written by Tamagnini 10/2008

c

c-----------------------------------------------------------------------------

c

implicit none

c

external matmul

c

integer n,m,nparms

c

double precision y(n),parms(nparms),deps(6)

double precision depsv,void

double precision F(n),De(6,6),dsig_e(6)

double precision one



c

data one/1.0d0/

c

call pzero(F,n)

c

c ... void ratio and volum. strain increment

c

void = y(13)

depsv=deps(1)+deps(2)+deps(3)

c

c ... elastic stiffness matrix

c

call el_stiff_DM(y,n,parms,nparms,De,


c

call matmul(De,deps,dsig_e,6,6,1)

c

F(1)=dsig_e(1)

F(2)=dsig_e(2)

F(3)=dsig_e(3)

F(4)=dsig_e(4)

F(5)=dsig_e(5)

F(6)=dsig_e(6)

c

F(13)=-(one+void)*depsv

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine f_plas_DM(y,n,nasvy,z,nz,parms,nparms,deps,kRK,nfev,



c-----------------------------------------------------------------------------

155

c calculate coefficient kRK from current state (stored in y and z) and

c strain increment deps

c


c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c







c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

c z(12) = not used

c z(13) = not used

c z(14) = not used

c


c-----------------------------------------------------------------------------

implicit none

c

integer n,nz,nasvy,nparms,i,nfev

c


c

double precision y(n),z(nz),kRK(n),parms(nparms),deps(6)

double precision F_sig(6),F_q(nasvy)


156


c


c


c


c

c ... update counter for the number of function f(y) evaluations

c

nfev=nfev+1

c

c ... initialize kRK

c

call pzero(kRK,n)

c

c ... build F_sig(6) and F_q(nasv) vectors and move them into kRK

c

call get_F_sig_q(y,n,nasvy,z,nz,parms,nparms,deps,F_sig,F_q,

& switch2,mario_DT_test,error)

if(switch2.gt.zero) return

if(error.eq.10) return

c

do i=1,6

kRK(i)=F_sig(i)

end do

c

do i=1,nasvy

kRK(6+i)=F_q(i)

end do

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine get_F_sig_q(y,n,nasvy,z,nz,parms,nparms,deps,F_sig,F_q,

& switch2,mario_DT_test,error)

c-----------------------------------------------------------------------------

c

c computes vectors F_sigma and F_q in F(y)


c


c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

157

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c







c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

c z(12) = not used

c z(13) = not used

c z(14) = not used

c


c-----------------------------------------------------------------------------

implicit none

external matmul

c


c

integer nparms,n,nasvy,nz

c

double precision y(n),z(nz),parms(nparms),deps(6)

double precision Dep(6,6),HH(nasvy,6),F_sig(6),F_q(nasvy)





c

c ... compute tangent operators

c

call get_tan_DM(y,n,nasvy,z,nz,parms,nparms,Dep,HH,switch2,

& mario_DT_test,


if(switch2.gt.zero) then

158

c write(*,*) 'get_tan - switch2>0'

return

endif

c

c ... compute F_sig=Dep*deps

c

call matmul(Dep,deps,F_sig,6,6,1)

c

c ... compute F_q=HH*deps

c

call matmul(HH,deps,F_q,nasvy,6,1)

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine get_tan_DM(y,ny,nasvy,z,nz,parms,nparms,Dep,Hep,



c-----------------------------------------------------------------------------

c computes matrices Dep and Hep


c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c






159


c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

c z(12) = not used

c z(13) = not used

c z(14) = not used

c

c material properties allocated in vector parms(nparms):

c


















c

c NOTE: stress and strain convention: compression positive

c


c-----------------------------------------------------------------------------

implicit none

external matmul

c

integer nparms,ny,nz,nasvy,i,j,switch2,iter,iter_max,switch4

integer mario_DT_test

c

double precision dot_vect,distance,psi_void_DM

c

double precision y(ny),z(nz),parms(nparms)

double precision De(6,6),Dep(6,6),Hep(nasvy,6),m(6)

double precision LL(6),LL1(6),RR(6),RR1(6),U(6),V(6)

c

double precision p_a,e0,lambda,xi,M_c,M_e,cM,mm

double precision G0,nu,h0,c_h,n_b

double precision A0,n_d,z_max,c_z,bulk_w

c

double precision sig(6),alpha(6),void,Fab(6)

double precision alpha_sr(6),alpha_b(6)

double precision s(6),tau(6),n(6)

double precision norm2,norm,I1,p,psi,cos3t,gth,dgdth

double precision b0,d_sr,hh,db

double precision Hplas,LDeR,Kp,Kpm1

double precision mtrR,brack_mtrR,tol_ff,tol_dil,Hvs

160

double precision h_alpha(6),h_fab(6),HH_alpha(6,6),HH_fab(6,6)

double precision yf_DM,ff0,chvoid

c

double precision zero,tiny,half,one,two,three,large,kappa

double precision onethird,twothird



c

parameter(zero=0.0d0,one=1.0d0,two=2.0d0,three=3.0d0)

parameter(tiny=1.0d-15,large=1.0e15)

c Heaviside function parameter

parameter(kappa=3.0d2)

c


c

data m/1.0d0,1.0d0,1.0d0,0.0d0,0.0d0,0.0d0/

c

switch2=zero

switch4=zero

iter=0

iter_max=1e3

c


c

onethird=one/three

twothird=two/three

half=one/two

c

call pzero(Dep,36)

call pzero(Hep,6*nasvy)

c


c

p_a=parms(1)

e0=parms(2)

lambda=parms(3)

xi=parms(4)

M_c=parms(5)

M_e=parms(6)

mm=parms(7)

G0=parms(8)

nu=parms(9)

h0=parms(10)

c_h=parms(11)

n_b=parms(12)

A0=parms(13)

n_d=parms(14)

z_max=parms(15)

c_z=parms(16)

bulk_w=parms(17)

c

cM=M_e/M_c

c


c

do i=1,6

161

sig(i)=y(i)

end do !i

c

do i=1,6

alpha(i)=y(6+i)

end do !i

c

void=y(13)

c

do i=1,6

Fab(i)=y(13+i)

end do !i

c

do i=1,6

alpha_sr(i)=z(i)

end do !i

c

c ... deviator stress and mean pressure

c

call deviator(sig,s,I1,p)

c

c ... stress ratio tensor and unit vector n

c

do i=1,6

tau(i)=s(i)-p*alpha(i)

end do ! i

c

norm2=dot_vect(1,tau,tau,6)

norm=dsqrt(norm2)

if(norm.lt.tiny) then

norm=tiny

endif

do i=1,6

n(i)=tau(i)/norm

c if(n(i).lt.tiny)then

c n(i)=zero

c endif

end do

c

c ... elastic stiffness

c

call el_stiff_DM(y,ny,parms,nparms,De,


c

c ... gradient of yield function L and tensor V=De*L

c

call grad_f_DM(y,ny,parms,nparms,LL,LL1)

call matmul(De,LL1,V,6,6,1)

c

c ... gradient of plastic potential R and tensor U=De*R

c

call grad_g_DM(y,ny,parms,nparms,RR,RR1)

call matmul(De,RR1,U,6,6,1)

c

c ... plastic modulus functions b0 and hh

c

if (dabs(p).gt.zero) then

162

chvoid=c_h*void

if(chvoid.ge.1) then

c error=3

chvoid=0.99999

end if

b0=G0*h0*(one-chvoid)/dsqrt(p/p_a)

else

b0=large

end if

c

d_sr=distance(alpha,alpha_sr,n)

if (d_sr.lt.zero) then

call push(alpha,alpha_sr,6)

c write(*,*) 'alpha updated'

end if

c

if (d_sr.lt.tiny) then

d_sr=tiny

end if

c

hh=b0/d_sr

c

c ... hardening function h_alpha

c

psi=psi_void_DM(void,p,parms,nparms)

call lode_DM(tau,cM,cos3t,gth,dgdth)

call alpha_th_DM(2,n,gth,psi,parms,nparms,alpha_b)

db=distance(alpha_b,alpha,n)

c

do i=1,6

h_alpha(i)=twothird*hh*(alpha_b(i)-alpha(i))

end do

c

c ... hardening function h_fab

c

mtrR=-RR(1)-RR(2)-RR(3)

brack_mtrR=half*(mtrR+dabs(mtrR))

c

c chiara heaviside function

c

cc tol_dil=tol_ff*A0

c

c if((-tol_dil.lt.mtrR).and.(tol_dil.gt.mtrR)) then

c Hvs=one/(1+exp(two*mtrR*kappa))

c bracK_mtrR=Hvs*mtrR

c endif

c

do i=1,6

h_fab(i)=-c_z*brack_mtrR*(z_max*n(i)+Fab(i))

end do

c

c ... plastic moduli Hplas and Kp

c

Hplas=twothird*hh*p*db

if(Hplas.gt.1e+15) then

163

c write(*,*)'Hplas'

endif

c

LDeR=dot_vect(1,LL1,U,6)

c

Kp=LDeR+Hplas

ff0=yf_DM(y,ny,parms,nparms)

c .......................................................................

if(mario_DT_test.eq.zero) then

if(LDeR.lt.zero) then

switch2=1

c write(*,*)'LDeR < 0'

return

endif

if(Kp.lt.zero) then

switch2=1

c write(*,*)'function get_tan: Kp < zero'

return

endif

else

if(LDeR.le.zero) then

switch2=1

c error=3

c write(*,*)'subroutine get_tan_DM: LDeR < 0'

return

endif

endif

if(Kp.lt.zero)then

c write(6,*)'function get_tan: Kp < 0'

error=3

return

endif

call push(alpha_sr,z,6)

Kpm1=one/Kp

c if (Kpm1.lt.tiny) then

c Kpm1=zero

c end if

c

c ... elastoplastic stiffness matrix

c

do i=1,6

do j=1,6

Dep(i,j)=De(i,j)-Kpm1*U(i)*V(j)

end do !j

164

end do !i

c

c ... hardening tensor H_alpha

do i=1,6

do j=1,6

HH_alpha(i,j)=Kpm1*h_alpha(i)*V(j)

end do !j

end do !i

c

c ... hardening tensor H_fab

c

do i=1,6

do j=1,6

HH_fab(i,j)=Kpm1*h_fab(i)*V(j)

end do !j

end do !i

c

c ... Build tangent evolution matrix Hep(nasv,6) row-wise

c

do j=1,6

c

Hep(1,j) =HH_alpha(1,j) ! alpha(1)






Hep(7,j) =-(one+void)*m(j) ! void

Hep(8,j) =HH_fab(1,j) ! Fab(1)

Hep(9,j) =HH_fab(2,j) ! Fab(2)

Hep(10,j)=HH_fab(3,j) ! Fab(3)




c

end do !j

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine grad_f_DM(y,ny,parms,nparms,gradf,gradf1)

c------------------------------------------------------------------------------

c ... subroutine to compute the gradient of yield function at state y

c Dafalias & Manzari (2004) SNAISAND model for sands

c

c P(i) = grad(f) stress-like vector in Voigt notation

c P1(i) = grad(f) strain-like vector in Voigt notation

c

c ... variables allocated in parms

c



165
















c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c



c------------------------------------------------------------------------------

implicit none

c


c

integer ny,nparms,i

c

double precision parms(nparms),y(ny),gradf(6),gradf1(6),del(6)

double precision mm,sig(6),s(6),r(6),I1,p

double precision alpha(6),tau(6),n(6)

double precision norm,norm2,v,vv

double precision one,two,three,sqrt23,onethird,small

double precision n1,n2

c

parameter(one=1.0d0,two=2.0d0,three=3.0d0)

parameter(small=1.0d-10)

166

parameter(n1=0.816496580927739,n2=-0.40824829046385)

c

data del/1.0d0,1.0d0,1.0d0,0.0d0,0.0d0,0.0d0/

c


onethird=one/three

c

call pzero(n,6)

c


c

mm=parms(7)

c


c

sig(1)=y(1)

sig(2)=y(2)

sig(3)=y(3)

sig(4)=y(4)

sig(5)=y(5)

sig(6)=y(6)

c

alpha(1)=y(7)

alpha(2)=y(8)

alpha(3)=y(9)

alpha(4)=y(10)

alpha(5)=y(11)

alpha(6)=y(12)

c


c


c

c ... reduced stress tensor and unit vector

c

do i=1,6


end do ! i

c


norm=dsqrt(norm2)

c

if(norm.lt.small) then

norm=small

endif

c

do i=1,6

n(i)=tau(i)/norm

enddo

c

c norm_n=dot_vect(1,n,n,6)

c

c if(norm2.lt.small) then

c n(1)=n1

c n(2)=n2

c n(3)=n2

c endif

167

c

c

c ... coefficient V

c

if(dabs(p).lt.small) then

do i=1,6

r(i)=s(i)/small

enddo

else

do i=1,6

r(i)=s(i)/p

enddo

endif

v=dot_vect(1,r,n,6)

vv=-onethird*v

c

c ... gradient of f

c

do i=1,6

gradf(i)=n(i)+vv*del(i)

if(i.le.3) then

gradf1(i)=gradf(i)

else

gradf1(i)=two*gradf(i)

endif

enddo

c

return

end

c

c


c

c

c-----------------------------------------------------------------------------

subroutine grad_g_DM(y,ny,parms,nparms,gradg,gradg1)

c------------------------------------------------------------------------------

c ... subroutine to compute the gradient of plastic potential at state y


c

c gradg(i) = grad(g) stress-like vector in Voigt notation

c gradg1(i) = grad(g) strain-like vector in Voigt notation

c


c













168






c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1) (stress--like)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c



c------------------------------------------------------------------------------

implicit none

c

double precision dot_vect,distance,psi_void,psi_void_DM

c

integer ny,nparms,i

c

double precision M_c,M_e,cM,A0

c

double precision parms(nparms),y(ny),gradg(6),gradg1(6)

double precision sig(6),s(6),alpha(6),Fab(6),I1,p

double precision n(6),n2(6),tau(6),Rdev(6)

double precision Ad,alpha_d(6),dd

double precision cos3t,gth,dgdth

double precision void,psi,dil,dil3

c

double precision temp1,temp2,temp3,temp4

double precision norm,norm2

double precision zero,one,two,three,six

double precision half,sqrt6,onethird,small,del(6)

c

integer chiara

c

parameter(half=0.5d0,one=1.0d0,two=2.0d0,three=3.0d0,six=6.0d0)

parameter(zero=0.0d0,small=1.0d-10)

c

169

data del/1.0d0,1.0d0,1.0d0,0.0d0,0.0d0,0.0d0/

c

sqrt6=dsqrt(six)

onethird=one/three

c

call pzero(n,6)

c


c

M_c=parms(5)

M_e=parms(6)

A0=parms(13)

c

cM=M_e/M_c

c


c

sig(1)=y(1)

sig(2)=y(2)

sig(3)=y(3)

sig(4)=y(4)

sig(5)=y(5)

sig(6)=y(6)

c

alpha(1)=y(7)

alpha(2)=y(8)

alpha(3)=y(9)

alpha(4)=y(10)

alpha(5)=y(11)

alpha(6)=y(12)

c

void=y(13)

c

Fab(1)=y(14)

Fab(2)=y(15)

Fab(3)=y(16)

Fab(4)=y(17)

Fab(5)=y(18)

Fab(6)=y(19)

c


c


c

c ... stress ratio tensor, unit tensor n and tensor n^2

c

do i=1,6


end do ! i

c


norm=dsqrt(norm2)

c

if(norm.lt.small) then

norm=small

endif

do i=1,6

170

n(i)=tau(i)/norm

enddo

c

n2(1)=n(1)*n(1)+n(4)*n(4)+n(5)*n(5)

n2(2)=n(4)*n(4)+n(2)*n(2)+n(6)*n(6)

n2(3)=n(6)*n(6)+n(5)*n(5)+n(3)*n(3)

n2(4)=n(1)*n(4)+n(4)*n(2)+n(6)*n(5)

n2(5)=n(5)*n(1)+n(6)*n(4)+n(3)*n(5)

n2(6)=n(4)*n(5)+n(2)*n(6)+n(6)*n(3)

c

c ... state parameter psi

c


c

c ... Lode angle; functions g(theta) and (1/g)dg/dtheta

c


c

c ... vector Rdev

c C&M

temp1=one+three*cos3t*dgdth

temp2=-three*sqrt6*dgdth

do i=1,6

Rdev(i)=temp1*n(i)+temp2*(n2(i)-onethird*del(i))

enddo

c

c ... dilatancy function

c

temp3=dot_vect(1,Fab,n,6)

temp4=half*(temp3+dabs(temp3))

Ad=A0*(one+temp4)

c

call alpha_th_DM(3,n,gth,psi,parms,nparms,alpha_d)

dd = distance(alpha_d,alpha,n)

c Chiara

if((psi.gt.zero).and.(dd.lt.zero)) then

dd=zero

endif

c end Chiara

c

dil=Ad*dd

dil3=onethird*dil

c

do i=1,6

gradg(i)=Rdev(i)+dil3*del(i)

if(i.le.3) then

gradg1(i)=gradg(i)

else

gradg1(i)=two*gradg(i)

endif

enddo

c

return

end

c

c-----------------------------------------------------------------------------

subroutine iniyz(y,nydim,z,nzdim,qq1,nasvy,qq2,nasvz,sig,ntens)

171

c-----------------------------------------------------------------------------

c initializes the vectors of state variables

c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c







c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

c z(12) = not used

c z(13) = not used

c z(14) = not used

c

c-----------------------------------------------------------------------------

implicit none

c

integer i,nydim,nzdim,nasvy,nasvz,ntens

c

double precision y(nydim),z(nzdim)

double precision qq1(nasvy),qq2(nasvz),sig(ntens)

c

call pzero(y,nydim)

call pzero(z,nzdim)

c

do i=1,ntens

y(i) = sig(i)

enddo

172

c

do i=1,nasvy

y(6+i) = qq1(i)

enddo

c

do i=1,nasvz

z(i) = qq2(i)

enddo

c

return

end

c-----------------------------------------------------------------------------

subroutine intersect_DM(y0,y1,y_star,n,parms,nparms,tol_ff,

& xi,


c-----------------------------------------------------------------------------

c

c ... finds the intersection point between the stress path

c and the yield surface (Papadimitriou & Bouckovalas model)

c using Newton method

c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c







c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

173

c z(12) = not used

c z(13) = not used

c z(14) = not used

c


c


















c

c Tamagnini 10/2008

c

c-----------------------------------------------------------------------------

c

implicit none

c

integer n,nparms,maxiter,kiter,i,kiter_bis,bisect

c

double precision yf_DM,dot_vect

c

double precision parms(nparms),y0(n),y1(n),y_star(n),y05(n)

double precision tol_ff,fy_star,err,dfdxi,dfdxi_m1,xi,fy05

double precision dxi, xip1

double precision sig0(6),sig1(6),dsig(6),P_star(6),P1_star(6)

double precision zero,one,half,three,onethird

double precision pp_star,low, fy11, fy00, xi_max, xi_i, pp05

double precision y00(n),y11(n)



c

parameter(zero=0.0d0,one=1.0d0,half=0.5d0,three=3.0d0)

parameter(low=1.0d-10)

c


c

xi=one

maxiter=5000

kiter=0

bisect=0

kiter_bis=0

c

do i=1,6

174

sig0(i)=y0(i)

sig1(i)=y1(i)

dsig(i)=sig1(i)-sig0(i)

end do !i

c

call push(y1,y_star,n)

c

fy_star=yf_DM(y_star,n,parms,nparms)

onethird=one/three

pp_star=(y_star(1)+y_star(2)+y_star(3))*onethird

err=dabs(fy_star/pp_star)

if(pp_star.gt.one) err=dabs(fy_star)

c

c

c

if(bisect.eq.0) then

c

c

c ... start Newton iteration

c

do while ((err.gt.tol_ff).and.(bisect.eq.0))

c

kiter=kiter+1

c

call grad_f_DM(y_star,n,parms,nparms,P_star,P1_star)

c

dfdxi=dot_vect(1,P_star,dsig,6)

if (dfdxi.lt.low) then

ccc dfdxi=low

ccc write(*,*)'dfdxi.lt.zero'

bisect=1

endif

dfdxi_m1=one/dfdxi

c

c ... search direction

c

dxi=-dfdxi_m1*fy_star

xip1=xi+dxi

c

c Line search

c

do while ((xip1.lt.zero).or.(xip1.gt.one))

dxi=half*dxi

xip1=xi+dxi

end do

c

xi=xip1

c

c End Line search

c

c ... find new intersection point and yield function value

c

do i=1,n

y_star(i)=y0(i)+xi*(y1(i)-y0(i))

end do !i

c

fy_star=yf_DM(y_star,n,parms,nparms)

175

if (fy_star.lt.zero) then

c write(*,*)'fy_star.lt.zero'

bisect=1

else

onethird=one/three

pp_star=(y_star(1)+y_star(2)+y_star(3))*onethird

err=dabs(fy_star/pp_star)

if(pp_star.gt.one) err=dabs(fy_star)

cccc err=dabs(fy_star)

endif

c

if (kiter.gt.maxiter+1) then

write(6,*) 'ERROR: max no. of iterations exceeded'

write(6,*) 'Subroutine INTERSECT_DM'

write(6,*) 'err = ',err

c error=10

c error=3

c return

err=0

end if

end do ! bottom of Newton iteration

c

c ... check that 0 < xi < 1 (intersection point between initial and final states)

c

if((xi.lt.zero).and.(xi.gt.one)) then

c

write(6,*) 'ERROR: the intersection point found lies'

write(6,*) ' outside the line connecting initial'

write(6,*) ' and trial stress states'

write(6,*) 'Subroutine INTERSECT_DM'

write(6,*) 'xi = ',xi

xi = zero

c error=10

c error=3

return

c

endif

endif

c

if(bisect.eq.1) then

c ... start bisection method

ccc write(*,*) 'bisection method'

c

c find f((a+b)/2)

do i=1,n

y00(i)=y0(i)

y11(i)=y1(i)

enddo

fy00 =yf_DM(y00,n,parms,nparms)


do i=1,n

y05(i)=y0(i)

enddo

176

pp05=(y05(1)+y05(2)+y05(3))*onethird


cccc err = dabs(fy05)

err=abs(fy05/pp05)

if(pp05.gt.one) err=dabs(fy05)

do while(err.gt.tol_ff)

kiter_bis=kiter_bis+1

c

do i=1,6

y05(i)=half*(y00(i)+y11(i))

enddo


pp05=(y05(1)+y05(2)+y05(3))*onethird

cccc err = dabs(fy05)

err=abs(fy05/pp05)

if(pp05.gt.one) err=dabs(fy05)

if(fy05.lt.zero) then

call push(y05,y00,n)

else

call push(y05,y11,n)

endif

if (kiter_bis.gt.maxiter+1) then

write(6,*) 'ERROR: max no. of iterations exceeded'

write(6,*) 'Subroutine INTERSECT_DM - bisection'

write(6,*) 'err = ',err

err=0

c error=10

c error=3

c return

endif

enddo

do i=1,n

y_star(i)=y05(i)

enddo

c xi= (y05(1)-y0(1))/(y1(1)-y0(1))

xi_max=zero

do i=1,6

if((y1(i)-y0(i)).ne.zero) then

xi_i= (y05(i)-y0(i))/(y1(i)-y0(i))

if(xi_i.gt.xi_max) then

xi_max = xi_i

endif

endif

enddo

xi = xi_max

c ... end bisection method

c

endif

177

c

return

end

c

c------------------------------------------------------------------------------

subroutine inv_sig(sig,pp,qq,cos3t)

c------------------------------------------------------------------------------

c calculate invariants of stress tensor

c Dafalias &Manzari SANISAND model (2004)

c


c

c 11 -> 1

c 22 -> 2

c 33 -> 3

c 12 -> 4

c 13 -> 5

c 23 -> 6

c

c------------------------------------------------------------------------------

c

implicit none

c

double precision sig(6),sdev(6),s2(6)

double precision I1,J2bar,J2bar_sq,J3bar,trs2,trs3

double precision pp,qq,cos3t,numer,denom

c

double precision zero,one,two,three

double precision onethird,half,onept5,sqrt3,tiny

c


c

data zero,one,two,three/0.0d0,1.0d0,2.0d0,3.0d0/

data tiny/1.0d-15/

c


c

onethird=one/three

half=one/two

onept5=three/two

sqrt3=dsqrt(three)

c

c ... trace and mean stress

c

I1=sig(1)+sig(2)+sig(3)

pp=onethird*I1

c


c

sdev(1)=sig(1)-pp

sdev(2)=sig(2)-pp

sdev(3)=sig(3)-pp

sdev(4)=sig(4)

sdev(5)=sig(5)

sdev(6)=sig(6)

c

178

c ... second invariants

c

trs2=dot_vect(1,sdev,sdev,6)

J2bar=half*trs2

qq=dsqrt(onept5*trs2)

c

c ... components of (sdev_ij)(sdev_jk) (stress-like Voigt vector)

c

s2(1)=sdev(1)*sdev(1)+sdev(4)*sdev(4)+sdev(5)*sdev(5)






c

c ... Lode angle

c

if(trs2.lt.tiny) then

c

cos3t=one

c

else

c

trs3=dot_vect(1,sdev,s2,6)

c

J3bar=onethird*trs3

J2bar_sq=dsqrt(J2bar)

numer=three*sqrt3*J3bar

denom=two*(J2bar_sq**3)

cos3t=numer/denom

if(dabs(cos3t).gt.one) then

cos3t=cos3t/dabs(cos3t)

end if

c

end if

c

return

end

c

c------------------------------------------------------------------------------

subroutine lode_DM(r,cM,cos3t,gth,dgdth)

c------------------------------------------------------------------------------

c calculate cos(3*theta) from deviatoric stress ratio tensor

c tau(i) = s(i)-p*alpha(i) stored in vector r(6)

c

c computes functions g(theta) and (1/g)dg/dtheta from:

c a) Argyris function (Argyris=1) (as in the original paper)

c b) Van Eekelen function (Argyris=0) (more appropriate for high friction angles)

c

c gth = g(theta)

c dgdth = (1/g)dg/dtheta

c

c NOTE: Voigt notation is used with the following index conversion (ABAQUS)

c

c 11 -> 1

c 22 -> 2

c 33 -> 3

179

c 12 -> 4

c 13 -> 5

c 23 -> 6

c

c SM stress convention: compression positive

c

c------------------------------------------------------------------------------

c

implicit none

c

integer Argyris

c

double precision r(6),r2(6)

double precision trr2,trr3,J2bar,J3bar,J2bar_sq

double precision cM,n_VE,n_VEm1,numer,denom,cos3t

c

double precision tmp1,tmp2,tmp3,tmp4,tmp5,tmp6

double precision alpha,beta,gth,dgdth

double precision one,two,three

double precision onethird,half,sqrt3,tiny

c


c

data one,two,three/1.0d0,2.0d0,3.0d0/

data tiny,n_VE/1.0d-15,-0.25d0/

data Argyris/0/

c


c

onethird=one/three

half=one/two

sqrt3=dsqrt(three)

c

c ... second invariant

c

trr2=dot_vect(1,r,r,6)

J2bar=half*trr2

c

c ... components of (r_ij)(r_jk) (stress-like Voigt vector)

c

r2(1)=r(1)*r(1)+r(4)*r(4)+r(5)*r(5)

r2(2)=r(4)*r(4)+r(2)*r(2)+r(6)*r(6)

r2(3)=r(6)*r(6)+r(5)*r(5)+r(3)*r(3)

r2(4)=r(1)*r(4)+r(4)*r(2)+r(6)*r(5)

r2(5)=r(5)*r(1)+r(6)*r(4)+r(3)*r(5)

r2(6)=r(4)*r(5)+r(2)*r(6)+r(6)*r(3)

c

c ... Lode angle

c

if(trr2.lt.tiny) then

c

cos3t=one

c

else

c

trr3=dot_vect(1,r,r2,6)

c

180

J3bar=onethird*trr3

J2bar_sq=dsqrt(J2bar)

numer=three*sqrt3*J3bar

denom=two*(J2bar_sq**3)

cos3t=numer/denom



end if

c

end if

c

c ... g function and its derivative

c

if (Argyris.ne.0) then

c

c ... Argyris function

c

gth=two*cM/((one+cM)-(one-cM)*cos3t)

dgdth=(1-cM)*gth/(two*cM)

c

else

c

c ... Van Eekelen function

c

n_VEm1=one/n_VE

c

tmp1=one/(two**n_VE)

tmp2=cM**n_VEm1

tmp3=one+tmp2

tmp4=one-tmp2

c

alpha=tmp1*(tmp3**n_VE)

beta=tmp4/tmp3

c

tmp5=(one+beta*cos3t)**n_VE

tmp6=one+beta*cos3t

c

gth=alpha*tmp5

dgdth=n_VE*beta/tmp6

c

end if

c

return

end

c

c------------------------------------------------------------------------------

subroutine matmul(a,b,c,l,m,n)

c------------------------------------------------------------------------------

c matrix multiplication

c------------------------------------------------------------------------------

implicit none

c

integer i,j,k,l,m,n

c

double precision a(l,m),b(m,n),c(l,n)

c

do i=1,l

181

do j=1,n

c(i,j) = 0.0d0

do k=1,m

c(i,j) = c(i,j) + a(i,k)*b(k,j)

enddo

enddo

enddo

c

return

end

c

c-----------------------------------------------------------------------------

subroutine move_eps(dstran,ntens,deps,depsv)

c-----------------------------------------------------------------------------

c Move strain/strain increment stran/dstran into eps/deps,

c computes volumetric strain/strain increment and switches

c sign convention from solid to soil mechanics

c

c NOTE:

c stran = strain tensor (extension positive)

c eps = strain tensor (compression positive)

c epsv = vol. strain (compression positive)

c dstran = strain increment tensor (extension positive)

c deps = strain increment tensor (compression positive)

c depsv = vol. strain increment (compression positive)

c

c eps/deps has always 6 components

c-----------------------------------------------------------------------------

implicit none

integer ntens,i

double precision deps(6),dstran(ntens),depsv

c

call pzero(deps,6)

c

do i=1,ntens

deps(i) = -dstran(i)

enddo

c

depsv=deps(1)+deps(2)+deps(3)

c

return

end

c

c-----------------------------------------------------------------------------

subroutine move_sig(stress,ntens,pore,sig)

c-----------------------------------------------------------------------------

c Computes effective stress from total stress (stress) and pore pressure (pore)

c and switches sign convention from solid to soil mechanics

c

c NOTE: stress = total stress tensor (tension positive)

c pore = exc. pore pressure (undrained conds., compression positive)

c sig = effective stress (compression positive)

c

c sig has always 6 components

c-----------------------------------------------------------------------------

implicit none

integer ntens,i

182

double precision sig(6),stress(ntens),pore

c

call pzero(sig,6)

c

do i=1,ntens

if(i.le.3) then

sig(i) = -stress(i)-pore

else

sig(i) = -stress(i)

end if

enddo

c

return

end

c

c-----------------------------------------------------------------------------

subroutine norm_res_DM(y_til,y_hat,ny,norm_R)

c-----------------------------------------------------------------------------

c evaluate norm of residual vector Res=||y_hat-y_til||


c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c-----------------------------------------------------------------------------

implicit none

c

integer ny,i

c

double precision y_til(ny),y_hat(ny)

double precision err(ny),norm_R2,norm_R

double precision sig_hat(6),sig_til(6),del_sig(6)

double precision alpha_hat(6),alpha_til(6),del_alpha(6)

double precision Fab_hat(6),Fab_til(6),del_Fab(6)

double precision void_hat,void_til,del_void

double precision norm_sig2,norm_alpha2,norm_Fab2

183

double precision norm_sig,norm_alp,norm_Fab

double precision dot_vect,zero

c


c

call pzero(err,ny)

c

c ... recover stress tensor and internal variables

c

do i=1,6

sig_hat(i)=y_hat(i)

sig_til(i)=y_til(i)

del_sig(i)=dabs(sig_hat(i)-sig_til(i))

end do

c

do i=1,6

alpha_hat(i)=y_hat(6+i)

alpha_til(i)=y_til(6+i)

del_alpha(i)=dabs(alpha_hat(i)-alpha_til(i))

end do

c

void_hat=y_hat(13)

void_til=y_til(13)

del_void=dabs(void_hat-void_til)

c

do i=1,6

Fab_hat(i)=y_hat(13+i)

Fab_til(i)=y_til(13+i)

del_Fab(i)=dabs(Fab_hat(i)-Fab_til(i))

end do

c

c ... relative error norms

c

norm_sig2=dot_vect(1,sig_hat,sig_hat,6)

norm_alpha2=dot_vect(1,alpha_hat,alpha_hat,6)

norm_Fab2=dot_vect(1,Fab_hat,Fab_hat,6)

norm_sig=dsqrt(norm_sig2)

norm_alp=dsqrt(norm_alpha2)

norm_Fab=dsqrt(norm_Fab2)

c

if(norm_sig.gt.zero) then

do i=1,6

err(i)=del_sig(i)/norm_sig

end do

end if

c

if(norm_alp.gt.zero) then

do i=1,6

err(6+i)=del_alpha(i)/norm_alp

end do

end if

c

err(13)=del_void/void_hat

c

c if(norm_Fab.gt.zero) then

c do i=1,6

c err(13+i)=del_Fab(i)/norm_Fab

184

c end do

c end if

c

c chiara 4 maggio 2010

c

do i=1,6

if((Fab_til(i).ne.zero).and.(norm_Fab.gt.zero)) then

err(13+i)=del_Fab(i)/norm_Fab

end if

end do

c

c ... global relative error norm

c

norm_R2=dot_vect(3,err,err,ny)

norm_R=dsqrt(norm_R2)

c

return

end

c

c-----------------------------------------------------------------------------

subroutine pert_DM(y_n,y_np1,z,n,nasvy,nasvz,err_tol,

& maxnint,DTmin,deps_np1,parms,

& nparms,nfev,elprsw,theta,ntens,DD,


c-----------------------------------------------------------------------------

c compute numerically consistent tangent stiffness

c Dafalias & Manzari (2004) SANISAND model for sand

c


c-----------------------------------------------------------------------------

implicit none

c

integer elprsw

c

integer ntens,jj,kk

integer n,nasvy,nasvz,nparms,nfev

integer maxnint,mario_DT_test

c

double precision y_n(n),y_np1(n),y_star(n),z(nasvz),parms(nparms)

double precision err_tol

double precision theta,DTmin

double precision deps_np1(6),deps_star(6)

double precision dsig(6),DD(6,6)

double precision zero,three



c

parameter(zero=0.0d0,three=3.0d0)

c


c common /z_plastic_flag/plastic

c

c ... initialize DD and y_star

c

call pzero(DD,36)

call pzero(y_star,n)

185

c

if(plastic.eq.0) then

c

c ... elastic process, DD = De (explicit)

c

call el_stiff_DM(y_np1,n,parms,nparms,DD,


c

else

c

c ... plastic process, DD computed using numerical perturbation

c loop over strain basis vectors

c

do jj=1,ntens

call push(y_n,y_star,n)

call push(deps_np1,deps_star,ntens)

c

c do jj=1,ntens

c

c ... perturbed strain increment

c

deps_star(jj)=deps_star(jj)+theta

c

c ... perturbed final state, stored in y_star

c

if(error.ne.10) then

call rkf23_upd_DM(y_star,z,n,nasvy,nasvz,err_tol,maxnint,

& DTmin,deps_star,parms,nparms,nfev,elprsw,

& mario_DT_test,


end if

c

c ... stiffness components in column jj (kk = row index, jj = column index)

c

do kk=1,ntens

dsig(kk)=y_star(kk)-y_np1(kk)

DD(kk,jj)=dsig(kk)/theta

end do !kk

c

end do !jj

c

end if

c

return

end

c

c-----------------------------------------------------------------------------

subroutine plast_mod_DM(y,ny,z,nz,parms,nparms,h_alpha,Kpm1,



c-----------------------------------------------------------------------------

c computes vector h_hard and plastic modulus Kp


c


c

c y(1) = sig(1)

186

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c







c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

c z(12) = not used

c z(13) = not used

c z(14) = not used

c

c material properties allocated in vector parms(nparms):

c


















c

187

c

c NOTE: stress and strain convention: compression positive

c


c-----------------------------------------------------------------------------

implicit none

external matmul

c

integer nparms,ny,nz,i,iter,switch2,mario_DT_test

c

double precision dot_vect,distance,ref_db,psi_void,psi_void_DM

c

double precision y(ny),z(nz),parms(nparms)

double precision De(6,6),h_alpha(6)

double precision LL(6),LL1(6),RR(6),RR1(6),U(6),V(6)

c

double precision p_a,e0,lambda,xi,M_c,M_e,cM,mm

double precision G0,nu,h0,c_h,n_b

double precision A0,n_d,z_max,c_z,bulk_w

c

double precision sig(6),alpha(6),void,Fab(6)

double precision alpha_sr(6),alpha_b(6)

double precision s(6),tau(6),n(6),I1,p,psi,cos3t

double precision b0,d_sr,hh,db,gth,dgdth

double precision HHp,LDeR,Kp,Kpm1,norm2,norm,chvoid

c

double precision zero,tiny,one,two,three,large

double precision twothird



c


parameter(tiny=1.0d-15,large=1.0e15)

c


c


c

twothird=two/three

c


c

p_a=parms(1)

e0=parms(2)

lambda=parms(3)

xi=parms(4)

M_c=parms(5)

M_e=parms(6)

mm=parms(7)

G0=parms(8)

nu=parms(9)

h0=parms(10)

c_h=parms(11)

n_b=parms(12)

A0=parms(13)

n_d=parms(14)

188

z_max=parms(15)

c_z=parms(16)

bulk_w=parms(17)

c

cM=M_e/M_c

c

switch2=zero

iter=0

c


c

do i=1,6

sig(i)=y(i)

end do !i

c

do i=1,6

alpha(i)=y(6+i)

end do !i

c

void=y(13)

c

do i=1,6

Fab(i)=y(13+i)

end do !i

c

do i=1,6

alpha_sr(i)=z(i)

end do !i

c


c


c

c ... stress ratio tensor and unit vector n

c

do i=1,6


end do ! i

c


norm=dsqrt(norm2)

if(norm.lt.tiny) then

norm=tiny

endif

do i=1,6

n(i)=tau(i)/norm

c if(n(i).lt.tiny)then

c n(i)=zero

c endif

end do

c

c ... elastic stiffness

c

call el_stiff_DM(y,ny,parms,nparms,De,


c

c ... gradient of yield function L and tensor V=De*L

189

c

call grad_f_DM(y,ny,parms,nparms,LL,LL1)

call matmul(De,LL1,V,6,6,1)

c

c ... gradient of plastic potential R and tensor U=De*R

c

call grad_g_DM(y,ny,parms,nparms,RR,RR1)

call matmul(De,RR1,U,6,6,1)

c

c ... plastic modulus functions b0 and hh

c

if (dabs(p).gt.zero) then

chvoid=c_h*void

if(chvoid.ge.1) then

c error=3

chvoid=0.99999

end if

b0=G0*h0*(one-chvoid)/dsqrt(p/p_a)

else

b0=large

end if

c

d_sr=distance(alpha,alpha_sr,n)

if (d_sr.lt.zero) then

call push(alpha,alpha_sr,6)

c write(*,*) 'alpha updated'

end if

c

if (d_sr.lt.tiny) then

d_sr=tiny

end if

c

hh=b0/d_sr

c

c ... hardening function h_alpha

c



call alpha_th_DM(2,n,gth,psi,parms,nparms,alpha_b)

db=distance(alpha_b,alpha,n)

c

do i=1,6

h_alpha(i)=twothird*hh*(alpha_b(i)-alpha(i))

end do

c

c ... plastic moduli HHp and Kp

c

HHp=twothird*hh*p*db

if(HHp.gt.1e+15) then

c write(*,*)'Hplas'

endif

c

LDeR=dot_vect(1,LL1,U,6)

c

190

Kp=LDeR+HHp

c .......................................................................

if(mario_DT_test.eq.zero) then

if(LDeR.lt.zero) then

switch2=1

c write(*,*)'LDeR < 0'

return

endif

if(Kp.lt.zero) then

switch2=1

c write(*,*)'function plast_mod: Kp < zero'

return

endif

else

if(LDeR.le.zero) then

switch2=1

c error=3

c write(*,*)'subroutine plast_mod_DM: LDeR < 0'

return

endif

endif

if(Kp.lt.zero)then

c write(6,*)'function plast_mod: Kp < zero'

error=3

return

endif

call push(alpha_sr,z,6)

Kpm1=one/Kp

c

return

end

c

c------------------------------------------------------------------------------

double precision function psi_void_DM(void,p,parms,nparms)

c------------------------------------------------------------------------------

c computes state parameter psi (pyknotropy factor)

191


c------------------------------------------------------------------------------

implicit none

c

integer nparms

c

double precision void,p,parms(nparms)

double precision p_a,e0,lambda,xi,ec

c


c

p_a=parms(1)

e0=parms(2)

lambda=parms(3)

xi=parms(4)

c

ec=e0-lambda*(p/p_a)**xi

psi_void_DM=void-ec

c

return

end

c

c-----------------------------------------------------------------------------

subroutine push(a,b,n)

c-----------------------------------------------------------------------------

c push vector a into vector b

c-----------------------------------------------------------------------------

implicit none

integer i,n

double precision a(n),b(n)

c

do i=1,n

b(i)=a(i)

enddo

c

return

end

c

subroutine pzero(v,nn)

c

c-----[--.----+----.----+----.-----------------------------------------]

c Purpose: Zero real array of data

c Inputs:

c nn - Length of array

c Outputs:

c v(*) - Array with zero values

c-----[--.----+----.----+----.-----------------------------------------]

implicit none

integer n,nn

double precision v(nn)

do n = 1,nn

v(n) = 0.0d0

end do ! n

192

end

c

c-----------------------------------------------------------------------------

subroutine rkf23_upd_DM(y,z,n,nasvy,nasvz,err_tol,maxnint,DTmin,

& deps_np1,parms,nparms,nfev,elprsw,

& mario_DT_test,


c-----------------------------------------------------------------------------


c

c numerical solution of y'=f(y)

c explicit, adapive RKF23 scheme with local time step extrapolation

c


c

c y(1) = sig(1)

c y(2) = sig(2)

c y(3) = sig(3)

c y(4) = sig(4)

c y(5) = sig(5)

c y(6) = sig(6)

c y(7) = alpha(1)

c y(8) = alpha(2)

c y(9) = alpha(3)

c y(10) = alpha(4)

c y(11) = alpha(5)

c y(12) = alpha(6)

c y(13) = void

c y(14) = Fab(1)

c y(15) = Fab(2)

c y(16) = Fab(3)

c y(17) = Fab(4)

c y(18) = Fab(5)

c y(19) = Fab(6)

c y(20) = not used

c


c







c z(7) = not used

c z(8) = not used

c z(9) = not used

c z(10) = not used

c z(11) = not used

c z(12) = not used

c z(13) = not used

c z(14) = not used

c

c-----------------------------------------------------------------------------

implicit none

193

c

integer elprsw,mario,switch2,switch3,mario2

integer mario_DT, mario_DT_test

c

integer n,nasvy,nasvz,nparms,i,ksubst,kreject,nfev

integer maxnint,attempt,maxnint_1



c

double precision y(n),z(nasvz),deps_np1(6)

double precision parms(nparms),DTmin,err_tol,err_tol_1, err_tol_n

double precision zero,half,one,two,three,four,six

double precision ptnine,one6,one3,two3,temp,prod,pt1

double precision z1(nasvz),deps_np1_star(6), z_k(nasvz)

c

double precision y_k(n),y_tr(n),y_star(n),yf_DM,y_k1(n)

double precision y_2(n),y_3(n),y_til(n),y_hat(n)

double precision p_atm,tol_ff,ff_tr,ff_k

double precision T_k,DT_k,xi

double precision kRK_1(n),kRK_2(n),kRK_3(n)

double precision norm_R,S_hull

double precision Fab(6),dev_fab(5),I1,f_p,absfp2

double precision pp,onethird,ptone,p_thres2,tol_ff1,pp_k,pp_tr

double precision ff_k_pp_k,ff_tr_pp_tr,pp_3,pp_2,pp_hat,ten,min_y_tr

double precision iter, pp_kk

c


parameter(four=4.0d0,six=6.0d0,half=0.5d0,ptnine=0.9d0)

parameter(pt1=1.0d-3,ptone=0.1d0,ten=1.0d1)

c parameter(drcor=1)

c



c common /z_tolerance/tol_f

c common /z_check_yield/check_ff

c common /z_drift_correction/drcor


c

c

c ... initialize y_k vector and other variables

c

call pzero(y_k,n)

c

one6=one/six

one3=one/three

two3=two/three

c

plastic=0

mario = 0

mario_DT=0

mario_DT_test=0

c

iter=iter+1

ccccc write(*,*)'iter =', iter

ccccc if(iter.gt.145) then

194

ccccc write(*,*)'stop'

ccccc endif

c

c ... start of update process

c

call push(y,y_k,n)

call push(z,z_k,nasvz)

c

c ... set tolerance for yield function

c

p_atm=parms(1)

tol_ff=tol_f*p_atm

c

c ... yield function at current state, T = 0.0

c

c if (check_ff) then

c

ff_k=yf_DM(y_k,n,parms,nparms)

onethird=one/three

pp_k=(y_k(1)+y_k(2)+y_k(3))*onethird

ff_k_pp_k=ff_k/pp_k

if(pp_k.gt.one) ff_k_pp_k=ff_k

c

c ... abort execution if initial state is outside the YS

c

c if (check_ff) then

c

if (ff_k_pp_k.gt.tol_ff) then

cccc if (ff_k.gt.tol_ff) then

c write(6,*) 'ERROR: initial state is outside the YS'

c write(6,*) 'Subroutine RKF23_UPDATE'

c write(*,*) 'f = ',ff_k_pp_k

c write(*,*) 'f = ',ff_k

c error=10

c error=3

c return

c

call drift_corr_DM(y_k,n,z1,nasvz,parms,nparms,tol_ff,



end if

c endif

c

c ... compute trial solution (single step) and update z

c

deps_np1_star=deps_np1

call trial_state(y_k,n,parms,nparms,deps_np1_star,y_tr,


pp_tr=(y_tr(1)+y_tr(2)+y_tr(3))*onethird

if((pp_k.gt.(p_thres+p_thres))) then

do while(pp_tr.le.p_thres)




195

deps_np1_star=deps_np1_star*half

c write(*,*) 'pp_k>p_thres'

end do

elseif((pp_k.le.(p_thres+p_thres))

&.and.(pp_tr.gt.(p_thres+p_thres))) then

deps_np1_star=deps_np1




c write(*,*) 'pp_k<=p_thres and p_tr>p_thres'

elseif((pp_k.le.(p_thres+p_thres))

&.and.(pp_tr.le.(p_thres+p_thres))) then

call push(y_k,y_tr,n)

write(*,*) 'pp_k<=p_thres and pp_tr<=p_thres'

endif

c

c ... yield function at trial state

c

ff_tr=yf_DM(y_tr,n,parms,nparms)


ff_tr_pp_tr=ff_tr/pp_tr

if(pp_tr.gt.one) ff_tr_pp_tr=ff_tr

c

c ... compute scalar product of dsig_tr and grad(f)

c to check if crossing of yl occurs

c

call check_crossing(y_k,y_tr,n,parms,nparms,prod)

c

c ... check whether plastic loading, elastic unloading or mixed ep loading

c

if (ff_tr_pp_tr.lt.tol_ff) then

cccc if (ff_tr.lt.tol_ff) then

c

c ... Case 1: Elastic unloading inside the yield function:

c trial state is the final state

c

call push(y_tr,y_k,n)

c

else

c

c ... Case 2: Some plastic loading occurs during the step

c

c ................................................................................................................

c

if(pp_tr.lt.p_thres) then

c (situation not admissible)

c

ccc p_thres2=ptone*p_thres

c

write(*,*) 'Low mean pressure, p=',pp_k

c

c

c........................................................................................................

c

196

c

else

c

if ((ff_k_pp_k.lt.(-tol_ff)).or.(prod.lt.zero)) then

cccc if ((ff_k.lt.(-tol_ff)).or.(prod.lt.zero)) then

c

c ... Case 2a: the initial part of the stress path is inside the YL;

c find the intersection point and update current state at the YL

call intersect_DM(y_k,y_tr,y_star,n,parms,nparms,tol_ff,xi,


call push(y_star,y_k,n)

c

c Chiara 27.08.09 beginning

c

plastic=1

c

c Chiara 27.08.09 end

c

else

c

c Case 2b: all the stress path lies outside the YL

c

xi=zero

c

c Chiara 27.08.09 beginning

c

plastic=1

c

c Chiara 27.08.09 end

c

end if

c

c

c initialize normalized time and normalized time step

c

c

T_k=xi

DT_k=(one-xi)

ksubst=0

kreject=0

nfev=0

attempt=1

maxnint_1=maxnint

err_tol_1=err_tol

err_tol_n=err_tol

switch3=0

c

c ... start substepping

c

ccc mario = zero

c

do while((T_k.lt.one).and.(mario.eq.zero)

&.and.(mario_DT.eq.zero)) !**********************************

c

ccc.and.(attempt.ne.3)

ksubst=ksubst+1

c

197

c ... write substepping info

c

cccc if(iter.gt.145)then

c if(ksubst.gt.82830) then

c write(*,*) ksubst,T_k,DT_k,pp_hat

c endif

cccc endif

c

c ... check for maximum number of substeps

c

c

if((ksubst.gt.maxnint_1).or.(switch3.eq.1)) then

c

if(attempt.eq.1) then

maxnint_1=2.0*maxnint

err_tol_1=1000.0*err_tol

write(*,*) 'I had to increase tolintT,', 'T_k=', T_k

c call push(y_hat,y_k,n)

c call push(z1,z,nasvz)

c call push(y_k,y,n)

attempt=2

DT_k=1-T_k

c return

c

elseif(attempt.eq.2) then

ccc write(*,*) 'number of substeps ',ksubst

ccc write(*,*) 'is too big, step rejected'

write(*,*) 'attempt number ',attempt

ccc write(*,*) 'T_k =',T_k

ccccc error=3

ccccc write(*,*) 'mario1 = one'

mario=one

call push(z1,z,nasvz)

call push(y_k,y,n)

ccccc attempt=3

return

endif

endif

c

c

c ... build RK functions

c

call push(z_k,z1,nasvz)

cc call push(y_k,y_k1,nasvy)

pp_kk=(y_k(1)+y_k(2)+y_k(3))*onethird

c

call f_plas_DM(y_k,n,nasvy,z1,nasvz,parms,nparms,

+ deps_np1,kRK_1,nfev,switch2,mario_DT_test,



if (switch2.eq.zero) then

c

c ... find y_2

198

c

temp=half*DT_k

c

do i=1,n

y_2(i)=y_k(i)+temp*kRK_1(i)

end do

c

pp_2=(y_2(1)+y_2(2)+y_2(3))*onethird

c

if(pp_2.gt.zero)then

cc if((y_2(1).gt.zero).and.(y_2(2).gt.zero).and.(y_2(3).gt.zero))then

c

call f_plas_DM(y_2,n,nasvy,z1,nasvz,parms,nparms,





c

c ... find y_3

c

do i=1,n

y_3(i)=y_k(i)-DT_k*kRK_1(i)+two*DT_k*kRK_2(i)

end do

c

pp_3=(y_3(1)+y_3(2)+y_3(3))*onethird

c

if(pp_3.gt.zero)then

cc if((y_3(1).gt.zero).and.(y_3(2).gt.zero).and.(y_3(3).gt.zero))then

c

call f_plas_DM(y_3,n,nasvy,z1,nasvz,parms,nparms,





c

c ... approx. solutions of 2nd (y_til) and 3rd (y_hat) order

c

do i=1,n

y_til(i)=y_k(i)+DT_k*kRK_2(i)

y_hat(i)=y_k(i)+DT_k*

& (one6*kRK_1(i)+two3*kRK_2(i)+one6*kRK_3(i))

end do

c

c ... local error estimate

c

call norm_res_DM(y_til,y_hat,n,norm_R)

c

c ... time step size estimator according to Hull

c

S_hull=ptnine*DT_k*(err_tol/norm_R)**one3

c

if(norm_R.eq.zero) then

ccc write(*,*)'RKF23: norm_R=0'

ccc error=3

endif

c

199

c

c

c

if ((norm_R.lt.err_tol).and.(attempt.ne.2).and.

& (attempt.ne.3)) then

c

c ... substep is accepted, update y_k and T_k and estimate new substep size DT_k

c

pp_hat=(y_hat(1)+y_hat(2)+y_hat(1))*onethird

if (pp_hat.lt.p_thres) then

write(*,*) 'mario_2 = one'

cccccc error =3

mario=one

endif

c

c ... correct drift from yield surface using Sloan algorithm

c

if(drcor.ne.0) then

call drift_corr_DM(y_hat,n,z1,nasvz,parms,nparms,tol_ff,



end if



call push(y_hat,y_k,n)

call push(z1,z_k,nasvz)

T_k=T_k+DT_k;

DT_k=min(four*DT_k,S_hull)

DT_k=min((one-T_k),DT_k)

endif

c

c

end if

c

c

if ((norm_R.lt.err_tol_1).and.(attempt.eq.2)

& .and.(switch2.eq.zero)) then

c


c



write(*,*) 'mario_3 = one'

mario=one

cccccc error =3

endif

c


c

cc if (switch2.eq.zero) then

200

if(drcor.ne.0) then




end if




T_k=T_k+DT_k;



endif !switch2.eq.zero

cc endif !switch2.eq.zero

c

c

end if !(norm_R.lt.err_tol_1).and.(attempt.eq.2).and.(switch2.eq.zero)

c

c

c

if((norm_R.gt.err_tol).and.(attempt.ne.3)

& .and.(switch2.eq.zero)) then

c

c ... substep is not accepted, recompute with new (smaller) substep size DT

c

c write(*,*) 'substep size ',DT_k,' norm_R =',norm_R

c

DT_k=max(DT_k/four,S_hull)

c

c ... check for minimum step size

c

if(DT_k.lt.DTmin) then

write(*,*) 'substep size ',DT_k,

& ' is too small, step rejected'

DT_k= one - T_k

mario2=1

cc err_tol_n=100.0*err_tol

write(*,*) 'err_tol_n=100*err_tol=',err_tol_n

switch3=1

c write(*,*) 'mario_4 = one'

c error =3

end if

c

end if

c

if((norm_R.lt.err_tol_n).and.(attempt.ne.3)

& .and.(switch2.eq.zero).and.(mario2.eq.one)) then

c


c



write(*,*) 'mario_31 = one; pp_hat<p_thres'

mario=one

201

cccccc error=3

endif

c


c

cc if (switch2.eq.zero) then

if(drcor.ne.0) then




end if




T_k=T_k+DT_k;



endif !switch2.eq.zero

cc endif !switch2.eq.zero

mario2=zero

cc

end if

c

c

if (attempt.eq.3) then

c

write(*,*) 'Third attempt: solution accepted'

c


c

if(drcor.ne.0) then

call drift_corr_DM(y_k,n,z_k,nasvz,parms,nparms,tol_ff,



endif


c

T_k=T_k+DT_k;

c

end if

c

c

c ... if switch2=1 (if switch2=1 occurs in accepted solution part)

if (switch2.ne.zero) then

DT_k=DT_k/four


DT_k= one - T_k

ccccccccc mario_DT=1

mario_DT_test=1

write(*,*) 'mario_DT - switch2>0, mario5 = one'

202

ccccccccc error =3

ccccccccc return

cccccc attempt=3

end if

endif

else

c ... if switch2=1 (in kRk_3)

DT_k=DT_k/four


DT_k= one - T_k


mario_DT_test=1

write(*,*) 'mario_DT - in kRk_3, mario6 = one, pp_3=',pp_3

ccccccccc error =3

ccccccccc return

cccccc attempt=3

end if

endif

else

cc ... if pp_3 lower than zero

DT_k=DT_k/four


DT_k= one - T_k


mario_DT_test=1

write(*,*) 'mario_DT - pp_3<0, mario7=one, pp_2=',pp_2

ccccccccc error =3

ccccccccc return

cccccc attempt=3

end if

endif

else


DT_k=DT_k/four


DT_k= one - T_k

cccccccc mario_DT=1

mario_DT_test=1

write(*,*) 'mario_DT - kRk_2, mario8=one, pp_2=',pp_2

cccccccc error =3

cccccccc return

cccccc attempt=3

end if

endif

else

c ... if pp_2 lower than zero

DT_k=DT_k/four


DT_k= one - T_k

ccccccc mario_DT=1

mario_DT_test=1

write(*,*) 'mario_DT - pp_2<0 - mario9=one, pp_kk=',pp_kk

203

cccccccc error =3

cccccccc return

cccccc attempt=3

end if

endif

c

else


DT_k=DT_k/four


DT_k= one - T_k

cccccccc mario_DT=1

mario_DT_test=1

write(*,*) 'mario_DT - kRk_1, mario10=one, pp_kk=',pp_kk

c

c

ccccccccc error =3

ccccccccc return

cccccc attempt=3

end if

endif

c

c

c

c ... bottom of while loop

c

end do !*****************************************************

c

end if

end if

c

if(mario.eq.1) then !stop solution, keep current configuration

cccccc DT_k=1-T_k

write(*,*) 'attempt number ',attempt

write(*,*) 'accept solution'

call push(z_k,z,nasvz)

call push(y_k,y,n)

c error=3

ccc attempt=1

else if(mario_DT.eq.1) then !abort solution, keep previous configuration

write(*,*) 'mario_DT'

c call push(z1,z,nasvz)

c call push(y_k,y,n)

else

c

call push(y_k,y,n)

call push(z_k,z,nasvz)

c

endif

c

if(drcor.ne.0) then

call drift_corr_DM(y,n,z,nasvz,parms,nparms,tol_ff,



endif

if (switch2.ne.zero) then

204

write(*,*) 'RKF23 - drift_corr - switch2>0'

cccccc error=3

return

endif

c

return

c

2000 format('Substep no.',i4,'- T_k=',d12.4,'- DT_k=',d12.4,

&' -pp_hat=',d12.4)

c

end

c

!-----------------------------------------------------------------------------

subroutine solout(stress,ntens,asv1,nasvy,asv2,nasvz,ddsdde,

+ y,nydim,z,pore,depsv_np1,parms,nparms,DD)

!-----------------------------------------------------------------------------

! copy the vector of state variables to umat output

! Dafalias &Manzari SANISAND model (2004)

!

! modified 4/2008 (Tamagnini)

!

! NOTE: solid mechanics convention for stress and strain components

! pore is always positive in compression

!

! depsv_np1 = vol. strain increment, compression positive

! y(1:6) = effective stress, compression positive

! pore = excess pore pressure, compression positive

! stress = total stress, tension positive

!

!-----------------------------------------------------------------------------

implicit none

!

integer nydim,nasvy,nasvz,nparms,ntens,i,j

!

double precision y(nydim),z(nasvz),asv1(nasvy),asv2(nasvz)

double precision stress(ntens),ddsdde(ntens,ntens),DD(6,6)

double precision parms(nparms),bulk_w,pore,depsv_np1

!

bulk_w=parms(17)

!

! ... update excess pore pressure (if undrained cond.), compression positive

!

pore=pore+bulk_w*depsv_np1

!

! updated total stresses (effective stresses stored in y(1:6))

!

do i=1,ntens

if (i.le.3) then

stress(i) = -y(i)-pore

else

stress(i) = -y(i)

end if

enddo

!

! additional state variables

!

do i=1,nasvy

205

asv1(i) = y(6+i)

enddo

!

do i=1,nasvz

asv2(i) = z(i)

enddo

!

! consistent tangent stiffness

!

do j=1,ntens

do i=1,ntens

if((i.le.3).and.(j.le.3)) then

ddsdde(i,j) = DD(i,j)+bulk_w

else

ddsdde(i,j) = DD(i,j)

end if

end do

enddo

!

return

end

c

c-----------------------------------------------------------------------------

subroutine tang_stiff(y,z,n,nasvy,nasvz,parms,nparms,DD,cons_lin,


c-----------------------------------------------------------------------------

c compute continuum tangent stiffness at the end of the step

c Dafalias & Manzari (2004) SANISAND model for sand

c


c-----------------------------------------------------------------------------

implicit none

c


c

integer n,nasvy,nasvz,nparms,cons_lin

c

double precision y(n),z(nasvz),parms(nparms)

double precision DD(6,6),HH(nasvy,6)

double precision zero,three



parameter(zero=0.0d0,three=3.0d0)

c



c

c ... initialize DD

c

call pzero(DD,36)

c

if(plastic.eq.1 .and. cons_lin.eq.1) then

c

c ... plastic process

c

206

call get_tan_DM(y,n,nasvy,z,nasvz,parms,nparms,DD,HH,switch2,

& mario_DT_test,


else

call el_stiff_DM(y,n,parms,nparms,DD,


c

c ... if switch2=0 (LDeR<0) try Elastic tangent matrix

c

! if(switch2.ne.zero) then

c error=3

! write(*,*) 'tang_stiff - switch2=1 - elastoplastic_matrix=0'

! endif

c

end if

c

return

end

c

c-----------------------------------------------------------------------------

subroutine trial_state(y_k,n,parms,nparms,deps,y_tr,


c-----------------------------------------------------------------------------

c

c ... computes the trial stress state (freezing plastic flow)

c

c eps = strain at the beginning of the step

c deps = strain increment for this step

c

c NOTE:

c

c mode = 1 single step Runge-Kutta 3rd order

c mode = 2 single step forward Euler

c

c-----------------------------------------------------------------------------

c

implicit none

c

integer n,m,nparms,mode,i

c

double precision y_k(n),parms(nparms),deps(6)

double precision y_2(n),y_3(n),y_tr(n)

double precision one,two,three,six

double precision kRK_1(n),kRK_2(n),kRK_3(n)

double precision DT_k,DTk05,DTk2,DTk6,DTk23



c

parameter(mode=1)

c

data one,two,three,six/1.0d0,2.0d0,3.0d0,6.0d0/

c

DT_k=one

c

207

c ... compute F_el

c

call f_hypoelas_DM(y_k,n,parms,nparms,deps,kRK_1,


c

if (mode.eq.1) then

c

c ... 3rd order RK - build F function

c

DTk05=DT_k/two

DTk2=two*DT_k

DTk6=DT_k/six

DTk23=two*DT_k/three

c

do i=1,n

y_2(i)=y_k(i)+DTk05*kRK_1(i)

end do ! i

c

call f_hypoelas_DM(y_2,n,parms,nparms,deps,kRK_2,


c

do i=1,n

y_3(i)=y_k(i)-DT_k*kRK_1(i)+DTk2*kRK_2(i)

end do ! i

c

call f_hypoelas_DM(y_3,n,parms,nparms,deps,kRK_3,


c

do i=1,n

y_tr(i)=y_k(i)+DTk6*kRK_1(i)+DTk23*kRK_2(i)+DTk6*kRK_3(i)

end do ! i

c

else

c

c ... forward Euler

c

do i=1,n

y_tr(i)=y_k(i)+DT_k*kRK_1(i)

end do ! i

c

end if

c

return

end

c

c-----------------------------------------------------------------------------

subroutine wrista(mode,y,nydim,deps_np1,dtime,coords,statev,

& nstatv,parms,nparms,noel,npt,ndi,nshr,kstep,kinc)

c-----------------------------------------------------------------------------

c ... subroutine for managing output messages

c

c mode

c

c all = writes: kstep, kinc, noel, npt

c 2 = writes also: error message,coords(3),parms(nparms),ndi,

c nshr,stress(nstress),deps(nstress),dtime,statev(nstatv)

c 3 = writes also: stress(nstress),deps(nstress),dtime,statev(nstatv)

208

c

c-----------------------------------------------------------------------------

implicit none

c

integer mode,nydim,nstatv,nparms,noel,npt,ndi,nshr,kstep,kinc,i

c

double precision y(nydim),statev(nstatv),parms(nparms)

double precision deps_np1(6),coords(3),dtime

c

c ... writes for mode = 2

c

if (mode.eq.2) then

write(6,*) '==================================================='

write(6,*) ' ERROR: abaqus job failed during call of UMAT'

write(6,*) '==================================================='

write(6,*) ' state dump: '

write(6,*)

endif

c

c ... writes for all mode values

c

if(mode.ne.4) then

write(6,111) 'Step: ',kstep, 'increment: ',kinc,

& 'element: ', noel, 'Integration point: ',npt

write(6,*)

endif

c

c ... writes for mode = 2

c

if (mode.eq.2) then

write(6,*) 'Co-ordinates of material point:'

write(6,104) 'x1 = ',coords(1),' x2 = ',coords(2),' x3 = ',

& coords(3)

write(6,*)

write(6,*) 'Material parameters:'

write(6,*)

do i=1,nparms

write(6,105) 'prop(',i,') = ',parms(i)

enddo

write(6,*)

write(6,102) 'No. of mean components: ',ndi

write(6,102) 'No. of shear components: ',nshr

write(6,*)

endif

c

c ... writes for mode = 2 or 3

c

if ((mode.eq.2).or.(mode.eq.3)) then

write(6,*) 'Stresses:'

write(6,*)

write(6,101) 'sigma(1) = ',y(1)

write(6,101) 'sigma(2) = ',y(2)

write(6,101) 'sigma(3) = ',y(3)

write(6,101) 'sigma(4) = ',y(4)

write(6,101) 'sigma(5) = ',y(5)

write(6,101) 'sigma(6) = ',y(6)

write(6,*)

209

write(6,*) 'Strain increment:'

write(6,*)

write(6,101) 'deps_np1(1) = ',deps_np1(1)






write(6,*)

write(6,*) 'Time increment:'

write(6,*)

write(6,108) 'dtime = ',dtime

write(6,*)

write(6,*) 'Internal state variables:'

write(6,*)

write(6,109) 'alpha_11 = ',statev(1)






write(6,109) 'void = ',statev(7)

write(6,109) 'Fab_11 = ',statev(8)






write(6,109) 'dummy = ',statev(14)

write(6,109) 'alpha_sr_11 = ',statev(15)














write(6,*)

write(6,*) '==================================================='

c

endif

if (mode.eq.4) then

write(6,111) 'Step: ',kstep, 'increment: ',kinc,

& 'element: ', noel, 'Integration point: ',npt

write(6,104) 'DT<DTmin:x1=',coords(1),' x2 = ',

& coords(2),' x3 = ',coords(3)

c write(6,*)

endif

c

101 format(1X,a10,f50.44)

210

102 format(1X,a25,i1)

103 format(1X,a7,i5)

104 format(1X,3(a12,f10.4,2X))

105 format(1X,a5,i2,a4,f20.3)

106 format(1X,3(a9,f12.4,2X))

107 format(1X,3(a10,f12.4,2X))

108 format(1X,a8,f12.4)

109 format(1X,a10,f50.44)

110 format(1X,a5,f10.4)

111 format(1X,a6,i4,2X,a11,i4,2X,a9,i10,2X,a19,i4)

c

return

end

c

c-----------------------------------------------------------------------------

double precision function yf_DM(y,ny,parms,nparms)

c-----------------------------------------------------------------------------

c compute yield function of Dafalias & Manzari (2004)

c SANISAND model for sands

c-----------------------------------------------------------------------------

c

implicit none

c

integer ny,nparms

c


double precision y(ny),parms(nparms)

double precision mm,zero,one,two,three,sqrt23,norm2

double precision sig(6),s(6),trace,p,alpha(6),sbar(6)

c


c

c ... some constants and material parameters

c


mm=parms(7)

c


c

sig(1)=y(1)

sig(2)=y(2)

sig(3)=y(3)

sig(4)=y(4)

sig(5)=y(5)

sig(6)=y(6)

c

alpha(1)=y(7)

alpha(2)=y(8)

alpha(3)=y(9)

alpha(4)=y(10)

alpha(5)=y(11)

alpha(6)=y(12)

c

c ... mean stress and deviator stress

c

call deviator(sig,s,trace,p)

c

211

sbar(1)=s(1)-p*alpha(1)






c

c ... compute yield function

c

norm2=dot_vect(1,sbar,sbar,6)

yf_DM=dsqrt(norm2)-sqrt23*mm*p

c

return

end

c

c.....MODIFICATO

c

subroutine xit_DM()

stop

return

end

c

c------------------------------------------------------------------------------

subroutine inv_sig_full(sig,pp,qq,cos3t,I1,I2,I3)

c------------------------------------------------------------------------------

c calculate invariants of stress tensor

c


c

c 11 -> 1

c 22 -> 2

c 33 -> 3

c 12 -> 4

c 13 -> 5

c 23 -> 6

c

c------------------------------------------------------------------------------

c

implicit none

c

double precision sig(6),sdev(6)

double precision eta(6),eta_d(6),eta_d2(6)

double precision xmin1,xmin2,xmin3

double precision tretadev3,pp,qq,cos3t,I1,I2,I3

double precision norm2,norm2sig,norm2eta,numer,denom

c

double precision half,one,two,three,six

double precision onethird,threehalves,sqrt6,tiny

c


c

data half,one/0.5d0,1.0d0/

data two,three,six/2.0d0,3.0d0,6.0d0/

data tiny/1.0d-18/

c


212

c

onethird=one/three

threehalves=three/two

sqrt6=dsqrt(six)

c

c ... trace and mean stress

c

I1=sig(1)+sig(2)+sig(3)

pp=onethird*I1

c


c

sdev(1)=sig(1)-pp

sdev(2)=sig(2)-pp

sdev(3)=sig(3)-pp

sdev(4)=sig(4)

sdev(5)=sig(5)

sdev(6)=sig(6)

c

c ... normalized stress and dev. normalized stress

c

if(I1.ne.0) then

eta(1)=sig(1)/I1

eta(2)=sig(2)/I1

eta(3)=sig(3)/I1

eta(4)=sig(4)/I1

eta(5)=sig(5)/I1

eta(6)=sig(6)/I1

else

eta(1)=sig(1)/tiny

eta(2)=sig(2)/tiny

eta(3)=sig(3)/tiny

eta(4)=sig(4)/tiny

eta(5)=sig(5)/tiny

eta(6)=sig(6)/tiny

end if

c

eta_d(1)=eta(1)-onethird



eta_d(4)=eta(4)

eta_d(5)=eta(5)

eta_d(6)=eta(6)

c

c ... second invariants

c

norm2=dot_vect(1,sdev,sdev,6)

norm2sig=dot_vect(1,sig,sig,6)

norm2eta=dot_vect(1,eta_d,eta_d,6)

c

qq=dsqrt(threehalves*norm2)

I2=half*(norm2sig-I1*I1)

c

c ... components of (eta_d_ij)(eta_d_jk)

c

eta_d2(1)=eta_d(1)*eta_d(1)+eta_d(4)*eta_d(4)+eta_d(5)*eta_d(5)

213






c

c ... Lode angle

c

if(norm2eta.lt.tiny) then

c

cos3t=-one

c

else

c

tretadev3=dot_vect(1,eta_d,eta_d2,6)

c

numer=-sqrt6*tretadev3

denom=(dsqrt(norm2eta))**3

cos3t=numer/denom



end if

c

end if

c

c ... determinant

c

xmin1=sig(2)*sig(3)-sig(6)*sig(6)



c

I3=sig(1)*xmin1-sig(4)*xmin2+sig(5)*xmin3

c

return

end

c

c-----------------------------------------------------------------------------

subroutine check_RKF_DM(error_RKF,y,ny,nasv,parms,nparms)

c-----------------------------------------------------------------------------

c Checks is RKF23 solout vector y is OK for hypoplasticity

c-----------------------------------------------------------------------------

implicit none

c

integer error_RKF,ny,nasv,i,nparms,testnan,iopt

c

double precision y(ny),parms(nparms)

double precision sig(6),pmean,sig_star(6)

double precision I1,I2,I3,pp,qq,cos3t

double precision ptshift,minstress,sin2phim,tolerance

double precision OCR,omega,fSBS,sensit,cos2phic

double precision coparam,sin2phicco

c

c check for NAN

testnan=0

do i=1,ny

call umatisnan_DM(y(i),testnan)

214

end do

if(testnan.eq.1) error_RKF=1

c

return

end

c

c-----------------------------------------------------------------------------

subroutine umatisnan_DM(chcknum,testnan)

c-----------------------------------------------------------------------------

c

c checks whether number is NaN

c

c-----------------------------------------------------------------------------

double precision chcknum

integer testnan

if (.not.(chcknum .ge. 0. .OR. chcknum .lt. 0.)) testnan=1

if (chcknum .gt. 1.d30) testnan=1

if (chcknum .lt. -1.d30) testnan=1

if (chcknum .ne. chcknum) testnan=1

return

end

EFFECT OF IMPERFECT DIRECT SIMPLE SHEAR TEST ...

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