Constitutive Modelling of Soils under High Strain Rates Literature Review Prepared By: Kaiwen Xia (Ph.D.) Mohammadamin Jafari Naidong Wang Patrick Paskalis Kanopoulos Department of Civil Engineering, University of Toronto, Toronto, Ontario PWGSC Contract Number: W7701-135578/001/QCL CSA: Grant McIntosh, Defence Scientist, 418-844-4000 ext. 4278 The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of the Department of National Defence of Canada. Contract Report DRDC-RDDC-2015-C072 March 2015
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Constitutive Modelling of Soils under High Strain Rates Literature Review
Prepared By: Kaiwen Xia (Ph.D.) Mohammadamin Jafari Naidong Wang Patrick Paskalis Kanopoulos Department of Civil Engineering, University of Toronto, Toronto, Ontario PWGSC Contract Number: W7701-135578/001/QCL CSA: Grant McIntosh, Defence Scientist, 418-844-4000 ext. 4278
The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of the Department of National Defence of Canada.
Fig. 5-2 Typical finite-element mesh used in the analysis of tunnels (Higgins et al. 2013). ........ 60
Fig. 5-2 Typical finite-element mesh used in the analysis of tunnels (Higgins et al. 2013). ........ 60
Fig. 5-3 Typical temporal simulation of material deformation during landmine detonation: (a) in
case of dry sand; (b) in case of fully saturated sand (Grujicic et al. 2008). .................................. 61
Fig. 5-4 Finite-element mesh of one quarter model (An et al. 2011). .......................................... 62
Fig. 5-5 Comparison of soil ejecta heights: high speed video versus simulation: (a) at time =
420μs; (b) at time = 1040 μs (An et al. 2011). .............................................................................. 62
1
Introduction
In the study of soil behavior under extreme loading such as blasting, it is necessary to take into
account the effect of strain rate on soil behavior. The stain rate is of overriding importance in
such study, where the load is expected to act as a single pulse and stress produced by the blast
pressure increases so rapidly that the material behaves significantly different from that under
quasi-static loadings.
The modeling and experimentation of high strain rate phenomena are very important in
determining the behavior of soils under critical events. It is widely accepted in the solid
mechanics community that the strength, constitutive behavior, and overall response of soils are
highly dependent on the strain rate. The dynamic properties of soils have been extensively
investigated to predict the response of solids subjected to blasts, earthquakes, pile driving,
meteor impact and many other transient loads. Despite existing studies, the behavior of soils at
high strain rates and under various confining pressures, saturation, particle distribution, void
ratios and other pertinent conditions have yet to be fully characterized.
This report contains a summary of the published studies in the field of high-strain rate soil
dynamics. The contents of this report are organized as follows:
Chapter 1: General introduction on soil mechanics and deformation.
Chapter 2: Brief review about experiments used to investigate the response of soil
behaviors under high strain-rate conditions.
Chapter 3: Review of the constitutive models used to simulate the time-dependent
behavior of soils.
Chapter 4: Equation of state (EOS) used in the numerical simulation of soil under blast
loading.
Chapter 5: Review of the numerical simulation of soil behaviors under blast or explode
loading.
2
1.1 Phase relationship of soil
Soil is made up of solid particles, with spaces or voids in between. The assemblage of particles
in contact is usually referred to as the soil matrix or the soil skeleton. In conventional soil
mechanics, it is assumed that the voids are in general occupied partly by water and partly by air.
This means that the soil is a three-phase material, comprising some solid (the soil grains), some
liquid (the pore water) and some gas (the pore air). This is illustrated in Fig. 1-1, which shows
the relative volumes of solid, liquid and gas. The phase relationships are important in
characterizing the state of soil. They are described as follows.
The void ratio e is defined as the ratio of the volume of voids vV to the volume of solids sV , and
it is conventionally given as:
s
v
V
Ve (1-1)
The saturation ratio rS is defined as the ratio of the volume of water to the volume of voids, and
it is presented as:
v
wr V
VS (1-2)
The saturation ratio must lie in the range 10 rS , 0rS for dry condition and and 1rS for
fully saturated.
The water content w is defined as the ratio of the mass of water wm to the mass of soil solids sm ,
and it is given as:
3
s
w
m
mw (1-3)
The density of the soil grains s is defined as sss Vm , and it also can be expressed as
wss G where sG is the particle relative density. sG is the density of the soil particles relative
to that of water and w is the density of water.
Fig. 2-1 Soil as a three-phase material
1.2 Strain rate effect on soil behavior
Under static loading, the pore pressure and drainage can be specified as drained or undrained. In
rapid loading the drained condition is difficult to attain even in coarse grained soils, because the
short time loading does not permit complete dissipation of the excess hydrostatic pore pressure.
When sheared in the undrained condition, the excess hydrostatic pore pressure will be affected
by the strain rate. Generally, the pore pressure decreases with the increase of strain rate
(Whitman and Healy 1962, Richardson and Whitman 1963). It is suspected that the strain rate
affects the nature of particle movements during shear deformation. During rapid strain the
individual particles have less freedom to choose a path of least resistance than during slow strain.
Hence, more particles are forced to override neighboring particles and this leads to an increase in
dilation. As shown in Fig. 1-2, the pore pressure Pw has a lower magnitude in rapid loading than
in slow loading for common triaxial tests.
4
Fig. 2-2 Strain-rate effect on soil stress-strain curves
The effect of the strain rate on the stress-strain curve is also shown in Fig. 1-2. The dashed
curves represent the curve obtained in rapid loading and the solid curve from slow loading. In
general, the secant modulus and tangent modulus also increase with the increase of strain rate.
In confined compressive tests, soil specimens are loaded axially, but expansion in the radial
direction is limited. The axial stress σ1 is the major principal stress. If stress increment is applied
in large time interval, such test becomes a consolidation test. When loaded rapidly on the
contrary, a curve of different shape will be obtained as shown in Fig. 1-3 (Heierli 1962). The
value of the secant modulus increases rapidly with the increase of strain rate. The initial tangent
modulus in rapid loadings is also substantially higher than that in slow loading.
Fig. 2-3 Strain-rate effect on soil one-dimensional compression behaviors
ε1
σ1
StaticDynamic
o
5
Fig. 2-4 Strain-rate effect on the ratio of the radial stress to the axial stress
The general behavior of the radial stress σ3 during loading and unloading is similar to that in
static tests. As shown in Fig 1-4, the increase of the radial stress σ3 is proportional to the increase
of the axial stress σ1 during loading. However, when the axial stress σ1 is reduced after reaching
a maximum value, the decrease of radial stress σ3 is smaller than that of the axial stress, in other
way, the ratio of the radial stress to the axial stress (K=σ3/σ1) increases as σ1 is reducing. Since in
rapid loading conditions there is no opportunity for consolidation, the value of K also depend on
the saturation ratio of soils: for dry soils, K varies in the range between 1/3 and 1/2 during
loading (Wu 1971); and for saturated soils, the K value is nearly unity.
1.3 Soil deformation under high strain rate monotonic loading
To characterize the stress wave propagating in a medium, properties of the medium should be
studied first. The basic general characteristic of a medium is the relationship between pressure
and relative volume.
Soils can be divided into two main groups; cohesive and noncohesive, based on the bonding
condition between soil particles. Cohesive soils contains sufficient clay content to stick the mass
together and have an internal strength. Noncohesive soils, such as sand and gravel, have no
internal strength on their own. As a result in noncohesive soils, strength depends on friction
between particles.
6
Fig. 2-5 Relationship between pressure and deformation (Wang and Lu 2003)
Generally, there are two main deformation mechanisms in soils under hydrostatic loading
(Henrych 1979):
(a) Deformation of the soil skeleton: This deformation consists of elastic deformation of
the bonds at the contact surfaces of grains at low pressure, and plastic deformation due to failure
of bonds at high pressure,.
(b) Deformation of all the soil phases: this deformation is determined by the volume
compression of all three phases in the soil.
In the process of deformation, both of the mechanisms may occur simultaneously. At a certain
stage of loading process, however, one of the mechanisms will dominate.
The deformation process in a dry soil is more complex compared to a saturated soil. It has the
general form represented in Fig. 1-5(a) for dry soils and Fig.1-5(b) for saturated soils. Most of
the voids in a dry soil are filled with air that has a higher compressibility as compared to the
solid (minerals) and liquid (usually water) phases of the soil. Therefore, when a static or dynamic
load is applied on a dry soil, the soil skeleton will resist the load and so the first deformation
mechanism dominates. By increasing the pressure, the bonds between grains will start to fail
7
(point A in Fig. 1-5(a)). Within the stage between point A and B, large displacements of the
particles occur even with a slight change in pressure. The soil will be compressed and cease to
transfer shear stresses, thus its behavior is similar to that of a liquid (after point C). In this stage
the second mechanism becomes dominant and the first mechanism can be neglected.
In water-bearing soil (most of the voids are filled with water), the deformation mechanism of the
soil depends on the loading rate. With a rapid dynamic loading, they have a higher resistance
than the contact bonds of the skeleton grains. Under this loading condition, the second
mechanism of deformation will be dominant (Fig. 1-5(b)).
The deformation mechanisms in the noncohesive soils are similar to the cohesive soils; however,
at high stress level grains begin crushed and the skeleton can contract further more. Fig. 1-6
represents the stress- strain response of dry sand under uniaxial strain loading while lateral
strains are prohibited.
Fig. 2-6 Stress-strain curve of dry sand under uniaxial strain loading condition (Omidvar et al. 2012).
There are usually three main mechanisms which govern the response of dry sands in the uniaxial
strain loading condition: (1) elastic compaction of sand particles; (2) slippage and rearrangement
of grains; and (3) grain crushing. Based on these three mechanisms, one can divide the response
of dry sands into four distinct zones as presented in Fig. 1-6:
8
‐ Zone 1: In this zone, the applied stress is not enough to overcome the friction between individual particles and so all deformations corresponding to the elastic deformation between grains.
‐ Zone 2: The applied stress overcomes friction between grains and the soil particles start to slide and roll into the voids. In this stage the density of sand will change and inelastic deformation will commence.
‐ Zone 3: By increasing the applied stress, the sand grains rearrange and the contact points between grains will increase more. As a result, the soil will be more compacted and sliding and rolling of the grains will be more difficult. The hardening (lock-up) is thus observed in sand.
‐ Zone 4: In this stage, since the applied stress is really high, the grains begin to crush, leading to the hardening response of the material again.
In contrast to uniaxial compression tests, in a triaxial test the sample is allowed to deform
laterally. In common triaxial test with a constant confining pressure, the soil can reach the
critical state. Fig. 1-7 shows the difference between the response of sand in the uniaxial
compression and triaxial compression. In triaxial compression test, after sample reaches the
critical state the axial stress remains constant.
Fig. 2-7 Stress strain response of sand in uniaxial and triaxial compression tests
9
2 Experiments and tests
To apply high strain rate to the soil sample, several means such as a drop weight systems (Heierli
Figure 5-2 shows the typical FE mesh by William et al. (2013). In order to save on the
computation time, only one half of the actual domain was analyzed by imposing a symmetry
boundary condition along the left vertical boundary of the mesh. The top horizontal boundary
was free to displace, while the bottom horizontal boundary was restrained against both vertical
60
and horizontal displacements. Vertical displacements were allowed along the left and right
vertical boundaries but not horizontal displacements. The bottom horizontal boundary and the
right vertical boundary were located at sufficient distances so that they had no impact on the
results of the analysis the results were obtained at a time when the stress wave from the blast was
far from these boundaries. The mesh for the 5 m deep tunnel consists of 1624 elements and 1718
nodes. The blast was simulated using the JWL equation-of-state model. It was found that the
type and relative density of sand and the depth of tunnel influence the propagation of the blast-
induced stress waves through the soil. The wave speed was found to be greater in Fontainebleau
sand than in Ottawa sand. The rate of decrease of the maximum mean stress in soil with increase
in distance from the tunnel was comparable for both the sands while the decrease of the
maximum shear stress in soil was faster for Ottawa sand.
Grujicic et al. (2008) presented a procedure and the results of a CU-ARL sand model
parameterization analysis. The CU-ARL sand model was incorporated into a transient non-linear
dynamics computer program and its use in the simulation of a number of buried-land mine blast
scenarios along with experimental results were also given.
Fig. 5-2 Typical finite-element mesh used in the analysis of tunnels (Higgins et al. 2013).
61
Fig. 5-3 Typical temporal simulation of material deformation during landmine detonation: (a) in
case of dry sand; (b) in case of fully saturated sand (Grujicic et al. 2008).
Grujicic et al. (2008) also present the temporal simulation of material deformation during
landmine detonation in case of dry sand and fully saturated sand, separately (shown in Figure 5-
3). The software used in these work is AUTODYN.
The viscoplastic cap model developed by Tong and Tuan (2007) to address the high strain rate
effects on soil behavior, has been used by An et al. (2011) for finite-element analysis of blasts
due to explosives embedded in soil. The revised cap model comprises a Gruneisen equation of
state for each of the three phases: soil, water and air. The equations of state for solid, water, and
air can be integrated with the viscoplastic cap model to simulate behaviors of soil with different
degrees of water saturation. An et al. (2011) incorporated this model into the LS-DYNA software
as user-supplied subroutines for numerical simulations.
(a) (b)
62
Fig. 5-4 Finite-element mesh of one quarter model (An et al. 2011)
Fig. 5-5 Comparison of soil ejecta heights: high speed video versus simulation: (a) at time =
420μs; (b) at time = 1040 μs (An et al. 2011)
Taking advantage of symmetry, An et al. (2011) established only one quarter finite model of the
test setup (as shown in Fig. 5-4) using the LS-DYNA software. The finite-element model,
(a) (b)
63
containing a 110-cm air volume above and a 90-cm soil volume below the soil surface, meshed
with 6,400 eight-node solid elements. Fine mesh was generated for the explosive and for the air
and soil volumes surrounding the C4 where high strain gradients are anticipated. The soil ejecta
heights between high speed video and numerical simulation at time = 420 and 1040 μs since
detonation for tests in dry sand and in saturated sand are compared in Fig. 5-5(a) and (b),
respectively. It shows that the ejecta heights predicted by the revised cap model agree fairly well
with the experimental data. Meanwhile, it also shows that the revised cap model is adequate for
blast loading behavior simulations for soils.
Other researchers also obtained some interesting results of the numerical simulation for dynamic
behavior of soils, which can be seen from the works by Nagy et al. (2010), Yang et al. (2010), Lu
et al. (2005), Feldgun et al. (2008; 2008) and Karinski et al. (2009), etc.
64
Conclusion
This report has considered the fundamental aspects of the simulation of soils under blast loading.
Accurate and effective numerical modeling depends on several factors; namely, the selection of
an appropriate constitutive model and equation of state to define the stress-strain behaviour of
the soil in question, parameterization of the constitutive models and EOS using carefully
designed experiments over a range of strain rates, and numerical implementation of the selected
theoretical models in a computer sub-routine.
What is immediately apparent from the literature review is that there is no unifying constitutive
model that succeeds in capturing the behaviour of soils under all possible conditions. Saturation;
particle shape, size, and distribution; cohesion; and strain rate all effect the constitutive models
and their components (flow rule, yield surface, rate parameters). Other effects, such as particle
crushing under high strain rates, are even less understood.
Moving forward in the investigation of soil and its dynamic properties, the class of viscoplastic
constitutive models will be employed. The viscoplastic constitutive models have been
demonstrably successful in the modeling of complicated material behaviour problems that occur
over orders of magnitude in strain rates. The class of models is also versatile enough that it can
be combined with many of the other tenets of plasticity such as various flow rules and cap
models.
In the next major phase of the project, experimentation and theoretical modeling of the soil will
occur simultaneously. It is important to note that no specific constitutive model has been selected
a priori; rather, selection of a specific constitutive model will follow from the initial
experimental data obtained from the tests. Depending on the type of model selected, additional
physical testing or computer calibration may be required to establish additional parameters.
65
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