A cohesive damage–friction interface model accounting for water pressure on crack propagation

A cohesive damage-friction interface model

accounting for water pressure on crack propagation G. Alfano

*, S. Marfia

**, E. Sacco

**

*Dipartimento di Scienza delle Costruzioni

Università di Napoli, Via Claudio, 21 - 80125, Napoli, Italy

e-mail: [email protected] **

Dipartimento di Meccanica, Strutture, Ambiente e Territorio

Università di Cassino, Via G. di Biasio 43, 03043 Cassino, Italy

e-mail:[email protected], [email protected]

Abstract. Aim of the work is the study of crack propagation in concrete constructions, such as dams,

taking into account the water pressure effects and the damage-friction evolution with unilateral

contact. With this aim, an interface model, based on the cohesive fracture, is developed. In particular, a damage-friction model based on a new multiscale approach is proposed for the interface, which is

able to simulate crack propagation in mode I, mode II, and mixed mode. The criteria for crack

initiation and propagation and the closure and reopening of the crack are considered for the interface.

The friction and the dilatancy are characterized by a decreasing rate of friction and dilatancy angles, respectively, with the possibility of the dilatancy recovering during cyclic loading. The water pressure

effect is taken into account by considering a further static pressure acting on the real crack and in the

process zone. The value of this pressure is set as a function of the crack opening displacement and it varies from a prescribed initial pressure to the external water pressure. A numerical procedure is

developed in order to integrate the evolutive laws governing the interface model. The interface model

is implemented in finite element codes; some numerical simulations and a comparison with experimental data are performed in order to validate the proposed model and to investigate the

influence of the water pressure effects on the crack propagation. Finally, the mechanical response of a

concrete dam is studied.

Keywords: Cohesive crack, interface, damage, friction, dilatancy, water pressure, concrete dams.

1 INTRODUCTION

In the last decades, concrete dams have been the object of an increasing attention from the

international scientific community. In this respect, sophisticated mechanical models, suitable

numerical procedures and appropriate design methods have been developed for the evaluation

of the nonlinear response of concrete dams. In particular, the presence of joints generated by a

step-wise successive construction of layer concrete blocks and between the dam and the

foundation can play a fundamental role in the overall dam behavior; in fact, these interfaces

are the potential sites of crack growth which can lead to the weakening of the structure and to

the penetration of water that exerts uplift pressure [1]. Thus, the possible crack paths are

determined a priori and the study of crack propagation in dams can be based on the

development of interface models suitable for concrete material.

Several interface models have been proposed in literature to study crack propagation in

cementitious material and at bimaterial interfaces. Among others, Carol et al. [2] developed a

mixed-mode interface based on the definition of a hyperbolic activation function that provides

a good fitting with experimental data. Cĕrvenka et al. [3] proposed a nonlinear elastic model

based on an extension of Hilleborg’s model to study crack propagation at the rock/concrete

interface. They considered the evolution of the failure function based on a softening

parameter which is the norm of the inelastic displacement vector. Ruiz et al. [4] developed a

three dimensional cohesive interface model that takes into account tension-shear damage and

mixed-mode fracture. They developed a finite element calculations considering crack

G. Alfano, S. Marfia and E. Sacco

2

nucleation, microcracking and the development of macroscopic cracks. Alfano and Crisfield

[5] presented an interface model for mixed-mode fracture based on damage mechanics.

Marfia and Sacco [6] developed a numerical procedure in order to study fracture propagation

in concrete material considering a cohesive crack model. Cocchetti et al. [7] proposed a

plastic-softening multidissipative interface model taking into account frictional contact and

quasi brittle fracture. Recently, Puntel et al. [8] developed an experimental and numerical

investigation in order to study the cyclic behavior of concrete joints. They emphasized the

presence of the dilatancy effect, which is recovered during loading unloading cycles.

Some studies of the mechanical response of concrete dams based on fracture mechanics [9]

have been presented in the literature.

Among others, Carpinteri et al. [10] developed an experimental and numerical study of a

gravity dam model subjected to equivalent hydraulic and weight loading. The numerical

simulation was performed adopting a mixed-mode cohesive fracture model. Bolzon et al. [11]

proposed a numerical simulation of a mixed-mode crack propagation for the dam model,

experimentally analyzed in [10], adopting a cohesive fracture model, boundary elements and

finite elements procedures. Chavez and Fenves [12] developed a numerical procedure to

compute the earthquake response of gravity dams taking into account sliding at the interface

between dam base and foundation rock. Plizzari [13] proposed an analysis of concrete dams

based on linear elastic fracture mechanics, developing approximate expressions to determine

stress intensity factors for different loads acting on a gravity dam. Wang et al. [14] presented

a 2D numerical study of seismic fracture behavior of concrete gravity dams, adopting a mesh

size of the finite elements close to the characteristic size of the crack band of concrete

material and a technique of finite element remeshing at the crack front. Horii and Chen [15]

developed a dynamic study of crack growth in concrete using crack-embedded elements.

Sujatha and Chandra Kinshen [16] presented a study of crack propagation at the interface

between concrete dam and rock foundation including the effect of friction associated with

sliding of crack surfaces and computing the energy dissipated during crack propagation.

Water pressure plays a significantly role in the crack propagation in dams since the fluid

pressure distribution can promote further propagation. On the other hand, this hydraulic

loading depends on the degradation of the fracturing surfaces that increase the crack opening

and the permeability of the material. This interaction phenomena is difficult to model. To

determine the stress state and the crack propagation in concrete dams more accurately, uplift

pressures need to be suitably modeled [17]. The problem of uplift pressure in concrete dams

has been studied by Terzaghi [18] from an experimental point of view. Among others,

Brühwiler E. and Saouma [19,20] presented an experimental study pointing out that the static

uplift pressure inside a crack is a function of the crack opening and that along the process

zone the water pressure reduces from full reservoir pressure to the initial pressure.

Regarding the modeling of the effect of water pressure on crack propagation, among others,

Devey et al. [21] presented a study of the influence of water pressure on the determination of

the stress intensity factor. Considering a linearly decreasing uplift distribution along the crack,

they found that the stress intensity factor increases nearly of ten percent. Plizzari [22] studied

the uplif pressure effects in cracked concrete gravity dams, developing a parametric study of

the influence of the water pressure on the stress intensity factor and on the crack propagation

angle. Slowik and Saouma [23] developed an experimental and numerical investigation of the

influence of water pressure on crack propagation in concrete. From the experimental results

they pointed out that the crack opening rate significantly influences the water pressure

distribution. On the basis of the experimental results they proposed an interface model,

considering the fluid permeability as a function of the crack opening displacements. Bolzon

and Cocchetti [24] presented a frictional-cohesive interface model based on a coupled

G. Alfano, S. Marfia and E. Sacco

3

degradation of normal and tangential strength taking into account the water pressure. In

particular, the uplift pressure is considered as a function of the crack opening displacement.

Aim of the present work is the development of a cohesive interface model which takes into

account the damage-friction evolution, the dilatancy and the influence of water pressure on

crack propagation. In particular, the main novelties of the proposed model are:

the cohesive fracture and friction evolution obtained by a micromechanical analysis;

the friction and dilatancy angle degradation;

the reversible dilatancy effect during cyclic shear loading;

the water pressure influence on crack propagation;

the micro-macro multiscale approach in the development of structural analyses.

The nonlinear constitutive relationship of the interface, obtained applying a homogenization

technique [25,26], takes into account the criteria for crack initiation and propagation in Mode

I and II and the closure and reopening of the crack and the friction occurring in the damaged

part.

Moreover, the model is able to simulate the presence of the process zone at the crack tip,

which significantly influences the crack growth in cementitious materials.

The water pressure effect in the concrete cracks is introduced assuming that the undamaged

concrete is impermeable. The presence of an initial water pressure is assumed at the interface

also before cracking; the fracture propagation modify the value and the profile of the initial

water pressure.

The time integration of the evolutive equations governing the interface response is performed

adopting a backward-Euler algorithm. Each time step is solved developing a predictor-

corrector numerical procedure.

The interface model has been implemented in the finite element codes LUSAS [27] and FEAP

[28]; numerical studies are developed to illustrate the main features of the proposed interface

model, assessing its reliability and efficiency, and to study the mechanical response of a

concrete dam.

Within the present paper, only two-dimensional problems are considered, so that the interface

is a line; the study is developed in the framework of small deformation theory.

The outline of the paper is the following: initially, the interface model is described in Section

2; then, the algorithmic finite time step implementation is detailed in Section 3; its consistent

linearization is presented in Section 4; finally, the numerical results concerning the

constitutive response, a comparison with experimental results and the analysis of crack

propagation in a dam are reported and discussed in Section 5.

2 INTERFACE MODEL

A structural system embedded in a fluid is considered; it is constituted by two bodies bonded

along a line, the interface, which is subjected to a cohesive fracture process; the decohesion

phenomenon induces the fluid penetration in the fracture and in the microcracks present in the

process zone developing in the neighborhood of the crack tip.

At microscale level, the presence of microcracks and undamaged parts can be distinguished in

the process zone, as shown in figure 1. As declared above, the interface is considered as a line

and it is characterized by zero thickness, contrarily to the schematic representation reported in

figure 1, where a finite thickness appears only to make the picture clearer.

With reference to figure 1, a local coordinate system is pointwise introduced in the interface,

denoting by x1 the axis normal to the interface and x2 the axis direct along its tangent.

G. Alfano, S. Marfia and E. Sacco

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Figure 1: Representative elementary area of the interface.

In order to develop a material model able to reproduce the nonlinear mechanical behavior of

the interface, from the nucleation of the microcracks to the formation of a fracture, a

multiscale approach is proposed. To this end, a representative elementary area (REA) of the

interface is considered. A micromechanical analysis is developed on the REA in order to

evaluate the overall response, which is assumed as the macromechanical constitutive law in a

typical point of the interface.

The REA is partitioned in an ‘undamaged’ part uA and a ‘damaged’ part dA ; the REA is fully

bonded in the undamaged part while a unilateral contact with dry-fiction accompanied by

possible dilatancy, can occur in the ‘damaged’ part. Moreover, in the damaged part of the

REA, i.e. where a microcrack is present, the fluid surrounding the structural system can

penetrate and increase the initial value of the uplift pressure. Denoting by A the area of REA

and by D the ratio dA A the following relationships hold:

with 1u d u dA A A A D A A DA (1)

In figure 1, the three possible cases are illustrated:

in the part of the interface where no damage occurs, i.e. point A, the REA is constituted

only by the undamaged part, so that it results D=0;

in the process zone of the interface, i.e. point B, the REA is divided in the undamaged

and in the damaged parts, so that it results 0<D<1;

in the real crack of the interface where full damage occurs, i.e. point C, the REA is

constituted only by the damaged part, so that it results D=1.

In order to derive the constitutive law for the interface in a classical homogenization context,

the relative displacement s and the stress σ , defined at a typical point of the interface, are

regarded as average values on the REA. Thus, the overall stress-displacement relation is

obtained developing a micromechanical analysis which accounts for the local constitutive

laws of the undamaged and damaged part of the REA and for the possible crack propagation,

i.e. the decohesion evolution.

G. Alfano, S. Marfia and E. Sacco

5

Kinematics of the REA

The relative displacement s , with tangential and normal components s1 and s2, respectively,

defined in a typical point of the interface represents the average value of the local relative

displacement in the REA. Then, the micromechanical analysis is developed assuming a very

simple kinematics. In fact, it is considered that the local relative displacement in the REA is

constant along the whole REA. As a consequence, denoting by us and d

s the relative local

displacement vectors on the two parts uA and

dA , respectively, it is assumed:

u d s s s (2)

Stress state

The stress in the undamaged and in the damaged part of the REA are denoted as uσ and d

σ ,

respectively. The overall homogenized stress is obtained by averaging the two values of the

stresses uσ and d

σ according to the formula:

1 u dD D σ σ σ (3)

In the following the normal and tangential components of the stress σ , uσ and d

σ are

denoted as and u and u d and d , respectively.

Evolution of decohesion

The crack propagation in the REA is governed by the evolution law of the damage parameter

D , which is assumed to be a function of the history of the displacement of the undamaged

part u s s .

A simple model, based on linear softening for the two pure-mode decohesion laws, as

schematically depicted in figure 2, is adopted. Denoting as ciG , ois and oi the fracture

energy, the first cracking relative displacement component and the peak value of the traction

component associated to the i -th fracture mode, respectively, the critical relative

displacement is evaluated as 2ci ci ois G .

Figure 2: Traction - relative displacement relationships for mode I and mode II.

The ratios between the first cracking and critical displacements are introduced as:

1 1 2 21 2

1 1 2 22 2

o o o o o o

c c c c

s s s s

s G s G

(4)

On the basis of formulas (4), the equivalent relative displacement ratio is defined as:

G. Alfano, S. Marfia and E. Sacco

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2 2

21 221 2 1 21 with

s ss s

ss s

(5)

where the symbol is the positive part of . The damage parameter is given by the

relation:

maxhistory

D D (6)

with

1

max 0 min 11

D

(7)

and

2 2

1 2

1 2

1o o

s s

s s

(8)

Setting 1 2 it results 1 21 1 and the damage interface model proposed by

Alfano and Crisfield [5] is obtained. In order to clarify the physics of the damage evolution

law, the cases of pure mode I or mode II are briefly discussed; in fact, in these cases and

assume the following expressions:

ci oi i oi

ci oi

s s s s

s s

(9)

with i=1,2. Thus, the physical meaning of the parameters and clearly appears: the first

measures the ductility of the material, so that increasing the value of a less steep softening

branch is obtained; the second gives a measure of the actual displacement with respect to the

initial damage displacement threshold. Substituting the parameters, evaluated from formulas

(9), into equation (7) the damage evolution becomes:

max 0 min 1

ci i oi

i ci oi

s s sD

s s s

(10)

In figure 3 the damage parameter D is plotted versus the relative displacement si in case of

pure mode I or II.

Constitutive equations

The undamaged part of the REA behaves according to a linear elastic constitutive law; thus,

the stress acting on the undamaged part, uσ , is related to the relative displacement u s s by

the equation:

u σ Ks (11)

where [ ]idiag KK is a diagonal matrix which collects the stiffness values in all the modes.

G. Alfano, S. Marfia and E. Sacco

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Figure 3: Damage evolution law for pure mode I or II.

These stiffnesses can represent either material parameters, if the decohesion involves the

behavior of a finite thickness interface, or they can be interpreted as numerical penalty values.

In the damaged part of the REA, unilateral with dry-friction contact can occur; thus, an

inelastic relative displacement can arise along the part dA . As a consequence, the relative

displacement in the damaged part is decomposed into an elastic part des and an inelastic part

dis :

d de di s s s (12)

The interface stress on the damaged part is denoted by dσ and it results:

ˆd d σ σ p (13)

where p is the water pressure and

ˆ d di

σ H s s (14)

The constitutive matrix results as 1 1 1 2[(1 ( )) ]didiag h s s H H H , where 1H and 2H are

stiffness parameters, representing the penalty values due to the unilateral friction contact.

Moreover, ( )h is the Heaviside function, with 1 if 0h x x and 0 if 0h x x ,

so that 1 11 ( ) 1dih s s when 1 1 0dis s i.e. in compression, and 1 11 ( ) 0dih s s when

1 1 0dis s , i.e. in tension. In such a way, the unilateral nature of contact is taken into

account.

In the proposed model, a distribution of an uplift pressure is assumed at the interface also

before cracking. Indeed, this uplift pressure is due to the permeability of the joints or of the

soil foundation and its profile depends on different factors, as the presence of drains. In

particular, if no active drains are present a linear profile can be assumed. When a crack

propagates at the joints and at the base of the dam, the magnitude of the assigned uplift

pressure is modified [22].

In fact, once a crack start to propagate, a further static pressure, acting on the real crack and in

the process zone, is considered. The value of this pressure is a function of the crack opening

G. Alfano, S. Marfia and E. Sacco

8

displacement (COD) and it varies from the prescribed initial value to the maximum value

obtained as the sum of the hydrostatic pressure and of the applied overpressure.

In accordance with some experimental results [20], the water pressure is assumed as an

exponential function of the crack opening relative displacement according to the formula:

1 1

0 0

1with 1

0

dis s

Lp p p p e p

p (15)

In equation (15) Lp is the external full reservoir pressure and

0p is the prescribed pressure

distribution at the interface before cracking. The parameter modulates the value of pressure

as a function of the crack opening 1

di

ss s , with denoting the positive part of and

0 1 allowing to calibrate the dependence of the water pressure on the inelastic part of

the crack opening; in fact, if 1 the water pressure depends on the elastic part of the crack

opening and if 0 the water pressure depends on the total crack opening. The parameter

is set on the base of laboratory tests on concrete specimens subjected to hydraulic crack

propagation. It can be noted that the static water pressure is considered in the model; thus, the

proposed approach can be adopted to analyze the behavior of dam subjected to cyclic

variation of the water level of the reservoir; on the contrary, the model could be inadequate to

be used in seismic analyses where the dynamic effects can play a fundamental role.

The classical Coulomb friction law is assumed to hold on the damaged part of the REA:

ˆ ˆd d d

σ (16)

with the friction coefficient, which is assumed to vary according to the expression:

1 q

f i ie (17)

with governing the evolution of the friction parameter and 2

diq u , being q the plastic

accumulation, i and f are the initial and final values of the friction coefficient,

respectively.

It can be pointed out that the normal stress on the damaged part of the REA in equation (16)

results always non positive, i.e. ˆ 0d , because of equations (14) and the definition of the

constitutive matrix H .

The inelastic displacement dis is split in two parts:

di di di s u v (18)

The inelastic part diu accounts for the dilatancy effect, while the part di

v is able to recover

the dilatancy during the unloading phase. The evolution of diu is governed by a non-

associative relationship, obtained introducing the function:

ˆ ˆd d d

σ (19)

with:

0

qe (20)

and governing the evolution of the dilatancy parameter . From equation (20), the

dilatancy parameter starts from the value 0 and decreases as function of the plastic

accumulation q .

G. Alfano, S. Marfia and E. Sacco

9

The inelastic relative displacement diu evolves according to the rate equation:

ˆ d

ddi

d

u (21)

in the respect of the additional Kuhn-Tucker conditions:

0 0 0ˆ ˆd d (22)

For monotonic loading conditions, the inelastic slip dis can be set equal to di

u .

On the contrary, in order to reproduce the cyclic response of rough joints, characterized by

dilatation and contraction in forward and backward slip, respectively, the further inelastic

displacement div is considered. This latter displacement vector di

v has only the first

component different from zero and it is able to recover the dilatancy effect. To obtain a linear

recovering of the dilatancy, the following equation can be considered during the backward

slip:

1 11 2 1 1 2

2 2

di didi di di di di

di di

s ss s u v u

s s (23)

As consequence of equation (23), the evolution of div is governed by equation:

1 12 2 2 1 2 2

2 2

1 11 1

0 0

di di ddi di di di di didi

di di d

s sh u u u u h u u

s s

v (24)

It can be noted from equation (24) that the evolution of div occurs only when

2

diu and 2

diu

have opposite sign, i.e. during the unloading sliding phase.

Moreover, the mode II slip is not affected by the correction introduced to reproduce the cyclic

behavior, i.e. 2 2

di dis u .

For monotonic shear loading the dilatancy reaches asymptotically a maximum value; for

cyclic shear loading the initial value of decreases at each cycle, until it becomes equal to

zero.

It can be noted that, according to formula (15), the inelastic part of the opening 1

dis , due to the

dilatancy, can influence the value of water pressure in the crack. When crack propagates in

mode I, it results 1 0dis , i.e. the water pressure depends only on the total crack opening and

it is not influenced by the value of . On the contrary if crack propagates in mode II or in a

mixed mode, it results 1 0dis , i.e. the water pressure can be influenced by the inelastic crack

opening depending on the value of the parameter . In particular, in pure mode II if 1 ,

the value of water pressure does not depend on the dilatancy, while if 0 1 the water

pressure is also influenced by the dilatancy effect.

As final comment on the proposed damage-plastic interface model, it can be pointed out that

the damage related to the crack growth and the inelastic effect, due to friction phenomenon,

are governed by uncoupled equations.

G. Alfano, S. Marfia and E. Sacco

10

3 ALGORITHMIC IMPLEMENTATION

The nonlinear behavior of the proposed interface model is governed by equations (2) - (24); in

order to evaluate the damage parameter and the plastic relative displacement in the damaged

part of the REA, the time integration of the evolutive equations is required.

The time integration is performed by developing a numerical procedure based on an implicit

backward-Euler integration technique [29,30]. To this end, the equations are written in

discretized form; once the solution is determined at the time tn, the solution at the current time

t = tn + t is computed. In the following, the quantities with the subscript ‘n’ are related to the

time tn, while the ones with no subscript are referred to the current time t. Moreover,

indicates the variable increment in the time step t.

Finally, the following finite step relationships are obtained:

micro-macro stress equation

1 dD D σ Ks σ (25)

damage evolution equations

2 2

1 2

1 2

2 2

1 21 2

1

1

1max min 1

1

o o

n

s s

s s

s s

D D

s s (26)

friction and water pressure effect equations

1 1

0 0

2

12 2

2

11

0

ˆ ˆ0 0 0

1

1

di

n

n

n

s sd di

L

di di di

q q

o

di di dn

d

di

d d d d

q q

f i i

di dqdi di didi

n odi d

p p e p

e

q s

e

sh u u e

s

σ H s s

s u v

u u

vv 1

0

q

(27)

Formulas (25) - (27) represent a nonlinear system of equations, governing the damage and

friction evolution. The algorithm developed in the following is driven by the relative

displacement s .

G. Alfano, S. Marfia and E. Sacco

11

Initially, the damage evolution is evaluated solving equations (26); thus, the updated damage

parameter D is obtained. Then, the friction evolutionary problem is approached developing a

predictor-corrector algorithm.

The trial step is computed assuming frozen the evolution of the inelastic relative

displacement:

,1 1

,

, ,

, ,

0 0

, ,

ˆ

1ˆ 1

0

1

ˆ

di TR

n

di TR di

n

d TR di TR

s sd TR d TR

L

qTR

f i i

TR TR d TR d TR

p p e p

e

s s

σ H s s

σ σ (28)

If 0TR , the step is elastic and the trial represents the solution of the finite time step, i.e. ,di di TRs s , ,

ˆ ˆd d TR

, ,d d TR , ,d d TR , TR . Otherwise a return map algorithm is

developed in order to solve the corrector phase.

The equations to be solved are the following:

1 1

0 0

1 1 1 1 1

,

2 2 2,

1 1,

2 2,

11 1, 2 2

2

1ˆ

1ˆ

1

ˆ0

1

di

n

n

s sd dL

di did

d d TR di di

n

qdi di

n o

ddi di

n d

q

f i i

d d

di dqdi di di di

n odi d

p p e p

h s s H s s

H s s

u u e

u u

e

sv v h u u e

s

2 0

n

di

di di di

v

s u v (29)

It can be proved that when 2 2 0di diu u , i.e. when the second term of the right-hand side of

formula (29)8 does not vanish, it results 1 2 1, 2,/ /di di di di

n ns s s s .

In order to simplify the notation, in the following the Heaviside functions involved in the

model formulation are denoted as:

1 2

1 1 2 2

di di dih h s s h h u u (30)

From equations (29)3, (29)5, (29)9 and (29)10, it results:

G. Alfano, S. Marfia and E. Sacco

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,2 1 d d TR

d

H

(31)

whose absolute value of the two members allows to obtain:

,

2

d d TR H (32)

Combining equations (31) and (32), it results:

,

,

d TRd

d d TR

(33)

Taking into account the definition (30) and formulas (29)2, (29)6 and (32), the consistency

relation (29)7 is written in the following residual form:

1 ,

1 1 1 21 1 0nq di d TR

f i iR e h H s s H

(34)

In order to evaluate at each time step, the nonlinear residual equation (34) is solved

developing a Newton-Raphson technique. In the following, the superscript k denotes the

quantities at the k th iteration and the superscript 1k those at the current iteration. The

symbol indicates the quantities variation between two consecutive iterations, i.e.

1k k .

By linearization, the equation (34) becomes:

10 k k

k

dRR R

d

(35)

with

1,

1

,1,2, 2, ,

1 1, 2,2,

1 1

1

kn

kn

qk k

f i i

di d TRq ndi k k k d TR k

n o di d TRn

R e h H

ss s h e h H

s

(36)

1,

1

,1,2, 2,

2,2,

1,

1

,1,2, 2,

1 1, ,2,

1 1

1 1

1

1

kn

kn

kn

kn

q k

f i i

k

di d TRq nk k k

o di d TRn

q k

f i

di d TRq ndi k k k

n o di d TRn

dRe h H

d

sh e h H

s

e h H

ss s h e h

s

(37)

assuming the derivatives of 1,kh and 2,kh with respect to vanishing.

G. Alfano, S. Marfia and E. Sacco

13

The plastic multiplier increment is obtained solving equation (35). Then, the iterative

procedure goes on until a convergence test is satisfied, i.e. when the values of the residual 1kR is less then a prefixed tolerance. Once is evaluated, the normal and tangential

stresses can be computed from formulas (29).

4 CONSISTENT TANGENT OPERATOR

The consistent linearization of the finite-step relationships is obtained as the derivative σ s

of the micro-macro stress equation (25):

1d

d

t

D DD D

σ σK K Ks σ

s s s s (38)

The derivative D s is equal to zero when there is no damage evolution, i.e. nD D ;

otherwise, in the case of damage growth, i.e. nD D , it is evaluated differentiating equation

(26)3:

22

11

1

D D D

s s s s s (39)

Taking into account equations (26)1 and (26)2, it results:

1

2

1

2

2

2

1

1

o

o

s

s

s

s

s (40)

1 21

2 2 14

1

2 h s sss

s

s s

(41)

For what concerns the term /d σ s in equation (38), two cases can occur in the finite time

step:

no-evolution of the inelastic relative displacement, i.e. 0 ,

variation of the inelastic relative displacement, i.e. 0 .

In the first case, the derivative of equation (27)1 is evaluated considering di di

ns s :

1 1

0 1 1

1 0

0 0

dids s di

Lp p e h s s

σH

s (42)

In the case of inelastic evolution, differentiating equation (27)1 gives:

1 2

1 2

d d

d

d d

s s

s s

σ

s (43)

with

G. Alfano, S. Marfia and E. Sacco

14

1 1

1 1

1 11 0 1 1

1 1

1 11 0 1 1

2 2

22

1 1

22

2 2

1 1

1

1

di

di

dids s di

L

dids s di

L

did

did

sh H p p e h s s

s s

sh H p p e h s s

s s

sH

s s

sH

s s

(44)

The derivatives of the inelastic relative displacements are computed differentiating equations

(29)4, (29)5 and (29)8:

1 1 1 1 1 1

1 1 1 2 2 2

1 1

1 1 2 2

1 1

1 1 2 2

, ,

2 2 2 2

, ,1 1 1 2 2 2

1 2

di di di di di di

di di

di di

di di d TR di di d TR

d TR d TR

s u v s u v

s s s s s s

u uA A

s s s s

v vB B

s s s s

s u s u

s s s s s s

C Ds s

(45)

The parameters in equation (45) are the following:

,1,2

,2,

,1

1 2,

1 2 2

1 1 1, 2

,1,2

,2,

1 1

1

1

1

1

n

di d TRn

di d TRn

d TR

d TR

q di

f i n

di d TRn

di d TRn

sA B h A

s

C H h E D H E

Eh H h A F e s s h F H

sF h

s

(46)

5 NUMERICAL RESULTS

The interface model described in Sections 2-4 has been implemented in the finite-element

codes LUSAS [27] and FEAP [28] as a new constitutive law for 2D interface elements.

Several types of numerical studies are developed in the following: first, an investigation

devoted to illustrate the main features of the proposed interface model is presented, then, a

comparison with experimental results, available in literature, is performed; finally, the

mechanical response of a concrete dam is studied.

G. Alfano, S. Marfia and E. Sacco

15

Behavior of a typical point of the interface

As a first step towards the assessment of the reliability and the efficiency of the model,

numerical simulations have been conducted to derive the mechanical response in a typical

point of the interface considering different loading-unloading histories. The simulations are

performed at level point without studying any structural problem requiring a mesh

discretization.

The following interface parameters are considered:

1 o1 1 0

2 o2 2

0.35 N/mm 0.3 MPa 0.006 mm 0

0.09 N/mm 0.7 MPa 0.010 mm 1

tan 30 0

c o

c o

o

f i

G s p

G s

(47)

From the above input data, the stiffness parameters 1K and

2K are evaluated. Furthermore, it

has been set 1 1H K and

2 2H K .

Initially, in order to investigate on the effect of the uplift water pressure and on its

dependency on the interface opening relative displacement, a mode I tensile test is performed.

The hydrostatic pressure is set equal to 0.8 MPaLp , while different values of the parameter

are considered in the computations.

In figure 4, the interface normal stress versus the crack opening is plotted for -10, 0.1,1,10,100 mm . From the numerical results different softening behaviors of the

interface can be noted. For all the analyzed values of the uplift parameter , the maximum

normal stress and the final damage relative displacement are the same of those corresponding

to 0 , i.e. 0.0Lp . When is set equal to zero a linear softening is recovered, while,

when the uplift pressure is considered, the softening curve depends on the value of the

parameter : a steeper softening is obtained increasing the value of .

Figure 4: Interface normal stress for mode I opening.

G. Alfano, S. Marfia and E. Sacco

16

Then, the shear behavior of the proposed interface model is studied. A normal stress

2 MPa is assigned and taken constant, while the tangential relative displacement is

increased from 0 to 1.2 mm.

In figure 5, the shear stress versus the tangential relative displacement 2s is plotted for

three different values of the dilatancy parameter, 0 0, 0.1, 0.5 , with -10.8 mm . It can be

noted that the interface behavior is influenced by the value of the dilatancy parameter. In fact,

increasing the value of 0 an increment of the maximum shear stress is attained, while a

steeper softening branch occurs. This special interface shear response can be interpreted

looking at the micromechanics of the REA. As described in Section 2, according to the

proposed model, the shear stress is the sum of two terms: a first one due to the elasticity of the

undamaged part of the REA and a second one due to the friction, proportional to the local

normal stress occurring on the damaged part. Because of the ditatancy, a positive normal

relative displacement occurs in the REA, which induces a decrement of the absolute value of

the normal stress acting on the undamaged part of the REA; as the overall equilibrium of the

REA has to be satisfied, the local normal stress in the cracked part of the REA increases in

absolute value, leading to an increment of the friction stress.

Figure 5: Shear stress versus tangential displacement for different dilatancy parameters 0.

In figure 6, the overall stress at the interface and the local stresses in the damaged and

undamaged part of the REA weighted for their corresponding fraction of area, i.e. 1 uD

and dD , are plotted versus the tangential relative displacement 2s , for 0 0 and 0 0.5 .

It can be noted that the normal stress in the damaged part of the REA is higher when the

dilatancy effect is considered.

G. Alfano, S. Marfia and E. Sacco

17

Figure 6: Normal stresses at in interface: overall stress, stress on the undamaged part

and stress on the damaged part.

The steeper softening behavior in presence of dilatancy is due to the development of positive

normal relative displacement in the interface which, according to formulas (7) and (8), leads

to an increase in the rate of damage parameter evolution.

Then, a numerical study of the interface subjected to cyclic shear laoding-unloading history is

performed, considering the dilatancy parameter 0 0.5 and assuming the total normal stress

2 MPa constant during the cycles. In figure 7 the shear stress is plotted versus the

relative displacement 2s . Results are obtained through a displacement control. With reference

to figure 7 and figure 8, the interface response consists in the following steps:

OA, the tangential relative displacement 2s is increased from 0 to 1.0 mm; the shear

stress reaches its maximum and, then, a softening phase occurs until the complete

damage of the interface is reached; during this first step a dilatancy effect is obtained;

AB, the tangential displacement is decreased to 0; the shear stress attains a negative

constant value; the accumulated dilatancy normal displacement is recovered;

BC, the tangential displacement is decreased to -1.0 mm; the shear stress has a negative

constant value; the dilatancy normal displacement is obtained, characterized by a

reduced value respect to the first step;

CD, the tangential displacement is increased to 0; the shear stress attains a positive

constant value; the accumulated dilatancy normal displacement is again completely

recovered.

The shear behavior of the interface subjected to several loading-unloading cycles has been

studied according to the following horizontal displacement history:

2

[s] 0 1 2 3 4 5 6 7 8 9

[mm] 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1

t

s (48)

G. Alfano, S. Marfia and E. Sacco

18

Figure 7: Shear behavior for one complete loading cycle: shear stress vs horizontal

displacement.

Figure 8: Shear behavior for one complete loading cycle: dilatancy vs slip.

In figure 9, the shear stress versus the horizontal displacement is represented; a description of

the interface behavior in the time interval 0,2t is reported in detail:

OA, during the first loading phase, a partial damage occurs at the interface, with

frictional sliding at the damage part of the REA;

G. Alfano, S. Marfia and E. Sacco

19

AB, an unloading phase follows, characterized by a linear response with the initial

stiffness and for 0 a residual inelastic displacement is found; in this phase there is no

damage growth;

BC, the slope of the 2s curve changes because of the development of the inelastic

negative slip, without damage evolution;

CD, at point C, the absolute value of the horizontal displacement is equal to the one in

point A, thus the damage increases again beyond point C; the response of the interface is

characterized by inelastic slip and damage growth.

Figure 9: Shear behavior for several loading-unloading cycles: shear stress vs horizontal

displacement.

Similar considerations can be made for the subsequent cycles. The final reloading is

characterized by a flat 2s curve due only to the friction effect, as the interface is

completely damaged.

In figure 10, the dilatancy versus the slip is plotted, reporting also the time value in order to

better understand the global interface response. It can be noted that at each positive or

negative loading-unloading the dilatancy is recovered and the slope of the 1 2

di dis s curve

decreases during the cycles until the curve becomes flat, i.e. dilatancy is completely sutured.

From the numerical results concerning the mechanical response, the proposed interface model

appears able to reproduce the main features of the behavior of material discontinuities in

concrete constructions also in presence of uplift water pressure; in particular, the model can

be used to simulate the response of joint-interfaces resulting by the typical step-wise

construction process or by the interaction between rock foundation and the construction itself.

G. Alfano, S. Marfia and E. Sacco

20

Figure 10: Shear behavior for several loading-unloading cycles: dilatancy vs slip.

Comparison with experimental results

A comparison between the numerical results derived adopting the proposed interface model

with the experimental ones, obtained by Lee et al. [31], is performed. In particular, the

behavior of the interface between rock blocks of Hwangdeung granite is studied. In figure 11,

the scheme of the experiment developed in Ref. [31] is reported. Initially a normal load is

applied on the top of the upper block, then a shear loading cyclic history is performed keeping

constant the normal load. To model the experimental test a 2D analysis is carried out adopting

one element discretization for the upper block and a rigid support simulating the lower one.

The joint is modeled by the proposed interface finite element as schematically reported in

figure 11.

Figure 11: Geometry and loading of the joint.

The following interface parameters are considered:

G. Alfano, S. Marfia and E. Sacco

21

1 o1 1 0

2 o2 2

-1 -1

0

1.255 N/mm 1.97 MPa 1.25 mm 0

1.255 N/mm 1.97 MPa 1.25 mm 1

0.9 1.9 0.5 mm 0.162 0.002 mm

c o

c o

f i

G s p

G s

(49)

The joint is subjected to the loading history reported below:

1

2

[s] 0 1 2 3 4 5 6

[MPa] 0 1 1 1 1 1 1

[mm] 0 0 15 15 15 15 0

t

s

(50)

Each shear loading cycle is divided into four stages: forward advance (stage I), forward return

(stage II), backward advance (stage III), backward return (stage IV).

Figure 12: Shear behavior of rough granite joints for the first two cycles; comparisons

between experimental and numerical results: (a) first cycle; (b) second cycle.

In figure 12 the shear behavior of rough granite joints for the two cycles is reported putting in

comparisons the experimental and the numerical results. In figure 12 a) and b) the shear stress

versus the shear relative displacement is represented for the first and second cycles,

respectively. In both the cycles, it can be pointed out a good accordance in the evaluation of

the overall behavior of the joint. In particular, the peak shear strength and the plateau in the

forward advance and in the backward advance phases are very well determined by the

proposed model. The softening branch after the peak, due to the asperity degradation, is in a

good accordance with the simulation where it is modeled by the friction coefficient

G. Alfano, S. Marfia and E. Sacco

22

degradation. Some differences appear between the numerical and the experimental results

during the forward return and in the backward return phases, because the proposed interface

model does not consider the difference of frictional resistance during the advance and return

stages of shear displacement. As emphasized in Ref. [31], increasing the normal stress level

this effect becomes negligible. However, this effect could be included in the proposed model.

In figure 12 c) and d) the normal relative displacement versus the shear relative displacement

is represented for the first and second cycles, respectively. It can be noted that the numerical

and the experimental results are in a very good accordance. The proposed interface model is

able to simulate the dilatancy effects due to the asperity degradation occurring during the

cyclic shear test.

Mechanical response of a dam

The benchmark problem for dam analysis proposed by the ICOLD in Ref. [32] is considered.

The geometry of the dam is described in figure 13, and a non-proportional loading process is

assumed. The initial loads are given by the sum of the specific weight of the concrete, c =

24000 N m-3

, and of the hydrostatic pressure, qw (y) = 1000 · 9.81 · (80 - y) N m-2

. The

additional load consists of an overpressure, constant along y, equal to a reference value qo =

1.0 MPa multiplied by a loading factor . Hence, the external water pressure is given by:

w oq y q y q (51)

Concrete behavior is assumed to be linearly elastic, with Young modulus E = 24 GPa and

Poisson ratio = 0.15.

Figure 13: Geometry and loading of the dam.

A predefined ‘weak’ concrete-soil interface is considered at the base of the dam, while no

concrete joint-interfaces resulting by the typical step-wise construction process have been

modeled, for the sake of simplicity. The interface properties are reported in (47) and they have

been chosen in accordance with [24], with the dilatancy parameter 0 0 .

The finite-element mesh depicted in figure 14 is constructed with the aim of having a

sufficiently refined discretization in the neighborhood of the concrete-soil interface, and a

coarser mesh away from it. For the concrete bulk material, 434 4-noded and 50 3-noded

G. Alfano, S. Marfia and E. Sacco

23

plane-strain interface elements with enhanced modes have been used, while 64 4-noded

interface elements have been placed on the concrete-soil interface.

Several incremental, quasi-static analyses have been performed varying the value of the

parameter , which governs the diffusion of the external water pressure within the crack on

the interface. In all of them, in the first increment, the dead load and the hydrostatic water

pressure are assigned with their entire value, with the load multiplier set to zero.

Then, starting from the second increment, is increased following an automatic

incrementation procedure.

In order to follow the strongly nonlinear structural response, the local-control arc-length

scheme with line searches described in [33] is used.

Figure 14: Adopted finite-element mesh.

Figure 15: CSD- curves for different values of .

G. Alfano, S. Marfia and E. Sacco

24

Figure 16: COD- curves for different values of .

In figures 15 and 16, the crack-sliding-displacement (CSD) (the horizontal displacement

component ux of point P in figure 13), and the crack-opening-displacement (COD) (the

vertical displacement component uy of point P in figure 13) are plotted versus the load

multiplier , for different values of the parameter . All the curves begin with a steep,

increasing and almost linear branch, which does not exactly start from the origin of the axes

because of the initial elastic deformation of the interface for =0 due to the dead load and the

hydrostatic water pressure.

For = 0 the CSD monotonically increases with , with a decreasing slope of the curve, until

the maximum value = 0.492 is reached for a value ux of about 0.8 mm. This point represents

a limit point in the equilibrium path, which is followed by a softening, unstable part of the

load-displacement curve, until the value = 0.423 is attained. At this point the crack has

reached the end of the concrete-soil interface and the whole dam slides at a constant value of

the applied overpressure. In this part of the process the interface adhesion has been

completely lost and the interface tangential tractions are only due to friction.

The corresponding - COD curve for = 0 is characterized by an increasing part until the

maximum value of , at which a snap back occurs. Then, the COD decreases back to a value

of about 1.8 mm, which represents the constant COD value during the final sliding phase.

For increasing values of , two phenomena can be observed in the - CSD curves. Firstly,

both the maximum applied load and the final ‘sliding’ load decrease, as was expected.

Secondly, for values > o, with o somewhere between 0.05 and 0.10 mm-1

, the CSD does

not increase in a monotonic way, but the curves perform a sort of loop at a certain point. This

behavior can be explained by observing that the applied water pressure on the opened crack

determines a local state of compression on the bottom - left part of the dam, which in turn

results in a local transverse dilatation due to the Poisson ratio of the concrete, and then,

ultimately, in a decreasing value of the CSD. This occurs for a relatively small part of the

G. Alfano, S. Marfia and E. Sacco

25

loading process after which the CSD again increases because the ‘global’ deformation of the

whole structure becomes predominant with respect to the above described ‘local’ effect.

In figures 15 and 16, the results obtained setting the dilatancy parameter 0 0.5 , with

-10.8 mm and 0 , are reported. It can be emphasized the quite different mechanical

response due to the dilatancy effect. In particular, comparing the two - COD diagrams

obtained considering or neglecting the dilatancy, when 0 , a different snap-back response

can be noted.

In the - COD curves, for increasing values of , apart from the decrease in the maximum

value of , which is clearly a result already noticed from the - CSD curves, one can also

observe an increase in the maximum value of the COD, when varies from 0 to 0.50 mm-1

.

In figures 17 and 18, the normal and tangential stress profiles y and xy at y = 0, during the

final sliding phase of the process, are reported for different values of . In particular, figure

17 shows that the state of compression of the bottom-left part, which is due to the water

pressure acting on the open crack, increases with , while the compression of the bottom-right

part decreases. Accordingly, the shear stresses due to friction decrease with increasing in

the bottom-right part as is shown in figure 18.

Figure 17: y profile at y=0 for different during the final sliding phase.

In figure 19 the water pressure p profiles at y=0, during the final sliding phase process, are

represented for different values of . It can be pointed out that the water pressure profile is

almost linear for low values of while it results almost constant for high values of . This is

in accordance with the experimental finding that the actual uplift pressure profile varies

between a triangular and rectangular shape, as reported in figure 19.

It can be remarked that the numerical analysis of the dam has been performed assuming the

initial value of water pressure equal to zero, i.e. 0 0p , which corresponds to an ideal

G. Alfano, S. Marfia and E. Sacco

26

perfectly drained dam. This choice is done in order to emphasize the effect of the uplift water

pressure on the crack propagation and, consequently, on the overall response. Moreover, from

a computational point of view, this case results more complex than the case 0 0p , as when

0 0p the uplift water pressure increment is greater.

Figure 18:xy profile at y=0 for different during the final sliding phase.

Figure 19: Water pressure profile at y=0 for different during the final sliding phase.

G. Alfano, S. Marfia and E. Sacco

27

6 CONCLUSIONS

A multiscale interface model for the analysis of crack propagation in dams has been

presented. The formulation takes into account damage, friction and dilatancy occurring at the

interface as well as the influence of water pressure on the open part of a crack.

The inclusion of friction effects in the model is based on the additive decomposition of a

representative area of the interface (REA) into a damaged and an undamaged part, with a

unilateral dry-friction contact model adopted on the latter part. The degradation of the friction

and dilatancy coefficients are taken into account. The possible recovering of the dilatancy

effect occurring during cyclic loading is modeled.

The presence of water in the crack is accounted for by adding a further static pressure as an

exponential function of the opening relative displacement. Such pressure is equal to an initial

value for a perfectly closed crack and asymptotically reaches the external water pressure for

increasing values of the opening relative displacement, with a rate which increases with the

parameter .

From the numerical results developed to assess the efficiency of the proposed model, it results

that the interface model is able to reproduce the main features (damage, friction, recoverable

dilatancy effect) of crack propagation in concrete structures also considering the influence of

the uplift water pressure. A comparison between experimental and numerical results for a

cyclic test demonstrates the ability of the proposed model to reproduce the real behavior of

rough interfaces.

The numerical procedure is stable and robust and it shows good convergence properties.

A series of quasi-static, incremental simulations has been conducted for a benchmark problem

of dam analysis, concerning a concrete dam subjected to gravity load, hydrostatic pressure,

and an additional overpressure depending on a load factor.

The presented results show the ability of the model of well capturing most of the expected

features of the structural response, namely the increase in the mode-II interface strength due

to friction and the decrease of the failure load due to the presence of the water pressure.

ACKNOWLEDGEMENTS

The financial support of the Italian Ministry of Education, University and Research (MIUR),

the National Research Council (CNR) and the Regione Campania is gratefully acknowledged.

The first author is also grateful to FEA Ltd. for providing the finite-element code LUSAS.

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