A Cohesive Element Model for Mixed Mode Loading With Frictional Contact Capability 2013 International Journal for Numerical Methods in Engineering

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2013; 93:510526Published online 31 July 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4398

A cohesive element model for mixed mode loading with frictionalcontact capability

Leonardo Snozzi and Jean-Franois Molinari*,

cole Polytechnique Fdrale de Lausanne (EPFL), School of Architecture, Civil and Environmental Engineering(ENAC), Computational Solid Mechanics Laboratory (LSMS), Lausanne, Switzerland

SUMMARY

We present a model that combines interface debonding and frictional contact. The onset of fracture isexplicitly modeled using the well-known cohesive approach. Whereas the debonding process is controlledby a new extrinsic traction separation law, which accounts for mode mixity, and yields two separate values

for energy dissipation in mode I and mode II loading, the impenetrability condition is enforced with a contactalgorithm. We resort to the classical law of unilateral contact and Coulomb friction. The contact algorithmis coupled together to the cohesive approach in order to have a continuous transition from crack nucleationto the pure frictional state after complete decohesion. We validate our model by simulating a shear test ona masonry wallette and by reproducing an experimental test on a masonry wall loaded in compression andshear. Copyright 2012 John Wiley & Sons, Ltd.

Received 29 November 2011; Revised 15 May 2012; Accepted 1 July 2012

KEY WORDS: cohesive zone model; dynamic fracture; frictional contact; mixed mode loading; masonrywallette; numerical methods

1. INTRODUCTION

A fundamental understanding of the failure of heterogeneous brittle materials, such as concrete,

needs a careful treatment of both microcracking and frictional contact mechanisms. Indeed, fracture

of such materials involves the opening of local microcracks, which may propagate, coalesce, and

subsequently enter into contact. The contacting rough surfaces play an important role in the amount

of dissipated frictional energy, influencing the structural strength.

One of the most popular approaches, to model the onset of fracture, is to use cohesive zone

models. This method relates the displacement jumps at the crack tip with tractions. The approach

has been introduced by Dugdale [1] and Barenblatt [2] in the 1960s. The model describes fracture

as a separation process, which occurs at the crack tip. The debonding is assumed to be confined

in a small region of material called the cohesive zone, which represents the region where atomistic

separation occurs. A constitutive relationship called traction separation law (TSL) describes the

decohesion process. Several works have had recourse to this method (among them, [36], and more

recently, [79]).The transition from debonding to sliding frictional contact is an often overlooked (but key)

dissipative mechanism in cohesive zone approaches. One of the first attempts combining both

fracture and frictional contact phenomena has been introduced by Tvergaard [10]. In this approach,

the response of the surface is assumed to be frictionless until complete debonding of the cohesive

zone has occurred. Afterwards, other authors considered the onset of friction to start in conjunc-

tion with the onset of fracture, for instance, by coupling friction to decohesion [11, 12] or to

*Correspondence to: Jean-Franois Molinari, Btiment GC - A2, Station 18, CH 1015-Lausanne, Switzerland.E-mail: [email protected]

Copyright 2012 John Wiley & Sons, Ltd.


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A COHESIVE ZONE MODEL FOR MIXED MODE LOADING WITH FRICTIONAL CONTACT 511

adhesion [13, 14]. More recent works on this topic include those on fiber pullout/pushout

(e.g., [1519]) and applications to structural engineering (e.g., plane and reinforced concrete by

Cervenka et al [20] and Raous et al. [21] and masonry walls by Alfano and Sacco [22], Sacco andToti [23], Fouchal et al. [24], and Koutromanos and Shing [25]) and to brittle materials [26].

The aim of this paper is to present a new approach coupling cohesive zone modeling, to represent

damage, and a contact algorithm, for enforcing the impenetrability condition, in a 2D finite element

framework. The key features of the model are the following:

A novel initially rigid extrinsic TSL, which enables us to define two separate values for the

dissipated fracture energy in modes I and II. This formulation extends the work of van den

Bosch et al. [27], who developed such a model for TSL with an initial elasticity (intrinsicapproach); such intrinsic cohesive laws are known to alter the elastic properties of the

specimen [28].

A continuous transition from decohesion to the pure frictional state. Friction and fracture

occur simultaneously when a crack is forming under compression, avoiding numerical insta-

bilities due to a possible stress jump when transitioning from complete debonding to pure

frictional sliding.

An explicit dynamics analysis (second-order explicit version of the popular Newmark method

[29]). This approach is chosen to avoid convergence problems, which might arise when multiple

cracks are propagating and coalesce, offering therefore a suitable and simpler alternativecompared with a classical implicit scheme.

Section 2 describes the developed mixed mode cohesive approach and applies the proposed TSL to a

benchmark [27]. In Section 3, the adopted frictional contact enforcement algorithm is reported. The

model has been successfully implemented in a finite element program and validated by experimental

data at the material level and the structural level.

Section 4 validates the model by comparing the numerical results to experimental data at the

mesolevel and structural level. The first application consists of a shear test experiment on masonry

wallettes (a conventional test in order to determine the shear resistance of joints between bricks),

whereas the second application simulates a masonry wall loaded in compression and shear. Finally,

concluding remarks are reported in Section 5.

2. ADOPTED TRACTION SEPARATION LAW

2.1. Cohesive zone models

Within the cohesive zone approach, the damage and the discontinuity in the displacement field are

assumed to be concentrated at the crack interfaces. The constitutive relationship, which relates the

traction acting on the separating surfaces to the relative opening displacement vector (), is called

the TSL:

TD T./ (1)

Usually, the TSL is related to a potential (), which represents the stored fracture energy for a spe-cific interface. Because of this formulation, it is possible to obtain the components of the traction

vector (T) simply by differentiating the potential with respect to the opening displacement vector:

TD@

@(2)

In the computational framework, the cohesive zone is represented by interface elements (in which

the TSL is implemented). These elements, called cohesive elements, are placed at the boundaries

between bulk elements.

One can distinguish between two main categories of TSL depending on the response of the inter-

face before the softening behavior. With the extrinsic approach (see, for instance, [5]), which is

used in this work, the surface shows only a softening behavior (the response of the surface prior

crack initiation is assumed to be rigid), whereas, with the intrinsic approach (see, for instance, [3]),

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2013; 93:510526

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512 L. SNOZZI AND J.-F. MOLINARI

the TSL exhibits generally an initial linear elastic behavior prior to softening, which requires

cohesive elements to be present from the beginning of the simulation. This leads to an artificial

compliance of the uncracked body, which is avoided with the extrinsic approach where cohesive

elements are inserted dynamically.

2.2. CamachoOrtiz linear irreversible cohesive law

One of the most popular TSLs for the extrinsic approach was proposed by Camacho and Ortiz

[5] in 2D (and Pandoli and Ortiz [30] in 3D). There, the cohesive law is a linear decreasing

function (depicted in Figure 1) of the effective opening displacements and is derived from a free

potential energy.

./ D1

2c

2

c

(3)

where c represents the local material strength and c represents the effective relative displace-ment beyond which complete decohesion occurs. The potential is assumed to depend not directly

on the relative displacement vector, , but on an effective scalar displacement, which has the

following form:

Dq 2

2

t C 2n (4)

where n and t represent the normal and the tangential separation over the surface with unitoutward normal n and unit tangential vector t, respectively. The parameter accounts for the cou-pling between normal and tangential displacement. The derivation of the free potential energy with

respect to the opening displacement leads to the cohesive traction (2D):

TD@

@D

T

.2 ttCnn/ (5)

This traction in case of crack opening is given by the following:

T., max/ D c

1

c

for D max (6)

where max stores the maximal effective opening displacement attained. On the other hand, forcrack closure or reopening ( smaller than max), the functional form is assumed to have thefollowing form:

T., max/ DTmax

maxfor < max (7)

where Tmax is the value of the effective traction when is equal to max. Moreover, max alsoaccounts for the irreversibility of the law allowing successive loading/unloading steps.

Note that the definition of c and c implicitly establish the existence of an effective fractureenergy Gc, which corresponds to the area under the curve of Figure 1.

Figure 1. CamachoOrtiz linear decreasing cohesive law [5].


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Figure 2. Cohesive zone loaded with two different paths leading potentially to two distinct values ofdissipated energy.

2.3. Modified traction separation law for mixed mode loading

Because the CamachoOrtiz TSL is based on a potential, it is clear that the dissipated fracture energy

for a growing crack does not depend on the opening path. However, as reported in [27] by van den

Bosch et al., it can be argued that the dissipated fracture energy should be path dependent, becauseof microstructural details such as interface roughness that cannot be depicted without an extremely

fine discretization.

Hence, the TSL should allow for independent values of the dissipated energy depending on the

loading path as depicted in Figure 2. We propose an alternative cohesive law based on the clas-

sical model of Camacho and Ortiz by relaxing the hypothesis of a well-defined energy potential

(as previously done in [27] but for the intrinsic cohesive model of Xu and Needleman [3]). We have

preferred this option to a pseudo-potential formulation (for instance, that proposed in [31]) because

of its ease of formulation. This results in a law that is not anymore bounded by a free potential

energy (i.e., only the tractions are defined). A new independent parameter , which enables us todefine the ratio between Gc,I and Gc,II, is introduced.

DGc,II

Gc,I(8)

We set the work of separation for a cohesive zone opened completely under mode I ( Gc,I) to cor-respond to the one computed with the CamachoOrtiz model (Gc). Therefore, the cohesive normaltraction envelop has, in this case, an identical shape as in the CamachoOrtiz TSL. Conversely,

because the introduced parameter, , is linked with the tangential direction, the adjustment influ-

ences the shapes of the tractions, when the opening does not occur in pure normal direction. Ourmodel yields the following equation for the cohesive tractions in case of crack opening:

TD

2

ttCnn

c

1

c

(9)

The effective opening displacement, previously defined in Equation (4), needs to be redefined

as follows:

D

r2

22tC2

n(10)

Normal and tangential tractions are shown in Figure 3 as functions of the normalized normal and

tangential opening displacements. As postulated in [5], damage is considered to be an irreversible

process. Unloading or reloading takes place when < max. In this case, the tractions are calculatedwith the following:

Tn D nc

max

1

max

c

(11)

Tt D2

t

c

max

1

max

c

(12)

With these equations, the tractions are linearly interpolated between the origin and the tractions at

which the opening will be equal to the maximal attained effective opening displacement in the new

loading direction.


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(a) (b)

Figure 3. Cohesive traction in (a) normal direction and (b) tangential direction (with > 1).

1

30 60 90

(b)(a)

Figure 4. (a) Cohesive zone loaded under a constant angle as in [27] and (b) resulting dissipated work asfunction of the loading angle expressed in units ofGc,I (with > 1).

2.3.1. Work of separation under combined normal shear loading. If is set equal to one, the trac-tions and the dissipated energy match with those of the CamachoOrtiz TSL, whereas, if is setdifferent to one, the dissipated fracture energy becomes path dependent and therefore does not cor-

respond for every loading path to the mode-independent fracture toughness Gc. This energy can becomputed by integrating the tractions over the path. In order to depict the mode-dependent behav-

ior of the proposed TSL and to verify our model, we repeat the two different patch tests reported

in [27]. In this previous work, it was demonstrated that the classical XuNeedleman law failed

these patch tests, showing some unphysical/unexpected behavior in the computed amount of sep-

aration work. The first one consists in loading a surface monotonically with an imposed constant

loading angle until complete separation occurs (Figure 4(a); i.e., proportionality between tangential

and normal displacements). On the other hand, in the second case, the cohesive zone is loaded

first in one direction up to a maximal value and then broken completely in the other direction

(non-proportional loading).

For a separation associated with a loading angle , the work of separation can be expressed as

Wtot D

Zn,c0

Tn./dnC

Zt,c0

Tt./dt (13)


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where n,c and t,c represent the opening displacement values in normal and tangential direc-tions at which complete separation occurs. The evolution of Wtot in respect to the loading angle isdepicted in Figure 4(b). The graph has been normalized by Gc,I. Because we have chosen a highervalue of the fracture energy in mode II, the dissipated fracture energy increases monotonically by a

factor of between the two extremes 0 and 90, which correspond to the fracture energy for mode Iand mode II, respectively. Indeed, increasing the angle is equivalent to increasing the work done by

the tangential traction (Wt) and decreasing the portion corresponding to the normal traction (Wn).For the nonproportional loading, the surface is opened in the first case up to a maximal normal

displacement n,max and then broken completely in shear (Figure 5(a)). Conversely, in the secondcase, the zone is loaded first in tangential direction (up to the value t,max) and then separated innormal direction until complete separation occurs (Figure 5(b)).

The dissipated fracture energy for cases 1 and 2 can be expressed with

Wtot D

Zn,max0

Tn.n, t D 0/dnC

Zc,t0

Tt.n D n,max, t/dt (14)

Wtot DZt ,max0

Tt.n D 0, t/dt CZc,n0

Tn.n, t D t,max/dt (15)

The results of the first case are reported in Figure 6(a). Wtot is equal to Gc,I ifn,max is set equalc and corresponds to Gc,II if n,max is equal to zero. An analogous behavior can be observed inthe second case (Figure 6(b)). In both loading cases, it is possible to recognize that the transition of

the dissipated fracture energy as function of the maximal first opening is smooth and does not show

any kink or unphysical oscillation.

(a) (b)

Figure 5. Interface nonproportionally loaded: (a) first in normal direction until n,max and then broken inshear and (b) first in tangential direction until t,max and then broken in normal direction as in [27].

1

0.5 1

(a) (b)

0.5 1

1

Figure 6. Work of separation when loading first in (a) normal direction and in (b) tangential direction,respectively, expressed in units ofGc,I (with > 1).


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3. CONTACT ENFORCEMENT

When a crack is growing under compressive shear loading, we enforce the impenetrability condition

by a masterslave explicit contact algorithm. Because our goal is to be able to deal with multiple

cracking, which causes numerous asperities to enter into contact, we have preferred explicit contact

enforcement rather than a more traditional static or implicit dynamic formulation. Among this sec-

ond class of methods, augmented Lagrangian multipliers (for instance, [32]) and penalty methods(for instance, [33]) have been widely used. The first class is able to provide an accurate constrain

enforcement circumventing ill-conditioning. Nevertheless, the size of the problems might be lim-

ited because of the implicit system of equations that one needs to solve, which can rapidly increase

the computational cost with increasing system size. On the other hand, penalty methods allow a

little interpenetration of the contacting surfaces, which is inversely proportional to the value of the

penalty parameters. Nevertheless, a high value of these parameters can produce an ill-conditioned

system of equations, requiring a high number of iterations and leading to convergence problems.

Moreover, another drawback, in the explicit dynamic version of this method, is that the critical time

step decreases with increasing stiffness penalty parameter.

Thereby, in this work, we have decided to address the aforementioned difficulties enforcing

the impenetrability constrain by using a ballistic method called decomposition contact response

developed by Cirak and West [34]. This method is based on the conservation of linear and angu-

lar momentum while the impenetrability condition is guaranteed by acting on the displacements

directly, for example, by projecting the impacting nodes on the penetrated surface. Therefore, the

purpose of the method is not to resolve the impact time exactly for every node as depicted in

Figure 7(a) but with an approximation as illustrated in Figure 7(b). Thus, nodes, which are allowed to

penetrate the master surface within the predictor

ti

, must subsequently be projected back within

the same time step

tCi

. The equations governing the impact (beside the projection of the slave

nodes on the master surfaces) are

ptC

i pti D rxg

xtC

i

(16)

pTM1ptC

i

ti

D 0 (17)

where p D MPx represents the momentum vector of slave and master nodes (with M being themass matrix, which has the size 6 6 for edge-to-node contact), g is the gap function, and ascalar parameter. The approach modifies only the post-impact velocities of the contacting nodes

involved in contact (components in direction of the gap gradient). A closed form for the post impact

velocities can be derived using momentum decomposition. We will report here the main equations

for computing the momentum after collision (for a detailed derivation, see [34]). First, velocities

of all impacting nodes need to be decomposed into their normal (subscript n) and tangential

(subscript t) components giving

PxD Pxn C Pxt (18)

(a) (b)

Figure 7. (a) Impact time resolved exactly and (b) approximation of the decomposition contact response.


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where the normal Pxn components are defined as

Pxn D

.rg/T Px

.rg/TM1rg

M1rg (19)

With the aforementioned equation, one can compute the normal components of the velocity before

impact of the contacting triplets (node-to-segment approach). The normal components of thevelocity after impact can be thus corrected as follows:

PxCnD Px

n Px

n.1 C cres/ (20)

where cres represents the coefficient of restitution, which can range between 0 (completely inelasticcontact) and 1 (perfectly elastic contact), and the superscripts C and denote quantities beforeand after projection, respectively (in the application presented in this paper, we choose cres D 0).

3.1. Frictional impact

A simple Coulomb friction law can be included by computing the relative motion between the

contacting entities. To this end, the velocity needs to be decomposed into fixed and nonfixed

components:

PxD Pxnonfix C Pxfix (21)

where only the nonfixed components Pxnonfix lead to a relative motion between the bodies. In orderto derive the nonfixed component of the velocity, a separation vector between two impacting points

needs to be defined:

hD xL xR (22)

where xL and xR stand for the positions of the two impacting points. Consequently, the nonfixed

components can be obtained as follows:

Pxnonfix DM1.rh/T .rh/Px

.rh/M1.rh/T

(23)

Because the nonfixed velocity is the resultant between motion leading to interpenetration ( Pxn) andrelative tangential displacement (sliding), the slide components are given by

Pxslide D Pxnonfix Pxn (24)

According to the Coulomb model, one can compute the slip velocity as follows:

kPxT

nM1 Pxnk

kPxTslide

M1 PxslidekPxslide (25)

In order to account for stickslip, the authors [34] suggested to bound the maximal frictionalimpulse, which can be delivered during sliding with the minimum value between Equations (24)

and (25) (the stick and slip velocity, respectively):

PxCtD Px

tmin

1,

kPxTnM1 Pxnk

kPxTslide

M1 Pxslidek

Pxslide (26)

If the relative motion between two contacting bodies becomes too small, Equation (26) will cause

most of the nodes to get into stick, leading consequently to an insufficient frictional force between

the sliding bodies. Therefore, the limitation criterion of Equation (26) has been removed, allowing

bigger correction in the tangential quantity of motion within the predictor. Nevertheless, this can


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produce too large values in the delivered frictional impulse. Thus, the relative motion between the

sliding nodes needs to be verified after Newmarks corrector and, if necessary, corrected.

3.2. Coupling with cohesive zone model

As already mentioned before, the contact algorithm is coupled in parallel with the TSL. This implies

that friction and fracture occur simultaneously in the cohesive zone when a crack is growing undercompression. A first possibility, as illustrated in Figure 8, is to assume a full onset of friction at the

beginning of decohesion. This superimposition combined with the extrinsic approach gives a very

steep (rigid) initial behavior followed by the softening branch. Therefore, in order to reproduce the

experimental interfacial behavior, a different transition from the onset of fracture to pure frictional

sliding has been chosen (note that the accuracy of the transition might depend on the level of obser-

vation and material). Our second approach (depicted in Figure 9) considers a progressive shift to

friction during debonding. When the cohesive zone starts to damage, friction does not act on the

interface, but it raises gradually with increasing decohesion, giving a soft transition from debonding

Figure 8. Shear stresstangential opening displacement relationship for a growing crack in mode II with full

onset of friction.

Figure 9. Adopted shear stresstangential opening displacement relationship for a growing crack in mode IIwith increasing frictional capability.


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to the pure frictional stage. The function that regulates the increase in the friction force depends on

the amount of damage experienced by the cohesive zone and is given by the following:

1

1

c

q(27)

where the exponent q has been set in this work to 4. Note that the chosen quartic transition func-

tion implies a relatively high dissipation of frictional energy already during the decohesion process(and that the peak shear strength/traction remains essentially influenced by both the cohesive

strength and the amount of applied compressive force). To summarize, as depicted in Figure 9,

for a growing crack in mode II, the cohesive elements are inserted if the tangential stresses exceed

c. A fracture energy corresponding to Gc multiplied by the length of the cohesive zone willhave been dissipated when the tangential opening (t) attains the critical opening value (c=).During this process, the cohesive zone experiences a continuous transition from debonding to pure

frictional sliding.

4. NUMERICAL VALIDATION

As mentioned in Section 1, in order to validate our numerical model, we have selected two prob-

lems related to the masonry engineering field. The first application is a simple and representativeexperiment of a masonry wallette, which has been used to identify the interfaces parameters. These

parameters have then been used to predict the behavior of a masonry wall loaded in compression

and shear.

4.1. Masonry wallette

Masonry is one of the most widely used construction materials in civil engineering around the world.

Therefore, different tests have been designed in order to extract the properties of its components.

Among these, a common experiment in order to determine the ultimate shear strength of the mortar

joints is a shear test of a masonry wallette, which consists of three bricks linked with two mortar

joints (shear triplet) as depicted in Figure 10(a). For our paper, we refer to the experimental data

obtained by Beyer et al. [35, 36].

Experimentally, the masonry wallettes have been tested on a loading machine. During the exper-iment, the inner brick is supported at the upper edge with two rigid blocks, whereas the shear load

is introduced on the lower edge of the two outer bricks. Horizontal rods are responsible for the

horizontal axial force, which guarantees constant normal stress acting on the mortar joints during

the shearing.

The virtual experimental setup chosen for the numerical test consists in half of the symmetric

part of the experimental setup and is drawn in Figure 10(b) with the adopted finite element mesh.

(a) (b)

Figure 10. (a) Experimental test setup of the wallette (courtesy of Beyer [35, 36]) and (b) correspondingfinite element mesh, corresponding to one half of the experimental setup (all dimensions in millimeter).


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The mesh has been chosen fine enough in order to achieve numerical convergence leading to an ele-

ment size of about 5 mm (element edge) at the interbrick boundary. The boundary conditions for the

upper and lower edges have been simplified by extending the length for load introduction and sup-

port to the edge length. It should be pointed out that this change in the boundary condition has not

a significant influence on the global behavior of the wallette (in terms of shear stressdisplacement

relationship). The two bricks have been modeled explicitly, whereas the mortar layer and the two

interfaces between mortar and bricks are represented by means of a line of dynamically insertedelements with the proposed cohesive frictional capability.

Elastic constitutive behavior is assumed for the bricks elements. The elastic material parameters

are reported in Table I (and have been taken from [35] for the density and Youngs modulus and

from [22] for the Poisson ratio). The cohesive parameters (Table II) for the interfacial transition

zone between bricks have been fitted, in order to achieve a good agreement between the numerical

and experimental sheardisplacement relationship. The value for the critical stress is of the same

order as the values for the strength of the joints reported in [37,38], whereas the values for the frac-

ture energies are somewhat higher than the values suggested in these two references but on the same

order of magnitude as the ones given in [39]. Moreover, has been set in order to represent the usualratio between the fracture energies in modes II and I in masonry engineering. The fitting has been

achieved by varying the value of the cohesive strength and dissipated fracture energy. As depicted

in Figure 11(a), larger values ofc increase the peak shear resistance, whereas larger values ofGc,I,as depicted in Figure 11(b), shift the transition to the pure frictional state towards higher values

of the brick displacement. One can notice from Table II that only cohesive values for the inter-

face between mortar and brick are given. Indeed, as reported earlier, the propagation of cracks can

Table I. Material properties.

Material E (Gpa) () (kg/m3)

Brick 14.0 0.15 940

Table II. Cohesive properties for interface elements.

Interface c,I (MPa) Gc,I (J/m2) c,II (MPa) Gc,II (J/m2)

Brickmortar 0.2 0.3 0.4 75 125 200 c,I 6 Gc,I 0.77

Those in bold are assumed values for the shear test.

0 2 4

0

0.6

0.4

0.2

[MPa]

[mm]

= 0.25 MPa= 0.3 MPa= 0.35 MPa

(a)

0 2 4

0

[MPa]

[mm]

= 75 J/m2

= 125 J/m2

= 200 J/m20.6

0.4

0.2

(b)

Figure 11. Shear stress plotted against shear displacement: (a) for different values ofc (Gc,I kept constant)and (b) for different values of Gc,I (c kept constant).


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occur only at the interelement boundaries between brick and mortar, whereas bricks are assumed

to remain uncracked. Indeed, we have not adopted a fully mesoscale representation of masonry

(as, for instance, in [39]), and all the cracking is concentrated at the straight interfaces.

In order to have a smoother transition from uncracked to cracked regime, variations in the

value of the critical stresses following a normal distribution (with standard deviation in the order

of 0.05 MPa) have been adopted. Moreover, in order to reduce oscillation during sliding and to

reduce the amount of energy injected in the system (which is due to the projection of the slavenodes), material damping has been adopted. Figure 12 shows the influence of damping on the

computed stressdisplacement relationship and on the amount of kinetic energy in the specimen

for two different Rayleigh damping ratios [40]. As illustrated in Figure 12(b), the regularizationthrough damping allows to maintain the right level of kinetic energy (insufficient damping leads to

an increase of energy in the system and to oscillations in the stress displacement curve). For this

application, a damping ratio of 7% has been used.

As in the testing machine, the specimen has been loaded with an imposed displacement with a

constant normal pressure level of 0.4 MPa. The outer brick has been displaced at a constant initialvelocity (v) of 0.025 m/s in order to avoid rate effects (similar to [41]). Nevertheless, loading veloc-ities beyond 0.05 m/s increase the peak shear stress. Similarly to the experiments, the shearing hasbeen interrupted after a path of approximately 10 mm.

The comparison of the shear stressdisplacement relationship between experiments and simula-

tion is illustrated in Figure 13. The stress (y-axis) in the figure is obtained by dividing the shearforce acting on the brick by the area of the interface between mortar and brick.

(a) (b)

Figure 12. Regularization of damping on (a) the stressdisplacement curve and (b) the amount of kineticenergy in the system.

0 102 4 6 8

0

0.8

0.6

0.4

0.2

[MPa]

[mm]

v = 0,025 ms-1

Beyer etal.

Figure 13. Shear stress plotted against shear displacement: comparison between experimental envelope[35, 36] (dotted gray area) and simulation (continuous red line).


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In general, the experimental and numerical curves are in good agreement. The transition to the

pure frictional regime is smooth, and the model is able to depict friction correctly. This demon-

strates the robustness of our frictional cohesive model. The curved shape of the numerical shear

displacement relationship at the initial stage of loading can be traced back to the assumed transient

coupling between decohesion and friction.

4.2. Masonry wall

The second selected application for the interface model is the simulation of a masonry wall loaded

in shear and compression under low and moderate strain rates. For this virtual experiment, we

have chosen to reproduce one test unit of the experimental test program conducted by Ganz and

Thrlimann [42]. The geometry of the wall is represented in Figure 14. The specimen is 3.6 m wide

and 2 m high (which represents approximately 80% of a full-scale wall in a building). Experimen-

tally, each test unit has been loaded first with a normal compressive force (kept constant during the

shearing) and then displaced horizontally under a quasistatic regime. The bricks that compose the

wall are hollow clay bricks 29 cm long, 19 cm high, and 15 cm thick, and thus similar to the bricks

of the previous application. Because no experimental data of the interface properties were available,

we have used the parameters calibrated for the first application.

The mesh adopted for the virtual experimental setup is illustrated in Figure 15. The bricks

have been modeled explicitly, whereas the interfacial transition zone is represented (as before) by

dynamically inserted elements with the proposed cohesive frictional capability. Because cohesive

insertion is allowed only at the interfaces between bricks, bricks are assumed to remain uncracked

(this assumption can be considered adequate for low-to-moderate values of the horizontal wall dis-

placement, because little crushing of the bricks is observed experimentally). The bricks are modeled

with elastic elements, and their material properties are the same as those reported in Table I except

for the density that has been set to 855 kg/m3. The cohesive parameters of the interfaces are reported

in Table II.

The specimen has been loaded first with a vertical force of 415 kN. Little fluctuations in the

amount of vertical displacement of the nodes located on the top edge have been allowed in order

to maintain a constant level of compression during the shearing test. The shearing has been applied

to the top edge of the wall prescribing a displacement u. The test has been performed at two load-

ing velocities (v): a low one (0.05 m/s), which can be considered representative of a quasistaticregime, and a moderate one (0.5 m/s) in order to show the rate-dependent response of the tested unit.

415 kN

3600

20

00

415 kN

Figure 14. Experimental setup of the masonry wall tested in [42] (all dimensions in millimeter).


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Figure 15. Adopted finite element mesh.

0 205 10 15

0

300

200

100

[kN]

[mm]

v = 0.05 ms-1

v = 0.5 ms-1

Ganz and Thrlimann

Figure 16. Comparison of the horizontal forcedisplacement curves for the masonry wall between the finiteelement analysis (continuous red line and dotted blue line) and the experimental results (dashed gray line)

of Ganz and Thrlimann [42].

(a) (b)

Figure 17. Deformed mesh configuration (at u D 14.5 mm) for shear velocity of (a) 0.05 m/s and (b) 0.5 m/s(magnification factor 25).

The damping ratio has been set to 0.75%. The numerically computed and the experimentally

recorded forcedisplacement curves are compared in Figure 16. One can remark that the model

can successfully reproduce the experimental behavior. Although the initial response of the wall is

stiffer than the experimental one (probably because of the virtual setup and the assumed interface


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parameters), the peak strength and the corresponding displacement at peak strength almost match

the values reported by Ganz and Thrlimann. As depicted in the figure, the response of the wall

depends on the amount of horizontal velocity. An increasing velocity causes a delay in the dissipa-

tion of cohesive energy and a more diffuse cracking network that produces an overall increase in

strength. The crack pattern for the two different loading velocities is depicted in Figure 17. One can

notice that the higher applied velocity causes a higher number of cracks perpendicular to the load-

ing direction. The bigger amount of cracks is related to the intrinsic opening time of the cohesivezone and is responsible for the increase in strength, which is obtained despite the rate-insensitive

constitutive law used to model the interfacial behavior.

5. DISCUSSION AND CONCLUSIONS

In this paper, a new model that combines interface debonding and frictional contact is presented.

The model is implemented in a 2D finite element framework within explicit dynamics and allows

for large relative displacements.

The debonding process is controlled by a TSL based on the popular linear extrinsic irreversible

law proposed by Camacho and Ortiz [5]. The law has been modified in order to account for

path-dependent behavior and therefore to introduce a mode-dependent fracture energy. The impene-

trability condition is enforced by the decomposition contact response algorithm developed by Cirakand West [34]. We resort to the classical law of unilateral contact and Coulomb friction. The contact

algorithm is coupled together with the cohesive approach in order to have a continuous transition

from crack nucleation to the pure frictional state.

The model has been validated by simulating a representative shear test on a masonry wallette.

Furthermore, the calibrated values for the interfacial transition zone in the wallette application have

been used to reproduce a test on a masonry wall loaded in compression and shear. The numerical

results have shown the capability of the model to represent the physical process involved during

cracking in compression and to give a good prediction of the structural response of the tested units.

This is achieved with a relatively simple model, which does not involve many parameters.

For the future, we plan to apply our model to simulate compression and shear of a mesoscale

concrete specimen to capture the interlock between the generated cracked surfaces.

ACKNOWLEDGEMENTS

This material is based on the work supported by the Swiss National Foundation under grant no. 200021122046/1. The authors wish to thank Katrin Beyer for providing experimental data and helpful discussion.

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A Cohesive Element Model for Mixed Mode Loading With Frictional Contact Capability 2013 International Journal for Numerical Methods in Engineering