A thermodynamics-based cohesive model for interface debonding and friction Irene Guiamatsia a,⇑ , Giang D. Nguyen b a School of Civil Engineering J05, University of Sydney, Sydney, NSW 2006, Australia b School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, SA 5005, Australia article info Article history: Received 1 May 2013 Received in revised form 15 October 2013 Available online 11 November 2013 Keywords: Interface Constitutive model Friction Coupling Delamination Thermodynamics abstract A constitutive model for interface debonding is proposed which is able to account for mixed-mode cou- pled debonding and plasticity, as well as further coupling between debonding and friction including post- delamination friction. The work is an extension of a previous model which focuses on the coupling between mixed-mode delamination and plasticity. By distinguishing the interface into two parts, a cracked one where friction can occur and an integral one where further damage takes place, the coupling between frictional dissipation and energy loss through damage is seamlessly achieved. A simple frame- work for coupled dissipative processes is utilised to derive a single yield function which accurately cap- tures the evolution of interface strength with increasing damage, for both tensile and compressive regimes. The new material model is implemented as a user-defined interface element in the commercial package ABAQUS and is used to predict delamination under compressive loads in several test cases. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction One of the most conspicuous mechanisms of material damage is the formation of well defined regions where inelastic deformation localises and subsequently propagates. When the size of these re- gions is small enough (compared to the structural scale), it can be computationally advantageous to idealise the localised damage zone as an interface; this way, the full continuum constitutive law is replaced by a formulation relating the interfacial tractions to the separation between the two surfaces. Examples are laminated composite materials where the excess resin accumulated between consecutive plies usually fosters the initiation of delamination; in the Arctic sea ice, the fracture lines or leads separating blocks of ice that are hundreds of kilometres in size; in concrete, the width of the damage zone is usually of the order of only a few aggregates in size. In all these cases, the computational cost of the numerical model can be reduced by sev- eral orders of magnitude by using an interface constitutive formulation. The problem with this simplification is that traditional interface decohesive models (Hillerborg et al., 1976; Xu and Needleman, 1994; Scheider, 2001; Allix and Corigliano, 1996; Camanho et al., 2003; Davies et al., 2006) tend to focus on the loss of stiffness (modelled with a damage parameter) and are not capable of cor- rectly capturing the frictional dissipation and inelastic deformation that accompany the damage process. Drastic improvements in pre- dicting the permanent set observed experimentally for some mate- rials, especially polymeric and cementitious materials can be obtained by coupling damage and plasticity. While most of these models are based on plasticity theory with the loss of stiffness con- sidered secondary to the plastic-like behaviour (Tvergaard and Hutchinson, 1993; Su et al., 2004; Matzenmiller et al., 2010; Kolluri, 2011), it is our opinion that the reverse (loss of stiffness as the primary failure process) is more faithful to the physics of interface damage. An energy-based formulation has hence been proposed by the authors (Guiamatsia and Nguyen, 2012) to capture the coupling between damage and friction, in a simple and elegant way rooting from the partition of the total dissipation into plastic/frictional and damage ones. However, like many other damage–plasticity models, such a model does not have the capabil- ity to explain the apparent increase in strength and toughness of the material when the loading of the damaging area consists of the combination of friction and transverse compression. Such increase is well documented in a large body of experi- ments found in the literature. For instance, Wisnom and Jones (1996) pointed out and carried out experiments to investigate the role of friction in increasing mode II fracture energy of the interface of laminated composite; de Teresa et al. (2004), within the context of composite delamination, tested cylindrical speci- mens made of laminated composite materials under combined shear and compression loading and observed a clear increase of shear strength with increasing pressure; Lugovy et al. (2005) 0020-7683/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijsolstr.2013.10.032 ⇑ Corresponding author. Tel.: +61 9351 2134; fax: +61 2 9351 3343. E-mail addresses: [email protected], [email protected], [email protected](I. Guiamatsia). International Journal of Solids and Structures 51 (2014) 647–659 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Transcript
International Journal of Solids and Structures 51 (2014) 647–659
Contents lists available at ScienceDirect
International Journal of Solids and Structures
journal homepage: www.elsevier .com/locate / i jsols t r
A thermodynamics-based cohesive model for interfacedebonding and friction
0020-7683/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijsolstr.2013.10.032
a School of Civil Engineering J05, University of Sydney, Sydney, NSW 2006, Australiab School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, SA 5005, Australia
a r t i c l e i n f o
Article history:Received 1 May 2013Received in revised form 15 October 2013Available online 11 November 2013
A constitutive model for interface debonding is proposed which is able to account for mixed-mode cou-pled debonding and plasticity, as well as further coupling between debonding and friction including post-delamination friction. The work is an extension of a previous model which focuses on the couplingbetween mixed-mode delamination and plasticity. By distinguishing the interface into two parts, acracked one where friction can occur and an integral one where further damage takes place, the couplingbetween frictional dissipation and energy loss through damage is seamlessly achieved. A simple frame-work for coupled dissipative processes is utilised to derive a single yield function which accurately cap-tures the evolution of interface strength with increasing damage, for both tensile and compressiveregimes. The new material model is implemented as a user-defined interface element in the commercialpackage ABAQUS and is used to predict delamination under compressive loads in several test cases.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction
One of the most conspicuous mechanisms of material damage isthe formation of well defined regions where inelastic deformationlocalises and subsequently propagates. When the size of these re-gions is small enough (compared to the structural scale), it canbe computationally advantageous to idealise the localised damagezone as an interface; this way, the full continuum constitutive lawis replaced by a formulation relating the interfacial tractions to theseparation between the two surfaces.
Examples are laminated composite materials where the excessresin accumulated between consecutive plies usually fosters theinitiation of delamination; in the Arctic sea ice, the fracture linesor leads separating blocks of ice that are hundreds of kilometresin size; in concrete, the width of the damage zone is usually ofthe order of only a few aggregates in size. In all these cases, thecomputational cost of the numerical model can be reduced by sev-eral orders of magnitude by using an interface constitutiveformulation.
The problem with this simplification is that traditional interfacedecohesive models (Hillerborg et al., 1976; Xu and Needleman,1994; Scheider, 2001; Allix and Corigliano, 1996; Camanho et al.,2003; Davies et al., 2006) tend to focus on the loss of stiffness(modelled with a damage parameter) and are not capable of cor-
rectly capturing the frictional dissipation and inelastic deformationthat accompany the damage process. Drastic improvements in pre-dicting the permanent set observed experimentally for some mate-rials, especially polymeric and cementitious materials can beobtained by coupling damage and plasticity. While most of thesemodels are based on plasticity theory with the loss of stiffness con-sidered secondary to the plastic-like behaviour (Tvergaard andHutchinson, 1993; Su et al., 2004; Matzenmiller et al., 2010; Kolluri,2011), it is our opinion that the reverse (loss of stiffness as theprimary failure process) is more faithful to the physics of interfacedamage. An energy-based formulation has hence been proposed bythe authors (Guiamatsia and Nguyen, 2012) to capture thecoupling between damage and friction, in a simple and elegantway rooting from the partition of the total dissipation intoplastic/frictional and damage ones. However, like many otherdamage–plasticity models, such a model does not have the capabil-ity to explain the apparent increase in strength and toughness ofthe material when the loading of the damaging area consists ofthe combination of friction and transverse compression.
Such increase is well documented in a large body of experi-ments found in the literature. For instance, Wisnom and Jones(1996) pointed out and carried out experiments to investigatethe role of friction in increasing mode II fracture energy of theinterface of laminated composite; de Teresa et al. (2004), withinthe context of composite delamination, tested cylindrical speci-mens made of laminated composite materials under combinedshear and compression loading and observed a clear increase ofshear strength with increasing pressure; Lugovy et al. (2005)
648 I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659
measured increased values of fracture toughness for Si3N4 lami-nates under the same loading conditions; finally, Hallett and Li(2005) performed numerical experiments of the impact of cross-ply carbon fibre laminated composites and concluded that, toobtain a realistic prediction, both the fracture toughness and thestrength of delaminating interfaces under compression had to beartificially increased in the numerical model.
The easiest way to handle the frictional interaction betweendelaminating interfaces is to adopt the very popular two-stepapproach consisting of delamination first, then frictional contact.This is the approach taken by Tvergaard (1990) whose model fea-tures an unphysical unloading–reloading in the constitutiveresponse if the material is loaded to and past the failure point incompressive shear. This is somewhat similar to what can beobtained with standard tools in most commercial finite elementpackages: by using element deletion for example, post-damagefrictional contact of the newly created surfaces can be activated;but the technique is unable to capture the way in which frictionactually affects the damage evolution itself.
A variety of solutions have been proposed by researchers overthe years, two popular ones being (a) the enhancement of Tverg-aard model proposed by Chaboche et al. (1997), where the shearload smoothly decreases to the Mohr–Coulomb limit when undercompression, and (b) the (phenomenological) adjustment of theyield function and/or the fracture toughness to include a normaltraction term (Matzenmiller et al., 2010; Li et al., 2008; Houet al., 2001; Christensen and De Teresa, 2004). A comprehensive re-view can be found in Raous (2011).
In 2006, Alfano and Sacco (2006) proposed the representation ofthe interface as a two-phase material consisting of a damaged partand an integral part, hence defining the damage parameter as theproportion of fully damaged interface. Using subscripts c and ifor ‘cracked’ and ‘integral’, A for the interface area, the damage var-iable D is the area ratio as follows:
A ¼ Ac þ Ai; with D ¼ Ac
Að1Þ
Using the classical mixture theory, the constitutive law of this‘composite’ material is obtain through appropriate compatibilityrelations and mixed stress:
r ¼ Drc þ ð1� DÞri ð2Þ
It hence becomes straightforward to derive a constitutive modelthat seamlessly introduces the effect of compressive friction1 byusing a traditional frictional contact model for the damaged part ofthe material rc. In their model, Alfano and Sacco (2006) use anon-associative plasticity model with the Mohr–Coulomb cone asthe dissipative function. In our model, a fully coupled constitutiveformulation is entirely derived from standard thermodynamic prin-ciples. The approach is similar to that used in our previous coupleddamage/plasticity interface model, but this time, the expression ofthe free energy is expanded to include the behaviour of the two-phase material described above.
The model is unifying in the sense that it is capable of account-ing for damage and plasticity in mixed-mode loading conditions, aswell as frictional effects on both strength and toughness undertransverse compression. In Fig. 1, we graphically present variousscenarios that the new model is capable of capturing.
The new model is simple, requiring the calibration of only twoparameters in addition to the previous ‘tension-only’ mixed-modedelamination model (Guiamatsia and Nguyen, 2012): (a) acompressive (elastic) stiffness, Kf
s , being the slope of the shear
1 The frictional dissipation between two fully damaged surfaces in compression isdistinguished from that taking place at an integral interface, and which is modelledthrough as plastic deformation.
traction/shear displacement plot measured for the interface undercompression, and (b) the Mohr–Coulomb coefficient of friction de-noted l. It is noted that in the previous model being extended here(Guiamatsia and Nguyen, 2012), the term ‘‘friction’’ and ‘‘plastic-ity’’ were used interchangeably to denote the partition of the en-ergy dissipation resulting in the non-reversible deformation ofthe interface. In the present paper, there is a distinction betweenthat dissipative process linked to residual deformation of the inter-face and the sliding friction at microcracks which occurs only whenthe interface is subjected to combined compression and shear.Therefore, ‘‘plasticity’’ is used to refer to the former process, while‘‘friction’’ is reserved for only the second.
The presentation begins with the thermo-mechanical descrip-tion of the model and derivation of the yield function and evolutionof internal variables. This is followed by the description of thestress-return algorithm, as implemented in a user-defined inter-face element with the commercial package ABAQUS/Explicit(2010). Finally the model is applied to the analysis of a laminatedcomposite plate under low-velocity impact and the modelling of afibre push out test.
2. Thermo-mechanical formulation
The following expression of the Helmholtz energy potential, W,is considered for the two-phase integral/cracked material:
W ¼ 12ð1� DÞKn un � up
n
� �2 þ 12
D 1� H un � upn
� �� �Knðun � up
nÞ2
þ 12ð1� DÞKsðusi � up
siÞ2 þ 1
2D 1� H un � up
n
� �� �Kf
sðusc � ufscÞ
2
ð3Þ
For each loading direction (n or s), there are two terms correspond-ing to the sum of the cracked (D) and integral (1 � D) contributions.Here, D is a scalar variable representing the interface damage state;u is the vector of interfacial separation, with normal and shear com-ponents, respectively represented by subscripts n and s; Kn is theelastic stiffness corresponding to the normal or transverse direc-tion; Ks and Kf
s are elastic shear stiffness corresponding, respec-tively, to the tensile and compressive loading regimes; finally, H(�)is the Heaviside function, taking the value of unity if the argument(�) is positive, and zero otherwise. The superscripts p and f indicateplasticity and friction, respectively. In the above expression of theenergy potential, it is implicitly assumed that the normal stiffnessKn, once completely lost in tension, is fully recovered upon com-pression. However it is not the case with the shear stiffness Ks: onlya fraction can be recovered, e.g. Kf
s < Ks, depending on the roughnessof the cracked surface. Further details on Kf
s will be elaborated later.It is easy to visualise the compatibility relation between the
interfacial displacement jump at the integral part ui and that atthe cracked part, uc (also cf. Alfano and Sacco (2006)):
u ¼ ui ¼ uc )un ¼ uni ¼ unc
us ¼ usi ¼ usc
�ð4Þ
Since the distinction has already been established between theinelastic deformation of the integral part being referred to as ‘plas-tic’ and that of the cracked part as ‘frictional’, the indices i and c canbe dropped altogether, yielding:
W ¼ 12ð1� DÞKn un � up
n
� �2 þ 12
D 1� Hðun � upnÞ
� �Knðun � up
nÞ2
þ 12ð1� DÞKs us � up
s
� �2 þ 12
D 1� Hðun � upnÞ
� �Kf
s us � ufs
� �2
ð5Þ
In the above expression, the classical additive decomposition of thetotal jump into elastic and plastic components has been used, with
us
ts
us
ts
us
ts
(a) Damage and Plast ic i ty under t ransverse tension
(b) Damage under t ransverse compression
(c) Post-damage fr ict ion
Fig. 1. Range of constitutive responses covered by the proposed formulation.
2 The realistic value for Kfs is discussed again later in the manuscript.
I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659 649
ue = u � up being the elastic displacement jump. There are four inter-nal variables: the plastic normal jump up
n, the plastic shear jump ups ,
the frictional shear jump ufs and the damage variable D. The corre-
sponding generalised forces, interface tractions tn, tsi, tsc and damagedissipative force v are obtained as derivatives of the free energypotential (5) with respect to the associated internal variables:
tn ¼@W@ue
n; tsi ¼
@W@ue
si
; tsc ¼@W@ue
sc; v ¼ � @W
@Dð6Þ
Expanding,
tn ¼@W@ue
n¼ @W@ un � up
nð Þ¼ ð1� DÞKnðun � up
nÞ
þ DKnH � un � upn
� �� �un � up
n
� �þ 1
2Ddðun � up
nÞ Kn un � upn
� �2 þ Kfs us � uf
s
� �2h i
ð7Þ
tsi ¼@W@ue
si
¼ @W@ us � up
sð Þ¼ ð1� DÞKs us � up
s
� �ð8Þ
tsc ¼@W@ue
sc¼ @W
@ us � ufs
� � ¼ DKfsHð� un � up
n
� �Þ us � uf
s
� �ð9Þ
The last term in the expression of the normal traction involves theDirac delta, which is infinity when the elastic normal displacementjump vanishes, i.e. ue
n ¼ un � upn ¼ 0. Physically, as the loading re-
gime goes from tension to compression, the normal ‘elastic’ separa-tion will tend to zero, as should the term us � uf
s . These terms mustindeed vanish and remain zero throughout the tensile loading re-gime. The result, only at ue
n ¼ un � upn ¼ 0, is hence infinity times
zero which is indeterminate, but for practical purpose, it will be as-sumed that the contribution can be neglected. Consequently, thetractions can be expressed in the reduced form:
tn ¼ð1� DÞKnðun � up
nÞ; uen > 0
Knðun � upnÞ; ue
n 6 0
(ð10Þ
ts ¼tsi ¼ ð1� DÞKsðus � up
s Þ; uen > 0
tsi þ tsc ¼ ð1� DÞKsðus � ups Þ þ DKf
sðus � ufsÞ; ue
n 6 0
(ð11Þ
The damage energy is:
v ¼ � @W@D¼ 1
2Knðun � up
nÞ2 � 1
21� Hðun � up
n� �
Knðun � upnÞ
2�
þ 12
Ksðus � ups Þ
2�
� 12
1� Hðun � upnÞ
� �Kf
s us � ufs
� �2�
¼ vn þ vsi � vsc ð12Þ
where
vn ¼12 Kn un � up
nð Þ2 ¼ t2n
2ð1�DÞ2Kn; un � up
nð Þ > 0
0; ðun � upnÞ 6 0
(ð13Þ
vsi ¼12
Ks us � ups
� �2 ¼ t2si
2ð1� DÞ2Ks
ð14Þ
vsc ¼0; ðun � up
nÞ > 012 Kf
sðus � ufsÞ
2¼ t2
sc
2D2Kfs; ðun � up
nÞ 6 0
8<: ð15Þ
Therefore, three terms contribute to the energy driving the damageprocess, including a negative contribution from the damaged inter-face in friction. This is consistent with physical observations, as the(compressive) friction should have the effect of slowing down thedamage evolution. However, that term must remain smaller orequal to the sum of the first two terms, in such a way that thedamage energy is always positive, so as to fulfill the irreversibilitycondition. For this reason, Kf
s should be much smaller than Ks
(Kfs � Ks). This also makes general physical sense as the shear stiff-
ness of a cracked interface upon compression must be much smallerthan that of the intact material.2 Note that damage stops evolvingonce the critical value D = 1 corresponding to full loss of cohesionbeing reached. Thermodynamically, the damage energy as perEq. (12) ceases to exist at that moment.
Similar to the approach taken in Guiamatsia and Nguyen(2012), a strong coupling between damage, plasticity and frictionis assumed and formulated through the specification of damagepotentials as homogenous functions of the rates of internalvariables, after the concept described in Einav et al. (2007). The dis-sipation rate is assumed of the following quadratic form, using theindicial notation corresponding to the associated internal variable:
U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2
D þup2n þup2
s þuf 2s
qð16Þ
where different individual contributions uD, upn, up
s and ufs are de-
scribed in Table 1. We note the appearance of the additional termh�tni in these expressions to account for the increase of bothstrength and toughness due to compression.
The rate of frictional dissipation,
ufs ¼ lh�tni þ X½ �duf
s ð17Þ
includes the Mohr–coulomb shear strength, l h�tni, because we areseeking a yield in the form of the Mohr–Coulomb model for thecracked interface. There is also the traction-like term, X, which isutilised to impose a condition that the yield function should be
Table 1Comparison of the expressions for the dissipation rates between this model and aprevious friction-free model.
This model Guiamatsia and Nguyen(2012) – k is the mode ratio
650 I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659
continuous for tn = 0, at the transition of loading regimes, asdescribed in Appendix A.
The other dissipation rates (damage, plastic normal, plasticshear) feature the parameter rD ¼ Gcf =Gc; this is the partition ofthe energy dissipation that is allocated to the creation of newsurfaces (pure fracture), Gc and Gcf being, respectively, the totalinterface toughness and the total dissipation from pure fracture3;F(D) is a monotonic function controlling the damage dissipation thatverifies:Z D¼1
D¼0FðDÞdD ¼ Gc ð22Þ
For a given ratio of mode mixity k = vs/v defined as the ratio be-tween energy for shear and the total damage energy (see Eqs.(12)–(15)), the (mixed mode) dissipative process controlled byfunction F(D) is a combination of functions Fn(D) and Fs(D) in puremodes I and II, respectively (see Guiamatsia and Nguyen, 2012 fordetails on the mixed mode formulation and behaviour). In addition,for the effects of matrix plasticity, a simple averaging of pure mode I(rDI) and pure mode II (rDII) parameters is also assumed:
rD ¼ krDII þ ð1� kÞrDI ð23Þ
FðDÞ ¼ b½kFsðDÞ þ ð1� kÞFnðDÞ�
where
Fn;sðDÞ ¼
ðN;SÞ22Kn;s
1þ ðN;SÞ2
2Kn;srDðI;IIÞGðI;IIÞc�ðN;SÞ2
� �2
1� Dþ ðN;SÞ2
2Kn;srDðI;IIÞGðI;IIÞc�ðN;SÞ2ð Þ
� �2 ð24Þ
Here, classical notation is used for the interface fracture toughnessin normal (GIc) and shear (GIIc) modes, and the strength in normal(N) and shear (S) modes. It can be readily verified that the integralof the proposed damage function is equal to the fracture toughness,
3 This parameter is identified experimentally, for each pure mode, from a traction-separation plot as described in Guiamatsia and Nguyen (2012).
Z D¼1
D¼0Fn;sðDÞdD ¼ GðI;IIÞc ð25Þ
Therefore, the mode-mixity parameter b can be tuned with experi-ments performed with the ratio k held constant, like Reeder andCrews (1990) mixed-mode bending experiments, by using:
bðkÞ ¼ Gcðk;DÞ½kGIIc þ ð1� kÞGIc�
ð26Þ
In the model implementation, either a power-law interaction(Whitcomb, 1984) or the BK criterion for composite failure(Benzeggagh and Kenane, 1996) are used to determine Gc(k,D),but any other function can be utilised for the mixed-modetoughness, provided that it corresponds to constant mode-mixityexperiments. Further details on the mixed mode interaction canbe found in our earlier paper (Guiamatsia and Nguyen, 2012).
3. Model behaviour
3.1. Strength
With all the dissipative terms being homogeneous functions (interms of the rates of change of the internal variables) of order 1, theresulting loading function (or yield curve) is obtained, in general-ised stress space, as the quadratic expression, Einav et al. (2007):
y� ¼ v@uD@dD
!2
þ tn
@upn
@dupn
0@
1A
2
þ tsi
@ups
@dups
0@
1A
2
þ tsc
@ufs
@dufs
0B@
1CA
2
� 1 6 0 ð27Þ
From the thermodynamic formulation described in Einav et al.(2007), y⁄ is obtained from the Legendre transformation of the dis-sipation potential (16) and plays the role of the plastic potential inclassical plasticity. It gives the evolution rules for the internal vari-ables. Expanding and simplifying the above using functions definedin Table 1 gives the yield condition, denoted here as y, in stressspace as:
This can also be written separately for tensile (+) and compressive(�) loading as follows, expanding the damage forces in terms ofdamage and respective tractions:
y ¼ðþÞ :
12ð1�DÞ2
t2n
Knþ
t2s
Ks
� �FðDÞ � 1 ¼ 0
ð�Þ :t2
si
ð1�DÞffiffiffiffiffiffiffiffiffiffiffiffi2FðDÞKs
pþlh�tni
� �2 þ tsclh�tniþX
� �2� 1 ¼ 0
8>>><>>>:
ð29Þ
Continuity is then imposed for tn = 0 (cf. Appendix A) to obtain thefollowing expression for the term X that, interestingly, vanishes fora damage value of either 0 or 1:
Fig. 2 shows isodamage yield curves for an interface with propertiesas specified on the graph, obtained by imposing and keeping con-stant the normal traction tn and loading in shear until the yield con-dition is met. It can be verified that the curves are indeedcontinuous at the tension/compression transition. It is also notedthat the curves in the compressive regime are straight lines, andare parallel to one another, suggesting a variation of the yield shearstress with pressure that is more or less linear:
t�;yields � tþ;yield
s þ lh�tni ð31Þ
-60 -40 -20 0 20 40 600
20
40
60
80
100
D=1
ts
D=0
D=0.5
D=0.9
tn
N=60.; S=90.Kn=Ks=6000.
=0.2;Ksf =60.
Fig. 2. Yield locus for various damage levels.
4 With the current model, the damage variable appears to increase asymptoticallytowards the maximum value, and hence a threshold for total damage needs to beapplied. This is also shown in the subsequent section (one element test), with theasymptotic softening of the shear traction towards the Mohr–Coulomb limit.
I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659 651
3.2. Toughness
The rate of energy dissipation is obtained through multiplyingthe generalised forces by the rate of change of the associated inter-nal variables.
dU ¼ dUD þ dUpn þ dUp
s þ dUfs ¼ vdDþ tndup
n þ tsidups þ tscduf
s ð32Þ
Because of the strong coupling used here, the consequence ofwhich is a single loading function, the increments of each internalvariable are related to a single plastic-type multiplier dk throughthe following ’’flow rules’’ obtained from the loading function,Eq. (27):
In the tensile loading case, the expressions simplify, showing thatincrements of all other internal variables are proportional to thatof damage, yielding an analytical expression of the total energy dis-sipation in function of the interface properties. In compression,however, the expressions can only be further reduced as far asthe following:
The yield equation in compression is used in Eq. (37). From theabove, the rate of total energy loss becomes:
d/ ¼ vsdDrD
lh�tni þ X½ �2
lh�tni þ X½ �2 � t2sc
" #ð41Þ
In reference to our previous model without friction (Guiamatsia andNguyen, 2012), the calculated rate of energy dissipation wassimply:
d/ ¼ vsdDrD¼ FðDÞdD
rDð42Þ
Therefore, the new expression clearly shows that the rate of dissipa-
tion predicted by this model is higher, with the ratio ½lh�tniþX�2
½lh�tniþX�2�t2sc
being larger than 1. It is, however, not possible to simplify theexpression further, meaning that the rate of dissipation is alsodependent, in this case, on the loading path. Considering the sameloading paths utilised to obtain the yield curves of Fig. 2, the totalenergy dissipation is numerically integrated, assuming total inter-face damage for D = 0.9999.4 For the purpose of comparison, we re-fer to the work of Li et al. (2008), who studied the effect ofcompressive delamination with traction–separation laws that wereenhanced with the transverse pressure. Two of their proposed mod-els (A and B) are shown in Table 2, along with the corresponding for-mula for the damage dissipation. The total energy loss, i.e. the energydissipated to bring the compressed interface to complete failure,D = 1, is calculated numerically for several values of transverse pres-sure and compared with models A and B in Fig. 3. The sensitivities ofthe predicted total dissipation with respect to (1) the coefficient offriction and (2) the relative stiffness of the fully damaged partition
a ¼ Kfs=Ks are investigated; in this simulation rD = 1.
As seen in Fig. 3(a), the total dissipation does not appear to de-pend on the stiffness ratio a, although the partitioning betweendamage dissipation and frictional dissipation does, as perFig. 3(b). This is physically reasonable, as smaller a, e.g. lower Kf
s ,leads to lower tsc (see Eq. (9)) and hence delayed frictional slidingfor the same shear displacement under same normal stress. In suchcases the elastic strain energy vsc will also decrease correspond-ingly, resulting in higher damage energy driving the debondingprocess (see Eq. (12)) and hence increasing damage dissipation.Therefore it can be said that the ratio a controls the energy parti-tion between the damage/plasticity dissipation on one side andthe frictional dissipation on the other, analogous to rD which con-trols the partition between pure damage and plasticity. Therefore,the only parameter that controls the total energy dissipated is thecoefficient of friction l. In other words, if l can be properly mea-sured, the model can predict the effects of compression on the in-crease of both strength and toughness of the interface without theneed of any unphysical tuning parameters. Practically, this frictionparameter l can normally be calibrated experimentally based onyield locii in the compressive section of the stress space such asthat obtained in the experiments of De Teresa et al. (2004).
Fig. 3(a) also shows that the interface toughness in compressionpredicted by the current model is higher than that predicted bymodels A and B, however this increase remains within the sameorder of magnitude. The key point is that our model relies onlyon physical basis of debonding and friction upon compression,embedded in an energy consistent formulation. The increase in
Table 2Two compression-enhanced laws considered in Li et al. (2008).
Model Shape of the compression-enhancedtraction separation law
G�c
A
S*
S Gc*
Gc 1� l tnS
� �
B
S*
S Gc*
Gc 1� 2l tnS þ l tn
S
� �2� �
652 I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659
both strength and toughness is left to the model prediction,without having to impose any phenomenological rule on thestrength and fracture properties of the material model.
(a)
-60 -40 -20 00
5
10
15
tn
Φ
a= 1., μ=0.2a=0.001, 0.1, 1.; μ=0.4
Model B, μ=0.4
Model A, μ=0.4
Φ
Fig. 3. Variation of the energy dissipation with transverse compre
12
3
0.003
m
0.003 m
2.e-05m
Fig. 4. Single element with load (le
3.3. One element tests
The constitutive model was implemented in ABAQUS/Explicit(2010) as a user-defined interface element with the stress-returnalgorithm provided in Appendix B. This is an 8-node directionalelement (4-node in two dimensions), with upper and lower facesclearly specified through node numbering. Nodal integrationscheme was used, to be consistent with ABAQUS generic elementsfor the purpose of comparison in the numerical examples. Singleinterface element tests with a fully constrained lower face anddisplacement loading on the upper face (cf. Fig. 4) were used forthe validation of the constitutive model implementation.
3.3.1. Cyclic loading test 1The interface used here had the following properties:
Kn = Ks = 0.5 1014 N/m3; GIc = 281 N m/m2; GIIc = 800 N m/m2; S =N = 5 � 107 Pa; rDI = rDII = 1.0; Kf
s = 0.5 � 1012 N/m3; l = 0.3. Fig. 4also illustrates the loading profile consisting of initial compression(u2) that was kept at a constant value u2min while it is being loaded/unloaded/reloaded in shear (u1). Three different levels of compres-sion were applied, namely u2min = �3 � 10�7 m, �2 � 10�7 m and0 m, corresponding to normal tractions tn = �0.27 kN, �0.18 kNand 0 kN; the responses are reported in the traction–separationplots of Fig. 5(a).
Although the partition of damage dissipation is one in this case,i.e. there is no plastic deformation up
s linked to debonding, thecurve will not return to the origin upon unloading, because thefrictional deformation uf
s is also inelastic, as shown. Undercompression, the shear behaviour of the interface varies smoothlyfrom ‘debonding’- dominated to friction – dominated. Underincreasing shear loads, the interface shear traction progressivelyreduces and converges towards the critical shear traction
(b)
-60 -50 -40 -30 -20 -10 02
4
6
8
10
12
14
D
tn
a = 0.001, 0.1, 1.
ssion; (a) total dissipation, (b) damage dissipation for l = 0.4.
Fig. 5. Shear traction against shear separation for different interfacial pressureloads.
I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659 653
tsc ¼ lh�tni, as expected. What was less expected is the asymptoticcharacter of that convergence, observed in Fig. 5(a): in fact, whilstunder compression, the constitutive model predicts that completedamage (D = 1) will actually never happen and this is linked to theassumed form of the dissipative functions. However, since themain features of compressive damage are captured by the model,a practical solution to the ‘asymptotic’ damage evolution is settinga threshold value of the damage variable, for which the debondingis assumed to be complete. In the results presented here, thisthreshold value is 0.9999.
Fig. 5 also shows that the energy dissipated (area under curve)increases during the progression of coupled debonding/frictionfailure process, as expected, with the applied transverse pressure.The energy dissipation was also integrated numerically and valuesof damage dissipation and total dissipation are reported in Table 3.Note that, for this table and Fig. 6, only one stage of shear loading/unloading was applied at a maximum amplitude of separation of20 � 10�5 m instead of the maximum 5 � 10�5 m applied to obtainthe plot in Fig. 4; this was to allow the threshold damage value to
0
1000
2000
3000
4000
5000
6000
0 0.1 0.2 0.3
Compression UnloadingLoading
friction becomes significant
friction duringunloading
full damageN.m
m/m
m^2
Time (s)
ФD, Ф, tn = 0
Ф, tn = - 0.27kN
ФD, tn = - 0.27kN
Fig. 6. Total and damage partition of the energy dissipated for the loading scenario.
Table 3Energy dissipated up to interface debonding, for various levels of pressure (rD = 1).
be reached for all simulations. A further analysis of the time profilefor the case tn = �0.27 kN is compared to the frictionless case,tn = 0 kN, in Fig. 6.
Fig. 6 clearly shows that damage is the main source of energyloss at the interface during the initial stages of failure (little gap be-tween damage and total curves), but as the damaged area evolves,frictional effects become increasingly important and dominate bythe end of the failure process. It is also interesting to note thatdamage may continue to progress when the shear loading isinverted; this indeed happens for the compressive case shown inFig. 6, and the damage variable (and dissipation) also increases, al-beit, insignificantly in this occasion. This response seems sensibleas one would expect further damage as a result of continuedfriction at the incompletely delaminated interface; this is achievedhere thanks to the coupling between damage and friction.
The effect of varying the compressive stiffness (a ¼ Kfs=Ks) and
partition of damage dissipation (rD ¼ Gcf =Gc was examined in moredetail, for the case tn = �0.18 kN, and reported in Fig. 7(a) and (b),respectively.
As expected, increasing the plasticity (by reducing rD) causes anincrease of the permanent deformation. However, increasing thecompressive stiffness, via increasing ratio a, has a much moresignificant effect on increasing the permanent deformation (in thiscase it is uf
s), while the overall stress-separation curves seeminvariable (as seen in Fig. 7 for values of a sufficiently small to com-plete the analysis). This is consistent with the result presented inSection 3.2 where the frictional dissipation increases for a highervalue of the ratio a. Note must be made, however, that the param-eter a must not be too high as per the basic model assumptionKf
s � Ks. As a rule of thumb, Kfs should be at least two orders of
magnitude smaller than Ks and in Fig. 7(a), the analyses corre-sponding to a P 0.1 were not completed due to numericaldifficulties.
The reason for a small Kfs is, however, not merely numerical, but
is underpinned by the physical significance of our fundamentalassumption, being the additive decomposition of the interface asdamaged and undamaged parts. It is straightforward to visualisethat the apparent shear stiffness Kf
s at a fully debonded interfaceis, in fact, dependent on the applied pressure. If that pressure isrelatively small, the force needed to impose a relative slidingdisplacement between the surfaces will also be relatively small,depending on the surface roughness; hence a small Kf
s . On theother hand, applying a very large pressure between the two sur-faces crush surface asperities and result, in the limit, to remergingof the two materials, meaning an effective shear stiffness of theinterface that is of the same order of magnitude as the parentmaterial, i.e. Kf
s Ks (in Alfano and Sacco (2006), Kfs = Ks). This latter
scenario is, in our view, in conflict with the premise that damagedand undamaged zones at the interface can be distinguished, asillustrated in Fig. 8. Since the effects of loading on the magnitudeof Kf
s are not taken into account in this paper, a simple ruleproposed above is adopted to simplify the implementation. Sincetheoretically Ks is very high, we found that the current conditionon the magnitude of Kf
s is not restrictive, at least for the examplesin this paper.
3.3.2. Cyclic loading test 2Alfano and Sacco (2006) cyclic test is now used to further exam-
ine the capabilities of the current model; in this test, the interfacepressure is held constant at tn = �10 MPa and the interface proper-ties are: S = N = 3 MPa; Kn = Ks = 150 N/mm3; GIc = 0.3 N mm/mm2;GIIc = 0.3 N mm/mm2; l = 0.5. We have also used Kf
s = 0.15 N/mm3.It is noted that Alfano and Sacco’s model only utilises an inelas-
tic displacement jump in the damage part of the interface, whichcorresponds to our frictional separation uf
s . There is no equivalentfor our ‘plastic’ separation in which plastic/frictional deformation
(a) (b)
-200
-100
0
100
200
300
400
500
600
700
0.E+00 2.E-05 4.E-05
a=0.001a=0.01a=0.1a=1
shear separation (m)
shea
r tra
ctio
n (N
)
-200
-100
0
100
200
300
400
500
600
0.E+00 2.E-05 4.E-05
rD=1
rD=0.5
shear separation (m)
shea
r tra
ctio
n (N
)
Fig. 7. Shear traction against shear separation.
Fig. 8. Interface decomposition. Top: under low compressive stress, the interfacecan be partitioned into pristine (1 – D) and fully damaged (D); Bottom: highcompressive stress cause crushing of asperities to the extent that no separate fullydamaged partition can be isolated.
654 I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659
is coupled with damage during the delamination of the interface,e.g. due to plastic deformation of the resin or frictional sliding ofmicro cracks during delamination (Guiamatsia and Nguyen,2012). Hence the coupling coefficient is initially taken to berD = 1, such that, accordingly, no plasticity is present in the inter-face debonding process. With these settings, the response to thecyclic loading is shown in Fig. 9(a) for two values of the frictioncoefficient l. As expected, there is not much residual deformationas the compressive stiffness is very small. If however, plasticityduring delamination is coupled into the model, in addition to fric-tion from delaminated part of the interface, the prediction is thatproduced in Fig. 9(b), which shows significantly more residualdeformation. The prediction of our model in these two cases is
Fig. 9. Shear traction against shear separation of a single interface. (a) Effec
straightforward and very similar to that of a damage/plasticitymodel, with the addition of the increased yield point. In both cases,it is clear that in our model, the damage component (loss ofstiffness) is always present and accompany the other dissipativeprocesses, whereas the loading/unloading curves in Alfano andSacco’s prediction are more or less parallel to one another and tothe elastic loading curve, suggesting that all energy dissipation islinked to the inelastic frictional deformation.
4. Simulation of impact driven delamination
The new interface element is now utilised to predict the extentof delamination in laminated composite plates subjected to impactloading. The experiment considered is that of Aymerich et al.(2008) to which the reader is referred to for more details on theexperiment that was performed for several energy levels. For thesake of completeness, key material properties and model parame-ters are presented in Table 4.
The case chosen for the present simulation was the 5.1 J impact,which corresponds to an initial velocity of 2108.5 mm/s of the2.3 kg impactor. The finite element model, shown in Fig. 10, con-sidered only a quarter of the plate, taking advantage of the problemsymmetry; only the two 0/90 interfaces are modelled withinterface elements, as experimental observations showed theconfinement of delamination to these locations. These interfacesare designated here by ‘upper’ and ‘lower’ according to theirdistance from the impactor. The upper interface is expected to besubjected to compression while the lower one is loaded in tension
t of the friction coefficient, (b) effect of the damage/plasticity coupling.
Table 4Test parameters and material properties, from Aymerich et al. (2008).
CFRP [03,903]s, Plate dimensions: 45 mm � 67.5 mmImpactor: M = 2.3 kg, Diameter = 12.5 mm (spherical)
a These values of fracture toughness for matrix in-plane damage are higher thanthe interface toughness, but were found to yield better correlation with experi-mental results in our finite element analysis.
Fig. 10. ABAQUS quarter finite element model for the impact test.
I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659 655
from the bending of adjacent CFRP layers. A common practisewhen using standard commercial tools to model this sort of impactproblem is to enhance the interface under compression (the upper
0
1
2
3
4
5
6
7
8
9
0 5 10 15mm
mm
(b)
(a)
Fig. 11. Numerical predictions vs. experiment: delamination at the lower
one) by specifying higher values of shear strength and toughness.Such enhancement is purely artificial and the higher values utilisedare more or less arbitrarily chosen with the sole purpose of fittingthe experimental results available, hence the models are devoid ofany predictive capabilities.
Here, the following scenarios were examined:
(a) generic interface element utilised as is at both interfaces;(b) generic interface element with higher shear strength
(100 MPa vs. 80 MPa) and toughness (2.5 N mm/mm2 vs.0.97 N mm/mm2) at the upper interface, elements with nor-mal properties at the lower interface;
(c) interface elements with the current constitutive model,without friction activated (Kf
s = 0 MPa/mm, l = 0), used atboth interfaces;
(d) interface elements with the current constitutive model, withfriction activated, used at both interfaces (Kf
s = 40 MPa/0.02 mm, l – 0).
The predicted delaminated areas on the bottom and upperinterfaces are compared with the experimental findings (e) inFig. 11. When the upper interface is made artificially stronger withthe ABAQUS generic cohesive element (b), the prediction of thedelaminated area at the lower interface is also slightly smaller thanthe area predicted without such a fix. The new interface modelwithout friction activated (c) yields more or less the same delami-nated area as ABAQUS generic element, which is too large. By usingthe new model with friction activated (d), a smaller delaminatedarea is also predicted at both interfaces. If the coefficient of frictionis chosen to be l = 0.8, then an excellent match with the experi-mental findings is obtained, as seen in Fig. 11. It is noted that thepartition of damage to total dissipation rD is set to 1 since thereis no significant plasticity reported in coupon testing also reportedin Aymerich et al. (2008).
It is worth mentioning, however, that the parameters neededfor the frictional part of the model were not validated for thespecific material submitted to the impact test: l = 0.5 is roughlywithin the range of measurements by Schon (2000), but it wasnecessary here to use l = 0.8 in order to obtain a good match withexperimental results. For a truly predictive model, it is desirablethat the coefficient of friction be calibrated independently, forinstance, through combined compression–shear loading assuggested in Section 2, although it may also vary in function of
20 250 degree line
(e)
(d)
(c)
(a)
(b)
(c)
(d)
(e) Upper interface
Lower interface
(plots and damage map) and the upper (damage map only) interface.
Fiber
H
rf
rsrm
p
MAT
RIX
FIBR
E
Fig. 12. Fibre push out test: experimental setup (left) and axisymmetric finite element model (right).
656 I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659
the transverse compressive traction, which varies during loading.In addition, we emphasise that the aim of this demonstration isto achieve a better numerical modelling of the friction enhance-ment to delamination. We do not aim for a high-fidelity modellingof damage propagation in composite laminates, in which the inter-action between matrix cracking and delamination (Choi and Chang,1992; de Moura and Goncalves, 2004) must be taken into account.
5. Simulation of a fibre push out experiment
The experiment of pushing out a polyester fibre embedded in anepoxy resin was performed by Bechel and Sottos (1998) and stud-ied by other researchers (e.g. Hutchinson and Jensen, 1990; Linet al., 2001). Fig. 12 illustrates the experimental setup and the fi-nite element model used in this analysis, and the geometric andmaterial parameters are as follows:
rf = 0.95 mm; rs = 1.025 mm; rm = 4.3 mm; H = 5.36 mmEf = 2500 MPa; mf = 0.35; Em = 4000 MPa; mm = 0.33GIc = GIIc = 0.11 N/mm; Linear failure criterion g = 1.0S = N = 22 MPaKn = 8000/0.001 MPa/mmKs = 2000/0.001 MPa/mmKf
s = 20/0.001 MPa/mmrD = 1.0
To simulate the reported initial compressive matrix strain of�0.0022, an initial temperature field of �2.2 �C was imposed onto
Fig. 13. Deformed finite element model; left: ABAQUS generic element, right: User-defined element with new model.
the matrix which is assigned a coefficient of thermal expansion ofa = 10�3/�C, resulting in a compressive strain along most of thefibre/matrix interface. Then the displacement Dp is applied directlyonto the fibre nodes in contact with the punch. A relatively finemesh was utilised around the interfacial region in both the fibreand the matrix, resulting in a total of 9509 nodes and 9104elements.
The quasi static analysis was performed with the currentmodel with and without friction activated, as well as with theABAQUS/Explicit solver.5 Fig. 13 shows the final deformations ob-tained with both the ABAQUS generic interface element and theuser-defined interface element implementing the current constitu-tive model.
The standard (generic) decohesive model is compared withthe new unified constitutive model based on the predicted peakload. Fig. 14 shows the various predictions for the total reac-tion force recorded at the punch against its downwarddisplacement.
It can be noted that the prediction of the new model withno friction activated is in good agreement the prediction ofthe ABAQUS generic interface element; this is a good verifica-tion test. For these two analyses, the delamination is drivenonly by the relative shear displacement of the fibre respectiveto the matrix, without consideration of friction. The introduc-tion of friction through the coefficient l results in an increaseof the predicted peak load beyond the point of first decohesion(approximately 210 N). In this instance, it is found that takingthe value of l = 0.1 for the coefficient of friction yields the bestmatch to the experimental finding, with the peak load of about400 N predicted to occur at a punch displacement of 0.15 mm.The coefficient of friction was again tuned to obtain the bestprediction, similar to the impact-driven delamination test, butshould really be determined from independent experimentsthat were not available for the specific material system usedhere. In addition, as can be seen in Fig. 14, the effects of fric-tion on both the strength and toughness of the interface cannotbe captured using ABAQUS generic elements with post failurefriction. Our proposed model therefore provides a unified andconsistent approach to treat both pre failure (increase ofstrength and toughness in compression) and post failure fric-tional effects.
5 Note that, only in this fibre push-out example, for the simulation with theABAQUS generic element to proceed in the post-peak stage, it is necessary to deletethe failed cohesive elements, otherwise, severe distortion and hourglassing of thereduced integration continuum elements may cause the simulation to abortprematurely.
-300
-200
-100
0
100
200
300
400
500
600
700
0 0.05 0.1 0.15 0.2
Generic, No Fric�onNew Model, No Fric�onNew model, mu=0.1New Model, mu=0.3Exp
displacement (mm)
load
(N)
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 0.05 0.1 0.15 0.2
Generic, no fric�on
Generic, mu=0.1
Generic, mu=0.3
displacement (mm)
load
(N)
Fig. 14. Load against displacement for the fibre push out test. Experimental results from Lin et al. (2001).
I. Guiamatsia, G.D. Nguyen / International Journal of Solids and Structures 51 (2014) 647–659 657
6. Conclusion
This paper presented an interface constitutive model couplingdamage, plasticity and friction within a consistent thermodynamicframework. Two main ingredients constitute the basis of the mod-el: (1) an expression for the free energy that unifies the threesources of dissipation and couples the evolution of friction to thatof damage by separating a unit interface area into distinctdamaged area undergoing friction and integral area undergoingdamage; and (2) assumed functions of the damage dissipationpotentials that accounts for the effect of compression on the evolu-tion of all internal variables. These ingredients were introducedinto the existing framework developed by Einav et al. (2007), lead-ing to a set of constitutive equations that are able to capture theobserved behaviour of interfaces under a wide spectrum of loadingscenarios, in particular, the increased yield point and total dissipa-tion observed when the interface is loaded in combined compres-sion and shear.
Three key parameters control the response of the constitutivemodel:
– The friction coefficient, l, which directly determines the com-pressed interface shear strength. This total shear strength isequal to the sum of the shear strength under zero compressionand the equivalent critical shear strength of the Mohr–Coulombcone (�ltn).
– the compressive shear stiffness Kfs of the debonded part of the
interface, which controls the amount of inelastic frictionaldeformation and hence directly controls the partition of energydissipation by friction.
– The partition of fracture dissipation which is the ratio rD of puredamage (loss of stiffness) dissipation to the cumulative dissipa-tion due to damage and plasticity under tensile loading only. Inthis work, it was assumed that rD = 1 as the emphasis wasplaced on illustrating the ability of the interface model to cap-ture the response under compressive loading.
The constitutive model was implemented as a user-definedelement in a commercial finite element package and utilised forthe prediction of a number of test cases with readily availableexperimental data. It was shown that toggling on the frictionaldissipation capability resulted in predictions that were more faith-ful to experimental results and expected trends. The key advanta-ges of this model are (a) the thermodynamic framework used in itsdevelopment, from which all the constitutive relations are consis-tently derived with a single yield curve and (b) the meaningfulnessof the model parameters which have a clear physical significanceand can be straightforwardly calibrated from relatively simpleexperiments.
In the examples presented, the coefficient of friction was simplytuned until a good match with experiments was achieved. This isbecause the experimental results used were taken from the exist-ing literature and no coefficient of friction was provided for thespecific material systems. As the range for the coefficient of frictioncan be rather wide and strongly depend on the interfacing materi-als, an experimental programme needs to be developed for furthervalidation and we are aiming to address this aspect in future work.
We also acknowledge the strong assumption made in theformulation of this model, which is the lumping of the interfaceroughness into a single stiffness parameter Kf
s . While the demon-strations in this study show the usefulness and practicality of thisassumption, further development to properly take into account theeffects of (cracked) surface roughness on both compressive shearstiffness and strength, e.g. (Serpieri and Alfano, 2011) is alsoplanned for the next steps.
Acknowledgments
The authors gratefully acknowledge the support of fundingfrom the Australian Research Council discovery grant DP1093485.
Appendix A. Derivation of X(D)
Enforcing the continuity of the yield curve between tensile (+)and compressive (�) side of the stress space gives:
sc , as well as the damage D are historyvariables used to track the state of the interface and a flag Comp-Flag is used to determine whether the interface is in tension orcompression. Note the two components of shear traction in threedimensions. Given an incremental displacement of the formd = (dn, d1
s , d2s ), the stress is updated according to the following
algorithm.
B.1. Step 1: Elastic predictor
The normal (transverse) component is updated first to assessthe loading regime: tensile or compressive.
tn ¼ �tn þ ð1� �DÞKndn
Here, the prescript ‘‘�’’ indicates the sate variable at the previoustime step. The flag C for compression is set (to 1) if the predicted nor-mal traction is negative tn < 0, otherwise the flag is zero. The predic-tors for the three interface tractions (in 3D, j = 1, 2) are then furtherupdated and the shear resultants for the integral and cracked partsare computed as the norm of their individual components.
tn ¼ �tn þ ½ð1� CÞð1� �DÞ þ C�Kndn
tjsi ¼
�tjsi þ ð1�
�DÞKsdjs
tjsc ¼ �tj
sc þ C�DKfsd
js
and the damage energy is also calculated:
v ¼ ðtnÞ2
ð1� �DÞ2Ks
" #þ
t1si
� �2 þ ðt2siÞ
2
ð1� �DÞ2Ks
" #� C
t1sc
� �2 þ t2sc
� �2
ð�DÞ2Kfs
" #
B.2. Step 2: Yield verification and flow rules
The value of the yield, y, is then calculated using Eq. (28). If it isviolated, then the flow rules are calculated and the state variablesupdated using the increments of internal variables as per Eqs.(33)–(36) with the plastic multiplier given by:
dk ¼ �y
2
@y@Dþ
@y@tn
@tn@D þ
@y@tsi
@tsi@D þ
@y@tsc
@tsc@D
� �v
@/D=@dDð Þ2þ
@y@tn
@tn@up
s
tn
@/pn=@dup
nð Þ2þ @y
@tsi
@tsi@up
s
tsi
@/ps =@dup
sð Þ2þ @y
@tsc
@tsc
@ufs
tsc
@/fs=@duf
sð Þ2
8><>:
9>=>;
Finally, the state variables are updated:
kþ1tn ¼ ktn � ð1� CÞktn
1� DdDþ ð1� DÞKndup
n
�
kþ1tjsi ¼
ktjsi �
ktjsi
1� DdDþ ð1� DÞKsdupj
s
( )
kþ1tjsc ¼ ktj
sc þktj
sc
DdD� DKf
sdufjs
� kþ1D ¼ kDþ dD
where the individual components of increments of plastic and fric-tion shear separation are, with j = 1, 2:
dupjs ¼ dup
s
tjsi
tsi; dufj
s ¼ dufs
tjsc
tsc
Step 2 is repeated iteratively until the value of the yield is within acertain accuracy tolerance, |y| < tol.
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