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Int J Fract (2013) 181:177–188DOI 10.1007/s10704-013-9834-3

ORIGINAL PAPER

The cohesive band model: a cohesive surface formulationwith stress triaxiality

Joris J. C. Remmers · René de Borst ·Clemens V. Verhoosel · Alan Needleman

Received: 22 November 2012 / Accepted: 6 March 2013 / Published online: 21 March 2013© Springer Science+Business Media Dordrecht 2013

Abstract In the cohesive surface model cohesivetractions are transmitted across a two-dimensional sur-face, which is embedded in a three-dimensional con-tinuum. The relevant kinematic quantities are the localcrack opening displacement and the crack sliding dis-placement, but there is no kinematic quantity thatrepresents the stretching of the fracture plane. As aconsequence, in-plane stresses are absent, and fracturephenomena as splitting cracks in concrete and masonry,or crazing in polymers, which are governed by stresstriaxiality, cannot be represented properly. In this paperwe extend the cohesive surface model to includein-plane kinematic quantities. Since the full strain ten-sor is now available, a three-dimensional stress statecan be computed in a straightforward manner. The

J. J. C. Remmers · C. V. VerhooselDepartment of Mechanical Engineering, EindhovenUniversity of Technology, PO Box 513,5600 MB Eindhoven, The Netherlandse-mail: J.J.C.Remmers@tue.nl

C.V. Verhoosele-mail: C.V.Verhoosel@tue.nl

R. de Borst (B)School of Engineering, University of Glasgow, RankineBuilding, Oakfield Avenue, Glasgow G12 8LT, UKe-mail: Rene.DeBorst@glasgow.ac.uk

A. NeedlemanDepartment of Materials Science and Engineering Collegeof Engineering and Center for Advanced ScientificComputing and Modeling, University of North Texas,Denton, TX 76203, USAe-mail: needle@unt.edu

cohesive band model is regarded as a subgrid scale frac-ture model, which has a small, yet finite thickness at thesubgrid scale, but can be considered as having a zerothickness in the discretisation method that is used atthe macroscopic scale. The standard cohesive surfaceformulation is obtained when the cohesive band widthgoes to zero. In principle, any discretisation method thatcan capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometricfinite element analysis seem to have an advantage sincethey can naturally incorporate the continuum mechan-ics. When using interface finite elements, traction oscil-lations that can occur prior to the opening of a cohesivecrack, persist for the cohesive band model. Examplecalculations show that Poisson contraction influencesthe results, since there is a coupling between the crackopening and the in-plane normal strain in the cohe-sive band. This coupling holds promise for capturinga variety of fracture phenomena, such as delaminationbuckling and splitting cracks, that are difficult, if notimpossible, to describe within a conventional cohesivesurface model.

Keywords Discrete fracture · Discontinuities ·Stress triaxiality · Cohesive surface model ·Partition of unity method · Interface elements

1 Introduction

Fracture lies at the heart of many failure phenom-ena of man-made and natural structures. Since the

123

178 J. J. C. Remmers et al.

seminal work of Griffith (1920) and Irwin (1957)on brittle fracture a plethora of approaches to frac-ture have been developed, resulting in a rich litera-ture. For quasi-brittle and ductile fracture, where thelength of the fracture process zone is not small com-pared to a typical structural size, cohesive surfacemodels, originally proposed by Dugdale (1960) andBarenblatt (1962), and later by Hillerborg et al. (1976)for concrete fracture, have proven particularlysuccessful.

The cohesive surface model is very powerful, yetremarkable in its simplicity. It basically consists of afracture initiation criterion, and after nucleation, crackopening is governed by the work of separation or frac-ture toughness. The fracture process zone is lumpedinto a single plane ahead of the crack tip. Its open-ing is governed by the shape of the decohesion curve,which sets the relation between the normal and theshear tractions across the crack surfaces on one hand,and the relative displacements between these surfaceson the other hand. Fracture is then a natural outcome ofthe loading process.

In spite of its conceptual simplicity the incorpo-ration of cohesive surface models in simulation soft-ware such that cohesive crack propagation can besimulated in a predictive manner, free from the underly-ing discretisation, has proven a non-trivial task that hasbeen a main issue in computational mechanics for thepast thirty years. When the composition of the struc-ture clearly indicates the potential fracture planes, asin lamellar materials, or when the fracture plane isknown from experiments, a discrete formulation likethe cohesive surface model can be incorporated ininterface elements that a priori are inserted betweencontinuum elements at predefined locations, e.g. Rots(1991), Schellekens and de Borst (1993b), Needleman(1987)). This methodology has been generalised inXu and Needleman (1994), where interface elementswere placed between all interelement boundaries, thusallowing for a greater flexibility in the cohesive crackpath that can be obtained. Alternatively, a remesh-ing strategy has been proposed in Camacho and Ortiz(1996). More recently, the partition-of-unity propertyof finite element shape functions has been exploited toobtain a discretisation-independent path for cohesivecracks (Wells and Sluys 2001; Wells et al. 2002; Moësand Belytschko 2002; Remmers et al. 2003; de Borstet al. 2006). Furthermore, it has been shown that alsoisogeometric analysis provides an elegant and powerful

tool to implement cohesive surface models without dis-cretisation bias (Verhoosel et al. 2011).

The necessity, at least in earlier days, to align discon-tinuities with existing mesh lines, or to use remeshingstrategies for avoiding or ameliorating a mesh bias incomputations of the propagation of a discontinuity, hasprompted the search for methods in which the disconti-nuity was distributed, or smeared, over a finite domain.In finite element analyses, this was typically the tribu-tary area assigned to an integration point. Bažant andOh (1983) have proposed the Crack Band Model, inwhich the cohesive surface model was cast into a con-tinuum format, such that the zero-thickness interface inthe original approach was replaced by a finite widthw,in practice the size of the mentioned tributary area thatbelongs to an integration point. In this way, ‘smeared-crack’ analyses can be carried out for a fixed mesh.A further development along this line is to refine thekinematics at the element level such that the crack bandis properly represented at the element level. Startingfrom original ideas formulated in Ortiz et al. (1987),Belytschko et al. (1988) this approach has been fur-ther developed and has been cast into the frameworkof Enhanced Assumed Strain elements in Simo et al.(1993).

The above ‘smeared-crack’ approaches can be castwithin the framework of (anisotropic) continuum dam-age mechanics (de Borst and Gutiérrez 1999; de Borstet al. 2012), and share the disadvantage of continuumdamage models that they result in an ill-posed boundaryvalue problem beyond a certain threshold level of load-ing because of loss of ellipticity. Well-posedness canbe restored by nonlocal averaging schemes (Pijaudier-Cabot and Bažant 1987) or by adding spatial gra-dients to the material constitutive relation Peerlingset al. (1996). Continuum damage models are three-dimensional constitutive relations. This implies that thenormal strain parallel to the crack band is directly avail-able, and, via the constitutive relation, the normal stressin the crack band direction can be directly computed.Thus, failure modes in which stress triaxiality plays arole, i.e. when fracture depends on the hydrostatic stresslevel can be predicted in a natural manner using con-tinuum damage approaches, see for instance successfulcomputations for ductile failure of porous metals usingthe modified Gurson model (Gurson 1977; Tvergaardand Needleman 1984).

In Huespe et al. (2009, 2012) a finite thickness bandmethod was presented to model circumstances where

123123

The cohesive band model 179

a weak discontinuity precedes a loss of the stress car-rying capacity as, for example, occurs in modellingductile fracture using a rate independent constitutiverelation. In that formulation, a finite thickness band isintroduced when loss of ellipticity occurs at a materialpoint (an integration point in a finite element imple-mentation). The band thickness is regarded as a mater-ial parameter. Consistent with the kinematics of a weakdiscontinuity, see for example (Hill 1962; Rice 1976),the displacements vary linearly accross the band. Also,the tractions are continuous accross the band. The post-localisation material response in the band is governedby the pre-localisation constitutive relation togetherwith the constraint imposed by the weak discontinuitykinematics, which can permit the tractions to vanish,creating new free surface, thus giving a transition froma weak to a strong discontinuity. In this formulation, theband thickness serves as a regularisation parameter.

A conventional zero thickness cohesive surface for-mulation involves a relation between tractions and dis-placement jumps across a surface. Stress componentsthat do not affect the tractions are not accounted forin the cohesive constitutive relation and neither aredeformation components that only involve displace-ments and gradients parallel to the surface. This limitsthe modeling capability in a variety of circumstances,including ductile failure of metals where stress triaxi-ality plays an important role and the prediction of split-ting cracks in concrete or masonry structures where alarge compressive stress creates cracks that are alignedwith this normal stress (Rots 1991). One approachthat has been proposed to overcome this limitation isto insert the normal stress from a neighbouring inte-gration point in the continuum into the cohesive sur-face relation (Keller et al. 1999; Tijssens et al. 2000;Siegmund and Brocks 2000). Another approach, asnoted previously, is the finite band method of Huespeet al. (2009, 2012).

Here, we introduce a cohesive surface thickness todirectly model the evolution of fracture, which thenstraightforwardly allows for a dependence on all stressand deformation components. As a consequence, theapproach here differs from that in Huespe et al. (2009,2012) in several significant aspects. First, the in-bandresponse is taken to augment the cohesive surface rela-tion so that, as in the usual cohesive surface formula-tion, the band constitutive relation is independent ofthe volumetric material relation. Indeed—and this isthe second difference—the cohesive band model can

be conceived as a subgrid scale fracture model, withthe band thickness a numerical parameter, rather than amaterial parameter, and the formulation is such that asthe band thickness goes to zero, a conventional cohesivesurface formulation is recovered. Finally, the presentapproach is fully discrete, with continuity of the dis-continuity gap at element boundaries.

2 Band kinematics and virtual work

Attention is confined to small deformations and we con-sider the cohesive crack depicted in Fig. 1. The thicklines are the cohesive surfaces Γ −

d and Γ +d , charac-

terised by the normals nΓ −d

and nΓ +d

, respectively, seeFig. 2. The thickness of the cohesive bandΩb betweenthe surfaces Γ −

d and Γ +d is denoted by h. The bulk

ΩB = Ω\Ωb consists of the sub-domain Ω− that

Γd+

Γd

Fig. 1 A cohesive crack

Γd

bΩ

Ω+

Ω

Γ

Γd

d+

h

x

y

s

n

Fig. 2 A cohesive band model

123

180 J. J. C. Remmers et al.

borders the cohesive surface Γ −d , and the sub-domain

Ω+ that borders the cohesive surface Γ +d , Fig. 2.

In the cohesive surface methodology a relation isassumed between the normal crack opening vn and thecrack sliding components vs and vt , assembled in arelative displacement vector v,

vT = (vn, vs, vt )

and the normal traction tn and the shear tractions ts andtt , assembled in the traction vector t, which is expressedin the n, s, t local reference frame:

tT = (tn, ts, tt )

For consistency v and t must be decomposed in thesame coordinate system.

The displacement u(x) of a material point in thebody Ω can be expressed as:

u(x) = u(x)+ HΓd u(x) (1)

with HΓd the Heaviside function centered at themid-surface of the cohesive band, Γd . Then, the dis-placement jump v equals the value of the additionaldisplacement field at the discontinuity plane:

v(x) = u(x) ∀ x ∈ Γd (2)

The displacement jump v is expressed in the globalcoordinate system. The transformation

v = Tv (3)

between the relative displacements in the current localcoordinate system with the unit vectors en, es, et andthe displacement jump in the global coordinate systemwith unit vectors ex , ey, ez is achieved using the trans-formation matrix T, with components:

Ti j = ei · e j , where i = [n, s, t] , j = [x, y, z](4)

which is constructed using the unit vectors of the globalcoordinate system and those of the local coordinatesystem in the current configuration.

The strain tensor ε in the bulk ΩB = Ω\Ωb is nowderived in a standard manner:

ε = 1

2

(∇u + ∇uT

)∀ x ∈ ΩB (5)

We further define the strain tensor in the cohesive band,expressed in the n, s, t local frame of reference of theband:

E =⎡⎣

Enn Ens Ent

Esn Ess Est

Etn Ets Et t

⎤⎦ ∀ x ∈ Ωb (6)

The components of this matrix are based on the magni-tude of the relative displacements and on the in-planestrains in the band. The strain tensor E can be trans-formed to the local frame of reference using the trans-formation matrix T:

E = TETT (7)

with E containing the components Exx etc. in the globalx, y, z coordinate system.

We now define a (small) band width h0 as the valueof the crack opening, h, in a reference state. Using theassumptions that the strains in the band are piecewiseconstant at either side of the discontinuity Γd in then-direction, and that the normal strain component Enn ,and the shear strain components Ens and Ent are com-pletely determined by the crack opening vn , and thecrack sliding components vs and vt , respectively, wecan define:

Enn = vn

h0(8)

and

Ens = vs

2h0(9)

Ent = vt

2h0(10)

In a standard manner the virtual strain components canbe derived as

δEnn = δvn

h0(11)

and

δEns = δvs

2h0(12)

δEnt = δvt

2h0(13)

The in-plane terms of the strain tensor in the band,Ess, Et t and Est = Ets are independent of the magnitudeof the displacement jump. They represent the normalstrain components in the s- and t-directions, respec-tively, and the in-plane shear strain. In view of theassumption that the strains in the band are piecewiseconstant at either side of the discontinuity Γd in then-direction, and enforcing continuity for the in-planestrain components acrossΓ −

d andΓ +d these strain com-

ponents are defined as:

Ess = 1

2

(Ess |Γ −d

+ Ess |Γ +d

)

Et t = 1

2

(Et t |Γ −d

+ Et t |Γ +d

)(14)

Est = 1

2

(Est |Γ −d

+ Est |Γ +d

)

123123

The cohesive band model 181

The internal virtual work of the solid can beexpressed in terms of the stress tensor σ and the vari-ation of the strain tensor. In the bulk of the domain,ΩB , we denote the variation of the strain tensor by δε,while in the cohesive band, Ωb, we have δE denotingthe variation of the strain tensor and S the band stresses,so that:

δWint =∫

ΩB

σ : δεdΩ +∫

Ωb

S : δEdΩ (15)

This expression is formally identical to Eq. (30) ofHuespe et al. (2009), but, as alluded to in the Introduc-tion, the interpretation of the second term is different.Herein, it strictly relates to the energy that is dissi-pated by the cohesive tractions and by the in-plane bandstresses, and in the limiting case of a band with zerothickness, the energy expended by the cohesive trac-tions is retained. This is different from the approach inHuespe et al. (2009, 2012), where the energy dissipa-tion vanishes when the band width is zero.

The second term in Eq. (15), which represents thecontribution of the cohesive band, can be rewritten as:

δWint∣∣Ωb =

∫

Γd

h02∫

− h02

S : δE dndΓ (16)

Again using the assumption that the deformation inthe cohesive band is constant in the n-direction, weintegrate analytically in the thickness direction:

δWint∣∣Ωb = h0

∫

Γd

S : δE dΓ (17)

or written in terms of the individual components:

δWint∣∣Ωb = h0

∫

Γd

(SnnδEnn + SssδEss + St tδEt t

+ 2SnsδEns + 2SntδEnt + 2SstδEst ) dΓ

(18)

which relation holds irrespective of the value of thecohesive band width h0. Substitution of the expressionsfor the virtual strains derived in Eqs. (11), (12) and (14)gives:

δWint∣∣Ωb =

∫

Γd

(Snnδvn + h0SssδEss + h0St tδEt t

+ Snsδvs + Sntδvt + 2h0SstδEst ) dΓ

(19)

In the limit, i.e. when h0 → 0, this expressionreduces to:

δWint∣∣Ωb =

∫

Γd

(Snnδvn + Snsδvs + Sntδvt ) dΓ (20)

or replacing the stress components Snn,Sns and Snt bythe tractions tn, ts and ts , we obtain the usual cohesivesurface relation:

δWint∣∣Ωb =

∫

Γd

(tnδvn + tsδvs + ttδvt ) dΓ (21)

The effect of the in-plane strains in the cohesive band,Ess, Et t and Est , has now disappeared, as it should. Wewill come back to this in the example of Sect. 5.

To further elucidate how the tractions behave in thelimit when the band width h0 goes to zero, we considerthe case that only the normal components across theband, Snn and Enn are non-zero. Then, Eq. (18) reducesto:

δWint∣∣Ωb = h0

∫

Γd

SnnδEnndΓ (22)

From Eq. (8) we recall that

Enn = vn

h0

so that Eq. (22) can be rewritten as:

δWint∣∣Ωb =

∫

Γd

SnnδvndΓ (23)

For simplicity, but without loss of generality, since anyclassical constitutive relation could have been used viaintegration of a rate relation, we suppose that the mater-ial in the band obeys a linear elastic constitutive relationwith a Young’s modulus in the band denoted by Eb:

Snn = EbEnn = Ebvn

h0(24)

Equation (23) can now be written as:

δWint∣∣Ωb =

∫

Γd

Ebvn

h0δvndΓ (25)

We next take the limit h0 → 0. In this limit vn alsogoes to zero, so it is a singular limit. However, equi-librium across the finite band cohesive surface requirescontinuity of tractions, so

Ebvn

h0= tn

123

182 J. J. C. Remmers et al.

where tn is the traction given by the constitutive relationoutside the band. This must be satisfied for all h0 andin particular in the limit h0 → 0. Hence,

δWint∣∣Ωb =

∫

Γd

tnδvndΓ (26)

It is finally noted that a similar approach, in whicha discontinuity has been modelled as a zero-thicknessinterface at the macroscopic scale, while a small, butfinite thickness has been used for the modelling at asubgrid scale, has been used for modelling fluid flow incracks or shear bands that are embedded in a surround-ing porous medium (de Borst et al. 2006; Réthoré et al.2007, 2008).

3 Discretisation

As discussed in the Introduction, cohesive surface mod-els can be discretised in a variety of ways, startingfrom interface elements, to partition-of-unity basedfinite element methods (Wells and Sluys 2001; Wellset al. 2002; Moës and Belytschko 2002; Remmers et al.2003; de Borst et al. 2006) and isogeometric analysis(Verhoosel et al. 2011). This holds also for the cohesiveband model presented in the previous section, since thekinematic quantities known in this element in principleallow for the computation of the in-plane strains Ess , Et t

and Est . However, unlike interface elements, partition-of-unity based finite element methods naturally inheritthe kinematics of the underlying continuum, also atthe discontinuity Γd . For this reason we will adopt thepartition-of-unity based finite element technology forembedding the cohesive band model developed in thepreceding section. We note, however, that for the lim-iting case that the cohesive surface coincides with theedge of an element in a partition-of-unity approach, thestructure of an interface element is recovered (Simone2004; de Borst 2006). In particular, if the partition-of-unity approach is applied such that the discontinuity isdefined a priori to coincide with the element edges, itinherits disadvantageous features such as traction oscil-lations which can occur prior to the opening of thediscontinuity. In the next section we will investigateto which extent this also holds for the cohesive bandapproach.

For a set of shape functions φk that satisfy thepartition-of-unity property, a field u can be interpolatedas follows (Babuska and Melenk 1997):

u =n∑

k=1

φk

(ak +

m∑l=1

ψl akl

)(27)

with ak the ‘regular’ nodal degrees-of-freedom, ψl theenhanced basis terms, and akl the additional degrees-of-freedom at node k, which represent the amplitudes ofthe l th enhanced basis term ψl . A basic requirement ofthe enhanced basis terms ψl is that they are linearly-independent, mutually, but also with respect to the set offunctions φk . In a conventional finite element notationwe thus interpolate a displacement field as:

u = �(a + �a) (28)

where � contains the standard shape functions and �

the enhanced basis terms. The arrays a and a collect thestandard and the additional nodal degrees-of-freedom,respectively. A displacement field that contains a singlediscontinuity can be represented by taking (Wells andSluys 2001; Wells et al. 2002; Moës and Belytschko2002; Remmers et al. 2003; de Borst et al. 2006;Belytschko and Black 1999):

� = HΓd I (29)

Substitution into Eq. (28) gives:

u = �a︸︷︷︸u

+HΓd �a︸︷︷︸u

(30)

Identifying the continuous fields u = �a and u = �awe observe that Eq. (30) exactly describes a displace-ment field that is crossed by a discontinuity Γd , butis otherwise continuous. Accordingly, the partition-of-unity property of finite element shape functions canbe used in a straightforward fashion to incorporate dis-continuities in a continuum such that their discontinu-ous character is preserved.

To derive the discretised set of equations we take theinternal virtual work, Eq. (15), as point of departure,but we replace the second term by the expression ofEq. (17), which results after integration over the thick-ness of the band, and assume henceforth for simplicityof notation that the local and global coordinate sys-tems coincide. Evidently, in the actual implementationone has to take care that the rotations are carried outproperly. This results in:

δWint =∫

ΩB

σ : δεdΩ + h0

∫

Γd

S : δE dΓ (31)

In a Bubnov–Galerkin sense we assume that thetest functions are taken from the same space as the

123123

The cohesive band model 183

trial functions modulo inhomogeneous boundary con-ditions, so that in view of Eq. (30):

δu = �δa + HΓd �δa (32)

Substitution of Eq. (32) into Eq. (31) and requiring thatthe result holds for arbitrary δa and δa yields the follow-ing set of coupled equations in matrix-vector notation:

f aint =

∫

ΩB

BTσdΩ + h0

∫

Γd

BTSdΓ (33a)

and

f aint =

∫

Ω+BTSdΩ + 1

2h0

∫

Γd

BTSdΓ (33b)

where the Heaviside function has been eliminated fromthe volume integrals by a change of the integrationdomain from ΩB to Ω+. In the bulk, B = L�, thestrain-nodal displacement matrix, with L an operatormatrix, cf de Borst et al. (2012)—Chapter 2. Orderingthe strains in the cohesive band as

ET = (Enn, Ess, Et t , Ens, Ent , Est )

the matrices B and B read:

B = L� (34)

and

B = L� (35)

with the operator matrices

L =

⎡⎢⎢⎢⎢⎢⎢⎣

0 0 00 ∂

∂s 00 0 ∂

∂t0 0 00 0 00 ∂

∂t∂∂s

⎤⎥⎥⎥⎥⎥⎥⎦

(36)

and

L =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1h0

0 00 1

2∂∂s 0

0 0 12∂∂t

0 12h0

00 0 1

2h0

0 12∂∂t

12∂∂s

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(37)

respectively.Allowing for a wide range of cohesive relations, we

postulate a tangential relation between the stress ratein the band

ST = (Snn, Sss, St t , Sns, Snt , Sst )

and the strain rate in the band, E:

S = DbE (38)

We assume that the tangential stiffness matrix Db inthe band has a transversely isotropic structure, and isobtained by differentiating the cohesive relation

S = S (E, κ, Eb, νb) (39)

with κ an array of one or more internal variables, andEb and νb the Young’s modulus and the Poisson’sratio in the band, respectively. For the general three-dimensional case, a closed-form expression for Db canbe rather complicated. For this reason, a complianceformat is sometimes preferred:

D−1b =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1kn

− νbEb

− νbEb

0 0 0− νb

Eb

1Eb

− νbEb

0 0 0− νb

Eb− νb

Eb

1Eb

0 0 00 0 0 1

ks0 0

0 0 0 0 1kt

0

0 0 0 0 0 2(1+νb)Eb

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(40)

with

kn = ∂Snn

∂Enn

the stiffness that derives from the cohesive relation formode-I behaviour, and with

ks = ∂Sns

∂Ens

and

kt = ∂Snt

∂Ent

the tangential shear stiffnesses in the s- and t-directions,respectively. For plane-stress conditions, however, anexplicit expression for the tangential stiffness matrixcan easily be derived:

Db =

⎡⎢⎢⎣

Eb

Ebk−1n −ν2

b

νb Eb

Ebk−1n −ν2

b0

νb Eb

Ebk−1n −ν2

b

Eb

1−ν2b E−1

b kn0

0 0 ks

⎤⎥⎥⎦ (41)

We observe that the standard cohesive stiffnessesbetween the tractions and the relative displacementsare incorporated, but that the matrix also includes thein-plane stiffness and the coupling between the normalrelative displacement and the stretching of the fractureplane via the Poisson ratio νb in the band. The assumedtransversely isotropic structure of Eq. (40) has limi-tations, in particular when ductile fracture processes

123

184 J. J. C. Remmers et al.

are considered which involve metals, ductile poly-mers, or adhesives. An appropriate band constitutiverelation could then involve a significant shear-normalstress/strain coupling and the response cannot be char-acterised by an isotropic constitutive relation.

4 Aspects of numerical integration

As stipulated in the preceding section the spatial numer-ical integration is an important issue in conventionalinterface elements when applied in the context of cohe-sive surface models, as they can suffer from spuri-ous traction oscillations, in particular in quasi-brittlefracture where there is no compliant interface prior toreaching the tensile strength. The magnitude of theseoscillations increases with an increasing dummy stiff-ness, which is used prior to the opening of the discon-tinuity in order to ensure continuity (Schellekens andde Borst 1993a). A solution is to abandon Gauss inte-gration and to resort to Newton-Cotes integration or tolumped integration techniques.

We will now investigate whether the interface ele-ments equipped with a cohesive band model inheritthis deficiency, which plagues interface elements thatincorporate a cohesive surface model. For this purpose,we employ a notched three-point bending beam, shownin Fig. 3, and used before in Schellekens and de Borst(1993a). The dimensions of the beam are w=125 mmand h = 100 mm, and is made of an elastic, is otropicmaterial with Young’s modulus E = 20 000 MPa anda Poisson’s ratio ν = 0.2. The length of the notch isa =20 mm. The applied load is equal to P =1000 N.

Fig. 3 Geometry and boundary conditions of a notched beam ina three-point bending test

Fig. 4 Cohesive band stress Snn as a function of the positionat the interface for different magnitudes of the Young’s modulusEb in the cohesive band using a Gauss integration scheme

Fig. 5 Cohesive band stress Snn as a function of the position atthe interface for different magnitudes of the Young’s modulus Ebin the cohesive band using a Newton-Cotes integration scheme

The finite element model consists of a structured gridof 51 × 20 four-noded bilinear elements. The inter-face is represented by a cohesive band. The notch,0< y<20 mm, is traction free, i.e. the tractions and thetangent stiffness matrix vanish, irrespective of the mag-nitude of the strain field. In the cohesive band, i.e. when20< y< 100 mm, a linear-elastic, plane-strain consti-tutive relation is used. Calculations have been carriedout for different magnitudes of the Young’s modulus Eb

in the cohesive band. The spatial integration along thecohesive band is done using either Gauss or Newton-Cotes integration. The traction profiles at the interfaceare shown in Figs. 4 and 5.

The results for the cohesive band model con-firm those obtained for a cohesive surface model(Schellekens and de Borst 1993a) in the sense that trac-tion oscillations are present when a Gauss integrationscheme is used, and increase for larger values of theYoung’s modulus Eb in the band. Similarly, the trac-

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The cohesive band model 185

Fig. 6 Cohesive band stress Snn as a function of the position atthe interface for different Newton-Cotes integration schemes

Fig. 7 Numerical integration of a quadrilateral and a triangu-lar element. The triangulation of the sub-domains Ω+ and Ω−is denoted by the dashed lines; the corresponding integrationpoints are denoted by the ⊗ symbols. The discontinuity Γd , rep-resented by the bold line, is integrated by a two point Newton-Cotes scheme. These integration points are represented by the ⊕symbols

tion oscillations disappear when a lumped integrationscheme is used, Fig. 5, but reappear when over-integration is used, Fig. 6. In sum, standard inter-face elements show exactly the same behaviour withrespect to spatial integration irrespective whether theyare equipped with a cohesive surface model or with acohesive band model.

The contributions of the bulk parts to the linearmomentum equations of an element that is crossed bya cohesive band are integrated in a similar fashion asin Wells and Sluys (2001). Both sub-domains Ω+ andΩ− are triangulated as shown in Fig. 7. In the caseof linear elements, each triangle is integrated by a sin-gle Gauss-point, denoted by the ⊗ sign. In order toensure that the sum of the areas of the two bulk sub-domains and the cohesive band is equal to the areaof the original undeformed element, the width of thecohesive band is taken into account during the triangu-lation of Ω+ and Ω−. Note that in the case of quadri-

Fig. 8 Geometry and boundary conditions of a double cantileverpeel test

lateral elements in combination with a structured mesh,the area of the cohesive band is equal to the length ofthe line Γd times the width h0, see Fig. 7. In the caseof an unstructured mesh or triangular elements a smallnumerical error is introduced here. However, this erroris negligible for small values of the band thickness h0.

5 Double cantilever peel test

We next consider the double cantilever test shown inFig. 8. The structure with length l = 10 mm consistsof two layers with the same thickness h = 0.5 mm andwith the same (isotropic) material properties: a Young’smodulus E = 100 MPa and a Poisson ratio ν = 0.3.The two layers are connected through an adhesive witha tensile strength tmax = 1 MPa and an interfacial frac-ture toughness Gc = 0.1 N/mm. The initial delamina-tion extends over a = 1 mm. An external load P isapplied at the tip of both layers.

The specimen has been analysed with four-nodedquadrilateral elements: 100 elements in the horizontaldirection and 11 elements in the vertical direction. Theelements in the centre of the specimen, i.e. the elementsthat are crossed by the discontinuity, are square withdimensions le × le = 0.1×0.1 mm. The solutions havebeen obtained using the energy dissipation arc-lengthmethod (Verhoosel et al. 2009).

The constitutive behaviour of the cohesive band isgoverned by an isotropic, plane-strain continuum dam-age relation:

S = (1 − ω)DebE (42)

where Deb is the plane-strain elastic stiffness matrix, that

is constructed using the Young’s modulus Eb and the

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186 J. J. C. Remmers et al.

Fig. 9 Effect of Poisson’s ratio νb on the load-displacementcurve for a cohesive band model

Poisson’s ratio νb in the band. The damage parameterωis function of the history parameter κ , which is equal tothe highest value of the principal strain locally obtainedduring the loading:

ω =

⎧⎪⎪⎨⎪⎪⎩

0 if κ < κ0κc

κ

κ − κ0

κc − κ0if κ0 < κ < κc

1 if κ > κc

(43)

In this relation, κ0 and κc are defined as functions of thetensile strength tmax and a ‘volumetric’ fracture tough-ness gc:

κ0 = tmax

Eb; κc = 2gc

tmax(44)

The relation between the classical, interfacial fracturetoughness Gc and the volumetric fracture toughness is:

gc = Gc

h0(45)

The results of the simulations for different valuesof the Poisson’s ratio in the band, νb, are comparedwith a standard cohesive surface model in Fig. 9. Weclearly observe the effect of the in-plane strains, whichare generated through the coupling to the crack open-ing displacement through νb, the Poisson ratio in theband. The additional strains and ensuing stresses giverise to an additional term in the internal virtual work,thus resulting in a higher peak load and a more ductilebehaviour. Evidently, the effect diminishes for smallervalues of the Poisson’s ratio, and disappears for νb = 0,when the results of the standard cohesive surface modelare retrieved.

Next, the effect of the band thickness h0 is investi-gated. To this end, the simulations have been repeatedfor three different ratios h0/ le = 0.1, 0.2 and 0.4. The

Fig. 10 Effect of the band thickness h0 on the load-displacementcurve for a cohesive band model. The effect is shown for twovalues of Poisson’s ratio νb: 0.0 and 0.3

Fig. 11 The ratio of the stretch over the mode-I contributionsto the elastic energy in the cohesive band as a function of the tipdisplacement u

results are shown in Fig. 10. Note that the mechanicalbehaviour is almost independent of the choice of cohe-sive band width h0. For νb = 0.0 the curve coincideswith results for the standard cohesive surface modelwhen h0 is small. But even for non-zero values of Pois-son ratio the results are almost independent of the bandwidth.

The contribution of the stretch term Ess in the cohe-sive band becomes evident when we observe the contri-butions of all strain components to the internal energy.The ratio of the stretch over the normal (mode-I) con-tributions to the elastic energy is shown in Fig. 11 as afunction of the tip-displacement u. Evidently, the con-tribution of the relative magnitude of the stretch termto the elastic energy increases for an increasing ratio

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The cohesive band model 187

h0/ le, and is more pronounced for larger values of thePoisson ratio in the band, νb.

6 Concluding remarks

In this paper the cohesive band method has been pre-sented as an extension of the cohesive surface model.At the macroscopic scale it resembles a standard cohe-sive surface model in the sense that fracture occursover a discrete plane with zero thickness. Indeed, atthis scale discretisation methods that are commonlyused to incorporate cohesive surface formulations con-tinue to be applicable. Also, anomalies that reside incertain discretisation methods, e.g. the traction oscil-lations that occur in conventional interface elementsequipped with cohesive surface formulations and a highdummy stiffness to represent a non-compliant interfaceprior to reaching the tensile strength, persist, as hasbeen shown for a classical example (Schellekens andde Borst 1993a).

The cohesive band model deviates from standardcohesive surface formulations in the sense that a sub-grid scale fracture model is conceived at the loca-tion of the discontinuity, which has a finite thickness,and which features a full three-dimensional strain andstress state. In the present implementation a trans-versely isotropic constitutive relation has been assumedwithin the band, which would focus on quasi-brittlefracture, rather than on ductile fracture, where shear-normal stress/strain couplings can become significant,and an anisotropic constitutive relation within the bandmay then be required. Along the same line, the isotropiccontinuum damage formalism that has been used inthe example, is insufficient to model ductile fracture,where fracture is often preceded by plastic localisation.However, the constitutive relation for the band can bestraightforwardly extended to incorporate anisotropyand plasticity. With appropriate constitutive relationsthe cohesive band formulation holds promise for cap-turing fracture phenomena such as splitting cracks inconcrete and masonry under compressive axial stresses,crazing in polymers, and crack growth in porous met-als, which all depend on stress triaxiality.

An important property of the cohesive band modelis that it is consistent with standard cohesive surfaceformulations. Indeed, in the cohesive band model thestrength and the ductility depend, in the constitutive for-mulation used here, on the Poisson ratio in the band,

since the coupling between the crack opening displace-ment and the in-plane normal strains causes an addi-tional term in the virtual work equation. However, wehave shown that the cohesive band model reduces to thestandard cohesive surface model for a vanishing bandwidth. This is corroborated by numerical experiments,which show that the results from a standard cohesivesurface model are obtained when the Poisson ratio inthe band is set to zero, thus decoupling the in-planenormal strains from the crack opening displacement.The vanishing of the in-plane strains then implies thatno longer additional work is expended, and the load-displacement curve becomes identical to that obtainedfor a standard cohesive surface model.

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