Cohesive-zone models, higher-order continuum theories and

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2004; 60:289–315 (DOI: 10.1002/nme.963)

Cohesive-zone models, higher-order continuum theories andreliability methods for computational failure analysis‡

René de Borst∗,†, Miguel A. Gutiérrez, Garth N. Wells,Joris J. C. Remmers and Harm Askes

Koiter Institute Delft, Delft University of Technology, P.O. Box 5058, NL-2600 GB Delft, The Netherlands

SUMMARY

A concise overview is given of various numerical methods that can be used to analyse localizationand failure in engineering materials. The importance of the cohesive-zone approach is emphasized andvarious ways to incorporate the cohesive-zone methodology in discretization methods are discussed.Numerical representations of cohesive-zone models suffer from a certain mesh bias. For discreterepresentations this is caused by the initial mesh design, while for smeared representations it isrooted in the ill-posedness of the rate boundary value problem that arises upon the introduction ofdecohesion. A proper representation of the discrete character of cohesive-zone formulations whichavoids any mesh bias can be obtained elegantly when exploiting the partition-of-unity property offinite element shape functions. The effectiveness of the approach is demonstrated for some examplesat different scales. Moreover, examples are shown how this concept can be used to obtain a propertransition from a plastifying or damaging continuum to a shear band with gross sliding or to a fullyopen crack (true discontinuum). When adhering to a continuum description of failure, higher-ordercontinuum models must be used. Meshless methods are ideally suited to assess the importance of thehigher-order gradient terms, as will be shown. Finally, regularized strain-softening models are usedin finite element reliability analyses to quantify the probability of the emergence of various possiblefailure modes. Copyright 2004 John Wiley & Sons, Ltd.

KEY WORDS: localization; failure; cohesive zones; discontinuities; higher-order continua; partitionsof unity; meshless methods; reliability methods; imperfections

1. INTRODUCTION

Failure in most engineering materials is preceded by the emergence of narrow zones of intensestraining. During this phase of the so-called strain localization, the deformation pattern in abody rather suddenly changes from relatively smooth into one in which thin zones of highly

∗Correspondence to: René de Borst, Koiter Institute Delft, Delft University of Technology, P.O. Box 5058,NL-2600 GB Delft, The Netherlands.

†E-mail:[email protected]‡Revised version of semi-plenary lecture presented at the Fifth World Congress on Computational Mechanicsunder the original title ‘A Précis of Some Recent Developments in Computational Failure Mechanics’.

Received 10 May 2003Copyright 2004 John Wiley & Sons, Ltd. Accepted 25 July 2003

290 R. DE BORST ET AL.

strained material dominate. In fact, these strain localization zones act as a precursor to ultimatefracture and failure. Thus, in order to accurately analyse the failure behaviour of materials itis of pivotal importance that the strain localization phase is modelled in a physically consistentand mathematically correct manner and that proper numerical tools are used.

Until the mid-1980s analyses of localization phenomena in materials were commonly carriedout using standard, rate-independent continuum models. This is reasonable when the principalaim is to determine the behaviour in the pre-localization regime and some properties at incipientlocalization. However, there is a major difficulty in the post-localization regime, since localiza-tion in standard, rate-independent solids is intimately related to a local change in the characterof the governing set of partial differential equations. Upon this change of character, the rateboundary value problem becomes ill-posed and numerical solutions suffer from spurious meshsensitivity.

To remedy this deficiency, one must either introduce higher-order terms in the continuumrepresentation or take into account the inherent viscosity of most engineering materials. Analternative possibility is to by-pass the strain localization phase and to directly incorporate thediscontinuity that arises as an outcome of the strain localization process. The latter possibility ispursued with the so-called cohesive-zone models. We will start by describing them and discusshow they can be introduced in a numerical context. We will argue that finite elements with‘embedded’ localization zones do not rigorously incorporate discontinuities in finite elementmodels. Conversely, finite element formulations that exploit the partition-of-unity property of theshape functions [1] can, as will be discussed. Indeed, this concept also enables the modelling ofa gradual transition from a (higher-order) continuum description to a genuine discontinuum in anumerical context. This is very powerful, since now the entire failure process, from small-scaleyielding or the initiation of voids and micro-cracks up to the formation of a macroscopicallyobservable crack, can be simulated in a consistent and natural fashion.

Like for finite element methods, the shape functions of meshless discretizations, e.g. theelement-free Galerkin method [2], form partitions of unity. Meshless methods have a tremen-dous advantage for models in which higher-order terms are incorporated, since they inherentlyprovide for the required higher-order continuity. Accordingly, enhanced continuum models canbe implemented easily and the importance of the higher-order gradients can be assessed.

Finally, we will indicate how the heterogeneous character of materials at a macroscopicscale can be incorporated in numerical analyses of inelastic solids. In particular, the effect ofstochastically distributed imperfections on the failure load in inelastic solids will be quantifiedin the framework of the finite element reliability method.

2. COHESIVE-ZONE MODELS

2.1. Formulation

An important issue when considering failure is the observation that most engineering materialsare not perfectly brittle in the Griffith sense, but display some ductility after reaching thestrength limit. In fact, there exists a small zone in front of the crack tip, in which small-scale yielding, micro-cracking and void initiation, growth and coalescence take place. If thisfracture process zone is sufficiently small compared to the structural dimensions, linear-elasticfracture mechanics concepts apply. However, if this is not the case, the cohesive forces that

Copyright 2004 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 60:289–315

COMPUTATIONAL FAILURE ANALYSIS 291

cohesive zone

Figure 1. Schematic representation of a cohesive zone.

σ

u

f t

σf t

u

Figure 2. Stress–displacement curves for a ductile solid (left) and a quasi-brittle solid (right).

exist in this fracture process zone must be taken into account, and cohesive-zone models mustbe used, which were introduced by Barenblatt [3] and Dugdale [4] for elastic–plastic fracture inductile metals, and for quasi-brittle materials by Hillerborg et al. [5] in his so-called fictitiouscrack model.

In cohesive-zone models, the degrading mechanisms in front of the actual crack tip arelumped into a discrete line or plane (Figure 1) and a stress–displacement relationship acrossthis line or plane represents the degrading mechanisms in the fracture process zone. Figure 2shows some commonly used decohesion relations, one for ductile fracture (left) [6] and onefor quasi-brittle fracture (right) [7]. For ductile fracture, the most important parameters of thecohesive zone model appear to be the tensile strength ft and the work of separation or fractureenergy Gc [8], which is the work needed to create a unit area of fully developed crack. It hasthe dimensions J/m2 and is formally defined as

Gc =∫

du (1)

with and u being the stress and the displacement across the fracture process zone. For morebrittle decohesion relations, as shown for instance in the right part of Figure 2 (i.e. whenthe decohesion law stems from micro-cracking as in concrete or ceramics), the shape of thestress–separation relation plays a much bigger role and is sometimes even more important thanthe value of the tensile strength ft [9, 10].



Figure 3. Geometry of single-edged notched (SEN) beam.

Figure 4. Deformed configuration of the SEN-beam that results from an analysis where inter-face elements equipped with a quasi-brittle cohesive-zone model have been placed a priori at the

experimentally known crack path [12].

2.2. Discrete numerical representations of cohesive-zone models

In the past two decades, cohesive-zone models have been recognized to be an important toolfor describing fracture in engineering materials. Especially when the crack path is known inadvance, either from experimental evidence, or because of the structure of the material (suchas in laminated composites), cohesive-zone models have been used with great success. In thosecases, the mesh can be constructed such that the crack path a priori coincides with the elementboundaries. By inserting interface elements between continuum elements along the potentialcrack path, a cohesive crack can be modelled exactly. Figures 3 and 4 show this for mixed-modefracture in a single-edge notched (SEN) concrete beam [11]. In this example the mesh has beendesigned such that the interface elements, which are equipped with a quasi-brittle cohesive-zone model, are exactly located at the position of the experimentally observed crack path [12].Another good example where the potential of cohesive-zone models can be exploited fully usingtraditional discrete interface elements, e.g. Reference [13], is the analysis of delamination inlayered composite materials [14–17]. Since the propagation of delaminations is then restricted



to the interfaces between the plies, inserting interface elements at these locations permits anexact simulation of the failure mode.

To allow for a more arbitrary direction of crack propagation, Xu and Needleman [18]have inserted interface elements equipped with a cohesive-zone model between all continuumelements. A related method, using remeshing, was proposed by Camacho and Ortiz [19].Although analyses with this approach provide much insight, see also References [20, 21], theysuffer from a certain mesh bias, since the direction of crack propagation is not entirely free, butis restricted to interelement boundaries. Another drawback is that the method is not suitable forlarge-scale analyses. For these two reasons smeared numerical representations of cohesive-zonemodels have appeared, including the emergence of some, initially unforeseen mathematicaldifficulties, which can only be overcome in a rigorous fashion by resorting to higher-ordercontinuum models.

2.3. Smeared numerical representations of cohesive-zone models

Although the cohesive-zone model is essentially a discrete concept, it can be transformed intoa continuum or smeared formulation by distributing the work of separation or fracture energyGc over the full width of an element [22, 23]. Since the fracture energy is now smeared outover the width of the area in which the crack localizes, we obtain

Gc =∫∫

d(x) dx (2)

with x being the co-ordinate orthogonal to the crack direction. For low-order elements thestrains are constant over the width of the element w and we obtain Gc = wgc, with gc beingthe energy dissipated per unit volume of fully damaged material:

gc =∫

d (3)

Evidently, a length scale, namely the element size w, is now introduced in the numerical model.When doing so, the global load–displacement response can become almost insensitive to

the discretization. An example of smeared-crack calculations is given in Figures 5 and 6 [24].Figure 5 shows numerical results for various decohesion relations (linear and exponential)and a linear relation between the shear traction and the relative sliding in the cohesive zone(characterized by a coefficient as the fraction of the elastic shear modulus). Clearly, thecomputational results match the experimental data, since they all fall within the experimentalscatter. Nevertheless, the deformed structure (Figure 6) reveals that there is a strong tendency ofthe fracture process zone to propagate parallel to interelement boundaries. This is very apparentfor the coarse, structured mesh that has been used, but the same observation can be made—albeit less clearly—for finer and unstructured meshes [25]. It is a direct consequence of theill-posedness of the rate boundary value problem that arises upon the onset of decohesion andis inherent in smeared representations of cohesive-zone models. When adhering to a continuumconcept, the only solution to avoid this ill-posedness is to introduce higher-order gradients inspace (non-local models) or in time (rate-dependent or viscous models). This will be discussedin more detail in the section on higher-order continuum theories.

Finite element models with so-called embedded discontinuities provide a more elegant ap-proach to implement cohesive-zone models in a smeared context [26–31]. There are two



Figure 5. Load–CMSD curves for the SEN-beam that result from analyses with asmeared cohesive-zone model [24].

Figure 6. Deformed configuration of the SEN-beam that results from an analysiswith a smeared cohesive-zone model [24].

versions of these models, namely the strong discontinuity approach and the weak discontinuityapproach. We will depart from the latter approach and consider an element within which aband is defined where the strains are different in magnitude from the strains in the remainderof the element:

−ij = ij + −

2(nimj + njmi) (4)

and

+ij = ij + +

2(nimj + njmi) (5)

with n a vector normal to the band and m related to the deformation mode, e.g. m = nfor mode-I behaviour and n orthogonal to m for mode-II behaviour. − and + are scalars



indicating the magnitude of the strain inside and outside of the band, respectively, measuredrelative to the average, continuous strain ij in the element. The enhanced strain modes (secondpart of Equations (4) and (5)) are discontinuous across element boundaries. Consequently, theycan be solved for at element level.

The stress–strain relation in the band can be specified independently from that in the bulk ofthe element. Typically, a softening relation is prescribed which results in an energy dissipationper unit volume gc upon complete loss of material coherence. For a band width w, which isincorporated in the finite element formulation, we thus retrieve the fracture energy Gc that isdissipated for the creation of a unit area of fully developed crack.

A problem resides in the determination of the length of the crack band, lelem in a specificelement. For a given length lelem and an element thickness t , the total energy dissipation in anelement reads

Delem = t lelemGc (6)

If the crack length in an element is estimated incorrectly, the energy that is dissipated in eachelement is wrong, and so will be the load–displacement diagram [31]. Different possibilitiesexist to calculate lelem, e.g. to relate lelem to the area of the element Aelem, to assume thatthe enhanced mode passes through the element midpoint and to calculate the band lengthaccordingly, or to let the band connect at the element boundaries and to compute the bandlength in this fashion.

While the above considerations have been set up for the so-called weak discontinuity ap-proach, in which the displacement is continuous, it is also possible to let the enhanced strainmodes be unbounded. This so-called strong discontinuity approach can be conceived as a lim-iting case of the weak discontinuity approach for the band width h → 0 [32]. The strain thenlocally attains the form of a Dirac function and the displacement becomes discontinuous acrossa single discrete plane. Nevertheless, the integral over time of the product of the traction andthe difference in velocities between both sides of the plane still equals the fracture energy.

The embedded discontinuity approaches enhance the deformational capabilities of the ele-ments, especially when the standard Bubnov–Galerkin approach is replaced by a Petrov–Galerkinmethod, which properly incorporates the discontinuity kinematics [32]. At the expense of ob-taining a non-symmetric stiffness matrix, the high local strain gradients inside localization bandsare then better captured. However, a true discontinuity is not obtained because the kinematics ofEquations (4) and (5) are diffused over the element when the governing equations are cast in aweak format, either via a Bubnov–Galerkin or via a Petrov–Galerkin procedure. Indeed, severalauthors [33, 34] have proven the equivalence between embedded discontinuity approaches andclassical smeared-crack models in which the fracture energy is smeared out over the elementwidth in the sense of Equation (2), e.g. References [12, 35]. Accordingly, the embedded dis-continuity approaches inherit many of the disadvantages of conventional smeared-crack models,including the sensitivity of crack propagation to the direction of the mesh lines.

2.4. Exact numerical representation of discontinuities

Considering the mathematical difficulties that are inherent in smeared representations ofcohesive-zone models, the search has continued for a proper representation of the true dis-crete character of cohesive-zone models, while allowing for an arbitrary direction of crackpropagation, not hampered by the initial mesh design. Originally, meshless methods [2] were



thought as the answer to this problem, but they appear to be less robust than traditional finiteelement methods, they are computationally more demanding and the implementation in threedimensions appears to be less straightforward. However, out of this research, a method hasemerged recently, in which a discontinuity in the displacement field is captured exactly. It hasthe added benefit that it can be used advantageously at different scales, from microscopic tomacroscopic analyses.

The method makes use of the partition-of-unity property of finite element shape functions[1, 36, 37]. A collection of functions i , associated with nodes i, form a partition of unity if

n∑i=1

i (x) = 1 (7)

with n being the number of discrete nodal points. For a set of functions i that satisfy Equation(7), a field u can be interpolated as follows:

u(x) =n∑

i=1i (x)

(ai +

m∑j=1

j (x)aij

)(8)

with ai the ‘regular’ nodal degrees-of-freedom, j (x) the enhanced basis terms, and aij theadditional degrees-of-freedom at node i which represent the amplitudes of the j -th enhancedbasis term j (x).

A piecewise continuous displacement field u which incorporates a discontinuity with a unitnormal vector n pointing in an arbitrary, but fixed direction can be described by

u(x) = u(x) + Hd(x)u(x) (9)

with u the standard, continuous displacement field on which the discontinuity has been super-imposed. The discontinuous field is represented by the continuous field u and the Heavisidefunction Hd , centred at the discontinuity plane d. The displacement decomposition in Equa-tion (9) has a structure similar to the interpolation of Equation (8). Accordingly, the partition-of-unity property of finite element shape functions can be used in a straightforward fashionto incorporate discontinuities, and thus, cohesive-zone models in a manner that preserves theirtruly discontinuous character. Indeed, in conventional finite element notation, the displacementfield of an element that contains a single discontinuity can be represented as

u = N(a + Hd a) = Na + HdNa = u + Hd u (10)

where N contains the standard shape functions, and a and a collect the conventional andthe additional nodal degrees-of-freedom, respectively. The numerical development now followsstandard lines by casting the balance of momentum in a weak format, and, in the spiritof a standard Bubnov–Galerkin method, taking a decomposition as in Equation (9) also forthe test function. For small displacement gradients a complete derivation can be found inReference [38], while the extension to large displacement gradients is given in Reference [39].It is interesting to note that, at variance with requirements for the ‘embedded’ displacementdiscontinuity approaches, the field u does not have to be constant. The only requirement thatis imposed is continuity. Therefore, a distinction can be made between the normals n−

dand

n+d

at both sides of the discontinuity (Figure 7). It is this distinction that allows for capturing



nΓd

n

+

−Γd

Figure 7. Normals n−d

and n+d

at both sides of the discontinuity.

local buckling at the interface. An example involving the combination of delamination andlocal buckling will be given below.

It is emphasized that in this concept, the additional degrees-of-freedom cannot be condensedat element level, because, at variance with the ‘embedded’ displacement discontinuity approach,it is node-oriented and not element-oriented. It is this property which makes it possible torepresent a discontinuity in a rigorous manner.

From Equations (8) and (10) we infer that the partition-of-unity concept can naturally beconceived as a multiscale approach [40]. We can formally show this by decomposing u(x) as

u(x) = uC(x) + uF(x) (11)

with

uC(x) =n∑

i=1i (x)ai (12)

representing the coarse scale and

uF(x) =n∑

i=1

m∑j=1

i (x)j (x)aij (13)

representing the fine scale.The objectivity of the formulation with respect to mesh refinement is demonstrated for the

three-point bending beam of unit thickness shown in Figure 8. The beam is loaded quasi-statically by means of an imposed displacement at the centre of the beam on the top edge.Six-noded triangular elements are used as the underlying base elements. The following materialproperties have been used: Young’s modulus E = 100 MPa, Poisson’s ratio = 0.0, tensilestrength ft = 1.0 MPa and fracture energy Gc = 0.1 Jmm−2. For this example, the crack shearstiffness is set equal to zero. Since the crack shear stiffness vanishes, the top row of elementsof the beam have been prevented from cracking in order to avoid singularity of the systemwhen the crack has propagated through the entire beam.

Figure 9 shows the crack having propagated through the entire beam. Two meshes areshown, one with 523 elements and the other with 850 elements. Clearly, in both cases the



3

55

P

Figure 8. Geometry of a beam subjected to symmetric three-point bending (dimensions in mm).

Figure 9. Crack path at the final stage of loading for the coarse mesh (523 elements) and thefine mesh (850 elements) [38].

crack propagates from the centre at the bottom of the beam straight upwards towards the loadpoint, not influenced by the mesh structure.

The load–displacement responses of Figure 10 confirm objectivity with respect to meshrefinement. From the load–displacement response for the coarser mesh the energy dissipation iscalculated as 0.308 J, which only slightly exceeds the fracture energy multiplied by the depthand the thickness of the beam (0.3 J), giving further proof of mesh objectivity. It is furthernoted that the small irregularities that are observed in the load–displacement curve, especiallyfor the coarser mesh, are caused by the fact that in the current implementation a cohesivezone is inserted entirely in one element when the tensile strength has been exceeded. A more



Figure 10. Load–displacement diagrams for the analysis of the symmetrically loadedbeam using both meshes [38].

Figure 11. Crack path that results from the analysis of the SEN-beam using the partition-of-unityproperty of finite element shape functions [38].

sophisticated way is to advance the crack only in a part of the element, for instance using levelsets [41, 42]. However, the observation that these little bumps quickly disappear upon meshrefinement (Figure 10) casts doubt on the necessity to complicate the numerical model.

Usually, the requirement that the crack path is not biased by the direction of the mesh linesis even more demanding than the requirement of objectivity with respect to mesh refinement.Figure 11 shows that the approach also fully satisfies this requirement, since the numericallypredicted crack path of the single edge notched beam of Figure 3 is in excellent agreementwith experimental observations, e.g. Reference [11].

Even though traditional interface elements equipped with cohesive-zone models are able tocapture the failure behaviour of laminated composites accurately, the simulation of delami-nation using the partition-of-unity property of finite element shape functions offers some ad-vantages. Because the discontinuity does not have to be inserted a priori, no (dummy) stiffnessis needed in the elastic regime. Indeed, there does not have to be an elastic regime, sincethe discontinuity can be activated at the onset of cracking. Consequently, the issue of spurioustraction oscillations in the elastic phase becomes irrelevant. Also, the lines of the potential



Figure 12. Double cantilever beam with initial delamination under compression.

Debonding (coarse mesh)Debonding (dense mesh)

Perfect bond

P [N

]

1 2 3 4 5 6 7 8u (mm )

0

1

2

3

4

Figure 13. Load–displacement curves for delamination-buckling test [43].

delamination planes no longer have to coincide with element boundaries. They can lie atarbitrary locations inside elements. Accordingly, unstructured meshes can be used.

To exemplify the possibilities of this approach to model the combined failure mode of de-lamination growth and local buckling we consider the double cantilever beam of Figure 12 withan initial delamination length a. This case, in which failure is a consequence of a combinationof delamination growth and structural instability, has been analysed using conventional interfaceelements in Reference [17]. The beam is subjected to an axial compressive force 2P , whiletwo small perturbing forces P0 are applied to trigger the buckling mode. Two finite elementdiscretizations have been employed, a fine mesh with three elements over the thickness and250 elements along the length of the beam, and a coarse mesh with only one (!) element overthe thickness and 100 elements along the length. Figure 13 shows that the calculation withthe coarse mesh approaches the results for the fine mesh closely. For instance, the numeri-cally calculated buckling load is in good agreement with the analytical solution. Steady-statedelamination growth starts around a lateral displacement u = 4 mm. From this point onwards,delamination growth interacts with geometrical instability. Figure 14 presents the deformed



Figure 14. Deformation of coarse mesh after buckling and delamination growth (true scale) [43].

Figure 15. Load–displacement curve and deformations of shell model after buckling anddelamination growth (true scale) [43].

beam for the coarse mesh at a tip displacement u = 6 mm. Note that the displacements areplotted at true scale, but that the difference in displacement between the upper and lower partsof the beam is for the major part due to the delamination and that the strains remain small.

The excellent results obtained in this example for the coarse discretization has motivatedthe development of a layered plate/shell element in which delaminations can occur insidethe element between each of the layers [43]. Because of the absence of rotational degreesof freedom, a solid-like shell element [44] was taken as a point of departure. The numericalresults of Figure 15 show an excellent correspondence with the earlier two-dimensional resultsof Figures 13 and 14. This opens the possibility to analyse delamination-buckling in thin-walledcomposite structures at a structural scale.

In all examples discussed above, a single, continuous cohesive crack growth was simulated.Crack propagation in heterogeneous materials and also fast crack growth in more homogeneousmaterials is often characterized by the nucleation of (micro)cracks at several locations, which cangrow, branch and eventually link up to form macroscopically observable cracks. To accommodate



this observation, the concept of cohesive segments has been proposed [45], in which, againexploiting the partition-of-unity property of finite element shape functions, crack segmentsequipped with a cohesive law are placed over a patch of elements when a loading criterionis met at an integration point. Since the cohesive segments can overlap in elements, they canalso behave macroscopically as a single, dominant crack.

3. CONTINUUM–DISCONTINUUM TRANSITION

The above approach enables the gradual and consistent transition from a continuum to adiscontinuum description. In Reference [46] traction-free discontinuities have been inserted ina softening, viscoplastic Von Mises plasticity model when the equivalent plastic strain reachesa threshold level. Accordingly, the viscosity of the continuum has been used to ensure a mesh-independent result in the softening phase prior to local failure and the creation of a traction-freediscontinuity.

The three-point bending test of Figure 8 has been simulated again, but now the continuummodel is equipped with a Von Mises yield surface and linear softening after the onset ofyielding. Unless stated otherwise, the material properties are taken as: Young’s modulus E =1 × 102 MPa, Poisson’s ratio = 0.2, initial yield stress 0 = 1 MPa, viscosity = 2 sand the hardening modulus h = −200 Nmm−2. Traction-free discontinuities, which propagatefrom a 0.5 mm long initial cut, have been inserted using the partition-of-unity concept whenthe underlying, rate-independent Von Mises stress in the element is below 0.0001 × 0. Thedisplacement discontinuity is extended ‘in the direction in which the effective stress is largest’[46]. The evolution of the equivalent plastic strain field and displacement discontinuity areshown in Figure 16. The failure mode is mode-I dominated. A discontinuity propagates throughthe beam, with a plastic hinge forming at the final stage of failure.

The analysis has been repeated for a more ductile plastic behaviour: h = −20 Nmm−2,which is 10 times larger than before. The evolution of the equivalent plastic strain and thedisplacement discontinuity are shown in Figure 17. The failure mode for this example is clearlydifferent than that of the more brittle case and is mode-II dominated. The discontinuity extendsonly over a short distance and a large plastic hinge forms. Obviously, the interplay of plasticity,

Figure 16. Evolution of plastic zones and crack propagation in the symmetrically loaded three-pointbending beam of Figure 8 for a quasi-brittle softening plasticity model.



Figure 17. Evolution of plastic zones and crack propagation in the symmetrically loaded three-pointbending beam of Figure 8 for a more ductile softening plasticity model.

and in particular the plastic incompressibility constraint, and the relief that is provided for bythe insertion of the traction-free discontinuities, which do allow for inelastic volume changes,can give rise to failure modes that are completely different, depending on the choice of thematerial parameters [33].

4. HIGHER-ORDER CONTINUUM THEORIES

For many materials the viscosity is so low that its incorporation in the constitutive modelis not sufficient to restore well-posedness of the boundary value problem during the strainlocalization phase. Indeed, models that exploit the non-local interactions in the fracture processzone can be physically as well motivated and numerically more effective. Among these models,the gradient-enhanced models have shown to be computationally the most efficient, either ina plasticity-based format [47–49], a damage-based format [50, 51], or a combination of both[52]. Especially the gradient-enhanced damage model of Peerlings et al. [50, 51] has provento be robust and effective, not only for damage evolution under monotonic loading conditions,but also for fatigue loading [53].

The Peerlings gradient damage model is scalar-based and rooted in an injective relationbetween the stress tensor and the strain tensor :

= (1 − )De : (14)

De is the elastic stiffness tensor with the virgin elastic constants E (Young’s modulus) and (Poisson’s ratio), while is a monotically increasing damage parameter, with an initial value0 for the intact material, and an ultimate value 1, at complete loss of material coherence. It isa function of a history parameter : = (), with linked to a non-local strain measure via a loading function

f = − (15)

such that loading occurs if f = 0, f = 0 and < 1. In the original approach of Peerlingset al. [50, 51], the non-local strain measure is coupled to the local strain measure = () via:

− c1∇2 = (16)



Figure 18. Damage profiles for SEN-beam in one of the ultimate loading stages computed with agradient-enhanced damage model [55].

with c1 a material parameter with the dimension length squared. This model is referred toas the second-order implicit gradient damage model. A fourth-order implicit gradient damagemodel can be obtained as

− c1∇2 − c2∇4 = (17)

with c2 a material parameter with the dimension length to the power four, while a non-localdamage model in integral format [54] derives from

=∫

g(s + x)(s) ds (18)

with g a normalized weighting function. Finally, the non-local strain can also be postulated asan explicit function of the local strain measure :

= + c1∇2 (19)

As discussed before, higher-order continuum models, including gradient-enhanced models,can maintain well-posedness of the rate boundary value problem until well into the softeningregime. As a consequence, no spurious mesh sensitivity, neither an influence of the direction ofthe mesh lines will occur. This is exemplified in Figure 18, which shows the damage patternfor the single edge notched beam of Figure 3 at one of the ultimate stages of loading [55].

5. MESHLESS METHODS

A clear disadvantage of the use of higher-order continuum theories is the higher continuitythat may be required for the shape functions that are used in the interpolation of the non-localvariable. The second-order implicit gradient damage theory (Equations (14)–(16)) is such thatafter partial integration C0-interpolation polynomials suffice for the interpolation of . However,this no longer holds when the fourth-order term is retained. Then, and also because of themoving elastic–plastic boundary in gradient plasticity models, C1-continuous shape functionsare required, with all computational inconveniences that come to it. Here, meshless methods,which can easily be constructed such that they incorporate C∞-continuous shape functions,have a clear advantage. Below, we shall apply one such method, namely the element-freeGalerkin method [2], to the second- and the fourth-order implicit gradient damage models(Equations (16) and (17)) and to the second-order explicit gradient damage model (Equation



u

A =10 mm A = 9 mm

45 mm45 mm 10 mm

2 2

Figure 19. Bar with an imperfection subjected to an axial load.

0 0.02 0.04 0.06 0.08displacement [mm]

0

0.5

1

1.5

2

forc

e [N

]

Figure 20. Load–displacement curves for axially loaded bar using the fully non-local dam-age model (dash–dotted line), the second-order implicit gradient damage model (dottedline), the fourth-order implicit gradient damage model (dashed line) and the second-order

explicit gradient damage model (solid line) [56].

(19)). The simple one-dimensional bar with an imperfection in the centre (Figure 19) hasbeen analysed to make a quantitative assessment of the differences that emerge. The resultingload–displacement diagrams are given in Figure 20 and show that the fourth-order gradientdamage model approaches the solution of the non-local damage model in integral format veryclosely, with the second-order implicit gradient damage model also giving a good match. Thebehaviour of the second-order explicit gradient damage model is too ductile. In all cases, aquadratic convergence behaviour was obtained, except for the non-local damage model, wherea secant stiffness matrix was used due to the difficulty of obtaining a proper tangent matrix forsuch models. Indeed, the use of a secant stiffness matrix resulted in premature convergence,as shown in Figure 20.

Interestingly, the differences between the second-order and fourth-order gradient damagemodels are not very pronounced, and are only marginal until the later stages of loading. Thiscasts doubt on the necessity to include fourth and higher-order terms in the analysis. Whenconsidering a slightly more complicated example, a symmetrically loaded three-point bendingbeam (Figure 21), the differences between the load–displacement curves that are predicted byboth formulations almost vanish (Figure 22).

The higher-order continuity that is incorporated in meshless methods makes them well suitedfor localization and failure analyses using higher-order continuum models. In fact, this is a



u

200 mm 100

mm300

700 mm

mm

1000 mm

300

1000 mm

sym

met

ry

Figure 21. Three-point bending beam and node distribution (for a symmetric half of the beam).

0 0.5 1.0 1.5 2.0displacement [mm]

0

2000

4000

6000

forc

e [N

]

Figure 22. Load–displacement curves for three-point bending beam. Comparison betweenthe second-order implicit gradient damage model (dashed line) and the fourth-order

implicit gradient damage model (solid line) [56].

major advantage which meshless methods still have over finite element methods, even when inthe latter class of methods the partition-of-unity property of shape functions is exploited.

Also, the flexibility is increased compared to conventional finite element methods, sincethere is no direct connectivity, which makes placing additional nodes in regions with highstrain gradients particularly simple. This can also be achieved by finite element methods withspatial adaptivity. Originally, adaptivity for localization analyses was applied using standardcontinuum models [57]. Later, it became clear that the inherent loss of ellipticity preventederror estimators from working properly [58]. Contemporary approaches therefore apply mesh



adaptivity techniques for failure analyses in conjunction with cohesive-zone models or withregularized continuum models [58–60]. Similarly, the discretization itself cannot provide aregularization, neither for finite element methods, nor for meshless methods. Indeed, for thelatter class of methods the nodal spacing directly relates to the width of the localization zonethat is resolved if no regularization is provided for the continuum model.

6. RELIABILITY METHODS

So far, the discussion has concentrated on localization and the ensuing failure in solids whichhave uniform strength and stiffness properties. In reality, strength and stiffness have a randomdistribution over any structure. The distribution and the size of imperfections may have aprofound influence on the localization pattern and, therefore, on the ultimate failure load, aswas demonstrated more than half a century ago by Koiter in his landmark dissertation [61] onthe influence of imperfections in elastic solids. We may expect that this observation holds afortiori if material degradation plays a role.

Thus, for realistic analyses of localization and failure, material parameters like Young’smodulus, the tensile strength and the fracture energy should be considered as random fields,and the most probable realization(s) should be sought which lead to failure or violate a certainserviceability criterion. Indeed, in such analyses, not only the scatter in material parameters,but also the uncertainty in the boundary conditions should be considered. The simplest, butalso the most expensive method, would be to start a non-linear analysis from different randomdistributions and to obtain the statistics of the response by carrying out a sufficient numberof such Monte-Carlo simulations [62]. Evidently, this is very expensive and a more versatileapproach is to use the finite element reliability method [63–65].

The latter approach formally departs from a coupling between the random field of mate-rial parameters (and/or boundary conditions) and the equilibrium path of the body. Throughappropriate discretization techniques [66], the random fields can be discretized into a set ofuncorrelated standard normal variables that will be denoted as S. The simplest example of thiskind of discretization is given by the midpoint method, in which the domain of the randomfield is discretized into a set of finite element-like cells r. The random field, particularized atthe centroid xc of each cell, provides a set of correlated random variables V. These variablesare then converted into the set S of uncorrelated standard normally distributed variables throughNataf’s transformation [67]. The geometrical interpretation of this discretization is provided inFigure 23. The cells r do not need to coincide with the finite element discretization. However,the computer implementation is facilitated by choosing r coinciding with patches of finiteelements (Figure 24).

The equilibrium path of the considered body is related to the random variables S througha non-linear finite element algorithm. The statistical properties of any measure Q of theequilibrium path, like the peak load or the energy consumption, can then be studied in thestandard normal space expanded by S. In particular, the cumulative distribution of Q evaluatedat the realization q0 is expressed as

FQ(q0) =∫

q(s)<q0

d(s) (20)



Finite element discretisationΩr

Figure 23. Schematic representation of the mid-point discretization technique for random fields.

Figure 24. Selection of the domains r inrelation to the finite element mesh.

Figure 25. Schematization of the evaluation of the cumulative distribution of Q in the Q space (left)and in the uncorrelated standard normal space S (right).

where d(s) represents the uncorrelated standard normal probability measure. This is alsoschematized in Figure 25. The exact evaluation of the integral in the right hand side ofEquation (20) is still too expensive from a computational point of view. Alternatively, thesurface q(s) = q0 is approximated by a first-order or second-order surface. The most convenientpoint for approximation is given by the closest point to the origin on surface q(s) = q0. Thispoint, which represents the realization of S with the largest probability density, is referred to asdesign point or -point and represented by s. In the particular case that the surface q(s) = q0has been approximated by a hyperplane, the cumulative distribution of Q at q0 is approximated



Figure 26. Schematic representation of the hyperplane approximation procedure.

by the expression

FQ(q0) = (−) (21)

where represents the one-dimensional standard normal cumulative distribution function and is the distance from the origin to the approximating hyperplane. It is immediately observedthat

= ‖s‖ (22)

The hyperplane approximation procedure is summarized in Figure 26.The crucial step in the reliability method is the evaluation of the -point s. This is achieved

by solving a constrained minimization problem which is stated as

Minimize ‖s‖Subject to q(s) = q0

(23)

by means of a suitable optimization algorithm.When analysing solids which exhibit an unstable behaviour and, in particular, solids with

material instabilities, the minimization problem (23) can exhibit several local minima. Whilethis could be seen as a disadvantage of the method when the cumulative distribution of Q

should be evaluated as accurately as possible, it is actually this property that makes the methodespecially suitable to study the localization behaviour of solids with stochastic imperfectionpatterns. Indeed, each local minimum of (23) provides information on the contribution of thecorresponding localization mechanism to the distribution of Q.



Figure 27. Schematic representation of a tensile test on a double-edge-notchedspecimen. Dimensions in mm.

As an example a double-edge-notched specimen is considered (Figure 27). For detailedinformation regarding the parameters that have been used and the way in which the analysishas been carried out, the reader is referred to References [63, 65]. The considered characteristicproperty Q of the equilibrium path is the peak load. The cumulative distribution function ofQ can then be associated with the probability of failure, since it represents the probability thatthe peak load is lower than a given threshold.

Tensile tests on such specimens tend to be sensitive to the boundary conditions, in particularwhen the brittleness of the material increases. When an imperfection is not imposed in thematerial, nor an asymmetry in the boundary conditions, the deformations will remain symmetricthroughout the entire equilibrium path. However, if either of these occurs, asymmetric crackpropagation evolves from one of the notches at a generic stage in the loading process. Theprobability that either of these failure modes occurs can be simulated via the approach discussedabove where the tensile strength is randomized, while starting the solution algorithm from asymmetric as well as from an asymmetric realization. In particular, the influence of the boundaryconditions can be quantified. For instance, taking the upper loading platen of the double-edge-notched specimen fixed, the probability of failure was found to be Ps = 5.84×10−2, irrespectivewhether the algorithm was started from a symmetric or from an asymmetric realization. Indeed,the failure mode was purely symmetric. However, when the upper loading platen is allowed torotate freely, an asymmetric mode was found with a probability of failure that is significantlyhigher than that of the symmetric model, namely Pa = 0.41. The symmetric and the asymmetricmodes are represented in Figure 28. Next, the analysis was repeated for a longer specimen



Figure 28. Symmetric and asymmetric damage localization modes for thedouble-edge-notched specimen.

(L = 250 mm), while keeping the loading platens fixed. Not surprisingly, a symmetric failuremode was found with a probability of failure Ps which is almost the same as for the shorterspecimen. However, an asymmetric failure mode now also emerged, with a probability of failurePa = 0.13, which is purely a consequence of the increased rotational freedom of the longerspecimen.

It is emphasized that the inclusion of randomness of the material parameters in the analysisdoes not resolve the issue of loss of ellipticity at the onset of localization when standard,rate-independent continuum models are considered [62]. Here, the situation is similar to theuse of mesh adaptivity techniques, which are also unable to remedy the loss of well-posednesswhich is the fundamental cause of the spurious results which are obtained when analysinglocalization in standard, rate-independent continuum models.

7. CONCLUDING REMARKS

The importance of the cohesive-zone concept for describing fracture in a wide range of engi-neering materials has been recognized over the past few decades. However, its proper numericalimplementation has caused problems. Essentially, the cohesive-zone concept is a discrete modeland cannot be implemented readily in standard, continuum-based finite element methods. Whilethe development of special interface elements that are equipped with cohesive-zone models isfairly simple, the difficulty remains where to put these interface elements. Crack paths are



normally not known in advance. To partly circumvent this difficulty, proposals have beenmade to insert interface elements between all continuum elements, to carry out remeshingprocedures or to use mesh-free methods. Neither of these approaches is entirely satisfactoryand it seems that only recently, namely by using the partition-of-unity property of finite el-ement shape functions an elegant and powerful method has been developed to fully exploitthe potential of cohesive-zone models for arbitrary crack propagation. A consistent extensionto dynamics is possible as well as a transparent extension to multiphasic media. Among theadded benefits we list the ability to model delaminations independent of the underlying meshstructure, to achieve a consistent formulation for large strains and to obtain a smooth transi-tion from a plastifying or damaging continuum to a traction-free, macroscopically observablecrack.

With the exploitation of the partition-of-unity property of finite element shape functions,mesh-free methods have lost much of their attractiveness. However, they still retain the benefitof higher-order continuity, which can be exploited when quantifying the effect of higher-ordergradients which occur in gradient-enhanced damage or plasticity models for the description ofsoftening in solids.

As a final point, it is emphasized that materials and structures are never homogeneous.Imperfections play an important role in triggering failure modes. For this reason a method hasbeen discussed that can quantify the effect of the size and the location of imperfections insoftening solids.

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