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1 s e Borst, R., Remmers, J. J.C., an Verhoosel, C. V. (2014) Evolving iscontinuities an cohesive fracture. Proceia IUTAM, 10. pp ISSN Copyright 2014 Elsevier Lt. Deposite on: 06 January 2015 Enlighten Research publications by members of the University of Glasgow

2 Available online at ScienceDirect Proceia IUTAM 10 ( 2014 ) r International Congress of Theoretical an Applie Mechanics Evolving iscontinuities an cohesive fracture René e Borst a,, Joris J. C. Remmers b, Clemens V. Verhoosel b a School of Engineering, University of Glasgow, Oakfiel Avenue, Rankine Builing, Glasgow G12 8LT, UK b Department of Mechanical Engineering, Einhoven University of Technology, P.O. Box 513, 5600 MB Einhoven, Netherlans Abstract Multi-scale methos provie a new paraigm in many branches of sciences, incluing applie mechanics. However, at lower scales continuum mechanics can become less applicable, an more phenomena enter which involve iscontinuities. The two main approaches to the moelling of iscontinuities are briefly reviewe, followe by an in-epth iscussion of cohesive moels for fracture. In this iscussion emphasis is put on a novel approach to incorporate triaxiality into cohesive-zone moels, which enables for instance the moelling of crazing in polymers, or of splitting cracks in shear-critical concrete beams. This is followe by a iscussion on the representation of cohesive crack moels in a continuum format, where phase-fiel moels seem promising Publishe by Elsevier Lt. Open access uner CC BY-NC-ND license. c 2013 Publishe by Elsevier Lt. Selection an/or peer-review uner responsibility of Dick van Campen an Gengong Cheng Selection an/or peer-review uner responsibility of the Organizing Committee of The 23r International Congress of Theoretical an Applie Mechanics, ICTAM2012 Keywors: multi-scale methos, iscontinuities, fracture, cohesive surfaces, phase-fiel moels 1. Introuction The eman for new or improve materials has been one of the riving forces behin the evelopment of multiscale techniques. This class of methos aims at unerstaning the material behaviour at a lower level of observation, an can be a major tool for eveloping new materials. When consiering materials at a lower length scale, the classical concept of a continuum graually faes away. At the macroscopic level we have to take into account evolving or moving iscontinuities like cracks, shear bans, Lüers bans an Portevin-Le Chatelier bans, but at a lower level we also encounter grain bounaries in crystalline materials, soli-soli phase bounaries as in austenite-martensite transformations [1], an iscrete islocations [2]. Thus, the proper moelling of iscontinuities has a growing importance in material science. Classical iscretisation methos are not well amenable to capturing iscontinuities. Accoringly, next to multiscale moelling, the proper capturing of iscontinuities is a major challenge in contemporary computational mechanics of materials. In this contribution we focus on the capturing of iscontinuities. Basically, two methos exist to hanle them. One either istributes them over a finite istance, or hanles them as true iscontinuities. The first metho has been a subject of much research in the past two ecaes. It will be iscusse briefly at the beginning of this paper, an we will come back to a continuum approach within the context of phase-fiel Corresponing author. aress: Rene.DeBorst@glasgow.ac.uk Publishe by Elsevier Lt. Open access uner CC BY-NC-ND license. Selection an/or peer-review uner responsibility of the Organizing Committee of The 23r International Congress of Theoretical an Applie Mechanics, ICTAM2012 oi: /j.piutam

3 126 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) methos at the en of this paper. A major part of this paper focuses on an enhancement of the cohesivesurface moel for fracture to incorporate triaxiality effects. 2. Discrete vs. continuum representations of fracture When scaling own, iscontinuities arise which nee to be moelle in an explicit manner. When the iscontinuity has a stationary character, such as in grain bounaries, this is fairly straightforwar, since it is possible to aapt the iscretisation such that the iscontinuity, either in isplacements or in isplacement graients, is moelle explicitly. An evolving or moving iscontinuity, however, is more ifficult to hanle. A possibility is to aapt the mesh upon every change in the topology, as was one by Ingraffea an coworkers in the context of linear elastic fracture mechanics [3], an later by Camacho an Ortiz [4] for cohesive fracture. Another approach is to moel fracture within the framework of continuum mechanics. A funamental problem is then that stanar continuum moels o not furnish a length scale which is inispensible for escribing fracture, or, more precisely, they result in a zero length scale. Since the energy issipate in the failure process is given per unit area of material that has completely egrae, an since a vanishing internal length scale implies that the volume in which failure occurs goes to zero, the energy issipate in the failure process also tens to zero. Two approaches have been followe to avoi this physically unrealistic situation, namely via iscretisation an via regularisation of the continuum, see Fig. 1. Fig. 1. Application of regularisation an iscretisation methos to a iscontinuity In the first approach, researchers have let the spacing of the iscretisation take over the role of the missing internal length scale, so that the iscontinuity in the left part of Fig. 1 is replace by a isplacement istribution as in the right-lower part of this figure, where w is the spacing of the iscretisation. The iea is then to choose the iscretisation such that the spacing of the iscretisation coincies with the internal length scale that erives from the physics of the process. Eviently, a goo knowlege of the problem is require an solutions, incluing the proper choice for the iscretisation, are problem-epenent. Nevertheless, this approach has been use successfully to obtain insight in various issues in materials science [5]. Yet, this approach can not be calle a proper solution in the sense that the mathematical setting of the initial value problem remains unchange. Inee, the introuction of egraation of the material properties in a stanar, rate-inepenent continuum moel an therefore, the introuction of a stress strain curve with a escening slope can locally cause the governing ifferential equations to change character. Without special provisions such as the application of special interface conitions between both omains at which ifferent types of ifferential equations hol, the initial value problem becomes ill-pose. Numerically, this has the consequence that the solution becomes fully epenent upon the iscretisation [6, 7]. An example is shown in Figs. 2 an 3. It concerns a bar compose of a porous, flui-saturate material that is loae by an impulse loa at the left en. Upon reflection at the right bounary, the stress intensity oubles an the stress in the soli excees the yiel strength an enters a linear escening branch, Fig. 2. The results are shown in Fig. 3 in terms of the strain profile along the bar at iscrete time intervals. It is observe that

4 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) a Dirac-like strain istribution evelops immeiately upon wave reflection, inicating that the governing equations change locally from hyperbolic, as is normal in wave propagation, to elliptic, which implies that a staning wave evelops. To further strengthen this observation the analysis was repeate with a 25% refine mesh, which resulte in a marke increase of the localise strain (Fig. 3), an has been plotte on the same scale as the results of the original iscretisation. We remark that in ynamic calculations of softening meia without regularisation, not only the spatial iscretisation strongly influences the results, but also the time iscretisation [8]. σ σ 0 σ σ y t 0 t ε u ε Fig. 2. Top: Uniaxial bar subject to impact loa. Bottom: Applie stress as function of time (left) an local stress-strain iagram (right). 6 6 strain [x ] 5 4 strain [x ] x [m] x [m] Fig. 3. Strain profiles along the bar for 101 gri points (left) an 126 gri points (right) As a more rigorous solution, various regularisation methos have been propose, incluing nonlocal averaging, the aition of viscosity or rate epenency, or the inclusion of couple stresses or higher-orer strain graients, see [9] for an overview. The effect of these strategies is that the iscontinuity shown in the left sie of Fig. 1 is transforme into the continuous isplacement istribution shown in the right-upper part of this figure. In contrast to iscretisation strategies, the internal length scale w is now set by the constitutive moel for the soli material, an as soon as a sufficiently fine iscretisation has been aopte to properly capture this isplacement istribution, the numerically calculate results only change in a sense that is normally expecte upon mesh refinement. It is emphasise that the above observations for iscretisation an regularisation hol for any iscretisation metho, incluing finite element approaches, finite ifference methos, meshfree methos an finite volume methos [10]. The fact that iscretisation provies only a partial remey to the ill-poseness of the unerlying initial value problem, an that ifficulties that still persist with regularisation strategies notably the unresolve issue of aitional bounary conitions, the nee to use fine meshes in the zone of the regularise iscontinuity, an the nee to etermine aitional material parameters from tests that impose an inhomogeneous

5 128 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) eformation fiel has been a contributing factor to revisit the research into (more flexible) methos to capture arbitrary, evolving iscontinuities in a iscrete sense. At present, four such methos exist: zero-thickness interface elements [11], meshless or meshfree methos [12], the partition-of-unity metho which exploits the partition-of-unity property of finite element shape functions [13], also known as the extene finite element metho [14], an isogeometric analysis [15]. Meshfree methos were originally thought to behol a great promise for fracture analyses ue to the fact that this class of methos oes not require meshing, an remeshing upon crack propagation, but the ifficulties to properly reefine the support of a noe when it is partially cut by a crack, have le to a ecrease interest. However, out of the research into this class of methos, the analysis methos that exploit the partition-of-unity property of finite element shape functions have arisen, which is a powerful metho for large-scale fracture analyses. The most recent evelopment is to moel evolving iscontinuities, incluing cohesive cracks, using isogeometric analysis, where the use of knot insertion is believe to be the most elegant approach [15]. 3. Cohesive-surface moels Most engineering materials are not perfectly brittle in the Griffith sense, but isplay some uctility after reaching the strength limit. In fact, there exists a zone ahea of the crack tip, in which small-scale yieling, micro-cracking an voi initiation, growth an coalescence take place. If this fracture process zone is sufficiently small compare to the structural imensions, linear-elastic fracture mechanics concepts can apply. However, if this is not the case, the cohesive forces that exist in this fracture process zone must be taken into account. The most powerful an natural way is to use cohesive-surface moels, which were introuce in [16, 17] for elastic-plastic fracture in uctile metals, an for quasi-brittle materials in the so-calle fictitious crack moel [18]. In cohesive-surface moels, the egraing mechanisms in the wake of the crack tip are lumpe into a iscrete plane. The most important parameters of cohesive surface moels are the tensile strength f t an fracture energy G c [19], which is the work neee to create a unit area of fully evelope crack. It has the imensions J/m 2 an is formally efine as G c = u n =0 t n u n, (1) with t n the normal traction an u n the normal relative isplacement across the fracture process zone. The fracture energy introuces an internal length scale into the moel, since the quotient G c /E has the imension of length. For quasi-brittle fracture, also the shape of the ecohesion curve can play an important role [20]. The tractions at the iscontinuity, t, can be erive by ifferentiating the fracture energy G c with respect to the jump of the isplacement fiel, u, as follows t = G c u. (2) Most fracture problems are riven by crack opening (moe-i). However, in a number of cases, the sliing (moe-ii) components can become substantial. A possible way to inclue the sliing components is to reefine Eq. (1) as, cf. [21] G c = t ũ (3) with t = t( ũ ), where ũ = ũ =0 u n 2 + β ( u s 2 + u t 2) (4) an u s an u t the sliing components, β being a moe-mixity parameter that sets the ratio between the moe-i an the moe-ii components. The cohesive surface moel as outline in the preceing is a two-imensional moel embee in a three-imensional continuum, an only the crack opening an the crack sliing moes are available. The

6 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) Γ + + Γ + Γ Γ Γ * h b Fig. 4. A cohesive ban moel normal strain parallel to the cohesive surfaces is not available, an neither is the normal stress in this irection. This hampers the accurate computation of failure moes in metals where stress triaxiality plays a role, but also prevents the preiction of splitting cracks in concrete or masonry structures where a large compressive stress inuces cracks that are aligne with the normal stress [22]. To circumvent this eficiency of the cohesive surface moel it has been propose to take the normal stress from a neighbouring integration point in the continuum an to insert this stress component in the failure criterion for the cohesive surface moel [23, 24]. In the sequel a more rigorous solution is outline An extension: the cohesive-ban moel We consier the cohesive crack epicte in Fig. 4. The thick lines are the cohesive surfaces Γ an Γ+, characterise by the normals n Γ an n Γ +, respectively, see Fig. 4 right. The thickness of the cohesive ban b between the surfaces Γ an Γ+ is enote by h. The bulk B = \ b consists of the sub-omain that borers the cohesive surface Γ, an the sub-omain + that borers the cohesive surface Γ +, Fig. 4. For consistency the relative isplacement vector an the traction vector must be ecompose in the same coorinate system. There is some freeom in the choice of the vector that is normal to the fracture plane. A possible choice is n Γ = 1 2 (n Γ + n Γ + ), (5) as the normal vector of the plane Γ on which the cohesive tractions act, cf. [25], an on which the relative isplacement u n, u s an u t, an the tractions t n, t s an t t are efine. The position vector x of a material point in the boy can be expresse in terms of its position in the uneforme configuration ξ an two piecewise smooth isplacement fiels û(ξ) an ũ(ξ) where H Γ 0 graient can be erive as x(ξ) = ξ + û(ξ) + H Γ 0 ũ(ξ), (6) is the generalise Heavisie function centre at the iscontinuity Γ0. Then, the eformation F(ξ) = I + û ξ + H ũ Γ 0 ξ + δ Γũ 0 n Γ 0, (7) with I the unit matrix an δ Γ 0 the generalise Dirac function centre at Γ0. We note that this spatial erivative of the Heavisie function H 0 is non-zero only at the iscontinuity Γ0 which resies in the ban b. We can therefore efine the eformation graient at the sie of the iscontinuity, ξ,as while at the + sie of the iscontinuity, ξ +,wehave F = I + û ξ, (8) F + = I + û ξ + ũ ξ. (9)

7 130 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) Using Nanson s formula, the normals at the an at the + sies can be relate to that in the original configuration n Γ 0, as follows n Γ = et(f )n Γ 0 (F ) 1 Γ,0, The average normal n Γ then becomes Γ n Γ + = et(f + )n Γ 0 (F+ ) 1 Γ,0. Γ + (10) n Γ = et(f )n Γ 0 (F ) 1 Γ,0, (11) Γ where an F = 1 ( F + F +), (12) 2 Γ = 1 2 (Γ + Γ+ ). (13) In view of Eq. (6) the isplacement u(ξ) of a material point in the boy can be expresse as u(ξ) = û(ξ) + H Γ 0 ũ(ξ). (14) Then, the isplacement jump u equals the value of the aitional isplacement fiel at the iscontinuity plane u = ũ(ξ) ξ Γ 0. (15) The Green Lagrange strain E(ξ) in the bulk B = \ b is now erive from the eformation graient in a stanar manner E(ξ) = 1 ( F T (ξ) F(ξ) I ) ξ B. (16) 2 We next efine the Green Lagrange strain tensor in the cohesive ban, expresse in the n, s, t local frame of reference of the ban: E nn E ns E nt Ē = E sn E ss E st ξ b. (17) E tn E ts E tt The components of this matrix are base on the magnitue of the relative isplacements an on the in-plane Green Lagrange strains in the ban. Denoting h 0 as the value of h in a reference state, the component F nn of the eformation graient F reas F nn = h h 0 = 1 + u n h 0. (18) with u n the crack opening measure with respect to the reference state where h 0. The normal component of the Green Lagrange strain tensor within the ban can subsequently be erive as The corresponing shear components rea E nn = 1 ( F 2 2 nn 1) = u n + u n 2. (19) h 0 2h 2 0 E ns = u s 2h 0 + u s 2 2h 2 0 an E nt = u t + u t 2. (20) 2h 0 2h 2 0 In a stanar manner the virtual strain components can be erive as δe nn = δ u n h 0 + u n δ u n, (21) h 2 0

8 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) for the normal strain an δe ns = δ u s 2h 0 + u s δ u s h 2 0 an δe nt = δ u t 2h 0 + u t δ u t, (22) h 2 0 for the shear strains. Taking the current configuration as the reference configuration, so that h 0 = h an u n = u s = u t = 0, these expressions reuce to δe nn = δ u n h an δe ns = δ u s an δe nt = δ u t 2h 2h. (24) It is note that when h 0 is set equal to zero, the with h of the iscontinuity in the eforme configuration equals the normal opening of the iscontinuity u n (23) h = u n. (25) The in-plane terms of the strain tensor in the ban, E ss, E tt an E st = E ts are inepenent of the magnitue of the isplacement jump. They represent the Green Lagrange normal strain components in the s- an t-irections, respectively, an the Green Lagrange in-plane shear strain. Since h 0 is small compare to the in-plane imensions of the fracture plane, it is reasonable to assume that the in-plane strain components vary linearly in the n-irection of the ban b, so that E ss = 1 2( Ess Γ + E ss Γ + ) E tt = 1 2( Ett Γ + E tt Γ + ) an E st = 1 2( Est Γ + E st Γ + ). (26) The internal virtual work of the soli can be expresse in terms of the Cauchy stress tensor σ an the variation of the Green Lagrange strain tensor referre to the current configuration x. In the bulk of the omain, B, we enote the variation of the strain tensor by δɛ, while in the cohesive ban, b,wehaveδe enoting the variation of the strain tensor an S the Cauchy stresses, so that δw int = σ : δɛ + S : δe. (27) B b The secon term, which represents the contribution of the cohesive ban, can be rewritten as h 2 δw b int = S S S : δēēē nγ. (28) Γ Again using the assumption that the eformation in the cohesive ban is constant in the n-irection, we integrate analytically in the thickness irection δw b int = h S S S : δēēē Γ, (29) Γ or written in terms of the iniviual components δw b int = h (S nn δe nn + S ss δe ss + S tt δe tt + 2S ns δe ns + 2S nt δe nt + 2S st δe st ) Γ, (30) Γ which relation hols irrespective of the value of the cohesive ban with h. Substitution of the strain terms as erive in Eqs. (23), (24) an (26) gives δw b int = (S nn δ u n + hs ss δe ss + hs tt δe tt + S ns δ u s + S nt δ u t + 2hS st δe st ) Γ, (31) Γ h 2

9 132 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) Fig. 5. Geometry an bounary conitions of a three point bening test In the limit, i.e. when h 0, this expression reuces to δw b int = (S nn δ u n + S ns δ u s + S nt δ u t ) Γ, (32) Γ or replacing the stress components S nn, S ns an S nt by the tractions t n, t s an t s, we obtain the usual cohesive surface relation δw b int = (t n δ u n + t s δ u s + t t δ u t ) Γ. (33) Γ The effect of the in-plane strains in the cohesive ban has now isappeare, as shoul be. We finally note that a similar approach, in which a crack has been moelle as a zero-thickness interface at the macroscopic scale, while a small, but finite thickness has been use for the moelling at a subgri scale, has been use for moelling flui flow in cracks that are embee in a surrouning porous meium [26 28] Double cantilever peel test We next consier the ouble cantilever test shown in Fig. 5. The structure with length l = 10 mm consists of two layers with the same thickness h = 0.5 mm an with the same (isotropic) material properties: a Young s moulus E = MPa an a Poisson ratio ν = 0.3. The two layers are connecte through an ahesive with a normal strength t max = 1.0 MPa an a fracture toughness G c = 0.1N/mm. The initial elamination extens over a = 1.0 mm. An external loa P is applie at the tip of both layers. Delamination growth in a laminate with a symmetric lay-up can be moelle with a simple two-imensional amage moel with a loaing function f efine as f ( u n,κ) = u n κ. (34) The normal traction t n ecreases exponentially, accoring to ( t n = t max exp t ) max κ. (35) G c The specimen has been analyse with four-noe quarilateral elements. The results are compare to a moel with a stanar cohesive surface moel in Fig. 6. We clearly observe the effect of the in-planestrains, which are generate through the coupling to the crack opening isplacement through the Poisson ratio. The aitional strains an ensuing stresses give rise to an aitional term in the internal virtual work, thus resulting in a higher peak loa an a more uctile behaviour. Eviently, the effect iminishes for smaller values of the Poisson ratio, an isappears for ν = 0, when the results of the stanar cohesive surface moel are retrieve.

10 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) Fig. 6. Effect of Poisson s ratio ν on the loa-isplacement curve for a cohesive ban moel 4. Continuum representations of the cohesive moel The cohesive-zone moel is essentially a iscrete concept. However, it can be transforme into a continuum formulation by istributing the fracture energy G c over the thickness w of the volume in which the crack localises [29]. The isavantages of the formulation are that a pseuo-softening moulus is introuce which is inversely proportional to the number of elements, an, more importantly, that the bounary-value problem becomes ill-pose, implying a epenence on the iscretisation. As iscusse before, this anomaly is inherent in all smeare formulations without proper regularisation. A smeare formulation that is properly regularise, an therefore exhibits no mesh epenence, can be obtaine using the phase-fiel concept, an will be outline below Cohesive fracture an phase fiels As the starting point of the erivation of the phase fiel approximation to cohesive fracture we consier the internal energy W int = ψ e (ε)v + G( u,κ)a. (36) In this expression, the elastic energy ensity ψ e epens on the strain tensor ε. Uner the assumption of small isplacement graients, the infinitesimal strain tensor is an appropriate measure of the eformation of the boy, an is equal to the symmetric graient of the isplacement fiel u ε ij = u (i, j) = 1 ( ui + u ) j. (37) 2 x j x i The elastic energy ensity is expresse by Hooke s law for an isotropic linear elastic material as Γ ψ e (ε) = 1 2 λε iiε jj + με ij ε ij, (38) with λ an μ the Lamé constants. We now efine x n = (x x c ) n(x c ), with n(x) the unit vector normal to the fracture surface an x c the position vector of a point on the fracture surface. The Dirac function δ can then be use to relate the infinitesimal surface area A to the infinitesimal volume V of the surrouning boy A(x c ) = δ (x n ) V. (39)

11 134 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) Equation (39) allows for smeare escriptions of the fracture surface by an approximation of the Dirac function. As in [30, 31] we consier the approximate Dirac function δ ɛ (x n ) = 1 ( 2ɛ exp x ) n, (40) ɛ with ɛ>0 a length scale parameter. Eviently δ ɛ (x n )x n = 1 (41) for arbitrary ɛ. Clearly, the fracture surface is istribute over a larger volume for higher values of ɛ. The corresponing infinitesimal fracture surface area then follows from A ɛ (x c ) = δ ɛ (x n ) V. (42) Using the approximation to the Dirac function expresse in Eq. (40), the internal energy, Eq. (36), is approximate by ( W int,ɛ = ψ e (ε e ) ) + G(v,κ)δ ɛ V. (43) Note that, compare to Eq. (36), we have replace the infinitesimal strain tensor ε by the elastic infinitesimal strain tensor ε e, an the jump in the isplacement fiel u (x c ) by an auxiliary fiel v(x). This is necessary since in the phase-fiel formulation, the crack only exists in a smeare sense. Hence, the clear istinction between the bulk an interface kinematics, i.e. between the infinitesimal strain tensor, Eq. (37), an the crack opening u is lost. In the phase-fiel formulation for cohesive fracture it is crucial to erive kinematic relations that are consistent with the iscrete problem, in the sense that as the length scale parameter ɛ approaches zero, the kinematics of the iscrete problem are recovere. In orer to arrive at such relations, we first introuce the istribute internal iscontinuity Γ ɛ = {x (x) > tol} (44) with tol 1 being a small tolerance that truncates the support of the smeare crack, which provies the support for the auxiliary fiel v(x). Hence, the auxiliary fiel is only present at the smeare crack, an the kinematics away from the crack are governe by the isplacement fiel u(x). Next, we efine the normal to the smeare crack base on the point closest to the iscrete bounary Γ, enote by x c,as n ɛ (x) = n(x c ) with x c (x) = argmin ( y x ). (45) y Γ In the iscrete formulation, the isplacement jump is strictly efine at the internal iscontinuity Γ. In the phase-fiel approach the crack exists in smeare sense, an so oes the crack opening. Therefore, we approximate the iscrete isplacement jump at x c in terms of the auxiliary jump fiel v(x)as u (x c ) v(x)δ ɛ x n. (46) v Uner the conition that the auxiliary fiel is constant in the irection normal to the fracture, i.e. = 0, x n we have v(x) = v(x c + ζn) = v(x c ), (47) with n the vector normal to the crack, an ζ the coorinate along n. Back-substitution into Eq. (46) shows that v(x) reflects the crack opening at the closest point x c on the iscrete internal bounary. Consequently,

12 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) Fig. 7. Schematic representation of a uniaxial ro with a cohesive interface in Eq. (43) the crack opening u that appears as an argument of the fracture energy is irectly replace by the auxiliary fiel v(x). In the limiting case that the length scale parameter ɛ goes to zero, the smeare crack Γ ɛ coincies with the iscrete crack Γ an the auxiliary isplacement fiel consies with the iscrete isplacement jump. The requirement that the auxiliary jump fiel is constant in the irection normal to the crack is now enforce weakly through an aitional term in the internal energy functional ( W int,ɛ = ψ e (ε e ) + G(v,κ)δ ɛ α v x 2 ) V, (48) n with α a positive constant parameter. With the iscontinuity kinematics etermine through the auxiliary jump fiel, the elastic strain tensor ε e can be erive by requiring the auxiliary fiel not to exert any external power on the problem, such that the balance of power is given by ( σ e ij ε e ij + t ) i(v,κ)δ ɛ v i V = t i u i A. (49) Application of Gauss theorem to the right-han sie of this equation, an requiring the traction to be in equilibrium with the elastic stress, σ e ij n j = t i, yiels σ e ij ( ε e ij + δ ɛ sym ( v i n j )) V = σ e ij u (i, j)v. (50) From the balance of power it is irectly observe that the elastic infinitesimal strain tensor is relate to the isplacement fiel u(x) an auxiliary fiel v(x)as ε e ij = u (i, j) sym ( v i n j ) δɛ. (51) In this expression the first part is the symmetric part of the graient of the isplacement fiel. The secon part can be interprete as the strain cause by the isplacement jump. Hence the elastic strain is efine as the symmetric graient of the isplacement fiel, minus the strain cause by the crack opening. We immeiately note from Eq. (51) that away from the crack, i.e. for x Γ ɛ, the continuum expression of Eq. (37) is recovere. In the limiting case that ɛ goes to zero, the elastic equivalent strain is equal to the symmetric graient of the isplacement fiel at every point in the interior of the omain. The unboune strain term at the iscontinuity is cause by the jump of the isplacement fiel over this internal bounary Cohesive fracture of a ro As an example we consier the one-imensional bar loae in tension as shown in Fig. 7. The bar has a unit length an also the moulus of elasticity equals unity. The fracture toughness an fracture strength are taken as G c = 1 an f t = 0.75, respectively. Figure 8 shows the response obtaine using various mesh sizes for the phase-fiel moel an compares it with the exact solution to the iscrete problem. It is observe that upon mesh refinement the phase-fiel moel converges to the iscrete solution.

13 136 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) Fig. 8. Force-isplacement curve for the one-imensional cohesive zone problem 5. Concluing remarks Multi-scale moels have the potential to give preictions of material behaviour that are better roote in physical evience than purely phenomenological approaches. However, when scaling own, more an more iscontinuities are encountere. These iscontinuities can be moelle either using truly iscrete approaches, or by smearing them out an casting them in a continuum escription. A prototypical problem is (cohesive) crack propagation. Inee, the cohesive approach to fracture is enjoying an ever increasing popularity. Its successful use is limite by two factors. From the moelling point of view, it is eficient in the sense that in the classical cohesive surface moels the normal strain in the fracture plane is not inclue in the moel, thus precluing the proper capturing of triaxiality effects, or splitting cracks in shearcritical concrete beams. Regaring iscretisation, the proper moelling of propagating cracks along a priori unknown crack paths requires novel an flexible iscretisation methos, such as partition-of-unity base finite element methos or isogeometric analysis if the cohesive crack is moelle as a true iscontinuity. Alternatively, it can be smeare over a small, but finite with. This approach requires a proper regularisation, as for instance provie by phase-fiel approaches, or by graient amage moels [32]. References [1] Bhattacharya K. Microstructure of martensite: why it forms an how it gives rise to the shape-memory effect. Oxfor: Oxfor University Press; [2] van er Giessen E, Neeleman A. Discrete islocation plasticity - A simple planar moel. Moelling an Simulation in Materials Science an Engineering 1995; 3: [3] Ingraffea AR, Saouma V. Numerical moelling of iscrete crack propagation in reinforce an plain concrete. In Fracture Mechanics of Concrete. Martinus Nijhoff Publishers, Dorrecht, 1985; [4] Camacho GT, Ortiz M. Computational moeling of impact amage in brittle materials. International Journal of Solis an Structures 1996; 33: [5] Ruggieri C, Panontin TL, Dos RH. Numerical moeling of uctile crack growth in 3-D using computational cell elements. International Journal of Fracture 1996; 82: [6] Bažant ZP. Instability, uctility, an size effect in strain-softening concrete. ASCE Journal of the Engineering Mechanics Division 1976; 102: [7] e Borst R. Some recent issues in computational failure mechanics. International Journal for Numerical Methos in Engineering 2001; 52: [8] Abellan MA, e Borst R. Wave propagation an localisation in a softening two-phase meium. Computer Methos in Applie Mechanics an Engineering 2006; 195: [9] e Borst R. Damage, material instabilities, an failure. In Encyclopeia of Computational Mechanics. Stein E, e Borst R, Hughes TJR (es). Wiley, Chichester, 2004; Volume 2, Chapter 10. [10] Pamin J, Askes H, e Borst R. Two graient plasticity theories iscretize with the element-free Galerkin metho. Computer Methos in Applie Mechanics an Engineering 2003; 192: [11] Schellekens JCJ, e Borst R. On the numerical integration of interface elements. International Journal for Numerical Methos in Engineering 1993; 36:

14 René e Borst et al. / Proceia IUTAM 10 ( 2014 ) [12] Fleming M, Chu YA, Moran B, Belytschko T. Enriche element-free Galerkin methos for crack tip fiels. International Journal for Numerical Methos in Engineering 1997; 40: [13] Babuska I, Melenk JM. The partition of unity metho. International Journal for Numerical Methos in Engineering, 1997; 40: [14] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methos in Engineering, 1999; 45: [15] Verhoosel CV, Scott MA, e Borst R, Hughes TJR. An isogeometric approach to cohesive zone moeling, International Journal for Numerical Methos in Engineering 2011; 87: [16] Dugale DS. Yieling of steel sheets containing slits. Journal of the Mechanics an Physics of Solis 1960; 8: [17] Barenblatt GI. The mathematical theory of equilibrium cracks in brittle fracture. Avances in Applie Mechanics 1962; 7: [18] Hillerborg A, Moeér M, Petersson PE. Analysis of crack formation an crack growth in concrete by means of fracture mechanics an finite elements. Cement an Concrete Research 1976; 6: [19] Hutchinson JW, Evans AG. Mechanics of materials: top-own approaches to fracture. Acta Materialia 2000; 48: [20] Chanra N, Li H, Shet C, Ghonem H. Some issues in the application of cohesive zone moels for metal-ceramic interfaces. International Journal of Solis an Structures 2002; 39: [21] Tvergaar V, Hutchinson JW. The relation between crack growth resistance an fracture process parameters in elastic-plastic solis. Journal of the Mechanics an Physics of Solis, 1992; 41: [22] Rots JG. Smeare an iscrete representations of localize fracture. International Journal of Fracture 1991; 51: [23] Keller K, Weihe S, Siegmun T, Kröplin B. Generalize cohesive zone moel: incorporating triaxiality epenent failure mechanisms. Computational Materials Science 1999; 16: [24] Tijssens MGA, van er Giessen E, Sluys LJ. Moeling of crazing using a cohesive surface methoology. Mechanics of Materials 2000; 32: [25] Wells GN, e Borst R, Sluys LJ. A consistent geometrically non-linear approach for elamination. International Journal for Numerical Methos in Engineering 2002; 54: [26] e Borst R, Réthoré J, Abellan MA. A numerical approach for arbitrary cracks in a flui-saturate porous meium. Archive of Applie Mechanics 2006; 75: [27] Réthoré J, e Borst R, Abellan MA. A iscrete moel for the ynamic propagation of shear bans in flui-saturate meium. International Journal for Numerical an Analytical Methos in Geomechanics 2007; 31: [28] Réthoré J, e Borst R, Abellan MA. A two-scale moel for flui flow in an unsaturate porous meium with cohesive cracks. Computational Mechanics 2008; 42: [29] Bažant ZP, Oh B. Crack ban theory for fracture of concrete. RILEM Materials an Structures 1983; 16: [30] Miehe C, Hofacker M, Welschinger F. Thermoynamically consistent phase-fiel moels of fracture: Variational principles an multi-fiel FE implementations. International Journal for Numerical Methos in Engineering 2010; 83: [31] Miehe C, Hofacker M, Welschinger F. A phase fiel moel for rate-inepenent crack propagation: Robust algorithmic implementation base on operator splits. Computer Methos in Applie Mechanics an Engineering 2010; 199: [32] e Borst R, Crisfiel MA, Remmers JJC, Verhoosel CV. Non-linear Finite Element Analysis of Solis an Structures, 2n e. Wiley: Chichester, 2012.