Plastic model with non-local damage applied to concrete

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2006; 30:71–90Published online 10 October 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.479Plastic model with non-local damage applied to concretePeter Grassl 1,3 and Milan Jirásek 2,3,n,y1 Chalmers University of Technology, Go¨teborg, Sweden2 Czech Technical University in Prague, Czech Republic3 Swiss Federal Institute of Technology at Lausanne, SwitzerlandSUMMARYThe present paper deals with a certain class of regularized models for concrete failure, combining plasticitywith non-local damage. The plastic part is local and uses a yield condition formulated in the effective stressspace. Softening is incorporated through the damage part of the model. Damage is considered as isotropicand linked to the evolution of plastic strain. The regularized version of the model is based on weightedspatial averaging of the damage-driving variable. Conditions for a mesh-independent description of thelocalized zone of inelastic strains are studied using one- and two-dimensional examples solved by simplenon-local damage-plasticity models, in two dimensions with non-associated flow. Then, the framework isapplied to an advanced model for concrete. Structural examples with tensile and compressive failure arepresented. Copyright # 2005 John Wiley & Sons, Ltd.KEY WORDS:plasticity; damage mechanics; non-local continuum; concrete; failure; localization1. INTRODUCTIONDescription of failure processes in cohesive-frictional materials such as concrete, rock and soilsrequires constitutive laws with strain softening, i.e. with decreasing stress under increasingstrain. If these constitutive laws are local, i.e. the stress at a point depends uniquely on the strainhistory at that point only, strain softening leads to an ill-posed boundary value problem. In thecontext of the finite element method, this results in strong mesh dependence of the resolvedzones of highly localized inelastic strains and of the dissipation associated with the failureprocess.A remedy to the pathological mesh dependence are non-local constitutive laws, for which thestress at one point depends on the strain history not only at that point but also in itsneighbourhood or even in the entire body. In these models, a characteristic length isn Correspondence to: Milan Jirásek, Department of Mechanics, Faculty of Civil Engineering, Czech Technical Universityin Prague, Thákurova 7, 166 29 Prague, Czech Republic.y E-mail: Milan.Jirasek@fsv.cvut.czContract/grant sponsor: Swiss Commission for Technology and Innovation; contract/grant number: CTI 5501.1Contract/grant sponsor: Ministry of Education of the Czech Republic; contract/grant number: CEZ MSM 6840770003Copyright # 2005 John Wiley & Sons, Ltd.Received 28 September 2004Revised 22 July 2005Accepted 16 August 2005

72P. GRASSL AND M. JIRÁSEKincorporated, which allows mesh-independent description of energy dissipation in a localizedfailure process. Typically, the deformations remain continuous even in the localized zone. Forintegral-type non-local models, the interaction between neighbouring points is taken intoaccount by weighted spatial averaging. Such formulations were initially proposed in the contextof elasticity [1] and hardening plasticity [2], and they were applied as localization limiters for thefirst time by Pijaudier-Cabot and Bazˇ ant [3] in the context of damage mechanics. For softeningplasticity, integral-type non-local models were developed, for instance, by Bazˇ ant and Lin [4],Vermeer and Brinkgreve [5], Leblond et al. [6], Tvergaard and Needleman [7], or Stro¨mberg andRistinmaa [8]. Thermodynamic aspects were discussed by Svedberg and Runesson [9] andBorino et al. [10], and numerical implementation issues by Benvenuti and Tralli [11] andRolshoven and Jira´sek [12]. For an overview and discussion of various non-local plasticityformats, see Reference [13].With plasticity or damage mechanics alone, the complex failure process of cohesive-frictionalmaterials, which is characterized by stiffness degradation and irreversible deformations, cannotbe modelled satisfactorily. Plasticity is based on an elastic unloading stiffness, which is incontradiction to the stiffness degradation observed in experiments. Damage mechanics, on theother hand, is not suitable for the description of the irreversible deformations. A combination ofplasticity and damage mechanics can describe most of the important features of the failure ofcohesive-frictional materials. Combinations of stress-based plasticity and scalar damage weredeveloped, for example, by Ju [14], Lubliner et al. [15], Lee and Fenves [16], Gatuingt andPijaudier-Cabot [17], Bourgeois et al. [18], Kra¨tzig and Polling [19], Jason et al. [20, 21], Salariet al. [22], and Grassl and Jira´sek [23, 24]. Regularization of such models by gradient damageformulations was proposed by de Borst et al. [25] and Addessi et al. [26].A special group of integral-type non-local damage-plastic models, for which the damage partis driven by a non-local measure of the plastic strains, was studied analytically by Jira´sek andGrassl [27]. The present paper is a continuation of that work. The conclusions of the analyticalstudy of the non-local damage-plastic model are confirmed by one- and two-dimensionalexamples using simple versions of such model, in two dimensions with non-associated flow. Inparticular, the conditions for a mesh-independent description of the zone of inelastic strains arestudied. Then, the framework is applied to an advanced damage-plastic model for concrete,recently developed by the present authors [23, 24]. With this non-local damage-plastic model,structural examples of tensile and compressive failure are computed.2. BASIC EQUATIONS OF COMBINED PLASTICITY AND DAMAGEIn the present section, the framework for the combination of stress-based plasticity and strainbasedscalar damage is first presented in the local setting. The general stress–strain relation forthis class of models isr ¼ð1 oÞD e : ðe e p Þ¼ð1 oÞ%r ð1Þwhere r is the stress, o is the damage variable, D e is the isotropic elastic stiffness, e is the strain,e p is the plastic strain, and %r ¼ D e : ðe e p Þ is the effective stress. For o ¼ 0; Equation (1)reduces to the stress–strain relation of an elastic–plastic model, and for e p ¼ 0 to an elasticdamagemodel. Of course, the stress–strain equation (1) must be supplemented by certain lawsspecifying the evolution of plastic strain and damage. In paper [24], the present authors compareCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 73several options and propose an approach using a yield condition postulated in the effectivestress space and a damage law driven by the plastic strain.The plastic part of the model is based on the standard flow theory of plasticity, with all thebasic equations written in terms of the effective stress:’e p ¼ l ’ @g pð%r; k p Þð2Þ@%r’k p ¼ ’lk p ð%r; k p Þ ð3Þs y ¼ hðk p Þð4Þf p ð%r; s y Þ40; l50; ’ lfp ’ ð%r; s y Þ¼0 ð5ÞHere, a superimposed dot denotes the time derivative, l is the plastic multiplier, k p is theplastic hardening variable, g p is the plastic potential, k p is a scaling factor that relates the rateof the plastic multiplier to the rate of the hardening variable (this factor is often constantbut, in general, can depend on the current values of stress and hardening variable), s y isthe current yield stress, h is the hardening function, and f p is the yield function. No softeningis considered in the plastic model itself, i.e. the hardening function h is non-decreasing.Softening effects are attributed to cracking and taken into account by the damage part of themodel.The damage evolution is driven by a certain internal variable k d ; which typically remains zeroduring the initial stage of plastic hardening and starts growing when the yielding process attainsa critical level. The corresponding value of k p is denoted as k 0 : The subsequent evolution of k d isclosely linked to the plastic strain rate. For simple versions of the model, we set ’k d ¼ ’k p ; so thatk d can be identified with the difference k p k 0 : For the model that aims at realistic descriptionof concrete (to be presented in Section 6), ’k d is related to the plastic strain rate through a morecomplicated expression; see Equation (19). The damage variable o is a function of the internalvariable k d ; i.e.o ¼ g d ðk d Þð6Þwhere the damage function g d monotonically grows from zero to one.The algorithm for the stress update of this type of models is divided into two stages. First,the update of the effective stress for the plastic part is carried out by an implicit algorithm; then,the damage part is evaluated explicitly from the plastic strain increment obtained in thefirst stage.3. REGULARIZATION BY NON-LOCAL DAMAGEThe pathological mesh sensitivity of results obtained with local constitutive laws thatincorporate strain softening can be overcome by a suitable regularization technique, whichenforces mesh-independent dissipation associated with the failure process. Integral-type nonlocalmodels achieve this by weighted spatial averaging of a suitable state variable. Damagemodels of this kind were first proposed by Pijaudier-Cabot and Bazˇ ant [3, 28], but many otherformulations appeared in the literature; see Reference [29] for a systematic comparison. OneCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

74P. GRASSL AND M. JIRÁSEKefficient non-local damage formulation replaces k d in (6) by its non-local counterpartZ%k d ðxÞ ¼ aðx; sÞk d ðsÞ dsVHere, V is the spatial domain occupied by the body of interest, and aðx; sÞ is a non-local weightfunction that describes the strength of interaction between points x and s and decays withincreasing distance between these points. The weight function is usually non-negative andnormalized such that R Vaðx; sÞ ds ¼ 1 for all x 2 V: This can be achieved by settingaðx; sÞ ¼ a 0ðkx skÞRV a ð8Þ0ðkx tkÞ dtwhere a 0 ðrÞ is a function of the distance r ¼kx sk between points x and s: In the present studywe use the truncated polynomial function8r 2 2>< 1a 0 ðrÞ ¼ R 2 if r5Rð9Þ>:0 if r5RParameter R; called the radius of non-local interaction, sets the length scale of the model anddirectly affects the size of the localized inelastic zone.A simple non-local version of plasticity would be obtained by replacing k p in (4) by %k p ; i.e. bythe non-local plastic hardening variable computed from k p using an averaging operator similarto (7). For this type of non-local plasticity models, the plastic strain localizes into a set of zeromeasure, but the energy dissipation does not vanish, as shown by Planas et al. [30] in the onedimensionalsetting. Full regularization with a non-vanishing width of the plastic zone isobtained if the softening process is driven by a linear combination of the local and non-localplastic hardening variables, as proposed by Vermeer and Brinkgreve [5]. This type of non-localformulation is computationally quite expensive, because the plastic stress-return algorithmcannot be performed for each Gauss integration point of the finite element model separately andglobal iteration during stress evaluation is needed [8,11,12].In the present study, the non-local damage-plastic model is composed of a local plastic partand a non-local damage part. In analogy to the Vermeer–Brinkgreve [5] idea, we replace thelocal variable k d in (6) by#k d ¼ m%k d þð1 mÞk d ð10Þwhich is a linear combination of the local value k d and non-local value %k d : The role of theadditional model parameter m will be discussed later. For m ¼ 1; the combined variable #k dcoincides with the non-local variable %k d defined in (7), whereas for m ¼ 0 it reduces to the localvariable k d : In certain applications, values m > 1 are used, and this is why the formulation issometimes called ‘over-non-local’ [31].The present type of plastic model with non-local damage is computationally more efficientthan standard integral-type non-local plasticity models, since the plastic part, which is implicitand requires iteration, remains local, and non-local averaging affects only the explicitdamage part.The localization behaviour of plastic models with non-local damage has been studiedanalytically by the present authors. In the following, the main results from Reference [27]are summarized. The localization behaviour is governed by the value of the plasticð7ÞCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 75hardening modulus:H p ¼@f p dh@s y dk pð11ÞMore specifically, the actual value of H is compared to two critical values, H crit and H crit;p ;which are obtained by classical localization analysis of the corresponding local model based onthe acoustic tensor [32,33]. The value H crit is evaluated for the complete damage-plastic model,while H crit;p is evaluated for the plastic part only (with artificially ‘frozen’ damage). Thedifference between H crit and H crit;p ; denoted as DH crit;d ; is usually larger than zero. Anotherimportant characteristic value H dis is defined asH dis ¼ H crit;p þð1 mÞDH crit;d ð12ÞSuppose that the body deforms uniformly up to the current state, and the boundaryconditions admit a uniform solution of the rate problem. The question is whether this solution isunique, or whether there exists another solution with plastic yielding localized into a band.According to Jira´sek and Grassl [27], the following cases can be distinguished:* H crit 5H p }no localization possible;* H dis 5H p 4H crit }localization with continuous strain rate;* H p 5H dis }localization with discontinuous strain rate.The special case H p ¼ H dis requires a more refined discussion.In the finite element method, a discontinuous strain rate usually corresponds to a fullylocalized zone of plastic strains, which is related to the size of the finite element mesh. Incontrast to that, localization with a continuous strain rate leads to a mesh-independent zone ofinelastic strains. In the following sections, specific versions of the non-local damage-plasticmodel for cohesive-frictional materials are applied to structural problems with different failuremodes. In particular, the influence of parameters m and H crit;p is studied.4. ONE-DIMENSIONAL PLASTICITY COMBINED WITH NON-LOCAL DAMAGEThe first structural example is a bar subjected to uniaxial tension (Figure 1). In this simple case,we set ’k p ¼j’e p j: The plastic part of the model exhibits hardening until the cumulative plasticstrain k p attains a certain level k 0 ; after which the behaviour becomes perfectly plastic(i.e. H p ¼ 0). The hardening law (4) has the non-linear form(s y ¼ hðk p Þ¼ s i þðs 0 s i Þðk 2 p 3k p k 0 þ 3k 2 0 Þk p=k 3 0if k p 4k 0ð13Þs 0 if k p > k 0where s i is initial yield stress, s 0 is the ultimate (maximum) yield stress, k p is the cumulativeplastic strain, and k 0 is the plastic strain at peak. The damage evolution starts when themaximum yield stress is mobilized (which happens at k p ¼ k 0 ) and is driven by the variableCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

76P. GRASSL AND M. JIRÁSEKweakened sectionxuL/2L/2FFigure 1. Geometry and loading set-up of the one-dimensional bar under uniaxial tension.k d ¼ k p k 0 according to the exponential damage law(o ¼ g d ðk d Þ¼ 0 if k d40ð14Þ1 expð k d =e f Þ if k d 50Parameter e f controls the slope of the softening curve.During damage evolution, the plastic hardening modulus H p is equal to zero, which in thepresent one-dimensional case corresponds to the critical value H crit;p ¼ 0: The model responsesfor m ¼ 1 and 2 are presented in Figure 2 in terms of the load–displacement curves and evolvingprofiles of plastic strain and damage. For m ¼ 1; the plastic strain is fully localized, and damageis distributed in an interval of width 2R around the plastic zone (Figure 2, left). For m ¼ 2; theplastic strain is distributed continuously, and damage is uniform in the plastic zone anddecreases continuously to zero in intervals of width R on both sides of the plastic zone (Figure 2,right). In both cases, the load–displacement diagrams converge upon mesh refinement and thearea under them (dissipated energy) remains non-zero. For comparison, the dashed curves in theload–displacement diagrams show the unstable uniform solution.In the one-dimensional setting, H crit;p is equal to zero, so that the use of m ¼ 1 for the case ofperfect plasticity (H p ¼ 0) always leads to fully localized plastic strains. However, in a multidimensionalcase with non-associated flow, the critical hardening modulus depends on the stressstate and can be larger or smaller than zero. This will be demonstrated in the next section.5. NON-ASSOCIATED DRUCKER-PRAGER PLASTICITY COMBINED WITHNON-LOCAL SCALAR DAMAGEThe second structural example deals with two-dimensional analysis of a rectangular panelsubjected to compression under plane-strain conditions. The geometry, the loading set-up andthree finite element discretizations are shown in Figure 3. The plastic part of the model uses theDrucker-Prager yield condition with a non-associated flow rule. The yield function is defined aspf p ð%r; s y Þ¼c f I 1 ð%rÞþffiffiffiffiffiffiffiffiffiffiJ 2 ð%rÞ s yð15Þwhere I 1 ¼ d : %r is the first invariant (trace) of the effective stress tensor, d is the unit secondordertensor (Kronecker delta), J 2 ¼ 1 2%s : %s is the second invariant of the deviatoric effectivestress tensor %s ¼ %r dI 1 =3; and c f is the friction coefficient. The non-associated flow rule isderived from the plastic potentialpg p ð%rÞ ¼c c I 1 ð%rÞþffiffiffiffiffiffiffiffiffiffiJ 2 ð%rÞð16ÞCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 77Hp = 0 and m = 1 Hp = 0 and m = 2normalized load10.80.60.40.2normalized load10.80.60.40.200 1 2 3 4normalized displacement00 1 2 3 4normalized displacementnormalized plastic strain200150100500-2 -1 0 1 2normalized coordinate x/Rnormalized plastic strain14121086420-2 -1 0 1 2normalized coordinate x/R11damage0.5damage0.50-2 -1 0 1 2normalized coordinate x/R0-2 -1 0 1 2normalized coordinate x/RFigure 2. Load–displacement diagram and evolution of plastic strain and damage in a bar under uniaxialtension for the non-local damage-plastic model with H p ¼ 0 and different values of m:Figure 3. Geometry, loading set-up and finite element discretizations of therectangular panel under plane strain.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

78P. GRASSL AND M. JIRÁSEKwhere c c is the dilation coefficient. The plastic hardening variable is defined by the rate equationrffiffirffiffiffi2 2’k p ¼ jj’e p jj ¼ ’l @g p3 3 @%r ¼ l ’ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 þ 2c 2 cð17Þwhich means that the scaling factor from (3) is for this particular model constant,qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik p ¼ 1=3 þ 2c 2 c: Damage is assumed to start growing right at the onset of yielding, and sothe variable k d is identical with k p : The damage law has again the exponential form (14).In the first example, the plastic part of the model is designed to be perfectly plastic with aconstant yield stress in shear s y ¼ s 0 : The parameters are Young’s modulus E ¼ 30 GPa;Poisson’s ratio n ¼ 0:2; s 0 ¼ 20 MPa; c f ¼ 0:1; c c ¼ 0:05; e f ¼ 0:0057 and R ¼ 100 mm:The load–displacement curves for the three meshes from Figure 3 are almost the same; seeFigure 4. The evolution of the increment of the cumulative plastic strain k p is shown in Figure 5.The plastic zone initially has a certain width and later is reduced to a single element layer. Thisbehaviour is explained by the evolution of the critical plastic hardening modulus H crit;p : Theexpression for H crit;p has been derived analytically by Runesson et al. [33] and can be written forthe present case according to Rolshoven [34] as2! 3 2EH crit;p ¼ 44k p ð1 nÞ 2ðc Sf c c Þ 2 3ð1 nÞ pffiffiffiffiþ c f þ c c5 ð18ÞJ 2where S 3 is the out-of-plane principal effective deviatoric stress. In Figure 6, the evolution ofH crit;p is compared to the hardening modulus for the loading path that corresponds to theconditions of Figure 3. At early stages of the loading process, H crit;p is below H p ¼ 0; and so theplastic zone remains distributed. During plastic flow the out-of-plane effective stress componentincreases, which results in an increase of H crit;p : Eventually, H crit;p becomes positive, which leadsto a fully localized plastic zone. The evaluation of the critical plastic hardening modulus in (18)is based on the analysis of a bifurcation from a uniform state. Nevertheless, this idealized2.52mediumfinevery finenormalized load1.510.500 0.001 0.002 0.003normalized top displacementFigure 4. Load–displacement diagram for a panel loaded in compression under plane strain obtained withthe non-local damage-plastic model with H p ¼ 0; m ¼ 1 and R ¼ 0:1 m on three different meshes.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 79Figure 5. Increments of the plastic hardening variable at four stages of analysis (marked in Figure 4 byhollow circles) on the fine mesh; non-local damage-plastic model with H p ¼ 0; m ¼ 1 and R ¼ 0:1 m:0.01hardening moduli0-0.01-0.02-0.03Hcrit,pHp = 0Hp = 0.01-0.040 0.01 0.02 0.03plastic hardening variableFigure 6. Evolution of the critical hardening modulus of the plasticity model during yielding.analysis gives also some indication of the trend to be expected during the actual evolution of theplastic zone under non-uniform strain.There are two ways of preventing the formation of a fully localized, and therefore meshdependent, zone of plastic strains. Firstly, the parameter m in (10) can be increased, similar towhat was done in the one-dimensional examples in the preceding section. However, this helpsonly if the plastic modulus is exactly equal to the critical value, and not strictly smaller.Secondly, the plastic hardening modulus can be increased, so that it remains above the criticalvalue throughout the loading process, as shown in Figure 6 for H p ¼ 0:01:For the case of linear hardening with constant plastic modulus H p ¼ 0:01 and non-localformulation with m ¼ 1; the evolution of the increment of plastic hardening variable in asimulation on the fine mesh is depicted in Figure 7. The control parameter of the damage lawwas set to e f ¼ 0:0051; in order to obtain the same slope of the softening curve as in theCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

80P. GRASSL AND M. JIRÁSEKFigure 7. Increments of the plastic hardening variable at four stages of analysis on the fine mesh for thenon-local damage-plastic model with H p ¼ 0:01; m ¼ 1 and R ¼ 0:1 m:preceding example. The plastic zone decreases but does not fully localize. Instead, it convergesto a constant width, larger than the element size. This response is in agreement with the resultsshown in Figure 6, where the difference between the critical plastic hardening modulus and theactual plastic hardening modulus decreases and then remains constant. Thus, for analysis withm ¼ 1 the final size of the localized plastic zone is controlled by the interaction radius R and bythe difference H p H crit;p :6. CONCRETE PLASTICITY COMBINED WITH NON-LOCAL SCALAR DAMAGEIn the last two examples we use a triaxial plasticity model with non-local scalar damage insimulations of tensile and compressive failure of concrete. This regularized formulation is a nonlocalextension of the damage-plastic model developed by the authors [24], which is brieflyreviewed in the following.The plastic part of the model consists of a three-invariant yield condition, non-associated flowrule and pressure-dependent hardening. The yield surface, depicted in Figures 8 and 9, is basedon the Menétrey–Willam failure criterion [35], which depends on all three stress invariants(volumetric effective stress %s V ; norm of deviatoric effective stress %r and Lode angle y). Themeridians are parabolic and the deviatoric sections change from almost triangular shapes at lowconfinement to circular shapes at high confinement. The yield surface limits the stress inhydrostatic compression during hardening, which allows modelling of compaction under highlyconfined compression. The initial and intermediate yield surfaces have two vertices on thehydrostatic axis but the ultimate yield surface has only one vertex on the tensile part of thehydrostatic axis.The flow rule is non-associated, which is important for realistic modelling of the volumetricexpansion under low-confined compression. An associated flow rule for this type of yield surfacegives an unrealistically high volumetric expansion in compression. In the case of passiveconfinement, overestimation of the lateral expansion leads to an overestimated strength (peakstress), as shown by Grassl [36].Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 81432θ = 01ρ / fc0-1-2-3θ = π-4-4 -3 -2 -1 0σ V /f cFigure 8.Evolution of the meridional section of the yield surface during hardening.θ = 0θ = 4π / 3θ = 2π / 3Figure 9.Evolution of the deviatoric section of the yield surface during hardening for a constantvolumetric effective stress of %s V ¼ f c =3:The hardening law (13) is the same as for the one-dimensional example, with an initialincrease of the yield stress and perfectly plastic behaviour (no softening) after the ultimate yieldstress s 0 has been attained. The rate of plastic hardening variable is equal to the norm of theplastic strain rate scaled by a ductility measure based on the volumetric effective stress, whichaccounts for different hardening responses under tensile and compressive stress states.The damage growth is activated when the ultimate yield stress is mobilized, i.e. when k p ¼ k 0 :The damage law has again the exponential form (14). The rate of the damage-driving variable k dis proportional to the rate of volumetric plastic strain, ’e pV ; and is scaled by a ductilitymeasure x s that accounts for different amounts of dissipation associated with tensile andcompressive failures:’k d ¼ ’e pVx s ðR s ð%rÞÞð19ÞCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

82P. GRASSL AND M. JIRÁSEKwhere(x s ðR s Þ¼ 1 þ A sR 2 s if R s 51pffiffiffiffiffið20Þ1 3A s þ 4A s R s if R s 51A s is a model parameter determined from the softening response in uniaxial compression. Thedimensionless variable R s ¼ ’e pV =’e pV is defined as the ratio between the ‘negative’ volumetricplastic strain rate’e pV ¼ X3I¼1h ’e pI i ð21Þand the volumetric plastic strain rate ’e pV ¼ d : _e p : Since this ratio depends only on the flowdirection @g p =@%r; R s can be considered as a function of the effective stress. In (21), ’e pI are theprincipal components of the rate of plastic strains and hi denotes the McAuley brackets(positive-part operator). For uniaxial tension, for instance, all three principal plastic strain ratesare non-negative, and so ’e pV ¼ 0; R s ¼ 0 and x s ¼ 1: This means that under uniaxial tensileloading we have k d ¼ k p k 0 ; same as for the simple one-dimensional model from Section 4. Onthe other hand, under compressive stress states the negative principal plastic strain rates lead toa ductility measure x s larger than one and the evolution of damage is slowed down. It should beemphasized that the flow rule for this specific model is constructed such that the volumetric partof plastic strain rate at the ultimate yield surface cannot be negative.The damage-plastic model contains 15 parameters, but only 6 of them are actually calibratedfor different concrete types, namely Young’s modulus E; Poisson’s ratio n; tensile strength f t ;compressive strength f c ; parameter e f of the damage law (14), and parameter A s in the ductilitymeasure (20) of the damage model. The remaining parameters are set to their default valuesspecified in Reference [24].The response of the local model has been compared to several material tests in Reference [24].It has been shown that the model can describe the failure of concrete under tension and uniaxial,biaxial and triaxial compression. Here, only two selected examples are reproduced: uniaxialtension with unloading tested by Gopalaratnam and Shah [37] (Figure 10) and uniaxialcompression with unloading tested by Karsan and Jirsa [38] (Figure 11).The non-local extension of the model is straightforward. The local variable k d is in thedamage law (14) replaced by the ‘over-non-local’ variable #k d defined in (10). Note that,for m > 1; #k d can be negative, and so it is important to include in (14) the definition of g d fornegative arguments.6.1. Three-point bending testThe first structural example with the non-local damage-plastic model for concrete is a threepointbending test of a single-edge notched beam reported by Kormeling and Reinhardt [39]; seeFigure 12.The experiment is modelled by triangular plane-strain finite elements. For this type ofanalysis, local stress–strain relations with strain softening are known to result in meshdependentload–displacement curves. The capability of the non-local damage-plastic model toovercome this pathological dependence is assessed by three analyses on three meshes withelement sizes of 7.5, 5 and 2:5 mm; see Figure 12 for the coarse mesh.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 834constitutive responseexperimentsstress [MPa]32100 0.2 0.4strain [mm/m]Figure 10. Model response under uniaxial tension with unloading compared to experimental resultsreported by Gopalaratnam and Shah [37].0-10stress [MPa]-20-30constitutive responseexperiments-4 -3 -2 -1 0strain [mm/m]Figure 11. Model response under uniaxial compression with unloading compared to experimental resultsreported by Karsan and Jirsa [38].The material parameters are E ¼ 20 GPa; n ¼ 0:2; f t ¼ 2:4 MPa; f c ¼ 24 MPa; e f ¼ 0:001025and A s ¼ 2: The other parameters of the local damage-plastic model are set to their defaultvalues [24]. The non-local parameters are taken as R ¼ 25 mm and m ¼ 1:The load–displacement curve obtained on the fine mesh is in a reasonable agreement with theexperiment, cf. Figure 13, and the results do not suffer by pathological sensitivity to the meshsize, cf. Figure 14. The contour plots of the damage variable and the increment of the volumetricplastic strain at three stages of the analysis (marked by hollow circles in Figure 14) on the finemesh are presented in Figure 15. The plastic strain tends to fully localize and the damage isdistributed over distance R around the plastic zone.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

84P. GRASSL AND M. JIRÁSEKFigure 12. Geometry, loading set-up and finite element mesh for the three-point bending test.1.61.4fine meshexp bounds1.2load [kN]10.80.60.40.200 0.2 0.4 0.6 0.8 1displacement [mm]Figure 13. Comparison of the analysis of the three-point bending test on the fine meshwith the experimental bounds.A continuous plastic strain profile can be obtained by increasing either H p or m: Nevertheless,even the non-local model with localizing plastic strains is still advantageous over local modelsbased on the crack band concept [40], which are widely used in commercial finite element codes.For models based on the crack band concept, the global load–displacement curves are mademesh-independent by adjustment of the local softening curve with respect to the size of the zonein which plastic strain localizes. Under tensile loading, the size of this zone is dictated by theelement size. Under compressive loading, however, the size of the localized zone may beindependent of the element size. If the structural failure involves both local tensile andcompressive mechanisms, single-surface models like the present one based on the crack bandapproach would fail to describe the energy dissipation independently of the mesh. Such anexample is analysed by the non-local model in the next section.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 851.61.41.2coarse meshmedium meshfine meshload [kN]10.80.60.40.200 0.2 0.4 0.6 0.8 1displacement [mm]Figure 14. Comparison of the analyses of the three-point bending test on three different meshes.Figure 15. Contour plots of: (a) damage; and (b) increments of the volumetric plastic strain at three stagesof the analysis on the fine mesh.6.2. Eccentric compression testThe last example treats the failure of a concrete prism subjected to eccentric compression, testedby Debernardi and Taliano [41]. The geometry and loading set-up are shown in Figure 16(a).The specimen with a relatively large eccentricity of 36:8 mm is modelled by a thin layer of linearthree-dimensional elements, in order to circumvent the need for special stress-return algorithmsfor plane-stress conditions and at the same time keep the computational time low compared to aCopyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

86P. GRASSL AND M. JIRÁSEKFigure 16. (a) Geometry and loading set-up of the eccentric compression test;and (b) the coarse finite element mesh.full three-dimensional analysis. Again, three different meshes with element sizes of 7.5, 5 and2:5 mm were used (see Figure 16(b) for the coarse mesh).The model parameters were set to E ¼ 30 GPa; n ¼ 0:2; f t ¼ 4 MPa; f c ¼ 46 MPa; e f ¼0:001042; A s ¼ 2; R ¼ 15 mm and m ¼ 1: The other parameters of the local model were takenby their default values [24].The model response in terms of the overall load versus the mean compressive strain on thecompressed side (Figure 16(a)) obtained on the fine mesh is compared to the experimental resultin Figure 17. The load capacity and the strain at peak are underestimated by the model. Theoverall behaviour, however, is captured well.The comparison of the relations between the load and the average compressive strain foranalyses run on meshes of different sizes (Figure 18(a)) indicates that the description of this typeof compressive failure is nearly mesh independent. The evolution of the damage zone for theanalysis on the coarse mesh is depicted in Figure 19 for three stages of analyses (marked byhollow circles in Figure 18(a)). On the tensile side, several zones of localized damage form,whereas the failure pattern on the compressive side is characterized by a diffuse damage zone.Despite the complicated failure mode, the load–displacement curves are mesh independent,which confirms that the non-local model can describe both types of failure in an objective way.It is important to realize that, for such combined failure modes, the local model exhibits strongmesh sensitivity even if the softening law is adjusted depending on the element size. This isclearly documented by the load–average-strain diagrams in Figure 18(b). As the mesh is refined,Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

PLASTIC MODEL WITH NON-LOCAL DAMAGE 87400350fine meshexperiment300load [kN]250200150100500-8 -7 -6 -5 -4 -3 -2 -1 0average strain [mm/m]Figure 17. Comparison of the analysis of the eccentric compression test with the experiment.350300coarse meshmedium meshfine mesh350300coarse meshmedium meshfine mesh250250load [kN]200150load [kN]2001501001005050(a)0-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0average strain [mm/m](b)0-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0average strain [mm/m]Figure 18.Load–average-strain curves for the eccentric compression test simulated on three differentmeshes with: (a) non-local model; and (b) local model.the global response of the local model with adjustment of the softening modulus is getting moreductile. The reason is that the adjustment is based on the assumption that the failure pattern islocalized into one layer of elements, but this is the case only in the region of tensile cracking butnot in the region of compressive crushing.7. CONCLUSIONSIn the present study we have proposed a framework for non-local damage-plastic models,combining local plasticity with integral-type non-local damage. The damage evolution is drivenby the plastic strain.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

88P. GRASSL AND M. JIRÁSEKFigure 19.Contour plots of the damage variable at three stages of the analysis oncoarse mesh with non-local model.The examples of tensile, shear and compressive failure show that the dissipation associatedwith the failure process for this type of regularized damage-plastic models is mesh independent.Furthermore, we have analysed two approaches that enforce a continuous strain profile ofinelastic strain:* The plastic hardening modulus is larger than the critical plastic hardening modulus, andthe local damage-driving variable is replaced by its non-local counterpart.* The local damage-driving variable is replaced by a linear combination of its local and nonlocalvalue with a suitable value of parameter m (m > 1), as proposed in Reference [5].Finally, it has been shown that the non-local damage-plastic model for concrete can provide amesh-independent description of various combinations of tensile and compressive failure, evenin cases when the traditional approach based on a local formulation with fracture-energymotivated adjustment of the softening modulus fails.To keep the number of parameters limited, only one scalar damage variable was consideredhere. Clearly, this choice restricts the range of applicability of the model. Generalizations toisotropic formulations with several damage variables and to anisotropic formulations withdamage tensors should be explored in the future.ACKNOWLEDGEMENTSFinancial support of the Swiss Commission for Technology and Innovation under project CTI 5501.1 andof the Ministry of Education of the Czech Republic under project CEZ MSM 6840770003 is gratefullyacknowledged.REFERENCES1. Rogula D. Influence of spatial acoustic dispersion on dynamical properties of dislocations. I. Bulletin de l’Acade´miePolonaise des Sciences. Se´rie des sciences techniques 1965; 13:337–343.2. Eringen AC. On nonlocal plasticity. International Journal of Engineering Science 1981; 19:1461–1474.Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:71–90

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