Plastic model with non-local damage applied to concrete

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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
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