Plastic model with non-local damage applied to concrete

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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
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