Plastic model with non-local damage applied to concrete

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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
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Plastic model with non-local damage applied to concrete

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