Plastic model with non-local damage applied to concrete

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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
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Plastic model with non-local damage applied to concrete

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