GEOMETRICAL METHOD FOR LOCALIZATION ANALYSIS IN GRADIENT–DEPENDENT J2 PLASTICITY

Summary. The geometrical method for assessment of discontinuous bifurcation condi-tions is extended to encompass J−2 gradient-dependent plasticity. The gradient-dependent localization condition is cast in form of an elliptical envelope condition in the coordinates of Mohr, see Pijaudier-Cabot and Benallal (1993)3, Liebe and Willam (2001)2. The re-sults of the localization analysis geometrically and numerically demonstrate the capability of the thermodynamically consistent gradient-dependent model formulations to suppress localized failure modes of the related local plasticity formulations 1 J2 GRADIENT–DEPENDENT ELASTOPLASTICITY Under consideration of small strain kinematics, the free energy density of the strain gradient elastoplastic J2 continuum is expressed in the additive form ρΨ(εe, κ,∇κ) = ρΨe(εe) + ρΨp,loc(κ) + ρΨp,g(∇κ) (1) where ρ is the material density. The elastic free energy density is ρΨe(εe) = 1 2 εe: Ee: εe, with εe and Ee the elastic strain tensor and the fourth order elastic operator, respectively. The local and gradient free energy density contributions due to inelastic strains Ψp,loc and Ψp,g are expressed in terms of the scalar hardening/softening variable κ. ρΨp,loc = 1