Softening Gradient Plasticity: Analytical Study of Localization under Nonuniform Stress

Abstract

Localization of plastic strain induced by softening can be objectively described by a regularized plasticity model that postulates a dependence of the current yield stress on a nonlocal softening variable defined by a differential (gradient) expression. This paper presents analytical solutions of the one-dimensional localization problem under certain special nonuniform stress distributions. The one-dimensional problem can be interpreted as describing either a tensile bar with variable cross section, or a beam subjected to a nonuniform bending moment. Explicit as well as implicit gradient formulations are considered. The evolution of the plastic strain profile and the shape of the load-displacement diagram are investigated. It is shown that even if the local constitutive law exhibits softening right from the onset of yielding, the global loaddisplacement diagram has a hardening part. The interplay between the internal length scales characterizing the material and the geometry is discussed.

Unified finite element discretizations of coupled Darcy–Stokes flow

“…At the onset of yielding, the load parameter first steeply increases and only later decreases. Complete failure is attained at larger elongation in comparison with the non-variational formulation analyzed in Jirásek et al (2010). Several values λ g = {1.019, 2.037, 4.075, 8.150}m 1 are reported (from top to bottom), to reflect the effect of spatial variation of the sectional area; lower values of λ g correspond to a stronger variation of the sectional area and lead to higher peak loads.…”

“…In particular, the often questioned regularity conditions for internal variables at the elasto-plastic interface will emerge naturally in a consistent and unified way. Detailed solutions for four different test problems with various regularity of the yield stress and stress distributions will be presented and compared to results obtained from the standard non-variational gradient formulation available in Jirásek et al (2010). The present work can also be viewed as a continuation and extension of results provided in our previous paper (Jirásek et al, 2013), where the second-order gradient plasticity model was investigated.…”

“…(12) reduces to −HAl 4 κ IV , and the standard formulation of the fourth-order gradient plasticity model is recovered, cf. Jirásek et al (2010) …”

“…1. Conditions for the standard solution can be found in Jirásek et al (2010) and Jirásek and Rolshoven (2009b).…”

“…There, it will be shown that the structural response may, in a certain range, exhibit hardening due to the gradient enrichment despite the softening character of the material model. In Sections 5 and 6, the results for piecewise linear and quadratic stress field distributions will be compared to standard non-variational solutions available in Jirásek et al (2010). In spite of all the simplifications we will be forced to use numerical solutions.…”