Geometrical localization analysis of gradient-dependent parabolic Drucker -Prager elastoplasticity

Abstract

In this work the geometrical method for the analysis of the localization properties of the thermodynamically consistent gradient--dependent parabolic Drucker-Prager elastoplastic model is presented. From the analytical solution of the discontinuous bifurcation condition of small strain gradient-dependent elastoplasticity the elliptical envelope for localization is formulated in the coordinates of Mohr. The tangency condition of the localization ellipse with the major principal circle of Mohr defines the type of failure (diffuse or localized) and the critical directions for discontinuous bifurcation. The results of the geometrical localization analysis illustrate the capability of the gradient--dependent elastoplastic Drucker--Prager material to suppress the discontinuous bifurcations of the related local or classical elastoplastic model formulation that take place when the adopted hardening/softening modulus H equals the critical (maximum) one for localization Hc. On the other hand, the results in this work also demonstrate that the thermodynamically consistent gradient--dependent Drucker--Prager model may lead to discontinuous bifurcation not only when the characteristic length l turns zero but also when H < Hc.