Abstract
Damage-induced strain softening is of vital importance for the modeling of localized failure in frictional-cohesive materials. This paper addresses strain localization of damaging solids and the resulting consistent frictional-cohesive crack models. As a supplement to the framework recently established for stress-based continuum material models in rate form (Wu and Cervera 2015, 2016), several classical strain-based damage models, expressed usually in total and secant format, are considered. Upon strain localization of such damaging solids, Maxwell’s kinematics of a strong (or regularized) discontinuity has to be reproduced by the inelastic damage strains, which are defined by a bounded characteristic tensor and an unbounded scalar related to the damage variable. This kinematic constraint yields a set of nonlinear equations from which the discontinuity orientation and damage-type localized cohesive relations can be derived. It is found that for the “Simó and Ju 1987” isotropic damage model, the localization angles and the resulting cohesive model heavily depend on lateral deformations usually ignored in classical crack models for quasi-brittle solids. To remedy this inconsistency, a modified damage model is proposed. Its strain localization analysis naturally results in a consistent frictional-cohesive crack model of damage type, which can be regularized as a classical smeared crack model. The analytical results are numerically verified by the recently-proposed mixed stabilized finite element method, regarding a singly-perforated plate under uniaxial tension. Remarkably, for all of the damage models discussed in this work, the numerically-obtained localization angles agree almost exactly with the closed-form results. This agreement, on the one hand, consolidates the strain localization analysis based on Maxwell’s kinematics and, on the other hand, illustrates versatility of the mixed stabilized finite element method.
Highlights
It is well known that under certain circumstances, frictional-cohesive materials with a softening regime exhibit strain localization prior to the occurrence of macroscopic failure
Compared to the frictional-cohesive relations (56) resulting from the Simó and Ju [41] damage model, the longitudinal modulus M0 in Equation (57) is replaced here by Young’s modulus E0. This modification is consistent with the concept of a frictional-cohesive crack model in which only the strain components acting on the discontinuity surface are accounted for [47]
The above mixed stabilized strain-displacement element is applied to analyses of strain localization
Summary
It is well known that under certain circumstances, frictional-cohesive materials with a softening regime exhibit strain localization prior to the occurrence of macroscopic failure. Strain localization and the induced localized failure in frictional-cohesive materials can be characterized by either generalized continuum models or nonlinear crack/fracture models In the former approach, the effects of strain/displacement discontinuities are smoothed or smeared. It is not so straightforward to extend the aforesaid Maxwell’s kinematics-based strain localization to strain-based continuum models, since the resulting inelastic (damage) strain depends heavily on the lateral deformations induced by Poisson’s ratio This fact is inconsistent with the frictional-cohesive zone models, which generally neglect the strain and stress triaxiality. The dyadic product ‘’ and the symmetrized Kronecker product are defined as: A⊗B ijkl
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