Non-convex rate dependent strain gradient crystal plasticity and deformation patterning

Abstract

a b s t r a c tA rate dependent strain gradient crystal plasticity framework is presented where the displacement and the plastic slip fields are considered as primary variables. These coupled fields are determined on a global level by solving simultaneously the linear momentum balance and the slip evolution equation, which is derived in a thermodynamically consistent manner. The formulation is based on the 1D theory presented in Yalcinkaya et al. (2011), where the patterning of plastic slip is obtained in a system with non-convex energetic hardening through a phenomenological double-well plastic potential. In the current multidimensional multi-slip analysis the non-convexity enters the framework through a latent hardening potential presented in Ortiz and Repettto (1999) where the microstructure evolution is obtained explicitly via a lamination procedure. The current study aims the implicit evolution of deformation patterns due to the incorporated physically based non-convex potential.

Energy-based Modeling of Localization and Necking in Plasticity

“…On the other hand, a concave cohesive energy is required to capture a softening response accompanied by strain localization, where the fracture is obtained as the extreme localization on singular points. Analogous correlations between the shape of a cohesive functional and the evolution of fracture have been found in Del Truskinovsky (2009), Del Piero et al (2013a), Lancioni (2014), Del Piero et al (2013b while, in the rate-dependent strain-gradient plasticity model (see Yalcinkaya et al (2011), Yalçinkaya et al (2012, , , Yalcinkaya (2013)), the convex-concave shape of a plastic energy determines the formation and evolution of deformation patterns at different length scales.…”

“…The first framework, proposed in Yalcinkaya et al (2011), Yalçinkaya et al (2012 to simulate the formation of dislocation cell type microstructures, follows the principle of virtual work to get balance equations at macro (linear momentum balance) and at micro level. The dissipation inequality is exploited to get the microstress definitions, and the plastic evolution equation is obtained to satisfy the reduced dissipation inequality for thermodynamical consistency.…”

Energy-based non-local plasticity models for deformation patterning, localization and fracture

“…As, from (2.23) 1 , (u t −γ t ) is constant in (0, l), u t can be written in terms ofγ t as follows:u t =γ t +ε t − (1/l) l 0γ t dx. Substituting this expression in (2.23) 2 , we obtain an integro-differential equation forγ . To make explicit the dependence on the deformation rateε, we write τ = δε/ε in (2.23) 2 , with δε the deformation increment imposed in the step τ .…”

“…[1,2]) and strain localization [3] leading to failure of metallic materials through necking, as long as an appropriate non-convex energy is incorporated in a thermodynamically consistent manner. It is observed that during the localization of deformation and evolution of microstructures, the macroscopic stress-strain response shows a hardening-softening stress-plateau type of paper addresses the RI evolution of the microstructure without a relaxation step through the incremental minimization of the total energy.…”

Variational Gradient Plasticity: Local-Global Updates, Regularization and Laminate Microstructures in Single Crystals

“…A small strain single slip example for this approach in one dimension with non-convex self hardening is given in Yalcinkaya et al [YBG11] and very similarly in Klusemann et al [KBS12]. A two-dimensional small strain double slip approach with nonconvex latent hardening and an isotropic gradient term is outlined in Yalcinkaya et al [YBG12]. A model framework which combines strain-gradient plasticity and lamination by partially relaxing the free energy function and hence restoring existence and uniqueness of solutions has been recently proposed by Anguige & Dondl [AD14].…”