On the role of higher-order conditions in distortion gradient plasticity

Andrea Panteghini and Lorenzo Bardella

Department of Civil, Environmental, Architectural Engineering and Mathematics, University of Brescia, Via Branze, 4325123, Brescia, Italy

Abstract

We focus on Gurtin's gradient plasticity (GP) theory adopting Nye's tensor as primal higher-order (HO) kinematic variable, contributing to the free energy. In the absence of other HO kinematic variables, this framework is characterised by kinematic HO boundary conditions (BCs) which admit discontinuity in some components of the plastic distortion. This implies that the finite element (FE) implementation of this theory requires a "non-standard" approach. To address this issue, here, we develop a specific H(curl) FE. A "standard" FE implementation, in which the nodal degrees of freedom are all the displacement and plastic distortion components, is successful if the above theory is extended by including a dissipative HO stress, proportional to a material length L, conjugated to the gradient of the plastic distortion rate, such that the HO BCs must be imposed on each plastic distortion component. The foregoing switch in the kinematic HO BCs makes it difficult to obtain reliable FE solutions for the first theory by particularising the "standard" FE code for the enriched theory with the choice of a suitably small L. Beside showing this, in this work, we aim at shedding light on the distortion GP based on Nye's tensor only, in which also the use of an appropriately small rate-sensitivity parameter to obtain rate-independent solutions within a viscoplastic framework reveals interesting aspects. To this purpose, we consider three plane strain benchmarks, with focus on limit load capacity: the bending of thin foils, relevant to validate our novel implementation by comparison with literature results; the simple shear of a strip constrained between bodies impenetrable to dislocations, that we discuss on the basis of a new analytical solution for the rate-independent case; a composite problem approximating a polycrystal, whose main feature consists of internal grain boundaries in which we impose homogeneous kinematic HO conditions. This last benchmark shows that the computational model involving HO dissipation cannot predict a collapse mechanism predicted by the novel H(curl) FE model.

Author Keywords: Strain gradient plasticity; Dislocation density tensor; Size effect; Finite element method; Viscoplasticity; Rate-independent plasticity; Implicit time-integration