Homogenization of elasto-plastic plate equations with vanishing hardening

TL;DR

This paper derives a homogenized elasto-plastic plate model for thin heterogeneous materials, accommodating general yield surfaces and analyzing the limit as hardening vanishes, revealing interface dissipation effects.

Contribution

It extends plate models to heterogeneous materials with general hardening relations and characterizes interface dissipation in the zero-hardening limit.

Findings

Derived a heterogeneous plate model with isotropic and kinematic hardening.

Performed two-scale homogenization leading to an elasto-perfectly plastic model.

Characterized the interface dissipation potential as a non-local inf-convolution.

Abstract

We study the asymptotic behavior of thin heterogeneous elastoplastic plates in the framework of linearized elastoplasticity, focusing on the regime where the plate thickness vanishes much faster than the characteristic scale of the material's periodic microstructure. In contrast to earlier analyzes that required restrictive geometric assumptions on admissible yield surfaces, our approach accommodates general relations between phases without imposing any specific ordering. The analysis proceeds in two main steps. First, we rigorously derive a heterogeneous plate model with both isotropic and kinematic hardening through a dimension reduction procedure based on evolutionary $\Gamma$-convergence. This result extends existing plate models for homogeneous materials to the heterogeneous setting and allows for general forms of hardening and dissipation potentials. In the second step, we perform two-scale homogenization while simultaneously letting the hardening tend to zero. This process yields an effective elasto-perfectly plastic plate model and, crucially, provides a characterization of the dissipation potential at the interfaces between different phases. The resulting dissipation functional takes the form of a non-local inf-convolution of the traces of plastic strains on both sides of the interface, reflecting the Kirchhoff-Love structure of admissible displacements.