Homogenization of elasto-plastic plate equations with vanishing hardening
Marin Bužančić, Igor Velčić, Josip Žubrinić
TL;DR
This work develops a rigorous two-stage asymptotic analysis for thin heterogeneous elastoplastic plates in linearized theory, focusing on the regime where plate thickness vanishes much faster than the material microstructure. First, a dimension-reduction via evolutionary Γ-convergence yields a heterogeneous plate model with isotropic and kinematic hardening; second, simultaneous homogenization and vanishing hardening produce an elasto-perfectly plastic plate with a non-local, interfacial dissipation driven by the Kirchhoff–Love structure. The limiting model captures how plastic strains concentrate at phase interfaces and how the dissipation is non-local via an inf-convolution across interfaces, extending homogeneous plate results to multiphase composites with general dissipation and hardening laws. The approach leverages two-scale convergence, unfolding, and a stress–strain duality framework to establish global stability and energy balance at the limit, providing a robust route from 3D elasto-plasticity to effective 2D plate theories with multiphase interactions. The results clarify the role of interface dissipation in heterogeneous active layers and pave the way for accurate reduced models of composite plates in engineering applications.
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 $Γ$-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.