Homogenization of elasto-plastic plate equations with vanishing hardening

Technical Summary: Homogenization of Elasto-Plastic Plate Equations with Vanishing Hardening

Problem Statement
This work investigates the asymptotic behavior of thin heterogeneous elastoplastic plates within the framework of linearized elastoplasticity. The study focuses on a specific limiting regime where the plate thickness ($h$) vanishes significantly faster than the characteristic scale of the material's periodic microstructure ($\varepsilon_h$), corresponding to the ratio $\lim_{h \to 0} h/\varepsilon_h = 0$. The primary objective is to derive an effective, homogenized plate model for elasto-perfectly plastic materials (where hardening tends to zero) without imposing restrictive geometric assumptions on the admissible yield surfaces or specific ordering between material phases.

A critical mathematical challenge addressed is the definition of the dissipation potential at the interfaces between different material phases. In the absence of hardening, the plastic strain becomes a Radon measure that can concentrate at interfaces. Previous works (e.g., [4]) required restrictive geometric constraints on yield surfaces to ensure the lower semicontinuity of the dissipation potential in extreme regimes. This paper aims to remove such constraints and provide a rigorous characterization of the dissipation functional at phase interfaces for general heterogeneous materials.

Methodology
The analysis proceeds via a two-step evolutionary $\Gamma$-convergence approach:

1. Dimension Reduction with Hardening: The authors first derive a heterogeneous plate model that includes both isotropic and kinematic hardening. Starting from a three-dimensional elasto-plastic model on a thin domain $\Omega_h$, they perform a dimension reduction to a two-dimensional plate model. This step utilizes the energetic formulation of rate-independent systems, establishing the existence and uniqueness of quasistatic evolutions. The limit model is characterized by Kirchhoff–Love displacements and reduced elasticity tensors, extending existing homogeneous plate models to the heterogeneous setting with general hardening and dissipation potentials.

2. Simultaneous Homogenization and Vanishing Hardening: In the second step, the authors simultaneously let the hardening tensors tend to zero and the microstructure periodicity $\varepsilon$ tend to zero. This involves:

   • Two-Scale Convergence: Utilizing two-scale convergence and unfolding techniques for symmetrized gradients of functions with bounded deformation (BD) and Hessians of functions with bounded Hessian (BH).
   • Compactness and Lower Semicontinuity: Proving compactness results for sequences of displacements and strains. A key technical innovation is the derivation of lower semicontinuity results for the dissipation functional that do not rely on specific phase ordering.
   • Interface Characterization: Defining the dissipation potential at the interfaces between phases. The authors introduce a non-local inf-convolution of the traces of plastic strains from both sides of the interface. This structure reflects the Kirchhoff–Love kinematics, where the plastic strain measure decomposes into zeroth and first-order moments.
   • Stress-Strain Duality: Instead of constructing recovery sequences for global stability (a method used in previous works like [3, 4]), the authors employ a stress-strain approach. They define a set of admissible two-scale limit stresses and prove a lower bound for the plastic dissipation functional using duality arguments on the periodic cell $I \times Y$. This allows them to establish global stability for the limit model directly.

Key Contributions

• Generalized Heterogeneous Plate Model: The paper generalizes the plate model obtained in [27] to the heterogeneous case, accommodating both isotropic and kinematic hardening without assuming specific forms for the dissipation potential or hardening tensors.
• Removal of Geometric Constraints: Unlike previous analyses of the extreme regimes ($\gamma=0$ and $\gamma=+\infty$) in [4], this work derives the limit model without imposing restrictive geometric constraints on admissible yield surfaces.
• Characterization of Interface Dissipation: The paper provides a rigorous definition of the dissipation functional at phase interfaces for elasto-perfectly plastic materials. The resulting functional is a non-local inf-convolution of the traces of plastic strains, specifically accounting for the rank-one structure inherited from the Kirchhoff–Love displacement assumption.
• Stress-Strain Approach for Stability: The authors demonstrate that the limit model's global stability can be proven via a stress-strain duality approach (introduced in [19] and extended in [3, 4]), bypassing the need for explicit recovery sequence constructions for stability, which simplifies the analysis in the presence of complex interface conditions.

Main Results
The primary result is Theorem 4.40, which establishes the convergence of the quasistatic evolutions of the heterogeneous plates with hardening to a limiting two-scale quasistatic evolution in the context of perfect plasticity.

• Convergence: The sequence of displacements, elastic strains, and plastic strains converges (weakly* or two-scale weakly*) to a limit configuration $(u, E, P)$.
• Limit Model: The limit model is a two-scale elasto-perfectly plastic plate model. The plastic strain $P$ is a measure on $\Omega \times Y$ that satisfies a specific structure involving corrector spaces $X_0(\omega)$ and $\Upsilon_0(\omega)$.
• Energy Balance: The limit evolution satisfies an energy balance equation involving the homogenized elastic energy and the newly defined dissipation functional $R^{\text{hom}}_0$.
• Regularity: Corollary 4.41 establishes that the limit solution is absolutely continuous in time and that the convergence of the elastic strains is strong.

Significance and Claims
The authors claim that their approach allows for the derivation of the limit model and the appropriate definition of the dissipation potential at the interface of different phases, a feature not present in earlier works like [4] which relied on placing the "softer" dissipation potential at the interface. The resulting dissipation functional takes the form of a non-local inf-convolution, reflecting the specific kinematic constraints of the plate.

The paper notes that while the results are derived by first reducing dimension and then homogenizing, it is expected that the same model could be obtained by simultaneously letting the hardening, plate thickness, and material oscillations tend to zero from the three-dimensional equations, at least in regimes where hardening is not too small compared to the thickness. The work also highlights that the derived homogenized plate model with non-vanishing hardening is a byproduct of the analysis, representing a result not previously present in the literature.

The methodology relies heavily on tools from convex analysis of measures, disintegration of Radon measures, and evolutionary $\Gamma$-convergence, building upon the foundational works cited in the introduction (e.g., [3, 4, 9, 12, 19, 29]).