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Projection scheme for a perfect plasticity model with a time-dependent constraint set

Yoshiho Akagawa, Kazunori Matsui

TL;DR

This work develops a projection-based time discretization for a perfect plasticity model with a time-dependent yield surface, proving stability and using it to establish the existence of an exact solution and strong convergence of the scheme. The approach circumvents nonlinear solvers by projecting onto the evolving yield set $K(t)$ at each time step, while maintaining yield-consistent solutions. Key contributions include a priori estimates, existence/uniqueness, and strong convergence results under weaker assumptions than prior work, with potential extensions to general convex yield sets and fully discrete finite element implementations. The results have implications for robust and efficient numerical simulations of elastoplastic materials with time-variant yielding behavior.

Abstract

This paper introduces a new numerical scheme for a system that includes evolution equations describing a perfect plasticity model with a time-dependent yield surface. We demonstrate that the solution to the proposed scheme is stable under suitable norms. Moreover, the stability leads to the existence of an exact solution, and we also prove that the solution to the proposed scheme converges strongly to the exact solution under suitable norms.

Projection scheme for a perfect plasticity model with a time-dependent constraint set

TL;DR

This work develops a projection-based time discretization for a perfect plasticity model with a time-dependent yield surface, proving stability and using it to establish the existence of an exact solution and strong convergence of the scheme. The approach circumvents nonlinear solvers by projecting onto the evolving yield set at each time step, while maintaining yield-consistent solutions. Key contributions include a priori estimates, existence/uniqueness, and strong convergence results under weaker assumptions than prior work, with potential extensions to general convex yield sets and fully discrete finite element implementations. The results have implications for robust and efficient numerical simulations of elastoplastic materials with time-variant yielding behavior.

Abstract

This paper introduces a new numerical scheme for a system that includes evolution equations describing a perfect plasticity model with a time-dependent yield surface. We demonstrate that the solution to the proposed scheme is stable under suitable norms. Moreover, the stability leads to the existence of an exact solution, and we also prove that the solution to the proposed scheme converges strongly to the exact solution under suitable norms.
Paper Structure (16 sections, 12 theorems, 106 equations, 1 figure)

This paper contains 16 sections, 12 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem
  3. Notations and definition of a solution of Problem \ref{['Prob']}
  4. Notations
  5. Definition of a solution of Problem \ref{['Prob']}
  6. Numerical scheme
  7. Time discretization
  8. Proposed scheme
  9. Main results
  10. Proofs of main results
  11. Preliminary result
  12. Stability
  13. Existence and uniqueness of solution to Problem \ref{['Prob']}
  14. Strong convergence
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 2.2

Under the following three conditions, there exists a unique solution to Problem Prob.

Figures (1)

  • Figure 1: A schematic of the rheological model, showing Kelvin-Voigt and perfectly plastic elements arranged in parallel, used to derive the second, third, and fourth equations of \ref{['strong:original']}.

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2: AFK23
  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Theorem 4.5
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • ...and 14 more