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Numerical approximation of a thermodynamically complete rate-type model for the elastic--perfectly plastic response

Pablo Alexei Gazca-Orozco, Vít Průša, Karel Tůma

TL;DR

The paper addresses the numerical approximation of a thermodynamically complete rate-type model for elastic--perfectly plastic response with an accompanying temperature evolution, avoiding elastic--plastic decomposition and variational inequalities. It presents a low-order finite element discretization with a regularized, non-sharp yield condition and employs an implicit time-stepping scheme to study discrete solvability and energy stability. The authors prove existence of discrete solutions under a mesh-dependent smallness condition and establish a discrete energy balance that accounts for numerical dissipation, supplemented by computational experiments in 1D and 2D. Overall, the work demonstrates the numerical tractability of rate-type, thermodynamically consistent inelastic models and provides a foundation for extending the framework to more complex thermo-mechanical couplings.

Abstract

We analyse a numerical scheme for a system arising from a novel description of the standard elastic--perfectly plastic response. The elastic--perfectly plastic response is described via rate-type equations that do not make use of the standard elastic-plastic decomposition, and the model does not require the use of variational inequalities. Furthermore, the model naturally includes the evolution equation for temperature. We present a low order discretisation based on the finite element method. Under certain restrictions on the mesh we subsequently prove the existence of discrete solutions, and we discuss the stability properties of the numerical scheme. The analysis is supplemented with computational examples.

Numerical approximation of a thermodynamically complete rate-type model for the elastic--perfectly plastic response

TL;DR

The paper addresses the numerical approximation of a thermodynamically complete rate-type model for elastic--perfectly plastic response with an accompanying temperature evolution, avoiding elastic--plastic decomposition and variational inequalities. It presents a low-order finite element discretization with a regularized, non-sharp yield condition and employs an implicit time-stepping scheme to study discrete solvability and energy stability. The authors prove existence of discrete solutions under a mesh-dependent smallness condition and establish a discrete energy balance that accounts for numerical dissipation, supplemented by computational experiments in 1D and 2D. Overall, the work demonstrates the numerical tractability of rate-type, thermodynamically consistent inelastic models and provides a foundation for extending the framework to more complex thermo-mechanical couplings.

Abstract

We analyse a numerical scheme for a system arising from a novel description of the standard elastic--perfectly plastic response. The elastic--perfectly plastic response is described via rate-type equations that do not make use of the standard elastic-plastic decomposition, and the model does not require the use of variational inequalities. Furthermore, the model naturally includes the evolution equation for temperature. We present a low order discretisation based on the finite element method. Under certain restrictions on the mesh we subsequently prove the existence of discrete solutions, and we discuss the stability properties of the numerical scheme. The analysis is supplemented with computational examples.
Paper Structure (9 sections, 40 equations, 9 figures)

This paper contains 9 sections, 40 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Model description
  3. Discrete formulation
  4. Existence of discrete solutions
  5. Stability
  6. Numerical experiments
  7. One dimensional mechanical response
  8. 2D plate with an elliptical hole
  9. Concluding remarks

Figures (9)

  • Figure 1: Non-sharp yield condition obtained as a consequence of a regularised Heaviside function $H_\varepsilon$.
  • Figure 2: Mechanical response at $X=0.75$ m for the problem with $\kappa_\star = 10^7$ Pa.
  • Figure 3: Stress-strain response at $X=0.75$ m for the problem with $\kappa_\star = 80$ Pa.
  • Figure 4: Maximum stress over time for the problem with $\kappa_\star = 80$ Pa, and various approximations of the Heaviside function.
  • Figure 5: Convergence behaviour for the solution of the 1D mechanical problem.
  • ...and 4 more figures

Theorems & Definitions (8)

  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • remark 7
  • remark 8