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Quasistatic nonassociative plasticity at finite strains

Ulisse Stefanelli, Andreas Vikelis

TL;DR

This work advances finite-strain elastoplasticity in the nonassociative setting by formulating a variational quasistatic evolution and proving the existence of measure-valued energetic solutions. The authors regularize the dissipation via a space-time mollification and introduce a gradient term in the plastic strain to gain compactness, resulting in a rigorous time-discrete to continuous existence theory built on generalized Young measures. A parallel energetic function-valued formulation is shown to be attainable, and a precise correspondence between the measure-valued and function-valued approaches is established. The methodology extends quasistatic plasticity theory to nonassociative, nonlinear finite-strain regimes and provides a solid mathematical basis for nonlocal regularizations in gradient-plasticity models, with potential implications for numerical approximations and convergence analysis.

Abstract

We investigate finite-strain elastoplastic evolution in the nonassociative setting. The constitutive material model is formulated in variational terms and coupled with the quasistatic equilibrium system. We introduce measure-valued energetic solutions and prove their existence via a time discretization approach. The existence theory hinges on a suitable regularization of the dissipation term via a space-time mollification. Eventually, we discuss the possibility of solving the problem in the setting of functions, instead of measures.

Quasistatic nonassociative plasticity at finite strains

TL;DR

This work advances finite-strain elastoplasticity in the nonassociative setting by formulating a variational quasistatic evolution and proving the existence of measure-valued energetic solutions. The authors regularize the dissipation via a space-time mollification and introduce a gradient term in the plastic strain to gain compactness, resulting in a rigorous time-discrete to continuous existence theory built on generalized Young measures. A parallel energetic function-valued formulation is shown to be attainable, and a precise correspondence between the measure-valued and function-valued approaches is established. The methodology extends quasistatic plasticity theory to nonassociative, nonlinear finite-strain regimes and provides a solid mathematical basis for nonlocal regularizations in gradient-plasticity models, with potential implications for numerical approximations and convergence analysis.

Abstract

We investigate finite-strain elastoplastic evolution in the nonassociative setting. The constitutive material model is formulated in variational terms and coupled with the quasistatic equilibrium system. We introduce measure-valued energetic solutions and prove their existence via a time discretization approach. The existence theory hinges on a suitable regularization of the dissipation term via a space-time mollification. Eventually, we discuss the possibility of solving the problem in the setting of functions, instead of measures.

Paper Structure

This paper contains 30 sections, 8 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Tensors and function spaces
  4. A caveat on notation
  5. Constitutive model
  6. Energy
  7. Constitutive equations
  8. Flow rule
  9. An example
  10. Dissipativity
  11. Linearization
  12. Dissipation
  13. Quasistatic-evolution system and regularization
  14. Variational formulation and existence
  15. Assumptions
  16. ...and 15 more sections

Key Result

Lemma 5.1

Let the sequence $(F_n,P_n,G_n)_n$ be bounded in and generate the Young measure $\nu=\left(\nu_{x,t},\lambda,\nu_{x,t}^\infty\right)$ with $\nu_{x,t}\in L^\infty_{w\ast}(\Omega \times (0,T);\mathcal{M}({\mathbb R}^m))$, $\lambda \in \mathcal{M}^+(\overline \Omega \times [0,T])$, and $\nu_{x,t}^\infty\in L^\infty_{w\ast}(\Omega \times (0,T);\mathca and the concentration measure $\lambda$ admits a

Theorems & Definitions (12)

  • Lemma 5.1: Structural result, Brenier2011
  • Definition 5.1: Energetic measure-valued solution
  • Theorem 5.1: Existence
  • Proposition 5.1: Solutions as functions
  • Proposition 5.2: Correspondence of solutions
  • Lemma 6.1: Coercivity of the energy
  • Lemma 6.2: Existence of discrete solutions
  • proof
  • Theorem A.1: Extended Helly selection principle
  • proof
  • ...and 2 more