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Well-Posedness of Discretizations for Fractional Elasto-Plasticity

Michael Feischl, David Niederkofler, Barbara Wohlmuth

TL;DR

This work introduces a fractional elasto-plasticity model where the classical flow rule is replaced by a normalized Riesz--Caputo derivative with a finitely supported kernel, enabling nonlocal, history-dependent plasticity. It develops explicit and implicit space-time discretizations within a finite element framework and proves local well-posedness of the explicit discretization via semismooth analysis, together with semismoothness properties for the return-mapping and its implicit counterpart. The results rely on positivity of subdifferentials and a careful perturbation analysis to ensure computability of the update maps, with extensions to infinite-dimensional inputs. Numerical experiments in 2D and 3D demonstrate the influence of the fractional order and kernel size on plastic flow and displacement, confirming robust convergence of semismooth Newton iterations and highlighting the practical impact of nonlocality in fractional plasticity modeling.

Abstract

We consider a fractional plasticity model based on linear isotropic and kinematic hardening as well as a standard von-Mises yield function, where the flow rule is replaced by a Riesz--Caputo fractional derivative. The resulting mathematical model is typically non-local and non-smooth. Our numerical algorithm is based on the well-known radial return mapping and exploits that the kernel is finitely supported. We propose explicit and implicit discretizations of the model and show the well-posedness of the explicit in time discretization in combination with a standard finite element approach in space. Our numerical results in 2D and 3D illustrate the performance of the algorithm and the influence of the fractional parameter.

Well-Posedness of Discretizations for Fractional Elasto-Plasticity

TL;DR

This work introduces a fractional elasto-plasticity model where the classical flow rule is replaced by a normalized Riesz--Caputo derivative with a finitely supported kernel, enabling nonlocal, history-dependent plasticity. It develops explicit and implicit space-time discretizations within a finite element framework and proves local well-posedness of the explicit discretization via semismooth analysis, together with semismoothness properties for the return-mapping and its implicit counterpart. The results rely on positivity of subdifferentials and a careful perturbation analysis to ensure computability of the update maps, with extensions to infinite-dimensional inputs. Numerical experiments in 2D and 3D demonstrate the influence of the fractional order and kernel size on plastic flow and displacement, confirming robust convergence of semismooth Newton iterations and highlighting the practical impact of nonlocality in fractional plasticity modeling.

Abstract

We consider a fractional plasticity model based on linear isotropic and kinematic hardening as well as a standard von-Mises yield function, where the flow rule is replaced by a Riesz--Caputo fractional derivative. The resulting mathematical model is typically non-local and non-smooth. Our numerical algorithm is based on the well-known radial return mapping and exploits that the kernel is finitely supported. We propose explicit and implicit discretizations of the model and show the well-posedness of the explicit in time discretization in combination with a standard finite element approach in space. Our numerical results in 2D and 3D illustrate the performance of the algorithm and the influence of the fractional parameter.

Paper Structure

This paper contains 16 sections, 26 theorems, 132 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. The model problem
  4. Fractional derivatives
  5. The fractional flow rule
  6. Auxiliary results
  7. Discretization
  8. Analysis of the Return-Mapping
  9. Well-posedness of the space-time discretizations
  10. Well-posedness of explicit (non-fractional) space-time-discretization
  11. Well-posedness of fractional explicit space-time-discretization
  12. Generalizations
  13. Semismoothness of fractional implicit return-mapping
  14. Numerical Experiments
  15. Two-dimensional domain
  16. ...and 1 more sections

Key Result

Lemma 2.1

For $\alpha \in (0,1)$, $\boldsymbol{\sigma},\boldsymbol{\chi}^1 \in M^d$, $\boldsymbol{\Delta} \in \mathbb{R}^{d \times d}_+$ and ${\rm dev}(\boldsymbol{\sigma}+\boldsymbol{\chi}^1) \neq 0$, we find that the fractional derivative $\mathbf{D}_{\boldsymbol{\sigma}}^{\boldsymbol{\Delta},{\alpha}}f(\bo

Figures (13)

  • Figure 1: (A): The domain $\Omega \subset \mathbb{R}^2$, is anchored on the left and subject to a pulling force on the right. (B): Depiction of $\Omega$ and $\partial \Omega= \Gamma_D \cup \Gamma_N$. (C): Depiction of locations where displacement measurements $d_x$ and $d_y$ are taken.
  • Figure 2: Loading $\mathbf{t}_N=(t_1^N,0)$ over time.
  • Figure 3: The flow-vectors of the fractional flow rule for $\alpha=0.5$, compared to the flow-vectors for $\alpha \approx 1$, projected to the $\sigma_{11}-\sigma_{22}$ plane. (A): $d=3$, (B): $d=2$
  • Figure 4: Residual of Newton iterates for Newton step $k$. Convergence threshold was $10^{-8}$.
  • Figure 5: Number of necessary Newton steps for different values of (A):$\alpha$ and (B): $\boldsymbol{\Delta}$. Convergence threshold is $10^{-8}$.
  • ...and 8 more figures

Theorems & Definitions (57)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Corollary 2.3
  • Proof 3
  • Remark 3.1
  • Theorem 4.1
  • Proof 4
  • Remark 4.2
  • ...and 47 more