Inhomogeneous finitely-strained thermoplasticity with hardening by an Eulerian approach

Tomáš Roubíček; Giuseppe Tomassetti

doi:10.3934/dcdss.2023180

Article Contents

2024, Volume 17, Issue 1: 181-220. Doi: 10.3934/dcdss.2023180

This issue Previous Article Saddle points and Lagrange multipliers in Banach spaces Next Article Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions

Inhomogeneous finitely-strained thermoplasticity with hardening by an Eulerian approach

Tomáš Roubíček^{1, 2,} and
Giuseppe Tomassetti^3,

1.
Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czechia
2.
Institute of Thermomechanics, Czech Acad. Sci., Dolejškova 5, CZ-18200 Praha 8, Czechia
3.
Department of Industrial, Electronic and Mechanical Engineering, Roma Tre University, Via Vito Volterra 62, I-00146 Roma, Italy

Dedicated to Pierluigi Colli on the occasion of his sixtyfifth birthday.

Received: April 2023

Revised: July 2023

Early access: October 2023

Published: January 2024

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Abstract

A standard elasto-plasto-dynamic model at finite strains based on the Lie-Liu-Kröner multiplicative decomposition, formulated in rates, is here enhanced to cope with spatially inhomogeneous materials by using the reference (called also return) mapping. Also an isotropic hardening can be involved. Consistent thermodynamics is formulated, allowing for both the free and the dissipation energies temperature dependent. The model complies with the energy balance and entropy inequality. A multipolar Stokes-like viscosity and plastic rate gradient are used to allow for a rigorous analysis towards existence of weak solutions by a semi-Galerkin approximation.

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Mathematics Subject Classification: 35Q74, 35Q79, 74A15, 74A30, 74C20, 74J30, 80A20.

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Figure 1. A schematic 0-dimensional diagramme of the mixed rheology acting differently on spherical (volumetric) and the deviatoric parts. The Jeffreys rheology in the deviatoric part combines Stokes' and Maxwell rheologies in parallel and may degenerate to mere Stokes-type fluid if $ {M} $ vanishes within melting. Note that the dampers $ K_{\rm V} $ and $ G_{\rm V} $ correspond to a simplified (2.45a) with $ \nu_1 = 0 $ and $ p = 2 $, i.e. with $ \boldsymbol D = \nu_0 \boldsymbol e(\boldsymbol v) $, for a special choice $ K_{\rm V} = 2\nu_0/d $ and $ G_{\rm V} = \nu_0 $

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