The multiplicative deformation split for shells with application to growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity

TL;DR

This paper develops a unified theoretical framework for modeling coupled nonlinear elastic and inelastic deformations in curved shells using a multiplicative surface deformation gradient decomposition, applicable to various physical phenomena.

Contribution

It introduces a comprehensive multiplicative decomposition approach for shells, enabling consistent modeling of growth, swelling, thermoelasticity, viscoelasticity, and elastoplasticity within a unified theory.

Findings

Derived general constitutive relations based on thermodynamics.

Presented nonlinear examples demonstrating the theory's applicability.

Formulated the theory in curvilinear coordinates for clarity and elegance.

Abstract

This work presents a general unified theory for coupled nonlinear elastic and inelastic deformations of curved thin shells. The coupling is based on a multiplicative decomposition of the surface deformation gradient. The kinematics of this decomposition is examined in detail. In particular, the dependency of various kinematical quantities, such as area change and curvature, on the elastic and inelastic strains is discussed. This is essential for the development of general constitutive models. In order to fully explore the coupling between elastic and different inelastic deformations, the surface balance laws for mass, momentum, energy and entropy are examined in the context of the multiplicative decomposition. Based on the second law of thermodynamics, the general constitutive relations are then derived. Two cases are considered: Independent inelastic strains, and inelastic strains that are functions of temperature and concentration. The constitutive relations are illustrated by several nonlinear examples on growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity of shells. The formulation is fully expressed in curvilinear coordinates leading to compact and elegant expressions for the kinematics, balance laws and constitutive relations.