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

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. (183) and consider only inelastic dilatation (â αβ = J in A αβ ) according to Sec. 3.3. Contracting (184) with a αβ and using the relations from Sec. 3.3 thus yields the evolution laẇfor J in . For a generalized viscoelastic solid, see Fig. 2b, evolution law (184) needs to be solved forâ αβ

A multi-field model for charging and discharging of lithium-ion battery electrodes

“…The coupled constitutive relations are obtained through the thermodynamical approach of Coleman and Noll [12], based on the assumption that the local state of a system is entirely described by the deformation, the entropy, and a set of suitable internal variables defined within a small neighborhood of a material point. The proposed model extends previous models by [2,8,22,39,65] and others, by coupling the full electrochemical-thermomechanical process of charging and discharging with large reversible and irreversible mechanical deformations.…”

“…Figure 13 describes the dependence of the results on the coefficient κ, a gradient energy factor that enters the interface term |∇c| 2 in Eq. (39). Using the 12 μm long clamped elastic specimen, results have been obtained for the reference value of κ reported in Table 2 and for a 10 times larger value, i. e., κ 1 = 10κ.…”

A nonlinear thermomechanical formulation for anisotropic volume and surface continua

“…An intermediate configuration B T is introduced, 4 and the reference, intermediate, and current configurations, B 0 , B T , and B , are connected (see Figure 1). An additive decomposition of the strain tensor should not be used in finite deformations, unless the logarithmic strain tensor is used (see Appendix 5) or a special strain tensor in the intermediate configuration is introduced [89]. The limitation of the additive decomposition of strains is resolved by using a multiplicative decomposition of the deformation gradient.…”

“…Sahu et al [7] and Sauer et al [89] developed a thermomechanical Kirchhoff-Love shell formulation directly from the surface balance laws. Here, however, the conservation laws for surface continua are derived as a special case of their three-dimensional counterparts.…”

“…Sauer et al [89] present the multiplicative decomposition of the surface deformation gradient and examine its consequences on the kinematics, balance laws, and constitutive relations of curved surfaces. Their formulation uses a direct surface description in curvilinear coordinates that is not necessarily connected to an underlying three-dimensional volumetric formulation.…”