A nonlinear thermomechanical formulation for anisotropic volume and surface continua

TL;DR

This work develops a nonlinear thermomechanical formulation for anisotropic materials by employing a multiplicative decomposition of the deformation gradient, unifying 3D volumetric continua with surface continua through Kirchhoff–Love shell kinematics. It derives tensorial, coordinate-agnostic kinematics and balance laws for both 3D polar/non-polar continua and Kirchhoff–Love shells, and constructs weak forms suitable for curvilinear and Cartesian coordinates. The constitutive framework incorporates anisotropic thermal expansion, heat conduction, and radiation, and provides explicit expressions for the Helmholtz free energy and its derivatives in both bulk and surface contexts. The approach enables extraction of membrane and shell material models from 3D counterparts and sets the stage for extensions to thermo-electro-magneto-mechanical problems, with applications to anisotropic crystals and soft biological materials.

Abstract

A thermomechanical, polar continuum formulation under finite strains is proposed for anisotropic materials using a multiplicative decomposition of the deformation gradient. First, the kinematics and conservation laws for three dimensional, polar and non-polar continua are obtained. Next, these kinematics are connected to their corresponding counterparts for surface continua based on Kirchhoff-Love kinematics. Likewise, the conservation laws for Kirchhoff-Love shells are derived from their three dimensional counterparts. From this, the weak forms are obtained for three dimensional non-polar continua and Kirchhoff-Love shells. These formulations are expressed in tensorial form so that they can be used in both curvilinear and Cartesian coordinates. They can be used to model anisotropic crystals and soft biological materials, and they can be extended to other field equations, like Maxwell's equations to model thermo-electro-magneto-mechanical materials.

Figures (2)

• Figure 1: Multiplicative deformation decomposition of 3D volumetric continua. $\mathcal{B}_{0}$, $\mathcal{B}_\text{T}$ and $\mathcal{B}$ are the reference, intermediate and current configuration and their corresponding tangent vectors are $\boldsymbol{G}_{i}$, $\boldsymbol{\mathfrak{g}}_{i}$ and $\boldsymbol{g}_{i}$. ${\boldsymbol{F}}$, ${\boldsymbol{F}}_{\!\text{T}}$ and ${\boldsymbol{F}}_{\!\text{e}}$ are the total, thermal and elastic deformation gradient.
• Figure 2: Multiplicative deformation decomposition of layered surface continua. $\mathcal{S}_{0}$, $\mathcal{S}_\text{T}$ and $\mathcal{S}$ denote the mid-surface in reference, intermediate and current configuration and their corresponding tangent vectors are $\boldsymbol{A}_{i}$, $\boldsymbol{\mathfrak{a}}_{i}$ and $\boldsymbol{a}_{i}$. $\overset{\circledast}{\mathcal{S}}_{0}$, $\overset{\circledast}{\mathcal{S}}_{\text{T}}$ and $\overset{\circledast}{\mathcal{S}}$ are the reference, intermediate and current configuration of shell layers (through the thickness) and their corresponding tangent vectors are $\overset{\circledast}{\boldsymbol{g}}_{i}$, $\overset{\circledast}{\boldsymbol{\mathfrak{g}}}_{i}$ and $\overset{\circledast}{\boldsymbol{G}}_{i}$. $\boldsymbol{F}_{\!\text{s}}$, $\boldsymbol{F}_{\!\text{sT}}$ and $\boldsymbol{F}_{\!\text{se}}$ are total, thermal and elastic surface deformation gradients.