1. Introduction

The objective of this document is to give the expression of flexibility and Hooke matrices for orthotropic, transversely isotropic and cubic symmetric thermoelastic materials in 3D, 2D-plane stresses, 2D-plane deformations and axisymmetry cases.

In all seriousness, for linear thermoelastic materials, stresses are linear functions of deformations and of the temperature differential. We write:

\({\sigma }_{\mathit{ij}}={H}_{\mathit{ijkl}}\left({\epsilon }_{\mathit{kl}}-{\alpha }_{\mathit{kl}}\left(T-{T}_{\mathit{réf}}\right)\right)\)

We talk about Hooke « matrices » because, for the sake of simplification, we did not adopt the notation of a 4-order tensor for \(\mathrm{H}\), in favor of the W.Voigt notation, where second-order tensors are represented by 6-component vectors and 4-order tensors by \(6\mathrm{\times }6\) matrices.

The symmetric nature of \(\sigma\) and \(\epsilon\) and the adoption of a vector form for these 2nd order tensors makes it possible to write:

\(\left\{\sigma \right\}=\left[\mathrm{H}\right]\left(\left\{\epsilon \right\}-\left\{\alpha \right\}\left(T-{T}_{\mathit{réf}}\right)\right)\)

where \(\left\{\sigma \right\}\) and \(\left\{\epsilon \right\}\) are the vector representation of the 2nd order tensors \(\sigma\) and \(\epsilon\), and where \(\left[\mathrm{H}\right]\), the Hooke matrix, is a matrix \(6\times 6\), which is necessarily symmetric, i.e. a priory 21 independent coefficients, i.e. 21 independent coefficients, in the three-dimensional case:

\(\left[\mathrm{H}\right]=\left\{\begin{array}{cccccc}{H}_{\text{11}}& {H}_{\text{12}}& {H}_{\text{13}}& {H}_{\text{14}}& {H}_{\text{15}}& {H}_{\text{16}}\\ & {H}_{\text{22}}& {H}_{\text{23}}& {H}_{\text{24}}& {H}_{\text{25}}& {H}_{\text{26}}\\ & & {H}_{\text{33}}& {H}_{\text{34}}& {H}_{\text{35}}& {H}_{\text{36}}\\ \text{SYM}& & & {H}_{\text{44}}& {H}_{\text{45}}& {H}_{\text{46}}\\ & & & & {H}_{\text{55}}& {H}_{\text{56}}\\ & & & & & {H}_{\text{66}}\end{array}\right\}\)

Vector \(\left\{\alpha \right\}\) designates the vector representation of the 2nd order thermal expansion tensor, which is necessarily symmetric, i.e. a priory 6 independent coefficients, in the three-dimensional case.