Linear Elastic Materials

Linear elastic materials extend proportionally to the applied load and return to the original configuration when the load is removed. The stress-strain relationship for linear elastic materials is linear and is given by Hooke's law.

The following formulation is valid for elastic materials with Poisson's ratio $\nu \le 0.45$. For $\nu \to 0.5$, materials are considered incompressible and require a two-field formulation (see Nearly Incompressible Materials).

The linear elastic assumption, which is valid for small strains, assumes a stress-strain relationship of the form:

where $D$ is called the material tangent, ${\epsilon }_T$ is the thermal strain (see Eqn. (4452)), and $\sigma$ and $\epsilon$ are an energy-conjugate stress-strain pair (either Cauchy stress and Euler-Almansi strain, or 2nd Piola-Kirchhoff stress and Green-Lagrange strain. See Energy-Conjugate Stress-Strain Pairs).

Anisotropic Materials

In the most general case, materials have different material properties in different directions. Materials with this characteristic are called anisotropic.

It is convenient to express Eqn. (4503) in Voigt notation. Voigt notation is a convenient notation for symmetric tensors, as it reduces the order of the tensor by specifying only its independent components. For example, second-order symmetric tensors can be represented as 6-dimensional vectors. In Voigt notation, the tensors in Eqn. (4503) can be written as:

$$\begin{array}{cccc}\sigma =\left(\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{23}\\ {\sigma }_{13}\end{array}\right);& D=\left(\begin{array}{cccccc}{c}_{11}& c_{12}& c_{13}& c_{14}& c_{15}& c_{16}\\ & c_{22}& c_{23}& c_{24}& c_{25}& c_{26}\\ & & c_{33}& c_{34}& c_{35}& c_{36}\\ & & & c_{44}& c_{45}& c_{46}\\ & & & & c_{55}& c_{56}\\ & & & & & c_{66}\end{array}\right);& \epsilon =\left(\begin{array}{l}{\epsilon }_{11}\hfill \\ {\epsilon }_{22}\hfill \\ {\epsilon }_{33}\hfill \\ 2{\epsilon }_{12}\hfill \\ 2{\epsilon }_{23}\hfill \\ 2{\epsilon }_{13}\hfill \end{array}\right);& \alpha =\left(\begin{array}{c}{\alpha }_{11}\\ {\alpha }_{22}\\ {\alpha }_{33}\\ 2{\alpha }_{12}\\ 2{\alpha }_{23}\\ 2{\alpha }_{13}\end{array}\right)\end{array}$$

(4504)

The factor 2 in the shear components of $\epsilon$ ensures that the variation of strain energy due to stress (Eqn. (4453)) is the same in tensor notation and Voigt notation. The strain shear components ${\gamma }_{xy}=2{\epsilon }_{xy}$ , ${\gamma }_{yz}=2{\epsilon }_{yz}$ , and ${\gamma }_{xz}=2{\epsilon }_{xz}$ are also called engineering shear strain. The material tangent $D$ is symmetric, with 21 independent parameters $c_{ij}$. As $D$ is symmetric (that is, $c_{ij}=c_{ji}$), the bottom half of the matrix has been omitted.

Orthotropic Materials

Materials that have independent properties on three mutually-orthogonal directions are called orthotropic. Specifically, the material properties are constant along each axis, but independent from the values along the other axes. For orthotropic materials, the material tangent $D$ reduces to :

$$D=\left(\begin{array}{cccccc}{c}_{11}& c_{12}& c_{13}& 0& 0& 0\\ & c_{22}& c_{23}& 0& 0& 0\\ & & c_{33}& 0& 0& 0\\ & & & {\begin{array}{c}G\end{array}}_{12}& 0& 0\\ & & & & {\begin{array}{c}G\end{array}}_{23}& 0\\ & & & & & {\begin{array}{c}G\end{array}}_{13}\end{array}\right)$$

(4505)

The material tangent coefficients for orthotropic materials can therefore be completely defined in terms of nine independent constants:

$$\begin{array}{ll}{E}_1,E_2,E_3& \mathrm{Young\text{'}s\; Moduli}\\ {\nu }_{12},{\nu }_{23},{\nu }_{13}& \mathrm{Poisson\; Ratios}\\ G_{12},G_{23},G_{13}& \mathrm{Shear\; Moduli}\end{array}$$

(4506)

as:

$$\begin{array}{l}{c}_{11}=\frac{1-{\nu }_{23}{\nu }_{32}}{E_2{E}_3\Delta }\\ c_{22}=\frac{1-{\nu }_{31}{\nu }_{13}}{E_3{E}_1\Delta }\\ c_{33}=\frac{1-{\nu }_{12}{\nu }_{21}}{E_1{E}_2\Delta }\\ c_{12}=c_{21}=\frac{1}{E_2{E}_3\Delta }({\nu }_{21}+{\nu }_{31}{\nu }_{23})\\ c_{13}=c_{31}=\frac{1}{E_2{E}_3\Delta }({\nu }_{31}+{\nu }_{21}{\nu }_{32})\\ c_{23}=c_{32}=\frac{1}{E_1{E}_3\Delta }({\nu }_{32}+{\nu }_{31}{\nu }_{12})\end{array}$$

(4507)

where:

$$\begin{array}{c}\begin{array}{ccc}{\nu }_{21}={\nu }_{12}\frac{E_2}{E_1};& {\nu }_{32}={\nu }_{23}\frac{E_3}{E_2};& {\nu }_{31}={\nu }_{13}\frac{E_3}{E_1}\end{array}\end{array}$$

(4508)

and:

$$\begin{array}{c}\Delta =\frac{1-2{\nu }_{13}{\nu }_{21}{\nu }_{32}-{\nu }_{13}{\nu }_{31}-{\nu }_{23}{\nu }_{32}-{\nu }_{12}{\nu }_{21}}{E_1{E}_2{E}_3}\end{array}$$

(4509)

For orthotropic materials, the vector of thermal expansion coefficients $\alpha$ reduces to:

$$\begin{array}{c}\alpha =\left(\begin{array}{c}{\begin{array}{c}\alpha \end{array}}_{11}\\ {\begin{array}{c}\alpha \end{array}}_{22}\\ {\begin{array}{c}\alpha \end{array}}_{33}\\ 0\\ 0\\ 0\end{array}\right)\end{array}$$

(4510)

Isotropic Materials

Materials that have identical properties in all directions are called isotropic. For isotropic materials, the material tangent $D$ reduces to:

$$D=\left(\begin{array}{cccccc}{c}_{11}& c_{12}& c_{12}& 0& 0& 0\\ & c_{11}& c_{12}& 0& 0& 0\\ & & c_{11}& 0& 0& 0\\ & & & G& 0& 0\\ & & & & G& 0\\ & & & & & G\end{array}\right)$$

(4511)

with:

$$\begin{array}{l}{c}_{11}=\frac{4}{3}G+K\\ c_{12}=-\frac{2}{3}G+K\end{array}$$

(4512)

That is, the material tangent coefficients for isotropic materials can be completely defined in terms of two independent constants, $G$ and $K$, called the shear modulus and the bulk modulus. These constants are related to the Young's modulus, $E$, and Poisson's ratio, $\nu$, through:

$$\begin{array}{c}G=\frac{E}{2\left(1+\nu \right)}\\ K=\frac{E}{3\left(1-2\nu \right)}\end{array}$$

(4513)

For isotropic materials, the vector of thermal expansion coefficients $\alpha$ reduces to:

$$\begin{array}{c}\alpha =\alpha \left(\begin{array}{c}1\\ 1\\ 1\\ 0\\ 0\\ 0\end{array}\right)\end{array}$$

(4514)