II.1.3.2 Materials and constitutive equations

When a material is used in the definition of a laminate, assumptions are done about the axes defined in the laminate. Axes 1 and 2 are parallel to the laminate plane and axis 3 is orthogonal to the laminate.

The classical laminate analysis is based on the assumption that the relation between stress and strain tensors is linear. Then, as these two tensors are symmetric, a $6\times 6$ matrix contains all the elastic coefficients defining the material:

One shows that, because the peculiar choice of angular strain tensor components, the matrix $\left[C_{ijkl}\right]$ containing the elastic coefficients is symmetric. Therefore, the matrix has only 21 independent coefficients. $\left[C_{ijkl}\right]$ is the stiffness matrix of the material.

Equation (II.1.11) can be reversed as follows:

In expression (12), one added the thermo-elastic and moisture expansion terms in previous expression. They are characterized by CTE and CME tensors noted $\left\{{\alpha }_{kl}\right\}$ and $\left\{{\beta }_{kl}\right\}$ respectively. Note that shear components of these two tensors are angular components. Practically, it does not matter much as most materials have zero shear components for CTE or CME tensors. $\left[c_{ijkl}\right]$ is the compliance matrix of the material. Obviously $\left[c_{ijkl}\right]={\left[C_{ijkl}\right]}^{-1}$. One often defines laminates with orthotropic materials:

• For a fabric, 1 corresponds generally to the warp direction, and 2 to the weft direction. The corresponding tensile/compressive moduli are noted $E_1$ and $E_2$ respectively. $E_3$ denotes the out-of-plane tensile/compressive modulus.
• Correspondingly, one defines shear moduli noted $G_{12}$, $G_{23}$ and $G_{13}$.
• In general six Poisson coefficients can be defined: ${\nu }_{12}$, ${\nu }_{23}$, ${\nu }_{31}$, ${\nu }_{21}$, ${\nu }_{32}$, ${\nu }_{13}$. However, these coefficients are not independent. The relations

  $$\frac{{\nu }_{21}}{E_2}=\frac{{\nu }_{12}}{E_1},$$

  (II.1.13)

  $$\frac{{\nu }_{31}}{E_3}=\frac{{\nu }_{13}}{E_1},$$

  (II.1.14)

  $$\frac{{\nu }_{32}}{E_3}=\frac{{\nu }_{23}}{E_2}$$

  (II.1.15)

  allow to eliminate the coefficients ${\nu }_{21}$, ${\nu }_{32}$ and ${\nu }_{13}$ so that only the three Poisson coefficients ${\nu }_{12}$, ${\nu }_{23}$ and ${\nu }_{13}$ have to be introduced when defining a material.

• The constitutive equation of an orthotropic material is given by $$\begin{array}{cc}& \left(\begin{array}{c}\hfill {ϵ}_{11}\hfill \\ \hfill {ϵ}_{22}\hfill \\ \hfill {ϵ}_{33}\hfill \\ \hfill {\gamma }_{23}\hfill \\ \hfill {\gamma }_{13}\hfill \\ \hfill {\gamma }_{12}\hfill \end{array}\right)=\left(\begin{array}{cccccc}\hfill 1∕E_1\hfill & \hfill -{\nu }_{21}∕E_2\hfill & \hfill -{\nu }_{31}∕E_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -{\nu }_{12}∕E_1\hfill & \hfill 1∕E_2\hfill & \hfill -{\nu }_{32}∕E_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -{\nu }_{13}∕E_1\hfill & \hfill -{\nu }_{23}∕E_2\hfill & \hfill 1∕E_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1∕G_{23}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1∕G_{13}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1∕G_{12}\hfill \\ \hfill \hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill {\sigma }_{11}\hfill \\ \hfill {\sigma }_{22}\hfill \\ \hfill {\sigma }_{33}\hfill \\ \hfill {\tau }_{23}\hfill \\ \hfill {\tau }_{13}\hfill \\ \hfill {\tau }_{12}\hfill \end{array}\right)\\ & +\Delta T\left(\begin{array}{c}\hfill {\alpha }_{11}\hfill \\ \hfill {\alpha }_{22}\hfill \\ \hfill {\alpha }_{33}\hfill \\ \hfill {\alpha }_{23}\hfill \\ \hfill {\alpha }_{13}\hfill \\ \hfill {\alpha }_{12}\hfill \end{array}\right)+\Delta H\left(\begin{array}{c}\hfill {\beta }_{11}\hfill \\ \hfill {\beta }_{22}\hfill \\ \hfill {\beta }_{33}\hfill \\ \hfill {\beta }_{23}\hfill \\ \hfill {\beta }_{13}\hfill \\ \hfill {\beta }_{12}\hfill \end{array}\right).& \text{(II.1.16)}\end{array}$$

For an isotropic material, the definition of $E$ and either $G$ or $\nu$ is sufficient to characterize the material. Then one has:

$E$, $G$ and $\nu$ satisfy the following relation:

Finally, one introduces shorter notations that allow to rewrite expressions (II.1.11) and (12) respectively as follows:

One introduces also the “Mechanical Strain Tensor” estimated as follows:

This new strain tensor differs from the one defined by (II.1.17) by the fact that no thermo-elastic or hygro-elastic contribution is taken into account to estimate its components. It is the strain that corresponds to the actual material stress, when no thermo-elastic or hygro-elastic expansion is considered. This “Mechanical Strain Tensor” is also sometimes called “Equivalent Strain Tensor”.