II.1.3.3 In-plane properties

The ply properties in a plane parallel to the laminate are considered here. Then the constitutive equation (16) reduces to:

The indices in this notation are integers and indicate that the corresponding properties are given in ply coordinate system. The equation (II.1.19) is written more concisely as follows:

Expression (II.1.20) introduces the material in-plane compliance matrix ${\left[c\right]}_{\text{ply}}$. To avoid overly complicated notations, the same notation is used as for the $6\times 6$ full material compliance matrix introduced in (II.1.17). This will be done systematically for the in-plane matricial and vectorial quantities in the rest of the document ($\left[c\right]$, $\left[C\right]$, $\left\{\alpha \right\}$, $\left\{\beta \right\}$, $\left\{ϵ\right\}$,...

The inverse of expression (II.1.20) is noted:

Expression (II.1.21) introduces the in-plane stiffness matrix ${\left[C\right]}_{\text{ply}}={\left[c\right]}_{\text{ply}}^{-1}$.

Plies are characterized by their orientation in the laminate. Let $\xi$ be the angle of the ply in the laminate axes. Then, the laminate axes are obtained by rotating the ply axes by an angle $-\xi$. Equations (II.1.20) and (II.1.21) are expressed in the laminate coordinate system as follows:

This leads to the new expression in laminate axes:

where new notations are introduced for in-plane ply properties rotated by an angle $-\xi$ (in laminate axes):

When a matrix is transformed as in (II.1.23) or a vector as in (II.1.24), they are said to be rotated with the ${\left[T_{+}^{\prime }\right]}_{(-\xi )}$ rotation matrix.