# Stiffness of laminates

Aligned composites are very stiff along the fibre axis, but also very compliant in the transverse direction. Many applications equal stiffness in all directions within a plane and the solution to this is to stack together and bond plies with different fibre directions.

Having established the off-axis elastic constants for a single thin ply, we will now establish those for a stack of plies, a laminate. In this case there is a further complication in that while the fibres in each ply make an angle Φ with the reference x-y axes, the elastic constants of the laminate as a whole will also be affected by the angle Φ that the loading direction makes with the same reference x-y axes. The corresponding stress - strain relationship for a laminate is:

\[\left[ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\gamma _{xy}}} \end{array}} \right] = {\left[ {{{\overline S }_g}} \right]^{}}\left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\tau _{xy}}} \end{array}} \right] \;\;\;(16)\]

where the subscript g refers to global.

The average stress in the x-direction of the loading system is given by:

\[{\sigma _{x{,^{}}g}} = \frac{{\sum\limits_{k = 1}^n {({\sigma _{x{,^{}}k}}^{}{t_k})} }}{{\sum\limits_{k = 1}^n {{t_k}} }} = {\overline C _{11{,^{}}g}}^{}{\varepsilon _{x{,^{}}g}} + {\overline C _{12{,^{}}g}}^{}{\varepsilon _{y{,^{}}g}} + {\overline C _{16{,^{}}g}}^{}{\gamma _{xy{,^{}}g}}\] (17)

where t_{k} is the thickness of the k^{th}
ply. This is simply an expansion of equation 16.

And since the in-plane strains are the same for each ply, the stress in the k^{th} ply can be written as:

\[{\sigma _{x{,^{}}g}} = {\overline C _{11{,^{}}g}}^{}{\varepsilon _{x{,^{}}g}} + {\overline C _{12{,^{}}g}}^{}{\varepsilon _{y{,^{}}g}} + {\overline C _{16{,^{}}g}}^{}{\gamma _{xy{,^{}}g}}\]

This is an expansion of equation 15 for the k^{th} ply.

Substituting this into equation 17 and equating the coefficients of ε_{x,g} , we have:

\[{\overline C _{11,g}} = \frac{{\sum\limits_{k = 1}^n {({{\overline C }_{11,g}}^{}{t_k})} }}{{\sum\limits_{k = 1}^n {{t_k}} }}\]

In this way the other components of the stiffness tensor for the laminate can also be found.

The inverse of the stiffness tensor, the compliance tensor, is often obtained because its relationships with the elastic constants are simpler. Like before,

\[{E_x} = \frac{1}{{{{\overline S }_{11{,^{}}g}}}},\;\;\;{G_{xy}} = \frac{1}{{{{\overline S }_{66{,^{}}g}}}},\;\;\;{\nu _{xy}} = - {E_x}{\overline S _{12{,^{}}g}}^{}\]

Clearly the laminate will exhibit different elastic constants if the loading system were applied at an arbitrary angle, Φ, to the x-y coordinate system. Try constructing your own laminate, using the model below, and calculate the elastic constants for different loading angles.