# Failure of laminates and the Tsai–Hill criterion

Similar to the discussions in the page on Strength of long-fibre composites, the failure of laminae can be understood by the same three failure modes: axial, transverse and shear. A number of failure criteria have been proposed for separate plies subjected to **in-plane stress states **, with the assumption that coupling stresses are not present. We will introduce two here.

### Maximum Stress Criterion

This assumes no interaction between the modes of failure, i.e. the critical stress for one mode is unaffected by the stresses tending to cause the other modes. Failure then occurs when one of these critical values, σ_{1u} , σ_{2u} and τ_{12u} , is reached. These values refer to the laminar principal axes and can be resolved from the applied stress system by using the equation

\[\left[ {\begin{array}{*{20}{c}} {{\sigma _1}}\\ {{\sigma _2}}\\ {{\tau _{12}}} \end{array}} \right] = {\left[ T \right]^{}}\left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\tau _{xy}}} \end{array}} \right]\]

where \[\left[ T \right] = \left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\theta }&{{{\sin }^2}\theta }&{2\cos \theta \sin \theta }\\ {{{\sin }^2}\theta }&{{{\cos }^2}\theta }&{ - 2\cos \theta \sin \theta }\\ { - \cos \theta \sin \theta }&{\cos \theta \sin \theta }&{{{\cos }^2}\theta - {{\sin }^2}\theta } \end{array}} \right]\]

( *See page on Off-axis loading *)

It follows that under an applied **uniaxial tension ** ( σ_{y} = τ_{xy} = 0) the critical values of σ_{x} for each failure mode are:

\[{\sigma _{xu}} = \frac{{{\sigma _{1u}}}}{{{{\cos }^2}\theta }},\;\;\;{\sigma _{xu}} = \frac{{{\sigma _{2u}}}}{{{{\sin }^2}\theta }},\;\;\;{\sigma _{xu}} = \frac{{{\tau _{12u}}}}{{\sin \theta \cos \theta }}.\]

### Tsai-Hill Criterion

Other treatments that take into account the interactions between failure modes are mostly based on modifications of yield criteria for metals (See TLP on Theory of Metal Forming - Stress States and Yielding Criteria ). The most important of these is the Tsai-Hill Criterion, which is an adaptation of the von Mises Criterion.

von Mises Criterion for Metals: ( σ_{1} - σ_{2} )^{2} + ( σ_{2} - σ_{3} )^{2} + ( σ_{3} - σ_{1} )^{2} = 2 σ_{Y}^{2}

where σ_{Y} is the metal yield stress.

For in-plane stress states ( σ_{3} = 0) this reduces to

\[{\left( {\frac{{{\sigma _1}}}{{{\sigma _Y}}}} \right)^2} + {\left( {\frac{{\sigma {}_2}}{{{\sigma _Y}}}} \right)^2} - \frac{{{\sigma _1}{\sigma _2}}}{{{\sigma ^2}_Y}} = 1\]

This is then modified to take into account the anisotropy of composites and the different failure mechanisms to give the following expression.

\[{\left( {\frac{{{\sigma _1}}}{{{\sigma _{1Y}}}}} \right)^2} + {\left( {\frac{{\sigma {}_2}}{{{\sigma _{2Y}}}}} \right)^2} - \frac{{{\sigma _1}{\sigma _2}}}{{{\sigma ^2}_{1Y}}} - \frac{{{\sigma _1}{\sigma _2}}}{{{\sigma ^2}_{2Y}}} + \frac{{{\sigma _1}{\sigma _2}}}{{{\sigma ^2}_{3Y}}} + {\left( {\frac{{{\tau _{12}}}}{{{\tau _{12Y}}}}} \right)^2} = 1\]

The metal yield stresses can be regarded as composite failure stresses and since composites are transversely isotropic ( σ_{2u} = σ_{3u} ) we arrive at the **Tsai-Hill Criterion ** for composites.

\[{\left( {\frac{{{\sigma _1}}}{{{\sigma _{1u}}}}} \right)^2} + {\left( {\frac{{\sigma {}_2}}{{{\sigma _{2u}}}}} \right)^2} - \frac{{{\sigma _1}{\sigma _2}}}{{{\sigma ^2}_{1u}}} + {\left( {\frac{{{\tau _{12}}}}{{{\tau _{12u}}}}} \right)^2} = 1\]

Below, the Maximum Stress and the Tsai-Hill criteria are used to predict the dependence on the loading angle of the tensile stress required to cause failure of a single lamina.

### Failure of Laminates

The above treatments only apply to single isolated plies. So, in order to extend this to laminates, we must obtain the in-plane stresses in each ply of a laminate subjected to an arbitrary in-plane stress state.

From Equation 13 the stress tensor for the k^{th} ply is related to the strain tensor by:

\[\left[ {\begin{array}{*{20}{c}} {{\sigma _{1k}}}\\ {{\sigma _{2k}}}\\ {{\tau _{12k}}} \end{array}} \right] = {\left[ C \right]_k}^{}\left[ {\begin{array}{*{20}{c}} {{\varepsilon _{1k}}}\\ {{\varepsilon _{2k}}}\\ {{\gamma _{12k}}} \end{array}} \right]\]

The strain tensor of the k^{th} ply can be resolved from the strain tensor of the laminate by using Equation 14:

\[\left[ {\begin{array}{*{20}{c}} {{\varepsilon _{1k}}}\\ {{\varepsilon _{2k}}}\\ {{\gamma _{12k}}} \end{array}} \right] = {\left[ {T'} \right]_k}^{}\left[ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\gamma _{xy}}} \end{array}} \right]\]

Now, from Equation 16, the laminate strain tensor is related to the laminate stress tensor by:

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

Combining these three equations gives:

\[\left[ {\begin{array}{*{20}{c}} {{\sigma _{1k}}}\\ {{\sigma _{2k}}}\\ {{\tau _{12k}}} \end{array}} \right] = {\left[ C \right]_k}^{}{\left[ {T'} \right]_k}{\left[ {{{\overline S }_L}} \right]^{}}\left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\tau _{xy}}} \end{array}} \right]\]

An appropriate failure criterion is then applied and the onset of laminate failure is taken to be the point at which one of the plies fail.

Note that the Maximum Stress Criterion suggests possible modes of failure whereas the Tsai-Hill criterion does not.