Sheared woven fibers strength predictions

The strength properties of the material along the fibrous reinforcement orientations are assumed to remain unchanged. The stresses and strains obtained from the simulation must be converted to a coordinate system that has axes matching the warp and weft fiber orientations before applying the standard failure criteria.

When you analyze the strength of a laminate with at least one ply that points to a sheared orthotropic material or a woven ply material, the software computes the failure metrics in the sheared space.

\begin{array}\{{\mathit{x}}^{1}}=\frac{\xi{{x'}^{1}}-\eta{{x'}^{2}}}{\xi}\{{\mathit{x}}^{2}}=\frac{{{x'}^{2}}}{\xi}\{{\mathit{x}}^{3}}={{x'}^{3}}\end{array}\Leftrightarrow\begin{array}\x'^1=\frac{\xi{{x}^{1}}-\eta{{x}^{2}}}{\xi}\x'^2=\frac{{{x}^{2}}}{\xi}\x'^3={{x}^{3}}\end{array}

where,

\begin{array}\\eta=\sin\beta\\xi=\cos\beta\end{array}

Applying the standard coordinate system transformation for covariant components of a tensor of rank 2 leads to the following.

• For strains: \begin{align}\epsilon'{ij}&=\epsilon{mn}\frac{\partial{x^m}}{\partial{x'}^i}\frac{\partial{x^n}}{\partial{x'}^j}\&\Downarrow\\epsilon'{11}&=\epsilon{11}\\epsilon'{12}&=\epsilon'{21}=\eta\epsilon_{11}+\xi\epsilon_{12}\\epsilon'{22}&=\eta^2\epsilon{11}+2\eta\xi\epsilon_{12}+\xi^2\epsilon_{22}\\epsilon'{13}&=\epsilon'{31}=\epsilon_{13}\\epsilon'{23}&=\epsilon'{32}=\eta\epsilon_{13}+\xi\epsilon_{23}\\epsilon'{33}&=\epsilon{33}\end{align}Rewriting with engineering components:\begin{align}\epsilon'{11}&=\epsilon{11}\\epsilon'{12}&=\epsilon'{21}=\eta\epsilon_{11}+\frac{\xi\gamma_{12}}{2}\\epsilon'{22}&=\eta^2\epsilon{11}+\eta\xi\gamma_{12}+\xi^2\epsilon_{22}\\gamma'{13}&=\gamma'{31}=\gamma_{13}\\gamma'{23}&=\gamma'{32}=\eta\gamma_{13}+\xi\gamma_{23}\\gamma'{33}&=\gamma{33}\end{align}

• For stresses: \begin{align}\sigma'{ij}&=\sigma{mn}\frac{\partial{x^m}}{\partial{x'}^i}\frac{\partial{x^n}}{\partial{x'}^j}\&\Downarrow\\sigma'{11}&=\sigma{11}\\sigma'{12}&=\sigma'{21}=\eta\sigma_{11}+\xi\sigma_{12}\\sigma'{22}&=\eta^2\sigma{11}+2\eta\xi\sigma_{12}+\xi^2\sigma_{22}\\sigma'{13}&=\sigma'{31}=\sigma_{13}\\sigma'{23}&=\sigma'{32}=\eta\sigma_{13}+\xi\sigma_{23}\\sigma'{33}&=\sigma{33}\end{align}

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