Tsai-Wu criterion (3D)

II.1.10.15 Tsai-Wu criterion (3D)

The 3D version of Tsai-Wu failure criterion leads to the following expression of the failure index:

\begin{matrix} F I^{TsaiWu3D} = \frac{σ_{11}^{2}}{σ_{11}^{t,all} σ_{11}^{c,all}} + \frac{σ_{22}^{2}}{σ_{22}^{t,all} σ_{22}^{c,all}} + \frac{σ_{33}^{2}}{σ_{33}^{t,all} σ_{33}^{c,all}} + {(\frac{τ_{12}}{τ_{12}^{all}})}^{2} + {(\frac{τ_{23}}{τ_{23}^{all}})}^{2} + {(\frac{τ_{31}}{τ_{31}^{all}})}^{2} \\ + (\frac{1}{σ_{11}^{t,all}} - \frac{1}{σ_{11}^{c,all}}) σ_{11} + (\frac{1}{σ_{22}^{t,all}} - \frac{1}{σ_{22}^{c,all}}) σ_{22} + (\frac{1}{σ_{33}^{t,all}} - \frac{1}{σ_{33}^{c,all}}) σ_{33} \\ + 2 F_{12}^{TW} σ_{11} σ_{22} + 2 F_{23}^{TW} σ_{22} σ_{33} + 2 F_{31}^{TW} σ_{33} σ_{11} . & (II.1.129) \end{matrix}

The values of $F_{23}^{TW}$ and $F_{31}^{TW}$ are submitted to the same limitations as $F_{12}^{TW}$ in section II.1.10.14. Ply passes the failure criterion if

F I^{TsaiWu3D} \leq 1 .

The RF calculation is done as follows:

\begin{matrix} a = \frac{σ_{11}^{2}}{σ_{11}^{t,all} σ_{11}^{c,all}} + \frac{σ_{22}^{2}}{σ_{22}^{t,all} σ_{22}^{c,all}} + \frac{σ_{33}^{2}}{σ_{33}^{t,all} σ_{33}^{c,all}} + {(\frac{τ_{12}}{τ_{12}^{all}})}^{2} + {(\frac{τ_{23}}{τ_{23}^{all}})}^{2} + {(\frac{τ_{31}}{τ_{31}^{all}})}^{2} \\ + 2 F_{12}^{TW} σ_{11} σ_{22} + 2 F_{23}^{TW} σ_{22} σ_{33} + 2 F_{31}^{TW} σ_{33} σ_{11}, & (II.1.130) \end{matrix}

b = (\frac{1}{σ_{11}^{t,all}} - \frac{1}{σ_{11}^{c,all}}) σ_{11} + (\frac{1}{σ_{22}^{t,all}} - \frac{1}{σ_{22}^{c,all}}) σ_{22} + (\frac{1}{σ_{33}^{t,all}} - \frac{1}{σ_{33}^{c,all}}) σ_{33},

c = - 1,

R F^{TsaiWu3D} = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 \cdot F o S \cdot a},

and the strength ratio is given by:

S R^{TsaiWu3D} = \frac{1}{R F^{TsaiWu3D}} .

This criterion is referred to as “TsaiWu3D” criterion in FeResPost.

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