Tsai-Wu criterion

II.1.10.14 Tsai-Wu criterion

The Tsai-Wu criterion is a quadratic criterion. The 2D version of this criterion has a failure index calculated as follows:

\begin{matrix} F I^{TsaiWu} = \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{τ_{12}}{τ_{12}^{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} + 2 F_{12}^{TW} σ_{11} σ_{22} . & (II.1.124) \end{matrix}

In this expression, $F_{12}^{TW}$ is a material parameter to be obtained by characterization tests. This parameter must satisfy the following relation:

{(F_{12}^{TW})}^{2} < \frac{1}{σ_{11}^{t,all} σ_{11}^{c,all} σ_{22}^{t,all} σ_{22}^{c,all}} .

(II.1.125)

Its units are, for example, $[{Pa}^{- 2}]$ . Sometimes the corresponding dimensionless parameter is used instead:

F_{12}^{*} = F_{12}^{TW} \sqrt{σ_{11}^{t,all} σ_{11}^{c,all} σ_{22}^{t,all} σ_{22}^{c,all}} .

(II.1.126)

This dimensionless parameter must satisfy the relation:

- 1 \leq F_{12}^{*} \leq 1 .

(II.1.127)

The value $F_{12}^{*} = - 1 ∕ 2$ corresponds to a generalized Von Mises criterion. The value

F_{12}^{TW} = \frac{- 1}{2 σ_{11}^{t,all} σ_{11}^{c,all}}

(II.1.128)

leads to the Hoffman criterion discussed in section II.1.10.16. Ply passes the failure criterion if

F I^{TsaiWu} \leq 1 .

In some of the terms of expression (124) the components of Cauchy stress tensor appear linearly, and other terms, they appear quadratically. Therefore, the reserve factor expression is a little more complicated. FeResPost calculates it as follows:

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{τ_{12}}{τ_{12}^{all}})}^{2} + 2 F_{12}^{T W} σ_{11} σ_{22},

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},

c = - 1,

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

and the strength ratio is given by:

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

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

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