Tsai-Wu criterion

The Tsai-Wu criterion is one of the non-differentiating failure criteria or general criteria for fiber-plastic composites . It goes back to Stephen W. Tsai and Edward M. Wu , who published it in 1971 for plane states of tension.

These materials are characterized by their structural anisotropy compared to homogeneous materials, which have largely isotropic material behavior. A fiber composite material can therefore only be fully described if, in addition to the elasticities, the material strengths, i.e. the material failure, can also be described as a function of the direction.

Mathematical derivation

The Tsai-Wu failure criterion describes a failure body in a 6-dimensional stress space ( , , , , , ). Failure is predicted when the stress state is outside the fracture surface of the fracture body. All stress states within the fracture body are failure-free. Mathematically, any fracture body in the stress space can be represented by a function ${\displaystyle \sigma _{ij}}$ ${\displaystyle \sigma _{x}}$ ${\displaystyle \sigma _{y}}$ ${\displaystyle \sigma _{z}}$ ${\displaystyle \tau _{xz}}$ ${\displaystyle \tau _{yz}}$ ${\displaystyle \tau _{xy}}$

${\displaystyle F(\sigma _{ij})=1\,}$

to be discribed. The sometimes known criterion in Composite Structures of this type is that on a series expansion of all the voltage coefficient (the Tsai-Wu failure criterion and anisotropic) resistance coefficients ( , , , ) decreases. If the mathematical representation is restricted to Einstein's sum convention , then a series expansion in the form ${\displaystyle \sigma _{ij}}$ ${\displaystyle a_{ij}}$ ${\displaystyle b_{ijkl}}$ ${\displaystyle c_{ijklmn}}$ ${\displaystyle \ldots }$

${\displaystyle F(\sigma _{ij})=(a_{ij}\sigma _{ij})^{\alpha }+(b_{ijkl}\sigma _{ij}\sigma _{kl})^{\beta }+(c_{ijklmn}\sigma _{ij}\sigma _{kl}\sigma _{mn})^{\gamma }+\ldots =1}$

being represented. The Tsai-Wu criterion is now ultimately a special case in which only the first two terms of the series expansion are taken into account. If we continue to assume for the exponents , the Tsai-Wu criterion results in the form ${\displaystyle \alpha =\beta =1}$

${\displaystyle F(\sigma _{ij})=a_{ij}\sigma _{ij}+b_{ijkl}\sigma _{ij}\sigma _{kl}=1\,}$

Determination of strengths

The strength tensors and must be adapted to the anisotropic strength properties of fiber composites. If only the two-dimensional, flat material properties , i.e. only the indices and as well as for the strength coefficients and a transversely isotropic material symmetry, are taken into account, the equation is simplified to ${\displaystyle a_{ij}}$ ${\displaystyle b_{ijkl}}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle a_{ij}}$ ${\displaystyle b_{ijkl}}$

${\displaystyle F(\sigma _{ij})=a_{11}\,\sigma _{11}+a_{22}\,\sigma _{22}+b_{1111}\,\sigma _{11}^{2}+b_{2222}\,\sigma _{22}^{2}+2b_{1122}\,\sigma _{11}\sigma _{22}+4b_{1212}\,\sigma _{12}^{2}=1}$

The strength coefficients are now to be determined from standard tests (tension, pressure, shear), for this purpose the strengths are determined from uniaxial loads along ( ) and across ( ) to the fiber direction. With the compressive and tensile strengths in the fiber direction as well as the compressive and tensile strengths perpendicular to the fiber direction and the shear strengths , the coefficients of the tensile strength values ​​are calculated as follows: ${\displaystyle \sigma _{\parallel }}$ ${\displaystyle \sigma _{\perp }}$ ${\displaystyle R_{\parallel }^{(-)}}$ ${\displaystyle R_{\parallel }^{(+)}}$ ${\displaystyle R_{\perp }^{(-)}}$ ${\displaystyle R_{\perp }^{(+)}}$ ${\displaystyle R_{\parallel \perp }}$

${\displaystyle a_{11}={\frac {1}{R_{\parallel }^{(+)}}}-{\frac {1}{R_{\parallel }^{(-)}}}}$

${\displaystyle a_{22}={\frac {1}{R_{\perp }^{(+)}}}-{\frac {1}{R_{\perp }^{(-)}}}}$

${\displaystyle b_{1111}={\frac {1}{R_{\parallel }^{(+)}R_{\parallel }^{(-)}}}}$

${\displaystyle b_{2222}={\frac {1}{R_{\perp }^{(+)}R_{\perp }^{(-)}}}}$

${\displaystyle b_{1212}={\frac {1}{4R_{\perp \parallel }^{2}}}}$

${\displaystyle b_{1122}={\frac {F_{12}^{*}}{\sqrt {R_{\parallel }^{(+)}R_{\parallel }^{(-)}R_{\perp }^{(+)}R_{\perp }^{(-)}}}}}$

The coefficient of Tsai referred to as interaction coefficient, may consist of a combined biaxial normal strain with be determined. A large number of different criteria differ in just this coefficient. Tsai specifies a validity range of for this coefficient . ${\displaystyle F_{12}^{*}}$ ${\displaystyle \sigma _{\parallel }}$ ${\displaystyle \sigma _{\perp }}$ ${\displaystyle -1\leq F_{12}^{*}\leq 1}$

Tsai-Wu criterion

Written out, the strength criterion according to Tsai-Wu results in:

${\displaystyle F(\sigma _{ij})=\left({\frac {1}{R_{\parallel }^{(+)}}}-{\frac {1}{R_{\parallel }^{(-)}}}\right)\,\sigma _{\parallel }+\left({\frac {1}{R_{\perp }^{(+)}}}-{\frac {1}{R_{\perp }^{(-)}}}\right)\,\sigma _{\perp }+{\frac {1}{R_{\parallel }^{(+)}R_{\parallel }^{(-)}}}\,\sigma _{\parallel }^{2}+{\frac {1}{R_{\perp }^{(+)}R_{\perp }^{(-)}}}\,\sigma _{\perp }^{2}+{\frac {2F_{12}^{*}}{\sqrt {R_{\parallel }^{(+)}R_{\parallel }^{(-)}R_{\perp }^{(+)}R_{\perp }^{(-)}}}}\,\sigma _{\parallel }\sigma _{\perp }+{\frac {\tau _{\parallel \perp }^{2}}{R_{\perp \parallel }^{2}}}=1}$

With

${\displaystyle -1\leq F_{12}^{*}\leq 1}$

Conclusion, discussion

The Tsai-Wu criterion is a straightforward approach to take into account direction-dependent material strengths. This criterion is attractive because of its ease of implementation and clarity. In contrast to other criteria, the Tsai-Wu criterion has no information on various modes of fracture . Intermediate fiber breakage (Zfb), fiber breakage (Fb) and failure under compressive and tensile loads are determined in one criterion. This property of the criterion is viewed as a disadvantage, since the designer does not receive any information about the break mode . Accordingly, there is no information available on how to change the design of a component in order to avoid failure.

See also

• Failure criteria for fiber-reinforced plastics
• Elasticity theory
• Fiber composite
• Carbon fiber reinforced plastic

Publications and contributions

• A General Theory of Strength for Anisotropic Materials . In: Journal of Composite Materials , Vol. 5, No. 1, 58-80 (1971)
• Failure Criteria In Fiber Reinforced Polymer Composites: The World-Wide Failure Exercise . Editors: MJ Hinton, AS Kaddour, PD Soden. Elsevier 2004