The Tsai–Wu failure criterion is a phenomenological material failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression. The Tsai-Wu criterion predicts failure when the failure index in a laminate reaches 1. This failure criterion is a specialization of the general quadratic failure criterion proposed by Gol'denblat and Kopnov and can be expressed in the form
Fi~\sigmai+Fij~\sigmai~\sigmaj\le1
ij=1...6
Fi,Fij
\sigmai
Fij
FiiFjj-
2 | |
F | |
ij |
\ge0
Fii
For orthotropic materials with three planes of symmetry oriented with the coordinate directions, if we assume that
Fij=Fji
\begin{align} F1\sigma1+&F2\sigma2+F3\sigma3+F4\sigma4+F5\sigma5+F6\sigma6\\ &+F11
2 | |
\sigma | |
1 |
+F22
2 | |
\sigma | |
2 |
+F33
2 | |
\sigma | |
3 |
+F44
2 | |
\sigma | |
4 |
+F55
2 | |
\sigma | |
5 |
+F66
2 | |
\sigma | |
6 |
\\ & +2F12\sigma1\sigma2+2F13\sigma1\sigma3+2F23\sigma2\sigma3\le1 \end{align}
\sigma1t,\sigma1c,\sigma2t,\sigma2c,\sigma3t,\sigma3c
\tau23,\tau12,\tau31
\begin{align} F1=&\cfrac{1}{\sigma1t
F12,F13,F23
\sigma1=\sigma2=\sigmab12,\sigma1=\sigma3=\sigmab13,\sigma2=\sigma3=\sigmab23
\begin{align} F12&=
2}\left[1-\sigma | |
\cfrac{1}{2\sigma | |
b12 |
(F1+F2)-\sigma
2(F | |
11 |
+F22)\right]\\ F13&=
2}\left[1-\sigma | |
\cfrac{1}{2\sigma | |
b13 |
(F1+F3)-\sigma
2(F | |
11 |
+F33)\right]\\ F23&=
2}\left[1-\sigma | |
\cfrac{1}{2\sigma | |
b23 |
(F2+F3)-\sigma
2(F | |
22 |
+F33)\right]\end{align}
F12,F13,F23
It can be shown that the Tsai-Wu criterion is a particular case of the generalized Hill yield criterion.
For a transversely isotropic material, if the plane of isotropy is 1–2, then
F1=F2~;~~F4=F5=F6=0~;~~F11=F22~;~~F44=F55~;~~F13=F23~.
\begin{align} F2(\sigma1+\sigma2)&+F3\sigma3+F22
2 | |
(\sigma | |
1 |
+
2) | |
\sigma | |
2 |
+F33
2 | |
\sigma | |
3 |
+F44
2 | |
(\sigma | |
4 |
+
2) | |
\sigma | |
5 |
+F66
2 | |
\sigma | |
6 |
\\ & +2F12\sigma1\sigma2+2F23(\sigma1+\sigma2)\sigma3\le1 \end{align}
F66=2(F11-F12)
In order to maintain closed and ellipsoidal failure surfaces for all stress states, Tsai and Wu also proposed stability conditions which take the following form for transversely isotropic materials
F22~F33-
2 | |
F | |
23 |
\ge0~;~~
2 | |
F | |
12 |
\ge0~.
For the case of plane stress with
\sigma1=\sigma5=\sigma6=0
F2\sigma2+F3\sigma3+F22
2 | |
\sigma | |
2 |
+F33
2 | |
\sigma | |
3 |
+F44
2 | |
\sigma | |
4 |
+2F23\sigma2\sigma3\le1
Fi,Fij
\sigma1c
\sigma1t
\sigma3c
\sigma3t
\tau23
\tau12
The Tsai–Wu criterion for closed cell PVC foams under plane strain conditions may be expressed as
F2\sigma2+F3\sigma3+F22
2 | |
\sigma | |
2 |
+F33
2 | |
\sigma | |
3 |
+2F23\sigma2\sigma3=1-k2
F23=-\cfrac{1}{2}\sqrt{F22F33
\sigma2c=4.6
\sigma2t=7.3
\sigma3c=6.3
\sigma3t=10
For aluminum foams in plane stress, a simplified form of the Tsai–Wu criterion may be used if we assume that the tensile and compressive failure strengths are the same and that there are no shear effects on the failure strength. This criterion may be written as
3~\tilde{J}2+(η2-
2 | |
1)~\tilde{I} | |
1 |
=η2
\tilde{J}2:=
2} | |
\tfrac{1}{3}\left(\cfrac{\sigma | |
1c |
-\cfrac{\sigma1\sigma2}{\sigma1c\sigma2c
The Tsai–Wu failure criterion has also been applied to trabecular bone/cancellous bone with varying degrees of success. The quantity
F12