Tsai–Wu failure criterion

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 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 where i , j = 1 … 6 {displaystyle i,j=1dots 6} and repeated indices indicate summation, and F i , F i j {displaystyle F_{i},F_{ij}} are experimentally determined material strength parameters. The stresses σ i {displaystyle sigma _{i}} are expressed in Voigt notation. If the failure surface is to be closed and convex, the interaction terms F i j {displaystyle F_{ij}} must satisfy which implies that all the F i i {displaystyle F_{ii}} terms must be positive. For orthotropic materials with three planes of symmetry oriented with the coordinate directions, if we assume that F i j = F j i {displaystyle F_{ij}=F_{ji}} and that there is no coupling between the normal and shear stress terms (and between the shear terms), the general form of the Tsai–Wu failure criterion reduces to Let the failure strength in uniaxial tension and compression in the three directions of anisotropy be σ 1 t , σ 1 c , σ 2 t , σ 2 c , σ 3 t , σ 3 c {displaystyle sigma _{1t},sigma _{1c},sigma _{2t},sigma _{2c},sigma _{3t},sigma _{3c}} . Also, let us assume that the shear strengths in the three planes of symmetry are τ 23 , τ 12 , τ 31 {displaystyle au _{23}, au _{12}, au _{31}} (and have the same magnitude on a plane even if the signs are different). Then the coefficients of the orthotropic Tsai–Wu failure criterion are The coefficients F 12 , F 13 , F 23 {displaystyle F_{12},F_{13},F_{23}} can be determined using equibiaxial tests. If the failure strengths in equibiaxial tension are σ 1 = σ 2 = σ b 12 , σ 1 = σ 3 = σ b 13 , σ 2 = σ 3 = σ b 23 {displaystyle sigma _{1}=sigma _{2}=sigma _{b12},sigma _{1}=sigma _{3}=sigma _{b13},sigma _{2}=sigma _{3}=sigma _{b23}} then The near impossibility of performing these equibiaxial tests has led to there being a severe lack of experimental data on the parameters F 12 , F 13 , F 23 {displaystyle F_{12},F_{13},F_{23}} . It can be shown that the Tsai-Wu criterion is a particular case of the generalized Hill yield criterion.