Tsai-Wu tensor failure criterion

The preceding biaxial failure criteria suffer from various inadequacies in their representation of experimental data. One obvious way to improve the correlation between a criterion and experiment is to increase the number of terms in the prediction equation. This increase in curvefitting ability plus the added feature of representing the various strengths in tensor form was used by Tsai and Wu [2-26]. In the process, a new strength definition is required to represent the interaction between stresses in two directions. 

Tsai and Wu postulated that a failure surface in six-dimensional stress space exists in the form 

The terms that are linear in the stresses are useful in representing different strengths in tension and compression. The terms that are quadratic in the stresses are the more or less usual terms to represent an ellipsoid in stress space. However, the independent parameter F,2 is new and quite unlike the dependent coefficient 2H = 1/X in the Tsai-Hill failure criterion on the term involving interaction between normal stresses in the 1- and 2-directions. 

Most components of the strength tensors are defined in terms of the engineering strengths already discussed. For example, consider a uniaxial load on a specimen in the 1-direction. Under tensile load, the engineering strength is Xj, whereas under compressive load, it is (for example, Xg = -400 ksi (-2760 MPa) for boron-epoxy). Thus, under tensile load. 

Similar reasoning, along with our observation that the shear strength in principal material coordinates is independent of shear stress sign, leads to 

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Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

Figure 2-45 Tsai-Wu Tensor Failure Criterion (After Pipes and Cole [2-25])...

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