Tensor Polynomial Failure Criterion: Coefficient Limits Based on Convexity Requirements

Abstract

The tensor polynomial failure criterion is a widely used tool for predicting failure of unidirectional composite lamina. Typically, the tensor polynomial failure criterion is used in the quadratic form, commonly referred to as the Tsai-Wu failure criterion. Some recent work has presented the cubic form as an alternative to the widely used Tsai-Wu failure criterion. This paper presents limits on the coefficients of the tensor polynomial failure criterion for the quadratic, cubic, and quartic forms based on convexity of the failure surface in the stress domain. The results for the quadratic form of the polynomial include those commonly found in the literature and a further limit that is not normally found due to assumptions on the affect of shear. Further, the limits on the quadratic coefficients are used to prove that the quadratic tensor polynomial failure function is invariant with respect to coordinate system transformation. Results for the cubic tensor polynomial show that, when the entire stress space is considered, the cubic tensor polynomial cannot generate a convex failure surface, and is therefore not an acceptable failure criterion. However, using constraints identified in this paper, it is possible to determine regions in the stress domain where the failure criterion is convex and gives valid results. A further conclusion is that the "hybrid method" of Tennyson, MacDonald, and Nanyaro, used to determine the cubic interaction coefficients, yields results that violate convexity requirements and therefore should not be considered as a valid method. Results for the quartic form of the tensor polynomial failure criterion indicate the cubic terms may be included in the polynomial provided the appropriate quartic terms are present. The derivation of the limits on the coefficients is presented in an appendix and the limits are presented in the body. Illustratory examples of quartic tensor polynomial are presented.