Abstract
The quadratic function of the original Tsai–Wu failure criterion for transversely isotropic materials is re-examined in this paper. According to analytic geometry, two of the troublesome coefficients associated with the interactive terms—one between in-plane direct stresses and one between transverse direct stresses—can be determined based on mathematical and logical considerations. The analysis of the nature of the quadratic failure function in the context of analytic geometry enhances the consistency of the failure criterion based on it. It also reveals useful physical relationships as intrinsic properties of the quadratic failure function. Two clear statements can be drawn as the outcomes of the present investigation. Firstly, to maintain its basic consistency, a failure criterion based on a single quadratic failure function can only accommodate five independent strength properties, viz. the tensile and compressive strengths in the directions along fibres and transverse to fibres, and the in-plane shear strength. Secondly, amongst the three transverse strengths—tensile, compressive and shear—only two are independent.
Highlights
Ever since Tsai and Wu proposed their failure criterion [1] based on a quadratic failure function, the polynomials employed to construct failure criteria have been mostly kept to the second order
A sensible question does not seem to have ever been asked: ‘Has the quadratic failure function been understood well enough before increasing or abandoning any further efforts along this line of development?’ It is true that quadratic functions are well understood in analytic geometry as a branch of mathematics, and yet it will be revealed in the present paper that the established mathematics has not been properly utilised in the important subject of composite failure
Having achieved what was presented in [6], the objective of the present paper is to address some intrinsic relationships in the quadratic failure function that will have profound implications for the understanding of the strength of transversely isotropic materials in general
Summary
Ever since Tsai and Wu proposed their failure criterion [1] based on a quadratic failure function, the polynomials employed to construct failure criteria have been mostly kept to the second order. Whilst one could legitimately argue for higher orders of polynomials along the line, as was suggested in [2], or to partition the failure function according to the failure modes, as was attempted in [3], many are on the verge of abandoning all such theories developed on a phenomenological basis, having been discouraged by their unsatisfactory performance. The well-established conclusions of analytic geometry have not been appropriately recognised in the formulations of failure criteria for composites based on quadratic functions. Before this aspect has been appropriately examined and evaluated, it would surely be a premature decision to propose more complicated arrangements for the failure function or to ditch every account of phenomenological criteria based on macroscopic stresses or strains completely. Blaming the phenomenological nature alone for the deficiencies in the existing criteria is a far too quick and easy escape
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