Symbolic Algebra and Theorem Proving for Failure Criteria Reduction

The present paper reports on recent efforts of utilizing symbolic computing for identifying failure criteria cross reducibility from the perspective of theorem proving. Utilizing equational theorem proving algorithms and Gro¨bner Basis polynomial theorem provers implemented in Mathematica we have proven a number of interesting theorems related to the area of structural failure criteria for anisotropic and particularly orthotropic materials. The main contribution of this work is the demonstration of the tremendous utility of symbolic algebra for engineering applications as well as the demonstration of the idea that all failure criteria presented in the literature up to know can be proven under certain conditions to be special forms of general criteria relating to the strain energy density function associated with material continua. Two specific examples are presented and discussed along with a theorem proving the existence of a dual form of all stress space based criteria to equivalent one expressed in strain space.