Derivation of a general three-dimensional crack-propagation law: A generalization of the principle of local symmetry

We derive a general crack-propagation law for slow brittle cracking, in two and three dimensions, using discrete symmetries, gauge invariance, and gradient expansions. Our derivation provides explicit justification for the ‘‘principle of local symmetry,’’ which has been used extensively to describe two-dimensional crack growth, but goes beyond that principle to describe three-dimensional crack phenomena as well. We also find that there are materials properties needed to describe the growth of general cracks in three dimensions, besides the fracture toughness and elastic constants previously used to describe cracking.