Some axioms and theorems in damage mechanics and fatigue of materials

The treatments of problems in continuum damage mechanics (CDM) that appear in the scientific literature are based on a number of more or less explicit assumptions. In this paper, these assumptions are reformulated and shown to form a single axiomatic framework of both the “internal variables” and the “multifield continuum” schemes: the fundamental features of both descriptions are thus recognized and correlated with the general principles of mechanics. These developments are based on the general theory of actions developed by Coleman and Owen (Coleman, B. D. and Owen, D. R. (1974). A mathematical foundation for thermodynamics. Archives of Rational Mechanical Analysis54, 1–104; (1975). On thermodynamics and elastic plastic materials. Archives of Rational Mechanical Analysis59, 25–51; (1977). On thermodynamics of semi-systems with restrictions on the accessibility of states. Archives of Rational Mechanical Analysis66, 173–181), which is presented in Section 2 in a way slightly different from usual. Then, five axioms for CDM are introduced in Section 3, and related theorems and corollaries are proved. Finally, the PalmgrenMiner's rule of fatigue damage and its limitations are reformulated in the context of the established general framework.