Stochastic evolution of microcracks in continua

Abstract

A multifield description of continua endowed by diffused microcracks, recently proposed by the authors [Mariano and Augusti, Math. Mech. Sol. 3 (1998) 183–200] within a deterministic context, is extended to cover some of the stochastic aspects of microcrack distribution and evolution. The microcrack state of the continuum is described by a tensor field defined as the second order approximation of the microcrack density function M( P,n,t ) that represents the distribution of the number of microcracks in each direction n in a neighborhood ( mesodomain) of the point P : m( P,n,t ) is considered as a submartingale process in each point P , leading to a stochastic field over the continuum body. Generalized measures of internal actions represent the interactions voids-voids and void-matrix. They perform work in the variation of relevant conjugated fields. Invariance requirements on the overall power allows to deduce both the usual balance of forces and the balance of generalized internal actions, obtaining a model different from the classical internal variable models not only conceptually but also formally. The introduction of a damage entropy flux, whose divergence is the production of configurational entropy, allows to include damage criteria within the context of Clausius-Duhem inequality. The basic features of an appropriate finite-element discretization are formalized in the case of linear elastic brittle materials: the stochastic distribution of microcracks is considered through the stochastic nature of the elements of the stiffness matrix. Among other assets, the multifield approach overcomes the mesh dependence of the numerical results obtained on the basis of the more used internal variable schemes.