FE ANALYSIS OF STRESS AND STRAIN FIELDS IN FINITE RANDOM COMPOSITE BODIES: APPLICATION TO CRACK TIP FIELD

摘要

Allowance for non-local interactions between heterogeneities in microscopically heterogeneous materials is necessary when the spatial variation of the load or the dimensions of the body, relative to the scale of the microstructure, cannot be ignored. This is the case in the vicinity of any stress concentrator, such as a crack tip. If the microstructure is specified precisely (e.g., if it is known to be periodic), an exact calculation can be performed, at least in principle. If however, the microstructure is random, some other approach is necessary. The present authors (Luciano and Willis) developed an approach, based on a stochastic variational principle. It generated an integral equation, whose kernel was related to a Green's function defined for the body in question. Such a Green's function can only be found explicitly for simple geometries. Here, the problem is approached via finite elements. The stochastic variational principle is projected directly onto a finite-element basis from the outset. In this framework, there is no advantage to developing non-local effective relations per se, though it can be done; rather the formulation leads directly to equations that provide representations for the stress and strain fields in any realization of the medium, from which any statistical average or local quantities can be computed.