On the asymptotically stochastic computational modeling of microstructures

We consider a class of alloys and ceramics with equilibria described by non-attainable infima of non-quasiconvex variational integrals. Such situations frequently arise when atomic lattice structure plays an important role at the mesoscopic continuum level. We prove that standard variational approaches associated with gradient based relaxation of non-quasiconvex integrals in Banach or Hilbert spaces are not capable of generating relaxing sequences for problems with non-attainable structure. We introduce a variational principle suitable for the computational purposes of approaching non-attainable infima of variational integrals. We demonstrate that this principle is suitable for direct calculations of the Young Measures on a computational example in one dimension. The new variational principle provides the possibility to approximate crystalline microstructures using a Fokker–Planck equation at the meso-scale. We provide an example of such a construction. © 2003 Elsevier B.V. All rights reserved.