Generalized boundary conditions on representative volume elements and their use in determining the effective material properties

When determining an effective stress–strain law by means of the representative volume element (RVE) method, one needs to subject the RVE to the effective strains by appropriate boundary conditions (BC). Usually, classical BC that prescribe a homogeneous stress or strain field at the boundary or a periodic unit cell are used. In this work, we discuss generalized BC, which involve the partitioning of the RVE boundary into n parts. It is demonstrated that the classical BC are contained as special cases, and that the Hill–Mandel-condition holds for all partitionings. ore or less fine surface partitioning, the generalized BC allow for a smooth scaling between the extremal cases of homogeneous stress or homogeneous strain BC. Further, by an irregular surface partitioning, one can obtain stochastic BC with an elastic stiffness close to the periodic/antipodal BC, but with a higher resistance against localization. This has been demonstrated by examining a softening example material. A test of plausibility for a RVE is to apply it to a homogeneous microstructure. Then, the microscale material law should be conducted directly to the macroscale. In case of softening microscale materials, this test works only for homogeneous strain BC. For homogeneous stress- and periodic/antipodal BC, localization occurs, accompanied by a drastic deviation from the expected stress–strain curve. From the generalization, one can derive stochastic BC that combine the moderate elastic stiffness of periodic BC with the high resistance against localization of homogeneous strain BC.