Optimal design of composite structures for strength and stiffness: an inverse homogenization approach

We introduce a rigorously based numerical method for compliance minimization problems in the presence of point wise stress constraints. The method is based on new multiscale quantities that measure the amplification of the local stress due to the microstructure. The design method is illustrated for two different kinds of problems. The first identifies suitably graded distributions of fibers inside shaft cross sections that impart sufficient overall stiffness while at the same time adequately control the amplitude of the local stress at each point. The second set of problems are carried out in the context of plane strain. Here we recover a novel class of designs made from locally layered media for minimum compliance subject to point wise stress constraints. The stress constrained designs place the more compliant material in the neighborhood of stress concentrators associated with abrupt changes in boundary loading and reentrant corners.