A new type of optimal microgeometry with properties that cannot be mimicked by a hierachical laminate. by Graeme Milton JTB320, 3:20pm Monday, October 6, 1997 Abstract It has been an outstanding question for more than 15 years as to whether the hierarchical laminate macrostructures form a complete set of representative geometries. Specifically given a composite, can one always find a laminate microgeometry (possibly consisting of laminates of laminates) built from the same component materials as the composite, such that the effective (elasticity or conductivity) tensor of the laminate material is the same as that of the given composite? Here, we show that the answer is no, by exhibiting a 7-phase counterexample for three-dimensional elasticity. The counterexample itself is interesting, providing a new class of optimal microgeometry. The counterexample is based on a counterexample of Sverak who showed that quasiconvexity and rank-one convexity are not generally equivalent. Requests for preprints and reprints to: milton@math.utah.edu This source can be found at http://www.math.utah.edu/research/