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Applied Math Seminar
Applications of Gamma-convergence in Solid Mechanics
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A variety of problems in Solid Mechanics require minimizing an energy functional to find the equilibrium configuration of the system. Specific examples include the deformation of nonlinear elastic solids, homogenization of composite materials, the formulation of reduced kinematic theories such as shells and membranes, and the study of transitions between atomistic and continuum descriptions at essentially zero temperature. In most of these problems the functional shows some sort of nonconvexity that makes solutions nonexistent in a classical sense and the problems interesting. In this talk I will briefly overview some features of these problems and discuss basic concepts in the direct method of calculus of variations, such as Gamma-convergence, used to solve them. Then I will concentrate on a recent work on the coarse-graining of layers of atomic planes to obtain the macroscopic cohesive or fracture behavior of brittle solids. The macroscopic behavior is obtained by using a renormalization group operation and Gamma-convergence. |