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Derivation of a rod theory for biphase materials with dislocations at the interface. (English) Zbl 1274.74064
Summary: Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of $$SO(3)$$. First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of $$\Gamma$$-convergence.

##### MSC:
 74B20 Nonlinear elasticity 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74N05 Crystals in solids 49J45 Methods involving semicontinuity and convergence; relaxation 46E40 Spaces of vector- and operator-valued functions
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##### References:
 [1] Ertekin, E.; Greaney, P.A.; Chrzan, D.C.; Sands, T.D., Equilibrium limits of coherency in strained nanowire heterostructures, J. Appl. Phys., 97, 114325, (2005) [2] Friesecke, G.; James, R.D.; Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55, 1461-1506, (2002) · Zbl 1021.74024 [3] Mora, M.G.; Müller, S., Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differ. Equ., 28, 161-178, (2007) · Zbl 1105.74016 [4] Ortiz, M.: Lectures at the Vienna Summer School on Microstructures, Vienna, September 25-29 (2000)
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