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\(\Gamma \)-convergence for incompressible elastic plates. (English) Zbl 1161.74038
Summary: We derive a two-dimensional model for elastic plates as a \(\Gamma \)-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by G. Friesecke et al. [Comm. Pure Appl. Math. 55, No. 11, 1461–1506 (2002; Zbl 1021.74024)]. We extend their result to the case of \(p\) growth at infinity with \(p \in [1, 2)\), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in \(W^{2,2}\) isometries, which was obtained by M. R. Pakzad [J. Differ. Geom. 66, No. 1, 47–69 (2004; Zbl 1064.58009)] for convex domains and by P. Hornung [C. R. Acad. Sci., Paris, Sér. I, Math. 346, 189–192 (2008)] for piecewise \(C^{1}\) domains.

MSC:
74K20 Plates
74G65 Energy minimization in equilibrium problems in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)
49J45 Methods involving semicontinuity and convergence; relaxation
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