Γ-convergene e for a geometrically exact Cosserat shell-model of defective elastic crystals

  • Patrizio Neff
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 516)


I consider the Γ-limit to a three-dimensional Cosserat model as the aspect ratio h > 0 of a flat domain tends to zero. The bulk model involves already exact rotations as a second independent field intended to describe the rotations of the lattice in defective elastic crystals. The Γ-limit based on the natural scaling consists of a membrane like energy and a. transverse shear energy both scaling with h, augmented by a curvature energy due to the Cosserat bulk, also scaling with h. A technical difficulty is to establish equi-coercivity of the sequence of functional as the aspect ratio h tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption. While the three-dimensional problem is well-posed for the Cosserat couple modulus μc ≥0, equi-coercivity needs a. strictly positive μc > 0. Then the Γ-limit model determines the midsorfaee deformation mH 1,2 (ω, ℝ3). For the true defective crystal case, however, μc=0 is appropriate. Without equi-coercivity, we obtain first an estimate of the Γ-lim in and Γ-lim sup which can be strengthened to the Γ-convergence result. The Reissner-Mindlin model is “almost” the linearization of the Γ-limit for μc=0.


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© CISM, Udine 2010

Authors and Affiliations

  • Patrizio Neff
    • 1
  1. 1.University Duisburg-EssenEssenGermany

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