Elastic Plate Deformation with Transverse Variation of Microrotation, arXiv:0811.1534v1

Source: arXiv


The purpose of this paper is to present a new mathematical model for the deformation of thin Cosserat elastic plates. Our approach, which is based on a generalization of the classical Reissner plate theory, takes into account the transverse variation of microrotation of the plates. The model assumes polynomial approximations over the plate thickness of asymmet-ric stress, couple stress, displacement, and microrotation, which are con-sistent with the elastic equilibrium, boundary conditions and the consti-tutive relationships. Based on the generalized Hellinger-Prange -Reissner variational principle and strain-displacement relation we obtain the com-plete theory of Cosserat plate. We also proved the solution uniqueness for the plate boundary value problem.

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Available from: Lev Steinberg, Sep 30, 2015
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    ABSTRACT: We investigate weaker than usual constitutive assumptions in linear Cosserat theory still providing for existence and uniqueness and show a continuous dependence result for Cosserat couple modulus μc → 0.This result is needed when using Cosserat elasticity not as a physical model but as a numerical regularization device. Thereafter it is shown that the usually adopted material restrictions of uniform positivity for a linear Cosserat model cannot be consistent with experimental findings for continuous solids: the analytical solutions for both the torsion and the bending problem in general predict an unbounded stiffness for ever thinner samples. This unphysical behaviour can only be avoided for specific choices of parameters in the curvature energy expression. However, these choices do not satisfy the usual constitutive restrictions. We show that the possibly remaining linear elastic Cosserat problem is nevertheless well-posed but that it is impossible to determine the appearing curvature modulus independent of boundary conditions. This puts a doubt on the use of the linear elastic Cosserat model (or the geometrically exact model with μ>c > 0) for the physically consistent description of continuous solids like polycrystals in the framework of elasto-plasticity. The problem is avoided in geometrically exact Cosserat models if the Cosserat couple modulus μc is set to zero.
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    ABSTRACT: After discussing various variational statements of the three-dimensional problem, we describe the development of two-dimensional sixth-order theories by Bolle, Hencky, Mindlin, and Reissner which take account of the effect of transverse shear deformation. Additionally, we report on an early analysis by Levy, on a direct two-dimensional formulation of sixth-order theory, on constitutive coupling of bending and stretching of laminated plates, on higher than sixth-order theories.
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