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Internal constraints, reactive stresses, and the Timoshenko beam theory
No full textIt is shown that the kinematical assumptions of the Timoshenko theory for shearable beams can be regarded as internal constraints, some involving the first, others the second deformation gradient; such constrains are thought of as maintained by reaction stresses and hyperstresses of the type occurring in non-simple materials of grade 2. It is discussed how these reactions can be used to better the first approximation of the unknown equilibrium stress field in the three-dimensional body modelled by the Timoshenko beam, an appoximation which is based on the solution of the one-dimensional Timoshenko problem.
The equations of Reissner-Mindlin plates obtained by the method of internal constraints
No full textThe aim of this paper is to give the title theory of shearable plates a precise and exact position with
respect to three-dimensional linear elasticity. We assume that the Reissner-Mindlin representation of the displacement
field hold in a 3-D body in the shape of a plate, and discuss how a constitutive response consistent with such
a representation should be chosen. We find that the Reissner-Mindlin plate theory results from mere integration
over the thickness of the equilibrium equations of a cylindrical body made of a linearly elastic material which is
both transversely inextensible and transversely isotropic
Justification of the Reissner-Mindlin plate theory through variational convergence
No full textWe provide a justification of the Reissner–Mindlin plate theory, using linear three- dimensional elasticity as framework and Γ-convergence as technical tool. Essential to our developments is the selection of a transversely isotropic material class whose stored energy depends on (first and) second gradients of the displacement field. Our choices of a candidate Γ-limit and a scaling law of the basic energy functional in terms of a thinness parameter are guided by mechanical and formal arguments that our variational convergence theorem is meant to validate mathematically
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