Exact Equilibrium Solutions for Linearly Elastic Plate-Like Bodies

A three-dimensional, explicit and exact, equilibrium solution is derived for transversely isotropic, linearly elastic bodies in the form of a right cylinder simply supported over its lateral boundary and loaded over its end faces. This solution, which generalizes an earlier proposal of Levinson in various respects, is to serve as a benchmark for the corresponding plate theories. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Variational Approaches to Plate Modes

The three basic functionals of potential energy, complementary energy and Hellinger–Prange–Reissner are usedto obtain a rational derivation of Reissner–Mindlinplate models, starting from the three-dimensional theory.We show that the models so obtained are instances of the same plate theory; nevertheless, due to the different constitutive relations governing their response, they mimic the three-dimensional behaviour in three different manners.     

Saint-Venant's problem with Voigt's hypotheses for anisotropic solids

We seek for a solution of Saint-Venant's problem for inhomogeneous and anisotropic materials under the assumptions, introduced by Voigt, that the stress is either constant along the axis of the cylinder or depends linearly on the axial coordinate. We first prove the uniqueness of the solution in terms of resultants, then we exhibit an explicit formula for such a solution; we show finally how Clebsch's hypothesis, that the stress vector on axial planes is parallel to the axis, is compatible with Voigt's hypotheses provided that the symmetry group of the material comprising the cylinder contains the reflections on the cross-section.     

Constitutive identification of affine rods

We present an identification procedure based on a version of the de Saint-Venant semi-inverse method for transferring constitutive information from a two-dimensional Cauchy continuum model to a one-dimensional affine rod model of a two-dimensional slender body. To test and evaluate the accuracy of the identification procedure proposed, we compare the dispersion and frequency curves predicted by the two models for an isotropic material and for an anisotropic one.     

Marcus Integration Method for Shearable Plates

A method proposed by Marcus [5] to integrate the classical biharmonic equation of simply supported, unshearable plates with polygonal contour is extended to apply to shearable plates as well, provided the supporting device is of the ‘hard’ type.     

Exact Solutions for Linearly Electroelastic Plate-Like Bodies

A three-dimensional, explicit and exact solution is derived for a transversely isotropic, linearly electroelastic body in the form of a right cylinder of arbitrary cross section, being simply supported and connected to ground over its lateral boundary, and subject to an arbitrary distribution of force and charge over its end faces. When electric phenomena are ignored, this solution reduces to the solution given in [9] for linearly elastic plate-like bodies.     

On Korn's Inequality in the Second Case

A simple proof of Korn's inequality in the second case is given. Technicalities are minimized by using both the compatibility condition between the symmetric and skew parts of the gradient of a vector field and a result of Magenes and Stampacchia [10] and Necăs [11] asserting the closure of the range of the gradient operator on L²-functions. This revised version was published online in August 2006 with corrections to the Cover Date.