On deformations of compressible hyperelastic material.
PhD thesis, University of Glasgow.
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We consider the character of several finite deformations of compressible isotropic, nonlinear hyperelastic materials, specifically azimuthal shear of a thick-walled circular cylindrical tube, the bending deformation of a rectangular block and axial shear of a thick-walled circular cylindrical tube. For each problem the equilibrium equations are applied to the special case of isochoric deformation, and explicit necessary and sufficient conditions on the strain-energy function for the material to admit such a deformation are obtained. These conditions are examined for several strain-energy functions and in each case complete solutions of the equilibrium equations are obtained. The predictions of the shear response for different strain-energy functions are compared using numerical results to show the dependence of the applied shear stress on the resulting macroscopic deformation.
It is then shown how consideration of isochoric deformations in compressible elastic materials provides a means of generating classes of strain-energy functions for which closed-form solutions can be found for incompressible materials. For the problem of bending deformation we find that isochoric deformation is not possible in a compressible material. The conditions for a non-isochoric bending deformation to be admitted by the equilibrium equations are then examined for each of three classes of compressible isotropic materials. Explicit solutions for each case are then derived. Finally, we consider an incremental displacement superimposed on the azimuthal shear of a circular cylindrical tube. Numerical results are obtained to show the incremental displacement and nominal stresses for a special material when the internal boundary is subject to an incremental displacement.
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