Extension, inflation and torsion of a residually-stressed circular cylindrical tube

In this paper, we provide a new example of the solution of a finite deformation boundary-value problem for a residually stressed elastic body. Specifically, we analyse the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress, the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress

On deformations of compressible hyperelastic material

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

Likely equilibria of stochastic hyperelastic spherical shells and tubes

In large deformations, internally pressurised elastic spherical shells and tubes may undergo a limit-point, or inflation, instability manifested by a rapid transition in which their radii suddenly increase. The possible existence of such an instability depends on the material constitutive model. Here, we revisit this problem in the context of stochastic incompressible hyperelastic materials, and ask the question: what is the probability distribution of stable radially symmetric inflation, such that the internal pressure always increases as the radial stretch increases? For the classic elastic problem, involving isotropic incompressible materials, there is a critical parameter value that strictly separates the cases where inflation instability can occur or not. By contrast, for the stochastic problem, we show that the inherent variability of the probabilistic parameters implies that there is always competition between the two cases. To illustrate this, we draw on published experimental data for rubber, and derive the probability distribution of the corresponding random shear modulus to predict the inflation responses for a spherical shell and a cylindrical tube made of a material characterised by this parameter.Comment: arXiv admin note: text overlap with arXiv:1808.0126

Preface for Millard Beatty

Professor Beatty has contributed a wide variety of research papers and book articles on topics in finite elasticity, continuum mechanics and classical mechanics, including some fundamental experimental work. His works are clear and informative and expose a didactic quality. In the following, we briefly touch upon some of the highlights of his research involvement throughout the years