Small deformations superimposed upon the symmetrical expansion of a spherical shell

Abstract

The problem of small deformations of isotropic incompressible hyperelastic materials superimposed upon the symmetrical expansion of a spherical shell is re-examined making use of a recent new solution applying to the material with a 'modified' Varga strain-energy function. This particular strain-energy function has not been previously used in the literature and comprises the Varga strain-energy function with an additional term involving the reciprocals of the principal stretches. This new term enhances the physical applicability of the Varga strain-energy function and this aspect is briefly examined in the Appendix. Use of the new solution in the present context appears to conflict with previous work of the first author. This apparent conflict is fully resolved and it is shown that previously-used expansions for the pressure function and the response coefficients are only valid for restricted forms of the strain-energy function. Generally, an additional term arises in the expansion of these quantities, which appears not to have been identified by previous authors. This particular problem illustrates the need to exercise caution in formulating the correct structure of the expansions, which almost certainly is a relevant issue for many other small on large problems. The solutions obtained are utilized for the problem of the determination of the critical pressures for the buckling of thick-walled spherical shells under uniform external pressure and some illustrative numerical results are presented for a particular buckling mode.