Inflation, extension, torsion and shearing of an inhomogeneous compressible elastic right circular annular cylinder

We study the inflation, extension, torsion and shearing of an isotropic inhomogeneous compressible annular right circular cylinder. Current approaches to homogenization that appeal to an equivalence in the stored energies could lead to serious errors in the estimate for stresses in a inhomogeneous body as stresses depend on the derivatives of the stored energy with respect to the deformation gradient. This is a serious drawback as many a time failures are determined by the stresses. The study demonstrates that, in particular, great caution should be exercised in homogenization, especially if an inhomogeneous body is to be approximated by a homogeneous body belonging to the same class. Comparison of local measures, such as stresses, reveal that their values in the case of the inhomogeneous body and its homogeneous counterpart can be both qualitatively and quantitatively far apart. Even the differences in global measures like the axial load, torque, etc., are found to be significant between the inhomogeneous body and its homogeneous counterpart. It is also shown that the material parameters characterizing the homogenous approximation gleaned from correlations from different experiments, performed on the same inhomogeneous body, can be quite different.