A nonlinear elastic spherical inhomogeneity with a spring-type interface under a deviatoric far-field load

We study the three-dimensional problem associated with an incompressible nonlinear elastic spherical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to a uniform deviatoric load at infinity. The nonlinear elastic material can incorporate both power-law hardening and softening materials. The inhomogeneity-matrix interface is a spring-type imperfect interface characterized by a common interface parameter for both the normal and tangential directions. It is proved that the internal stresses and strains within the spherical inhomogeneity are unconditionally uniform. The original boundary value problem is reduced to a single non-linear equation which is proved rigorously to have a unique solution which can be found numerically. Furthermore, the neutrality of the imperfectly bonded nonlinear elastic spherical inhomogeneity is accomplished in an analytical manner. Finally, we prove the uniformity of the internal elastic field of stresses and strains inside an incompressible power-law hardening or softening nonlinear elastic ellipsoidal inhomogeneity perfectly bonded to an infinite linear isotropic elastic matrix subjected to uniform remote shear stresses and strains.