Analytical solution for stresses and deformations in and near an elastic spherical inclusion in an infinite elastic solid
Authors: Hans Amstutz, Michael Vormwald
Materials Mechanics Group, Technische Universität Darmstadt, Franziska-Braun-Straße 3, 64287 Darmstadt
In the present work, solutions are recapitulated according to the theory of elasticity for the deformations of spherical inclusion in an inﬁnite matrix under remote uniform axial and axial-symmetrical radial tension. Stress ﬁelds in the inclusion and at the interface in the matrix are provided, too. It is shown that the sphere is deformed to a spheroid under any of the loading cases considered. Due to the axial-symmetric setup of the problem, the deformation is fully described by the two displacement values at line segments on the principal axes of the spheroid. The displacements depend on the applied remote load and on two traction ﬁelds at the inclusion-matrix interface. For any combination of inclusion and matrix stiﬀness, the condition of compatibility of deformations yields a system of two linear equations with the two magnitudes of the tractions as unknowns. Thus, the problem is reduced to a formulation for solving a two-fold statically indetermined structure. The approach follows a procedure outlined in detail in reference  for solving a different problem of the Theory of Elasticity.
The system is solved and the exact solution of the general spherical inclusion problem with diﬀering stiﬀness in terms of Young’s moduli and Poisson’s ratios of inclusion and matrix is presented.
 Hans Amstutz, Michael Vormwald: Analysis of an elastic elliptical inclusion in an infinite elastic plate under uniform remote tension based on the solution of the corresponding cavity problem. J. of Strain Analysis for Engineering Design 52(8) 515-527, 2017