Instabilities of embedded cylindrical inclusions undergoing isotropic swelling or growth

Abstract Infinite circular cylindrical elastic inclusions, or rods, embedded in an unbounded elastic matrix display various modes of instability when they undergo sufficiently large expansion due to either swelling or volumetric growth. In this letter two modes of instability are examined: sinusoidal axisymmetric modes and sinusoidal bending modes. The rod and the matrix are neo-Hookean materials, and the full range of the modulus ratio of rod to matrix is considered. In the primary case examined, deformation is driven by an isotropic volumetric expansion, or transformation, of the rod. A three-dimensional bifurcation analysis of the rod constrained by the matrix reveals the onset of the critical instability mode as dependent on the modulus ratio. Comparisons with related results are discussed, including the compressive buckling of a stiff rod in a compliant matrix and the other limit when the modulus of the rod is very small compared to that of the matrix and behaves effectively as a fluid exerting pressure on the wall of the matrix cavity. 1