We consider a composite material containing an elastic elliptic inhomogeneity which is assumed to be perfectly bonded to a surrounding matrix of similar elastic material. In fact, both the inhomogeneity and the matrix belong to the same class of compressible hyperelastic materials of harmonic-type but each has its own distinct material properties. We consider finite plane deformations of the inhomogeneity-matrix system and obtain the complete solution when the system is subjected to classes of nonuniform remote (Piola) stress characterized by stress functions described by general polynomials of order n ≥ 1 in the corresponding complex variable z used to describe the matrix.
John, F.Plane Strain Problems for a Perfectly Elastic Material of Harmonic Type, Communications in Pure and Applied Mathematics, XIII239-290 (1960).
2.
Ru, C.-Q.On complex-variable formulation for finite plane elastostatics of harmonic materials. Acta Mechanica , 156219-234 (2002).
3.
Knowles, J.K. and Sternberg, E.On the singularity induced by certain mixed boundary value problems in linearized and nonlinear elastostatics. International Journal of Solids and Structures, 111173-1201 (1975).
4.
Varley, E. and Cumberbatch, E.Finite deformation of elastic materials surrounding cylindrical holes. Journal of Elasticity10, 341-405 (1980).
5.
Abeyaratne, R.Some finite elasticity problems involving crack tips. In Modelling Problems in Crack Tip Mechanics, pp. 3-24, ed. J. T. Pindera, University of Waterloo Press, Waterloo, Canada, 1983.
6.
Abeyaratne, R. and Horgan, C.O.The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials. International Journal of Solids and Structures, 20, 715-723 (1984).
7.
Jafari, A.H. , Abeyaratne, R. and Horgan, C.O.The finite deformation of a pressurized circular tube for a class of compressible materials . Journal of Applied Mathematics and Physics (ZAMP), 35, 227-246 (1984).
8.
Horgan, C.O.Some remarks on axisymmetric solutions in finite elastostatics for compressible materials. Proceedings of the Royal Irish Academy, Section A (invited paper in P. M. Quinlan Birthday issue), 89185-193 (1989).
9.
Li, X. and Steigmann, D.J.Finite plane twist of an annular membrane. Quarterly Journal of Mechanics and Applied Mathematics, 46601-625 (1993).
10.
Horgan, C.O.On axisymmetric solutions for compressible nonlinearly elastic solids. Journal of Applied Mathematics and Physics (ZAMP) (invited paper in P. M. Naghdi Birthday issue), 46S107-S125 (1995).
11.
Horgan, C.O.Equilibrium solutions for compressible nonlinearly elastic materials. Invited Chapter in Nonlinear Elasticity: Theory and Applications , London Mathematical Society Lecture Notes, Vol. 283, pp. 135-159, ed. R. W. Ogden and Y. Fu, Cambridge University Press, Cambridge, 2001.
12.
Wang, G.F., Schiavone, P. and Ru, C.-Q.Harmonic shapes in finite elasticity under non-uniform loading. ASME Journal of Applied Mechanics, 72691-694 (2005).
13.
Ogden, R.W. and Isherwood, D.A.Solution of some finite plane-strain problems for compressible elastic solids . Quarterly Journal of Mechanics and Applied Mathematics , 31219-249 (1978).
14.
Ru, C.-Q. , Schiavone, P., Sudak , L.J. and Mioduchowski , A.Uniformity of stresses inside an elliptic inhomogeneity in finite plane elastostatics , Int. J. Non-Linear Mech., 40281-287 (2005).
15.
Muskhelishvili, N.I.Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, The Netherlands, 1953.
