10 infinity. Two conditions given in Eq. (18) are derived that ensure that the internal field is a linear function of the two coordinates x and y. These two conditions can be considered as restrictions on the remote non-uniform stress field for given material and geometric parameters of the composite. The specific expressions of the two conditions, which are generally rather tedious, will become very concise for the four special cases: (i) the interphase and the matrix have the same elastic constants; (ii) the interphase layer is extremely compliant; (iii) the interphase layer is rather stiff; (iv) the three-phase inclusion is circular. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No: 11272121), the Innovation Program of Shanghai Municipal Education Commission, China (Grant No: 12ZZ058), and the National Basic Research Program of China (Grant No. 2009CB623204). References [1] C.Q. Ru, Effect of interphase layers on thermal stresses within an elliptical inclusion. J Appl Phys, 84 (1998) 48724879. [2] C.Q. Ru, Three-phase elliptical inclusions with internal uniform hydrostatic stresses. J Mech Phys Solids, 47 (1999) 259273. [3] J.C. Luo, C.F. Gao, Stress field of a coated arbitrary shaped inclusion. Meccanica, 46 (2011)10551071. [4] X. Wang, X.L. Gao, On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite. Z Angew Math Phys, 62 (2011)11011116. [5] X. Wang, P. Schiavone, Three-phase inclusions of arbitrary shape with internal uniform hydrostatic stresses in finite elasticity. ASME J Appl Mech, 79 (2012) art. no. 041012 (6 pages). [6] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Groningen, Noordhoff, 1953. [7] G.S. Bjorkman, R. Richards, Harmonic holesan inverse problem in elasticity. ASME J Appl Mech, 43 (1976) 414418. [8] L.T. Wheeler, The problem of minimizing stress concentrations at a rigid inclusion. ASME J Appl Mech, 52 (1985) 8386. [9] G.S. Bjorkman, R. Richards, Harmonic holes for nonconstant fields. ASME J Appl Mech, 46 (1979) 573576. [10] G.F. Wang, P. Schiavone, C.Q. Ru, Harmonic shapes in finite elasticity under nonuniform loading. ASME J Appl Mech, 72 (2005) 691694.
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