Green's Function and Eshelby's Fields in Couple-Stress Elasticity

Conventional micromechanical schemes for estimating effective properties of composite materials in the matrix-inclusion type have no dependence upon absolute sizes of inclusions. However, there has been more and more experimental evidence that severe strain-gradient may result in remarkable size effects to mechanical behavior of materials. The strain field of an unbounded isotropic homogeneous elastic body containing a spherical inclusion subject to a uniform farfield stress may have very sharp strain-gradient within a surrounding matrix region of the inclusion, whenever the inclusion size would be very small. Consequently, the strain field variation in the whole matrix region of a composite with highly concentrated very small inclusions would be violent. Therefore, it is necessary to develop a micromechanical scheme in which the matrix phase is treated as a nonconventional material, and both the inclusion phases and the composite itself as an effective medium are treated as conventional materials. Such a scheme has been reported, with interesting applications. This scheme is based on the results of Green's functions and Eshelby's fields in couple-stress elastic theory. A thorough derivation of these results is given in the present paper. The main reason for choosing the couple-stress theory among various nonconventional theories of elasticity is that it contains the least number of material constants, in order to establish a simplest possible micromechanical scheme for taking account of absolute sizes.