Strain gradient solution for the Eshelby-type polygonal inclusion problem

The Eshelby-type problem of an arbitrary-shape polygonal inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. The Eshelby tensor for a plane strain inclusion with an arbitrary polygonal cross section is derived in a general form in terms of three potential functions, two of which have the same definitions as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as area integrals over the polygonal cross section, are first converted to three line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson’s ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. For homogenization applications, the area average of the new Eshelby tensor over the polygonal cross section is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its area average, three types of polygonal inclusions are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for all inclusion shapes considered vary with both the position and the inclusion size. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size. By applying the newly derived solution, a homogenization analysis of a composite reinforced by polygon-shaped fibers is performed. The induced strain in the composite is evaluated, and the effective stiffness of the composite is predicted. It is found that the values of the induced strain components approach from below those in a corresponding composite reinforced by circular fibers when the fiber size or the number of sides of the polygonal fiber increases. Also, the analysis reveals that the components of the effective stiffness tensor vary slightly with the fiber shape, and thus the simpler solution for the circular inclusion problem may be used to estimate the effective stiffness of a composite reinforced by regular polygon-shaped fibers.