Numerical Verification Of Eshelby Tensor For Weak Non-circular Inclusion And Its Applications

The strain fieldε_ij induced by eigenstrainε_ij^* in an inclusion is one of classical mechanics problems. Their relation can be expressed as by a fourth-order tensor S_ijkl (i.e.,ε_ij = S_ijklε_kl^*), where S_ijkl is called as the Eshelby tensor. Eshelby (1957) gave the expression of S_ijkl for ellipsoidal inclusions. The expression of Eshelby is often used to determine the mechanics properties of heterogeneous materials by the self-consistent method. However, inclusions in heterogeneous materials are often non-ellipsoidal in shape. Herein, I will check an expression of the weakly non-circular inclusions by a finite element program (ANSYS). We apply the expression of the weakly non-circular inclusions to study the mechanics properties of heterogeneous materials. My main work is as follows:1. Eshelby gave the Eshelby tensor of ellipsoidal inclusions. Hill, Mori, and Tanaka et. al apply the Eshelby tensor to study the macroscopic and mesoscopic relations of heterogeneous materials. However, inclusions in heterogeneous materials are often non-ellipsoidal. Hence, we make use of the expression of the Eshelby tensor for any arbitrary weakly non-circular inclusion to give the stress field and strain field inside and outside the inclusion. We use a finite element program (ANSYS) to check the expressions of our stress field and strain field. We discuss the influence of inclusion's shapes on the stress field and the strain field.2. The average value of the Eshelby tensor on the inclusion in the inclusion is the average Eshelby tensor. The average Eshelby tensor is very simple in form. The Eshelby tensor depend only on the Poisson's ratio v and the shape coefficients a₂,a₄,b₂,b₄. We use ANSYS (a FEM program) to verify the validity of the average Eshelby tensor. The computation of ANSYS shows that the numerical simulation results are close to those of the expression of the average Eshelby tensor.3. By combining the average Eshelby tensor with the self-consistent method and the equivalent inclusion method, we give an expression of the macroscopic constitutive relation for heterogeneous materials whose inclusions are weakly non-circular in shape. The expression include the effects of the shape coefficients a₂,a₄,b₂,b₄. The results of our FEM show that, for weak non-circular inclusions, our average Eshelby tensor can be used to determine the macro-constitutive relation of the heterogeneous materials when their inclusions are weakly non-circular in shape.In this paper, by the FEM, self-consistent method, and the equivalent inclusion method, we verify the corrections and validities of our expressions on the Eshelby tensor, the stress and strain field and the average Eshelby tensor for weakly non-circular inclusions. We apply these expressions to study mechanics properties of heterogeneous materials.