An approximate 2-D solution for the shear-induced strain fields in eigenstrained cubic materials

Abstract

Summary. An approximate analytical 2-D solution for the strain field components Ell , s and e22 occurring in a cubic material due to a coherently bonded shear eigenstrained inclusion of cylindrical geometry was obtained by means of Continuous Fourier Transforms (CFT). A Discrete Fourier Transform (DFT) based numerical model was used in order to test the validity of the results. For the case where the cylindrical inclusion and the surrounding media are elastically homogeneous and the orientation of their principal crystal axes are the same, a correlation between the analytical and numerical models is demonstrated, both for strongly and weakly anisotropic materials. Moreover, the strain fields within the inclusion are shown to be of homogeneous isotropic type. Finally, an expression for the closed-form strain energy of two cylindrical inclusions at arbitrary radius and angle was derived, and then used to determine the minimum energy configuration for the system.

An approximate analytical solution for the stresses and strains in heterogeneous cubic materials

“…(5), the problem is reduced to the solution of an essentially linear system of equations for the quantity ε H ij which is solved analytically. Moreover the type of integrations which are left to be performed are from the same type as they already appeared in [3] and therefore, the problem is solved. The final solution, ε ij , is delivered as follows:…”

“…As outlined in detail in [2,3] one important feature of the application of the CFT is that the integral kernel, A H ijkl , given in Eqn. (4) can be expanded by a geometric series and therefore approximated by the quantityΓ H mnop , a homogeneous polynomial in the components of k. This approximation is crucial for the integrations which has to be performed later.…”

“…It allows to map the original problem onto an auxiliary problem, where as a simplification a homogeneous body is considered. [2,3]. For a particular geometry of the inhomogeneity it is illustrated how to derive a closed-form solution of this approximated IEQ.…”