The elastic field and the interaction energy of anisotropic precipitates in an anisotropic matrix are investigated. The matrix is subjected to an applied strain field or the precipitates undergo stress-free transformation strains. Using the tensor elastic Green function, an integral equation is formulated for anisotropic precipitates embedded in an infinite anisotropic matrix. This equation is reduced to a system of algebraic equations by expanding the strains in Taylor series about the arbitrary points in the precipitates, and the strain distributions inside the precipitate are obtained. The strains outside the precipitates can also be obtained since the integral equation is written in terms of generalized functions. Employing the procedure described above, the following problems are specifically solved:
ⅰ) Non-uniform problem of an ellipsoidal precipitate in an anisotropic matrix;
The expressions for the strain field throughout the entire region, very effective for numerical computations, are obtained. It is directly proved that the final state of the strain in an ellipsoidal precipitate is a polynomial of degree M in $x_i$, when an applied strain and/or a stress-free transformation strain are polynomials of degree M in $x_i$. Illustrations are made for the simple cases (M≤1). The discontinuities of the strains across the boundary between the precipitate and the matrix are represented explicitly, and the asymptotic behavior of the strains at the point far from the precipitate is obtained. It is shown that the strains at the points far form the presipitate ($\mid{x}\mid ≪ 1$) are of order ($\mid{x}\mid^{-3}$) or higher.
ⅱ) Approximation of strain field within two anisotropic spherical precipitates embedded in an anisotropic matrix;
Since the present method requires no symmetry condition between the two spherical precipitates, it is possible to obtain the strain distribution within the precipitates when the elastic constants and/or the sizes of the precipitates are different from each other. The strains are expanded about arbitrary points, giving more accurate results than those presented elsewhere.
ⅲ) Elastic interactions between two anisotropic spherical precipitates and their asymptotic behaviors;
It is confirmed that an interaction energy between two spherical anisotropic precipitates (not necessarily identical to each other) separated by a large distance $\mid{d}\mid$ is of order ($\mid{d}\mid^{-3}$) or higher. Numerical computations are performed for the case of two identical misfitting anisotropic precipitates in an anisotropic matrix of cubic symmetry. It is observed that far-field interactions ($\mid{d}$\mid≫ 2a$) depend on the anisotropic factor $H = C_{12} + 2C_{44} - C_{11}$ of the matrix. When the factor H is positive, the far-field interactions are attractive between the two precipitates aligned along 〈001〉 directions of the matrix, while repulsive along 〈111〉 directions. In contrast, the far-field interactions are repulsive along 〈001〉 directions, and attractive 〈111〉 directions, when the factor H of the matrix is negative. It appears that the near-field ($\mid{d}\mid\simeq 2a$) interactions are strongly dependent on the direction of the alignment of the precipitates and the difference between the elastic moduli of the precipitates and the matrix.
