# zbMATH — the first resource for mathematics

The problem of two plastic and heterogeneous inclusions in an anisotropic medium. (English) Zbl 0612.73010
We demonstrate an integral equation for the total local strain $$\epsilon^ T$$ in an anisotropic heterogeneous medium with incompatible strain $$\epsilon^ P$$ and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.
The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Born’s approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.

##### MSC:
 7.4e+06 Inhomogeneity in solid mechanics 7.4e+11 Anisotropy in solid mechanics
Full Text:
##### References:
 [1] Eshelby, J.D., Elastic inclusions and inhomogeneities, (), 89-140 [2] Kneer, G., Uber die berechnung der elastizitätsmoduin vielkristalliner aggregate mit textur, Phys. stat. sol., 9, 825-838, (1965) · Zbl 0151.46101 [3] Lin, S.C.; Mura, T., Elastic fields of inclusions in anisotropic media (II), Phys. stat. sol., A15, 281-285, (1973) [4] Willis, J.R., Anisotropic elastic inclusion problem, Q. J. mech. appl. math., 17, 157-174, (1964) · Zbl 0119.39602 [5] Sendeckyj, G.P., Ellipsoidal inhomogeneity problem, () [6] Moshovidis, Z.A., Two ellipsoidal inhomogeneities and related problems treated by the equivalent inclusion method, () [7] Sternberg, E.; Sadowsky, M.A., On the axisymmetric problem of the theory of elasticity for an infinite region containing two spherical cavities, J. appl. mech., 19, 19-27, (1952) · Zbl 0046.17307 [8] Chen, H.S.; Acrivos, A., The solution of the equations of linear elasticity for an infinite region containing two spherical inclusions, Int. J. solid struct., 14, 331-348, (1978) · Zbl 0377.73008 [9] Moshovidis, Z.A.; Mura, T., Two-ellipsoidal inhomogeneities by the equivalent inclusion method, J. appl. mech., 42, 847-852, (1975) · Zbl 0338.73015 [10] Berveiller, M.; Zaoui, A., Self consistent schemes for heterogeneous solid mechanics, () · Zbl 0395.73033 [11] Johnson, W.C., Elastic interaction of two precipitates subjected to an applied stress field, Met. trans., 14A, 2219-2227, (1983) [12] Berveiller, M.; Zaoui, A., Plasticité anisotrope des polycristaux métalliques, (), (In Press). · Zbl 0564.73041 [13] Zeller, R.; Dederichs, P.H., Elastic constants of polycrystals, Phys. stat. sol., B55, 831-842, (1973) [14] Mura, T., Displacement and plastic distortion fields produced by dislocations in anisotropic media, J. appl. mech., 38, 865-868, (1971) · Zbl 0241.73112 [15] Willis, J.R., Stress fields produced by dislocations in anisotropic media, Phil. mag., 21, 931-949, (1970) · Zbl 0198.58604 [16] Kröner, E., Bounds for effective elastic moduli of disordered materials, J. mech. phys. solids, 25, 137-155, (1977) · Zbl 0359.73020 [17] Laws, N., The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material, J. elasticity, 7, 91-97, (1977) · Zbl 0384.73011 [18] Ghahremani, F., Numerical evaluation of the stresses and strains in ellipsoidal inclusions in an anisotropic elastic material, Mech. res. commun., 4, 89-91, (1977) [19] Berveiller, M., Contributions à l’étude du comportement plastique et des textures de déformations des polycristaux métalliques, Thèse d’etat université Paris XIII villetaneuse, (1978) [20] Morris, P.R., Iterative scheme for calculating polycrystal elastic constants, Int. J. engng sci., 9, 917-920, (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.