Ellipsoidal Domain with Piecewise Nonuniform Eigenstrain Field in One of Joined Isotropic Half-Spaces
Consider an arbitrarily oriented ellipsoidal domain near the interface of an isotropic bimaterial space. It is assumed that a general class of piecewise nonuniform dilatational eigenstrain field is distributed within the ellipsoidal domain. Two theorems relevant to prediction of the nature of the induced displacement field for the interior and exterior points of the ellipsoidal domain are stated and proved. As a resultant the exact analytical expression of the elastic fields are obtained rigorously. In this work a new Eshelby-like tensor, A is introduced. In particular, the closed-form expressions for A associated with the interior points of spherical and cylindrical inclusion are derived. The stress field is presented for a single ellipsoidal inclusion which undergoes a Gaussian distribution of eigenstrain field and one of the principal axes of the domain is perpendicular to the interface. For the limiting case of spherical inclusion the closed-form solution is obtained and the associated strain energy is discussed. For further demonstration, two examples of two concentric spheres and three concentric cylinders with eigenstrain field distributions which are descriptive of the general class of functions defined in this paper. The effect of some parameters such as distance between the inclusion and the interface, and the ratio of the shear moduli of the two media on the induced elastic fields are examined.
KeywordsMicromechanics Bimaterial Nonuniform piecewise eigenstrain field Ellipsoidal domain
Mathematics Subject Classification (2000)74M25
Unable to display preview. Download preview PDF.
- 3.Eshelby, J.D.: Elastic inclusions and inhomogeneities. In: Sneddon, I.N., Hill, R. (eds.) Progress in Solid Mechanics 2, pp. 89–140. North-Holland, Amsterdam (1961) Google Scholar
- 4.Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Hague (1987) Google Scholar
- 6.Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Solids. Elsevier, New York (1999) Google Scholar
- 7.Sendeckyj, G.P.: Ellipsoidal Inhomogeneity Problem. Ph.D. dissertation, Northwestern University, Evanston, IL (1967) Google Scholar
- 15.Guell, D.L., Dunders, J.: Further results on centres of dilatation and residual stresses in joined elastic half spaces. In: Shaw, W.A. (ed.) Developments in Theoretical and Applied Mechanics, vol. 3, pp. 105–115. Pergamon, Oxford (1966) Google Scholar
- 22.Rongved, L.: Force interior to one of two jointed semi-infinite solids. In: Proc. 2nd Midwestern Conf. Solid Mech., pp. 1–13 (1955) Google Scholar
- 23.Ferres, N.M.: On the potentials of ellipsoids, ellipsoidal shells, elliptic laminae and elliptic rings of variable densities. Q. J. Pure Appl. Math. 14, 1–22 (1877) Google Scholar
- 24.Dyson, F.W.: The potentials of ellipsoids with variable density. Q. J. Pure Appl. Math. XXV, 259–288 (1891) Google Scholar
- 26.Mindlin, R.D.: Force at a point in the interior of a semi-infinite solid. In: Proceedings of First Midwestern Conference on Solid Mechanics, pp. 55–59 (1953) Google Scholar