We present a rigorous proof of polynomial eigenstress inducing polynomial strain of the same degree in an ellipsoidal inclusion. The coefficients of the induced polynomial strain are explicitly given in terms of elliptic integrals. The analogous Eshelby's tensor for polynomial eigenstress is also computed, and applied to solve the inhomogeneous problem as an example of applications.
EshelbyJD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc Lond, Ser A1957; 241: 376–396.
2.
EshelbyJD. Elastic inclusions and inhomogeneities. In: SneddonINHillR (eds). Progress in solid mechanics II. Amsterdam: North Holland, 1961, pp. 89–140.
3.
MuraT. Micromechanics of defects in solids. Martinus Nijhoff, 1987.
4.
Nemat-NasserSHoriM. Micromechanics: overall properties of heterogeneous materials. Pergamon Press, 1999.
5.
MoriTTanakaK. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Met1973; 21: 571–574.
6.
LiuLPJamesRDLeoPH. Periodic inclusion–-matrix microstructures with constant field inclusions. Met Mat Trans A2007; 38: 781–787.
7.
LiuLPJamesRDLeoPH. New extremal inclusions and their applications to two-phase composites. Accepted byArch Rational Mech Anal2008.
8.
LiuLP. Effective conductivities of two-phase composites with a singular phase. J Appl Phys2009; 105: 103503.
9.
LiuLP. Hashin–Shtrikman bounds and their attainability for multi-phase composites. Proc R Soc A2010; 466(2124): 3693–3713.
10.
NeumanKCNagyA. Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy. Nature Methods2008; 5(6): 491–505.
11.
GosseCCroquetteV. Magnetic tweezers: micromanipulation and force measurement at the molecular level. Biophys J2002; 82(6): 3314–3329.
PankhurstQAConnollyJJonesSK. Applications of magnetic nanoparticles in biomedicine. J Phys D – Appl Phys2003; 36(13): R167–R181.
14.
HoriHNemat-NasserS. Compression-induced microcrack growth in brittle solids – axial splitting and shear failure. J Geophys Res – Solid Earth Planets1985; 90(NB4): 3105–3125.
15.
HoriMNemat-NasserS. Interacting micro-cracks near the tip in the process zone of a macrocrack. J Mech Phys Solids1987; 35(5): 601–629.
16.
AsaroRJBarnettD. The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion. J Mech Phys Solids1975; 23: 77–83.
17.
MuraTKinoshitaN. The polynomial eigenstrain problem for an anisotropic ellipsoidal inclusion. Phys Stat Sol (a)1978; 48: 447–450.
18.
MoschovidisZAMuraT. Two-ellipsoidal inhomogeneities by the equivalent inclusion method. J Appl Mech1975; 42: 847–852.
19.
RodinGJHwangY-L. On the problem of linear elasticity for an infinite region containing a finite number of non-intersecting spherical inhomogeneities. Int J Solids Struc1991; 27(2): 145–159.
20.
HazewinkelM (ed.). Monomial representation. Encyclopedia of mathematics. Springer, 2001.
21.
RudinW. Functional analysis. New York: McGraw-Hill, 1991.
22.
RudinW. Real & complex analysis. New York: McGraw-Hill, 1987.
23.
GilbargDTrudingerNS. Elliptic partial differential equations of second order. New York: Springer-Verlag, 1983.
