Abstract

Eshelby conjectured that if for a given uniform loading the field inside an inclusion is uniform, then the inclusion must be an ellipse or an ellipsoid. This conjecture has been proved to be true in two and three dimensions provided that the inclusion is simply connected. In this paper we provide an alternative proof of Cherepanov's result that an inclusion with two components can be constructed inside which the field is uniform for any given uniform loading for two-dimensional conductivity or for antiplane elasticity. For planar elasticity, we show that the field inside the inclusion pair is uniform for certain loadings and not for others. We also show that the polarization tensor associated with the inclusion pair lies on the lower Hashin–Shtrikman bound, and hence the conjecture of Pólya and Szegö is not true among nonsimply connected inclusions. As a consequence, we construct a simply connected inclusion, which is nothing close to an ellipse, but in which the field is almost uniform.

MSC codes

  1. 74M25

Keywords

  1. Eshelby's conjecture
  2. Pólya–Szegö conjecture
  3. uniformity property
  4. inclusions with multiple components
  5. polarization tensor
  6. Weierstrass zeta function

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1978.
2.
G. Alessandrini and V. Isakov, Analyticity and uniqueness for the inverse conductivity problem, Rend. Istit. Mat. Univ. Trieste, 28 (1996), pp. 351–369.
3.
H. Ammari, Y. Capdeboscq, H. Kang, E. Kim, and M. Lim, Attainability by simply connected domains of optimal bounds for polarization tensors, European J. Appl. Math., 17 (2006), pp. 201–219.
4.
H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Math. 1846, Springer-Verlag, New York, 2004.
5.
H. Ammari, H. Kang, and M. Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann., 332 (2005), pp. 277–286.
6.
K. Astala and V. Nesi, Composites and quasiconformal mappings: New optimal bounds in two dimensions, Calc. Var. Partial Differential Equations, 18 (2003), pp. 335–355.
7.
Y. Capdeboscq and M. S. Vogelius, A review of some recent work on impedance imaging for inhomogeneities of low volume fraction, in Partial Differential Equations and Inverse Problems, Contemp. Math. 362, AMS, Providence, RI, 2004, pp. 69–87.
8.
G. P. Cherepanov, Inverse problems of the plane theory of elasticity, Prikl. Mat. Meh., 38 (1974), pp. 963–979 (in Russian).
9.
J. D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. London Ser. A, 241 (1957), pp. 376–396.
10.
J. D. Eshelby, Elastic inclusions and inhomogeneities, in Progress in Solid Mechanics, Vol. II, North–Holland, Amsterdam, 1961, pp. 87–140.
11.
L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 1998.
12.
Y. Grabovsky and R. Kohn, Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II. The Vigdergauz microstructure, J. Mech. Phys. Solids, 43 (1995), pp. 949–972.
13.
Y. Grabovsky, Bounds and extremal microstructures for two-component composites: A unified treatment based on the translation method, Proc. Roy. Soc. London Ser. A, 452 (1996), pp. 919–944.
14.
Z. Hashin, The elastic moduli of heterogeneous materials, Trans. ASME Ser. E J. Appl. Mech., 29 (1962), pp. 143–150.
15.
C. O. Horgan, Anti-plane shear deformations in linear and nonlinear solid mechanics, SIAM Rev., 37 (1995), pp. 53–81.
16.
V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems, 6 (1990), pp. 311–318.
17.
H. Kang and G. W. Milton, On Conjectures of Pólya–Szegö and Eshelby, in Inverse Problems, Multi-scale Analysis and Effective Medium Theory, Contemp. Math. 408, AMS, Providence, RI, 2006, pp. 75–80.
18.
H. Kang and G. W. Milton, Solutions to the Pólya–Szegö conjecture and the weak Eshelby conjecture, Arch. Ration. Mech. Anal., 188 (2008), pp. 93–116.
19.
J. B. Keller, A theorem on the conductivity of a composite medium, J. Math. Phys., 5 (1964), pp. 548–549.
20.
R. V. Kohn and G. W. Milton, On bounding the effective conductivity of anisotropic composites, in Homogenization and Effective Moduli of Materials and Media, J. L. Ericksen, D. Kinderlehrer, R. V. Kohn, and J. L. Lions, eds., IMA Vol. Math. Appl. 1, Springer-Verlag, New York, 1986, pp. 97–125.
21.
R. Lipton, Inequalities for electric and elastic polarization tensors with applications to random composites, J. Mech. Phys. Solids, 41 (1993), pp. 809–833.
22.
L. P. Liu, Solutions to the Eshelby conjectures, Proc. Roy. Soc. A Math. Phys. Engrg. Sci., 464 (2008), pp. 573–594.
23.
G. W. Milton, The Theory of Composites, Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, UK, 2002.
24.
N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, English translation, Noordhoff International Publishing, Leiden, 1977.
25.
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Stud. 27, Princeton University Press, Princeton, NJ, 1951.
26.
C.-Q. Ru and P. Schiavone, On the elliptic inclusion in anti-plane shear, Math. Mech. Solids, 1 (1996), pp. 327–333.
27.
R. Schoen and S. T. Yau, Lectures on Harmonic Maps, International Press, Boston, 1997.
28.
G. P. Sendeckyj, Elastic inclusion problems in plane elastostatics, Internat. J. Solids Structures, 6 (1970), pp. 1535–1543.
29.
S. B. Vigdergauz, Effective elastic parameters of a plate with a regular system of equal-strength holes, Inzhenernyi Zhurnal. Mekhnika Tverdogo Tela, 21 (1986), pp. 165–169.
30.
S. B. Vigdergauz, Two dimensional grained composites of extreme rigidity, J. Appl. Mech., 61 (1994), pp. 390–394.
31.
L. T. Wheeler, Stress minimum forms for elastic solids, AMR, 45 (1992), pp. 1–12.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 577 - 595
ISSN (online): 1095-712X

History

Submitted: 14 May 2007
Accepted: 4 August 2008
Published online: 3 December 2008

MSC codes

  1. 74M25

Keywords

  1. Eshelby's conjecture
  2. Pólya–Szegö conjecture
  3. uniformity property
  4. inclusions with multiple components
  5. polarization tensor
  6. Weierstrass zeta function

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media