Inclusion Pairs Satisfying Eshelby's Uniformity Property
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
Keywords
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
Copyright
Copyright © 2008 Society for Industrial and Applied Mathematics.
History
Submitted: 14 May 2007
Accepted: 4 August 2008
Published online: 3 December 2008
MSC codes
Keywords
Authors
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
- Uniform anti‐plane stress field within two nonlinear elastic non‐elliptical inhomogeneitiesZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 79 | 29 July 2023
- Inclusions with Uniform Stress in a Bounded Elastic DomainJournal of Elasticity, Vol. 241 | 7 July 2023
- Average Eshelby tensor of an arbitrarily shaped inclusion from convexity to non-convexity: Effective elastic properties of compositesInternational Journal of Solids and Structures, Vol. 269 | 1 May 2023
- A free boundary problem of elasticity in an angle and a vector Riemann–Hilbert problem on a torusPhysica D: Nonlinear Phenomena, Vol. 444 | 1 Feb 2023
- A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stressesMathematics and Mechanics of Solids, Vol. 27, No. 11 | 9 October 2021
- Uniformity of stresses inside a hypotrochoidal inhomogeneityActa Mechanica, Vol. 233, No. 3 | 28 February 2022
- A non-circular inhomogeneity near a non-elliptical inhomogeneity designed to admit an internal uniform hydrostatic stress fieldMechanics of Materials, Vol. 162 | 1 Nov 2021
- Riemann–Hilbert problem on a hyperelliptic surface and uniformly stressed inclusions embedded into a half-plane subjected to antiplane strainProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 477, No. 2252 | 18 August 2021
- A modified Laurent series for hole/inclusion problems in plane elasticityZeitschrift für angewandte Mathematik und Physik, Vol. 72, No. 3 | 19 May 2021
- Uniform stresses inside both a non-parabolic inhomogeneity and a non-elliptical inhomogeneity located in the vicinity of a circular Eshelby inclusionMathematics and Mechanics of Solids, Vol. 471 | 7 May 2021
- Need the Uniform Stress Field Inside Multiple Interacting Inclusions Be Hydrostatic?Journal of Elasticity, Vol. 143, No. 2 | 20 January 2021
- An extension of the Eshelby conjecture to domains of general shape in anti-plane elasticityJournal of Mathematical Analysis and Applications, Vol. 495, No. 2 | 1 Mar 2021
- A screw dislocation near one open inhomogeneity and another closed inhomogeneity both permitting constant interior stressesApplied Mathematics and Mechanics, Vol. 42, No. 2 | 8 January 2021
- A hole of irregular shape interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stressesMathematics and Mechanics of Solids | 6 December 2020
- The symmetry and loading-independency of multiple inclusions enclosing uniform stresses in an infinite elastic planeApplied Mathematics and Mechanics, Vol. 41, No. 10 | 7 October 2020
- A circular inhomogeneity interacting with an open inhomogeneity designed to admit internal uniform hydrostatic stressesMechanics of Materials, Vol. 148 | 1 Sep 2020
- Uniformity of stresses inside a non-elliptical inhomogeneity near an irregularly shaped hole in plane elasticityMechanics of Materials, Vol. 145 | 1 Jun 2020
- Uniform fields inside an anisotropic elastic open inhomogeneityArchive of Applied Mechanics, Vol. 90, No. 5 | 4 January 2020
- Uniform stresses inside a non-elliptical inhomogeneity and a nearby half-plane with locally wavy interfaceZeitschrift für angewandte Mathematik und Physik, Vol. 71, No. 2 | 9 March 2020
- Method of Riemann surfaces for an inverse antiplane problem in an n -connected domainComplex Variables and Elliptic Equations, Vol. 65, No. 3 | 23 May 2019
- A circular Eshelby inclusion interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stressesMathematics and Mechanics of Solids, Vol. 25, No. 3 | 12 December 2019
- Uniform fields inside two interacting non-parabolic and non-elliptical inhomogeneitiesZeitschrift für angewandte Mathematik und Physik, Vol. 71, No. 1 | 9 January 2020
- Uniform stress state inside a non-elliptical inhomogeneity near an irregularly shaped hole in antiplane shearMathematics and Mechanics of Solids, Vol. 25, No. 2 | 11 October 2019
- Inclusions of General Shapes Having Constant Field Inside the Core and NonElliptical Neutral Coated Inclusions With Anisotropic ConductivitySIAM Journal on Applied Mathematics, Vol. 80, No. 3 | 8 June 2020
- A screw dislocation near a non-elliptical piezoelectric inhomogeneity with internal uniform electroelastic fieldComptes Rendus Mécanique, Vol. 347, No. 10 | 1 Oct 2019
- A mode III Dugdale crack interacting with a non-elliptical inhomogeneity with internal uniform stressesEngineering Fracture Mechanics, Vol. 216 | 1 Jul 2019
- Effect of a circular Eshelby inclusion on the non‐elliptical shape of a coated neutral inhomogeneity with internal uniform stressesZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 99, No. 6 | 7 April 2019
- An elastic harmonic inclusion near a rigid harmonic inclusion loaded by a coupleIMA Journal of Applied Mathematics, Vol. 84, No. 3 | 7 February 2019
- Method of automorphic functions for an inverse problem of antiplane elasticityThe Quarterly Journal of Mechanics and Applied Mathematics, Vol. 72, No. 2 | 28 February 2019
- On Sets of Multiple Equally Strong Holes in an Infinite Elastic Plate: Parameterization and ExistenceSIAM Journal on Applied Mathematics, Vol. 79, No. 6 | 26 November 2019
- Two non‐elliptical decagonal quasicrystalline inclusions with internal uniform hydrostatic phonon stressesZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 98, No. 11 | 17 September 2018
- Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a circular Eshelby inclusion in anti-plane shearArchive of Applied Mechanics, Vol. 88, No. 10 | 31 May 2018
- Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a mode III crackProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 474, No. 2218 | 17 October 2018
- Two non-elliptical inhomogeneities with internal uniform stresses interacting with a mode-III crackComptes Rendus Mécanique, Vol. 346, No. 9 | 1 Sep 2018
- Uniform stress field inside an anisotropic non-elliptical inhomogeneity interacting with a screw dislocationEuropean Journal of Mechanics - A/Solids, Vol. 70 | 1 Jul 2018
- A concentrated couple near two non-elliptical inclusions with internal uniform hydrostatic stressesZeitschrift für angewandte Mathematik und Physik, Vol. 69, No. 1 | 11 December 2017
- Slit Maps in the Study of Equal-Strength Cavities in $n$-Connected Elastic Planar DomainsSIAM Journal on Applied Mathematics, Vol. 78, No. 1 | 30 January 2018
- The Design of Nano-Inhomogeneities with Uniform Internal Strain in Anti-Plane Shear Deformations of Composite SolidsMicromechanics and Nanomechanics of Composite Solids | 20 July 2017
- Uniform stress resultants inside two nonelliptical inhomogeneities in isotropic laminated platesJournal of Mechanics of Materials and Structures, Vol. 13, No. 4 | 13 December 2018
- Harmonic holes with surface tension in an elastic plane under uniform remote loadingMathematics and Mechanics of Solids, Vol. 22, No. 9 | 17 May 2016
- Criteria for guaranteed breakdown in two-phase inhomogeneous bodiesInverse Problems, Vol. 33, No. 8 | 27 June 2017
- Uniform strain fields inside multiple inclusions in an elastic infinite plane under anti-plane shearMathematics and Mechanics of Solids, Vol. 22, No. 1 | 6 August 2016
- Periodic inclusions with uniform internal hydrostatic stress in an infinite elastic planeZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 96, No. 11 | 21 April 2016
- Non-circular nano-inclusions with interface effects that achieve uniform internal strain fields in an elastic plane under anti-plane shearArchive of Applied Mechanics, Vol. 86, No. 7 | 31 December 2015
- Two inhomogeneities of irregular shape with internal uniform stress fields interacting with a screw dislocationComptes Rendus Mécanique, Vol. 344, No. 7 | 1 Jul 2016
- Uniform strain field inside a non-circular inhomogeneity with homogeneously imperfect interface in anisotropic anti-plane shearZeitschrift für angewandte Mathematik und Physik, Vol. 67, No. 3 | 20 April 2016
- Non-elliptical inclusions that achieve uniform internal strain fields in an elastic half-planeActa Mechanica, Vol. 226, No. 11 | 11 August 2015
- Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformationProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 471, No. 2177 | 8 May 2015
- A review of recent works on inclusionsMechanics of Materials, Vol. 60 | 1 Jul 2013
- Three-phase piezoelectric inclusions of arbitrary shape with internal uniform electroelastic fieldInternational Journal of Engineering Science, Vol. 63 | 1 Feb 2013
- Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation methodCalculus of Variations and Partial Differential Equations, Vol. 45, No. 3-4 | 3 November 2011
- Uniform fields inside two non-elliptical inclusionsMathematics and Mechanics of Solids, Vol. 17, No. 7 | 5 January 2012
- Conjectures of Pólya-szegö and Eshelby, and the Newtonian potential problem: A reviewMechanics of Materials, Vol. 41, No. 4 | 1 Apr 2009