Journal of Elasticity

, Volume 130, Issue 2, pp 239–269 | Cite as

Nonlinear Elastic Inclusions in Anisotropic Solids



In this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic. Similarly, the stress in a cylindrical inclusion contained in an incompressible orthotropic cylindrical bar is uniform hydrostatic if the radial and circumferential eigenstrains are equal and the axial stretch is equal to a value determined by the axial eigenstrain. We also prove that for a compressible isotropic spherical ball and a cylindrical bar containing a spherical and a cylindrical inclusion, respectively, with uniform eigenstrains the stress in the inclusion is uniform (and hydrostatic for the spherical inclusion) if the radial and circumferential eigenstrains are equal. For compressible transversely isotropic and orthotropic solids, we show that the stress field in an inclusion with uniform eigenstrain is not uniform, in general. Nevertheless, in some special cases the material can be designed in order to maintain a uniform stress field in the inclusion. As particular examples to investigate such special cases, we consider compressible Mooney-Rivlin and Blatz-Ko reinforced models and find analytical expressions for the stress field in the inclusion.


Transversely isotropic solids Orthotropic solids Finite eigenstrains Geometric mechanics Anisotropic inclusions Nonlinear elasticity 

Mathematics Subject Classification

74B20 70G45 74E10 15A72 74Fxx 



This work was partially supported by ARO W911NF-16-1-0064, AFOSR—Grant No. FA9550-12-1-0290 and NSF—Grant No. CMMI 1130856 and CMMI 1561578.


  1. 1.
    Amar, M.B., Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53(10), 2284–2319 (2005) ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Carroll, M.: Finite strain solutions in compressible isotropic elasticity. J. Elast. 20(1), 65–92 (1988) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Doyle, T., Ericksen, J.: Nonlinear elasticity. Adv. Appl. Mech. 4, 53–115 (1956) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957) ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Giordano, S., Palla, P., Colombo, L.: Nonlinear elasticity of composite materials. Eur. Phys. J. B 68(1), 89–101 (2009) ADSCrossRefGoogle Scholar
  6. 6.
    Golgoon, A., Yavari, A.: On the stress field of a nonlinear elastic solid torus with a toroidal inclusion. J. Elast. 19, 1–31 (2017) MathSciNetMATHGoogle Scholar
  7. 7.
    Golgoon, A., Sadik, S., Yavari, A.: Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges. Int. J. Non-Linear Mech. 84, 116–129 (2016) ADSCrossRefGoogle Scholar
  8. 8.
    Goriely, A., Moulton, D.E., Vandiver, R.: Elastic cavitation, tube hollowing, and differential growth in plants and biological tissues. Europhys. Lett. 91(1), 18001 (2010) ADSCrossRefGoogle Scholar
  9. 9.
    Jiang, X., Pan, E.: Exact solution for 2D polygonal inclusion problem in anisotropic magnetoelectroelastic full-, half-, and bimaterial-planes. Int. J. Solids Struct. 41(16), 4361–4382 (2004) CrossRefMATHGoogle Scholar
  10. 10.
    Kim, C., Schiavone, P.: A circular inhomogeneity subjected to non-uniform remote loading in finite plane elastostatics. Int. J. Non-Linear Mech. 42(8), 989–999 (2007) ADSCrossRefGoogle Scholar
  11. 11.
    Kim, C., Schiavone, P.: Designing an inhomogeneity with uniform interior stress in finite plane elastostatics. Acta Mech. 197(3–4), 285–299 (2008) CrossRefMATHGoogle Scholar
  12. 12.
    Kim, C., Vasudevan, M., Schiavone, P.: Eshelby’s conjecture in finite plane elastostatics. Q. J. Mech. Appl. Math. 61(1), 63–73 (2008) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kinoshita, N., Mura, T.: Elastic fields of inclusions in anisotropic media. Phys. Status Solidi A 5(3), 759–768 (1971) ADSCrossRefGoogle Scholar
  14. 14.
    Lee, Y.-G., Zou, W.-N., Pan, E.: Eshelby’s problem of polygonal inclusions with polynomial eigenstrains in an anisotropic magneto-electro-elastic full plane. Proc. R. Soc. Lond. A 471(2179), 20140827 (2015) ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Li, J.Y., Dunn, M.L.: Anisotropic coupled-field inclusion and inhomogeneity problems. Philos. Mag. A 77(5), 1341–1350 (1998) ADSCrossRefGoogle Scholar
  16. 16.
    Liu, I., et al.: On representations of anisotropic invariants. Int. J. Eng. Sci. 20(10), 1099–1109 (1982) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lu, J., Papadopoulos, P.: A covariant constitutive description of anisotropic non-linear elasticity. Z. Angew. Math. Phys. 51(2), 204–217 (2000) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Marsden, J.E., Hughes, T.: Mathematical Foundations of Elasticity. Dover Publications, New York (1994) MATHGoogle Scholar
  19. 19.
    Merodio, J., Ogden, R.: Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Arch. Mech. 54(5–6), 525–552 (2002) MathSciNetMATHGoogle Scholar
  20. 20.
    Merodio, J., Ogden, R.: Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Struct. 40(18), 4707–4727 (2003) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Merodio, J., Ogden, R.: Tensile instabilities and ellipticity in fiber-reinforced compressible non-linearly elastic solids. Int. J. Eng. Sci. 43(8), 697–706 (2005) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Moulton, D.E., Goriely, A.: Anticavitation and differential growth in elastic shells. J. Elast. 102(2), 117–132 (2011) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51(3), 032902 (2010) ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pan, E.: Eshelby problem of polygonal inclusions in anisotropic piezoelectric full-and half-planes. J. Mech. Phys. Solids 52(3), 567–589 (2004) ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pan, E.: Eshelby problem of polygonal inclusions in anisotropic piezoelectric bimaterials. Proc. R. Soc. Lond. A 460(2042), 537–559 (2004) ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pence, T.J., Tsai, H.: Swelling-induced microchannel formation in nonlinear elasticity. IMA J. Appl. Math. 70(1), 173–189 (2005) ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pence, T.J., Tsai, H.: Swelling-induced cavitation of elastic spheres. Math. Mech. Solids 11(5), 527–551 (2006) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Pence, T.J., Tsai, H.: Bulk cavitation and the possibility of localized interface deformation due to surface layer swelling. J. Elast. 87(2–3), 161–185 (2007) MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ru, C.-Q., Schiavone, P.: On the elliptic inclusion in anti-plane shear. Math. Mech. Solids 1(3), 327–333 (1996) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ru, C., Schiavone, P., Sudak, L., Mioduchowski, A.: Uniformity of stresses inside an elliptic inclusion in finite plane elastostatics. Int. J. Non-Linear Mech. 40(2), 281–287 (2005) ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids (2015). doi: 10.1177/1081286515599458 MATHGoogle Scholar
  32. 32.
    Sadik, S., Yavari, A.: Small-on-large geometric anelasticity. Proc. R. Soc. Lond. A 472, 2195 (2016) MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Sozio, F., Yavari, A.: Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies. J. Mech. Phys. Solids 98, 12–48 (2017) ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Spencer, A.: Continuum Physics. Part III. Theory of Invariants, pp. 239–353 (1971) Google Scholar
  35. 35.
    Spencer, A.: The formulation of constitutive equation for anisotropic solids. In: Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes, pp. 3–26. Springer, Berlin (1982) CrossRefGoogle Scholar
  36. 36.
    Stojanovic, R., Djuric, S., Vujosevic, L.: On finite thermal deformations. Arch. Mech. Stosow. 16, 103–108 (1964) MathSciNetGoogle Scholar
  37. 37.
    Truesdell, C.: The physical components of vectors and tensors. Z. Angew. Math. Mech. 33(10–11), 345–356 (1953) MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Vergori, L., Destrade, M., McGarry, P., Ogden, R.W.: On anisotropic elasticity and questions concerning its finite element implementation. Comput. Mech. 52(5), 1185–1197 (2013) CrossRefMATHGoogle Scholar
  39. 39.
    Willis, J.: Anisotropic elastic inclusion problems. Q. J. Mech. Appl. Math. 17(2), 157–174 (1964) MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    I. Wolfram Research, Mathematica. Vaersion 11.0. Wolfram Research, Inc. Champaign, Illinois (2016) Google Scholar
  41. 41.
    Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20, 781–830 (2010) ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Yavari, A.: On the wedge dispiration in an inhomogeneous isotropic nonlinear elastic solid. Mech. Res. Commun. 78, 55–59 (2016) CrossRefGoogle Scholar
  43. 43.
    Yavari, A., Goriely, A.: Nonlinear elastic inclusions in isotropic solids. Proc. R. Soc. A 469(2160), 20130415 (2013) ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Yavari, A., Goriely, A.: On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities. J. Mech. Phys. Solids 76, 325–337 (2015) ADSMathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Yavari, A., Goriely, A.: The twist-fit problem: finite torsional and shear eigenstrains in nonlinear elastic solids. Proc. R. Soc. A 471(2183), 20150596 (2015) ADSCrossRefGoogle Scholar
  46. 46.
    Yavari, A., Marsden, J.E., Ortiz, M.: On the spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 85–112 (2006) MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Yue, Y., Xu, K., Chen, Q., Pan, E.: Eshelby problem of an arbitrary polygonal inclusion in anisotropic piezoelectric media with quadratic eigenstrains. Acta Mech. 226(7), 2365–2378 (2015) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Zheng, Q.-S., Spencer, A.: Tensors which characterize anisotropies. Int. J. Eng. Sci. 31(5), 679–693 (1993) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Civil and Environmental EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.The George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations