Skip to main content
SpringerLink
Log in

Solution of the Eshelby problem in gradient elasticity for multilayer spherical inclusions

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

We consider gradient models of elasticity which permit taking into account the characteristic scale parameters of the material. We prove the Papkovich–Neuber theorems, which determine the general form of the gradient solution and the structure of scale effects. We derive the Eshelby integral formula for the gradient moduli of elasticity, which plays the role of the closing equation in the self-consistent three-phase method. In the gradient theory of deformations, we consider the fundamental Eshelby–Christensen problem of determining the effective elastic properties of dispersed composites with spherical inclusions; the exact solution of this problem for classical models was obtained in 1976.

This paper is the first to present the exact analytical solution of the Eshelby–Christensen problem for the gradient theory, which permits estimating the influence of scale effects on the stress state and the effective properties of the dispersed composites under study.We also analyze the influence of scale factors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. A. Tupin, “ElasticMaterials with Couple Stresses,” Arch. Rat.Mech. Anal. 11, 385–414 (1962).

    Article  Google Scholar 

  2. R. D. Mindlin, “Micro-Structure in Linear Elasticity,” Arch. Rat. Mech. Anal. 16 (1), 51–78 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  3. R. D. Mindlin, “Second Gradient of Strain and Surface-Tension in Linear Elasticity,” Int. J. Solids Struct. 1, 417–438 (1965).

    Article  Google Scholar 

  4. R.D. Mindlin and N. N. Eshel, “On First Strain-Gradient Theories in Linear Elasticity,” Int. J. Solids Struct. 4, 109–124 (1968).

    Article  MATH  Google Scholar 

  5. S. Kakunai, J. Masaki, R. Kuroda, et al., “Measurement of Apparent Young’s Modulus in the Bending of Cantilever Beam by Heterodyne Holographic Interferometry,” Exp. Mech. 25 (4), 408–412 (1985).

    Article  Google Scholar 

  6. D. C. C. Lam, F. Yang, A. C. M. Chong, et al., “Experiments and Theory in Strain Gradient Elasticity,” J. Mech. Phys. Solids 51, 1477–1508 (2003).

    Article  ADS  MATH  Google Scholar 

  7. A. W. McFarland and J. S. Colton, “Role of Material Microstructure in Plate Stiffness with Relevance to Microcantilever Sensors,” J. Micromech. Microengng 15, 1060–1067 (2005).

    Article  ADS  Google Scholar 

  8. E. C. Aifantis, “Gradient Effects at the Macro,Micro and Nano Scales,” J.Mech. Behav. Mater. 5 (3), 335–353 (1994).

    Article  Google Scholar 

  9. G. A. Maugin, V. I. Alshits, and H.O. K. Kirchner, “Elasticity in Multilayers: Properties of the Propagation Matrix and Some Applications,” Math. Mech. Solids 6, 481–502 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Lurie, P. Belov, D. B. Volkov-Bogorodsky, and N. Tuchkova, “Nanomechanical Modeling of the Nanostructures and Dispersed Composites,” Comp.Mater. Sci. 28 (3–4), 529–539 (2003).

    Article  Google Scholar 

  11. S. Lurie, P. Belov, and N. Tuchkova, “The Application of the Multiscale Models for Description of the Dispersed Composites,” Int. J.Comp.Mater. Sci. Ser. A 36 (2), 145–152 (2005).

    Google Scholar 

  12. S. Lurie, P. Belov, D. B. Volkov-Bogorodsky, and N. Tuchkova, “Interphase Layer Theory and Applications in the Mechanics of CompositeMaterials,” J.Mat. Sci. 41 (20), 6693–6707 (2006).

    Article  ADS  Google Scholar 

  13. S. Lurie, D. B. Volkov-Bogorodsky, V. Zubov, and N. Tuchkova, “Advanced Theoretical and Numerical Multiscale Modeling of Cohesion/Adhesion Interactions in Continuum Mechanics and Its Application for Filled Nanocomposites,” Comp. Mater. Sci. 45 (3), 709–714 (2009).

    Article  Google Scholar 

  14. P. A. Belov and S. A. Lurie, “A Continuum Model ofMicroheterogeneousMedia,” Prikl. Mat. Mekh. 73 (5), 833–848 (2009) [J. Appl.Math. Mech. (Engl. Transl.) 73 (5), 599–608 (2009)].

    MathSciNet  Google Scholar 

  15. S. A. Lurie and Yu. O. Solyaev, “Modeling of Mechanical Properties of Nanostructured Porous Ceramics,” Deform. Razrush.Mater., No. 1, 6–16 (2012).

    Google Scholar 

  16. S. A. Lurie, and Yu. O. Solyaev, “Gradient Model of Thermoelasticity and Its Applications to the Modeling of Thin-Layer Composite Structures,” Mekh. Komp. Mater. Konstr. 18 (3), 440–449 (2012).

    Google Scholar 

  17. S. A. Lurie, A. A. Kasimovskii, Yu. O. Solyaev, and D. D. Ivanova, “Methods for Predicting Effective Thermoelastic Properties of Composite Ceramics Reinforced with Carbon Nanostructures,” Int. J. Nanomech. Sci. Technol. 3 (1), 1–14 (2012).

    Article  Google Scholar 

  18. J. D. Eshelby, Continual Theory of Dislocations (Inostr. Lit-ra,Moscow, 1963) [in Russian].

    Google Scholar 

  19. T. Mori and K. Tanaka, “Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions,” Acta Metal. 21, 571–574 (1973).

    Article  Google Scholar 

  20. Y. Benveniste, “A New Approach to the Application ofMori-Tonaka Theory in CompositeMaterials,” Mech. Mater. 6, 147–157 (1987).

    Article  Google Scholar 

  21. R. M. Christensen, Introduction to Mechanics of Composite Materials (Wiley, New York, 1979; Mir, Moscow, 1982).

    Google Scholar 

  22. S. A. Lurie, D. B. Volkov-Bogorodsky, A. Leontiev, and E. C. Aifantis, “Eshelby’s Inclusion Problem in the Gradient Theory of Elasticity. Applications to Composite Materials,” Int. J. Engng Sci. 49, 1517–1525 (2011).

    Article  MathSciNet  Google Scholar 

  23. D. B. Volkov-Bogorodskii and S. A. Lurie, “Eshelby Integral Formulas in Gradient Elasticity,” Izv. Akad. Nauk.Mekh. Tverd. Tela, No. 4, 184–194 (2010) [Mech. Solids (Engl. Transl.) 45 (4), 648–656 (2010)].

    Google Scholar 

  24. R. M. Christensen and K. H. Lo, “Solutions for Effective Shear Properties in Three Phase Sphere and CylinderModels,” J.Mech. Phys. Solids 27, 315–330 (1979).

    Article  ADS  MATH  Google Scholar 

  25. E. Herve and A. Zaoui, “Elastic Behavior of Multiply Coated Fiber-Reinforced Composites,” Int. J. Engng Sci. 33 (10), 1419–1433 (1995).

    Article  MATH  Google Scholar 

  26. X.-L. Gao and S. K. Park, “Vibrational Formation of Simplified Strain Gradient Elasticity Theory and Its Application to a Pressurized Thick-Walled Cylinder Problem,” Int. J. Solids Struct. 44, 7486–7499 (2007).

    Article  MATH  Google Scholar 

  27. P. F. Papkovich, Theory of Elasticity (Oborongiz,Moscow–Leningrad, 1939) [in Russian].

    MATH  Google Scholar 

  28. W. Nowacki, Theory of Elasticity (PWN,Warszawa, 1970; Mir,Moscow, 1975).

    MATH  Google Scholar 

  29. Ph. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953; Inostr. Lit.,Moscow, 1958).

    MATH  Google Scholar 

  30. D. B. Volkov-Bogorodskii, “Approach to Problems of Interaction between Acoustic and Elastic Media by Using the Block Multifield Method,” in Dynamical and Technological Problems of Structural and Continuum Mechanics. Proc. 11th Intern. Symp., Vol. 2 (MAI,Moscow, 2005), pp. 17–23 [in Russian].

    Google Scholar 

  31. D. B. Volkov-Bogorodskii, “Application of Analytic Calculations Based on the Block Method in Coupled Problems of Continuum Mechanics,” in Proc. All-Russia Science-Practical Conference “Engineering Systems-2008” Moscow, April 7–11, 2008 (Izdat. RUDN,Moscow, 2008), pp. 123–138 [in Russian].

    Google Scholar 

  32. M. E. Gurtin and A. I. Murdoch, “Surface Stress in Solids,” Int. J. Solids Struct. 14 (6), 431–440 (1978).

    Article  MATH  Google Scholar 

  33. Y. Z. Povstenko, “Theoretical Investigation of Phenomena Caused by Heterogeneous Surface Tension in Solids,” J. Mech. Phys. Solids 41, 1499–1514 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. H. L. Duan, J. Wang, Z. P. Huang, and B. L. Karihaloo, “Size-Dependent Effective Elastic Constants of Solids Containing Nano-Inhomogeneities with Interface Stress,” J. Mech. Phys. Solids 53, 1574–1596 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. A. I. Murdoch, “Some Fundamental Aspects of SurfaceModeling,” J. Elasticity 80, 33–52 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  36. R. V. A. Gorodtsov, and K. V. Ustinov, “On the Construction of Surface Elasticity Theory for the Plane Boundary,” Fiz. Mezomekh. 16 (4), 75–83 (2013).

    Google Scholar 

  37. P. Belov and S. A. Lurie, Mathematical Theory of Defective Media. Gradient Elasticity. Formulations. Hierarchy. Comparative Analysis. Applications (Plenum Academic Publishing, 2014).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, Russian Academy of Sciences, Leninskii pr. 32A, Moscow, 117334, Russia

    D. B. Volkov-Bogorodskii & S. A. Lurie

  2. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526, Russia

    S. A. Lurie

Authors
  1. D. B. Volkov-Bogorodskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. Lurie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. B. Volkov-Bogorodskii.

Additional information

Original Russian Text © D.B. Volkov-Bogorodskii, S.A. Lurie, 2016, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2016, No. 2, pp. 32–50.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Volkov-Bogorodskii, D.B., Lurie, S.A. Solution of the Eshelby problem in gradient elasticity for multilayer spherical inclusions. Mech. Solids 51, 161–176 (2016). https://doi.org/10.3103/S0025654416020047

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654416020047

Keywords

Search

Navigation