Mechanics of Solids

, Volume 48, Issue 1, pp 92–118 | Cite as

On transient layers as new phase domains in composite materials

  • E. N. Vilchevskaya
  • R. A. Filippov
  • A. B. Freidin


A model describing the development of transient layers as new phase domains in compositematerials is constructed under the assumption that the transient layers around (nano)particles are layers of the matrix material changed by the phase transformation and increase the effective volume of inclusions which become compound and consist of the nucleus (original particle) and the shell (transient layer of the new phase). As a result, the inclusion volume fraction increases, which, in turn, increases the particle influence efficiency. An example of spherical particles is used to consider the new phase development around an isolated particle and then, in the effective field approximation, around interacting particles in the composite material. The dependence of the compound inclusion radius on the external (averaged) strain is obtained for isotropic phases. Stability of the interphase boundaries depending on the parameters of the original inclusion material and the matrix phase materials is studied. The energy variations and the stress redistribution owing to the development of the new phase domains are considered in detail. It is shown that, in the case of an isolated inclusion, the development of a new phase may lead to a local energy decrease near the inclusions and, as a consequence, to a decrease in the stress concentration. At the same time, the formation of transient layers due to the phase transformation can result in an increase in the bulk modulus of elasticity as the effective shear modulus decreases.


composite materials transient layers phase transformations effective moduli of elasticity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. George, K. T. Kashyap, R. Rahul, and S. Yamdagni, “Strengthening in Carbon Nanotube/Aluminum (CNT/Al) Composites,” Scripta Mater. 53(10), 1159–1163 (2005).CrossRefGoogle Scholar
  2. 2.
    X.-L. Xie, Y.-W. Mai, and X.-P. Zhoui, “Dispersion and Alignment of Carbon Nanotubes in Polymer Matrix; AReview,” Mat. Sci. Engng 49(4), 89–112 (2005).CrossRefGoogle Scholar
  3. 3.
    E. T. Thostenson, Z. Ren, T.-W. Choui, “Advance in the Science and Technology of Carbon Nanotubes and Their Composites: A Review,” Comp. Sci. Technol. 61(13), 1899–1912 (2001).CrossRefGoogle Scholar
  4. 4.
    Y. B. Tang, H. T. Cong, R. Zhong, and H. M. Cheng, “Thermal Expansion of a Composite of Single-Walled Carbon Nanotubes and Nanocrystalline Aluminum,” Carbon 42(15), 3251–3272 (2004).CrossRefGoogle Scholar
  5. 5.
    D. Qian, E. C. Dickey, R. Andrews, and T. Rantell, “Load Transfer and Deformation Mechanism in Carbon Nanotube-Polystyrene Composites,” Appl. Phys. Lett. 76(20), 2868–2870 (2000).ADSCrossRefGoogle Scholar
  6. 6.
    V. Skakalova, U. Dettlaff-Weglikowska, and S. Roth, “Electrical and Mechanical Properties of Nanocomposites of SingleWall Carbon Nanotubes with PMMA,” Synthetic Metals 152(1–3), 349–352 (2005).CrossRefGoogle Scholar
  7. 7.
    M. Cadek, J. N. Coleman, V. Barron, et al., “Morphological and Mechanical Properties of Carbon-Nanotube-Reinforced Semicrystallin and Amorphous Polymer Composites,” Appl. Phys. Lett. 81(27), 5123–5125 (2002).ADSCrossRefGoogle Scholar
  8. 8.
    E. Flahaut, A. Peigney, Ch. Laurent, et al., “Carbon Nanotube-Metal-Oxide Nanocomposites: Microstructure, Electrical Conductivity and Mechanical Properties,” Acta Mater. 48(14), 3803–3812 (2000).CrossRefGoogle Scholar
  9. 9.
    H. Wan, F. Delale, and L. Shen, “Effect of CNT Length and CNT-Matrix Interphase in Carbon Nanotube (CNT) Reinforces Composites,” Mech. Res. Communicat. 32(5), 481–489 (2005).zbMATHCrossRefGoogle Scholar
  10. 10.
    G. M. Odegard, T. S. Gates, K. E. Wise, et al., “Constitutive Modeling of Nanotube-Reinforces Polymer Composite Systems,” Comp. Sci. Technol. 63(11), 1671–1687 (2003).CrossRefGoogle Scholar
  11. 11.
    H. L. Duan, J. Wang, Z. P. Huang, and B. L. Karihaloo, “Eshelby Formalism for Nano-Inhomogeneities,” Proc. R. Soc. 461(2062), 3335–3353 (2005).MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    R. J. Arsenault and N. Shi, “Dislocation Generation due to Differences between the Coefficients of Thermal Expansion,” Mater. Sci. Engng 81, 175–187 (1986).CrossRefGoogle Scholar
  13. 13.
    F. Bondioli, V. Cannillo, E. Fabbri, and M. Messori, “Epoxy-Silica Nanocomposites: Preparation, Experimental Characterization, and Modeling,” J. Appl. Polym. Sci. 97(6), 2382–2386 (2005).CrossRefGoogle Scholar
  14. 14.
    R. V. Goldstein and K. B. Ustinov, Effect of Inclusions on the Effective Properties of Composites with the Influence of the Intermediate Phase Taken into Account Preprint No. 792 (Inst. Problem Mekhaniki RAN, Moscow, 2006) [in Russian].Google Scholar
  15. 15.
    S. Boutaleb, F. Zairi, A. Mesbah, et al., “Micromechanics-Based Modeling of Stiffness and Yield Stress for Silica/Polymer Nanocomposites,” Int. J. Solids Struct. 46(7–8), 1716–1726 (2009).zbMATHCrossRefGoogle Scholar
  16. 16.
    I. Sevostianov and M. Kachanov, “Effect of Interphase Layers on the Overall Elastic and Conductive Properties of Matrix Composites. Applications to Nanosize Inclusion,” Int. J. Solids Struct. 44(3–4), 1304–1315 (2007).zbMATHCrossRefGoogle Scholar
  17. 17.
    A. Akbari, J. P. Riviere, C. Templier, et al., “Hardness and Residual Stresses in TiN-Ni Nanocomposite Coating Deposited by Reactive Dual Ion Beam Sputtering,” Rev. Adv. Mater. Sci. 15, 111–117 (2007).Google Scholar
  18. 18.
    S. G. Roberts, “Thermal Shock of Ground and Polished Alumina and Al2O3/SiC Nanocomposites,” J. Europ. Ceramic Soc. 22(16), 2945–2956 (2002).CrossRefGoogle Scholar
  19. 19.
    S. Lurie, D. Volkov-Bogorodsky, V. Zubov, and N. Tuchkova, “Advanced Theoretical and Numerical Multiscale Modeling of Cohesion/Adhesion Interactions in Continuum Mechanics and Its Applications for Filled Nanocomposites,” Comput. Mater. Sci. 45(3), 709–714 (2009).CrossRefGoogle Scholar
  20. 20.
    S. Lurie and N. Tuchkova, “A Continual Adhesion Model of Solid Nanostructured Media,” Kompos. Nanostruct. 2(2), 25–43 (2009).Google Scholar
  21. 21.
    S. Lurie, P. Belov, D. Volkov-Bogorodsky, and N. Tuchkova, “Nanomechanical Modeling of the Nanostructures and Dispersed Composites,” Int. J. Comp. Mater. 28(3–4), 529–539 (2003).CrossRefGoogle Scholar
  22. 22.
    S. Lurie, P. Belov, D. Volkov-Bogorodsky, and N. Tuchkova, “Interphase Layer Theory and Application in the Mechanics of Composite Materials,” J. Mater. Sci. 41(20), 6693–6707 (2006).ADSCrossRefGoogle Scholar
  23. 23.
    D. Volkov-Bogorodsky, Yu. G. Evtushenko, and V. Zubov, “Calculation of Deformations in Nanocomposites Using the Block Multipole Method with the AnalyticalЦNumerical Account of the Scale Effects,” Zh. Vych. Mat. Mat. Fiz. 46(7), 1302–1311 (2006) [Comput. Math. Math. Phys. (Engl. Transl.) 46 (7), 1234–1253 (2006)].Google Scholar
  24. 24.
    S. K. Kanaun and V. M. Levin, Efficient Field Method in Mechanics of Composite Materials (Izd-VO Petrozav. Univ., Petrozavodsk, 1993) [in Russian].Google Scholar
  25. 25.
    S. K. Kanaun and V. M. Levin, Self-Consistent Methods for Composites, Vol. 1: Static Problems(Springer, 2007).Google Scholar
  26. 26.
    N. F. Morozov, I. R. Nazyrov, and A. B. Freidin, “One-Dimensional Problem of the Phase Transformation of an Elastic Sphere,” Dokl. Ross. Akad. Nauk 346(2), 188–191 (1996) [Dokl. Math. (Engl. Transl.) 41 (1), 40–43 (1996)].MathSciNetGoogle Scholar
  27. 27.
    I. R. Nazyrov and A. B. Freidin, “Phase Transformations in Deformation of Solids in a Model Problem of an Elastic Ball,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 52–71 (1998) [Mech. Solids (Engl. Transl.) 33 (5), 39–56 (1998)].Google Scholar
  28. 28.
    V. A. Eremeev, A. B. Freidin, and L. L. Sharipova, “The Stability of the Equilibrium of Two-Phase Elastic Solids,” Prikl. Mat. Mekh. 71(1), 66–92 (2007) [J. Appl. Math. Mech. (Engl. Transl.) 71 (1), 61–84 (2007)].MathSciNetzbMATHGoogle Scholar
  29. 29.
    E. N. Vilchevskaya and A. B. Freidin, “On Phase Transitions in a Domain of Material Inhomogeneity. I. Phase Transitions of an Inclusions in a Homogeneous External Field,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 208–228 (2007) [Mech. Solids (Engl. Transl.) 42 (5), 823–840 (2007)].Google Scholar
  30. 30.
    L. B. Kublanov and A. B. Freidin, “Solid Phase Seeds in a Deformable Material,” Prikl. Mat. Mekh. 52(3), 493–501 (1988) [J. Appl. Math. Mech. (Engl. Transl.) 52 (3), 382–389 (1988)].MathSciNetGoogle Scholar
  31. 31.
    N. F. Morozov and A. B. Freidin, “Zones of Phase Transitions and Phase Transformations in Elastic Bodies under Various Stress States,” Trudy Mat. Inst. Steklov 223, 220–232 (1998) [Proc. Steklov Inst. Math. (Engl. Transl.) 223, 219–232 (1998)].MathSciNetGoogle Scholar
  32. 32.
    A. B. Freidin, “Small-Strain Approximation in the Theory of Phase Transitions of Elastic Bodies under Deformation,” in Strength and Fracture of Materials and Structures. Intervuz. Collection of Papers, Vol. 18: Studies in Elasticity and Plasticity, Ed. by N. F. Morozov (Izd-vo St. Petersburg Univ., St. Petersburg, 1999), pp. 266–290 [in Russian].Google Scholar
  33. 33.
    A. B. Freidin, “On New Phase Inclusions in Elastic Solids,” ZAMM 87(2), 102–116 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    M. A. Grinfeld, “On Conditions of Thermodynamic Equilibrium of Phases of a Nonlinearly Elastic Material,” Dokl. Akad. Nauk SSSR 251(4), 824–827 (1980) [Soviet Math. Dokl. (Engl. Transl.)].Google Scholar
  35. 35.
    M. A. Grinfeld, Methods of Continuum Mechanics in Theory of Phase Transformations (Nauka, Moscow, 1990) [in Russian].Google Scholar
  36. 36.
    R. D. James, “Finite Deformations by Mechanical Twinning,” Arch. Rat. Mech. Anal. 77(2), 143–177 (1981).zbMATHCrossRefGoogle Scholar
  37. 37.
    M. E. Gurtin, “Two-Phase Deformations of Elastic Solids,” Arch. Rat. Mech. Anal. 84(1), 1–29 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    I. A. Kunin and E. G. Sosnina, “Stress Concentration on an Ellipsoidal Inhomogeneity in an Anisotropic Elastic Medium,” Prikl. Mat. Mekh. 37(2), 306–315 (1973) [J. Appl. Math. Mech. (Engl. Trabsl.) 37 (2), 287–296 (1973)].Google Scholar
  39. 39.
    I. A. Kunin, Elastic Media with Microstructure. II. Three-Dimensional Models, in Springer Series in Solid State Sciences, Vol. 44 (Springer-Verlag, Berlin, New-York, 1983).Google Scholar
  40. 40.
    A. B. Freidin and A. M. Chiskis, “Phase Transition Zones in Nonlinearly Elastic Isotropic Materials,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 91–109 (1994) (Part 1); No. 5, 49–61 (1994) (Part 2) [Mech. Solids (Engl. Transl.)].Google Scholar
  41. 41.
    A. B. Freidin, E. N. Vilchevskaya, and L. L. Sharipova, “Two-Phase Deformations within the Framework of Phase Transition Zones,” Theor. Appl. Mech. 28–29, 149–172 (2002).MathSciNetGoogle Scholar
  42. 42.
    A. B. Freidin and L. L. Sharipova, “On a Model of Heterogenous Deformation of Elastic Bodies by the Mechanism of Multiple Appearance of New Phase Layers,” Meccanica 41(3), 321–339 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    A. B. Freidin, Y. B. Fu, L. L. Sharipova, and E. N. Vilchevskaya, “Spherically Symmetric Two-Phase Deformations and Phase Transition Zones,” Int. J. Solids Struct. 43(14–15), 4484–4508 (2006).zbMATHCrossRefGoogle Scholar
  44. 44.
    A. B. Freidin, L. L. Sharipova, and E. N. Vilchevskaya, “Phase Transition Zones in Relations with Constitutive Equations of Elastic Solids,” in Proc. of the XXXII Summer School Actual Problems in Mechanics (APM-2004) (IPME RAS, St. Petersburg, 2004), pp. 140–150.Google Scholar
  45. 45.
    Y. Grabovsky and L. Truskinovsky, “Roughening Instability of Broken Extremals,” Arch. Rat. Mech. Anal. 200(1), 183–202 (2011).MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    V. A. Eremeev, A. B. Freidin, and L. L. Shapirova, “Nonuniqueness and Stability in Problems of Equilibrium of Elastic Two-Phase Bodies,” Dokl.Ross. Akad.Nauk 391(2), 189–193 (2003) [Dokl. Phys. (Engl. Transl.) 48 (7), 359–363 (2003)].Google Scholar
  47. 47.
    J. D. Eshelby, “The Determination of the Elastic Field on an Ellipsoidal Inclusion and Related Problems,” Proc. Roy. Soc. London. Ser. A 241(1226), 376–396 (1957).MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Y. B. Fu and A. B. Freidin, “Characterization and Stability of Two-Phase Piecewise-Homogeneous Deformation,” Proc. Roy. Soc. London. Ser. A 460(2051), 3065–3094 (2004).MathSciNetADSzbMATHCrossRefGoogle Scholar
  49. 49.
    M. A. Antimonov, A. V. Cherkaev, and A. B. Freidin, “In Transformation Surface Construction for Phase Transitions in Deformable Solids,” in Proc. XXXVIII Summer School-Conference’ Advanced Problems in Mechanics’ (APM 2010), St. Petersburg (Repino), July 1–5, 2010 (St. Petersburg, 2010), pp. 23–29, Scholar
  50. 50.
    I. A. Kunin, “Theory of Dislocations,” in J. A. Schouten, Tensor Analysis for Physicists (Nauka, Moscow, 1965), pp. 373–443 [in Russian].Google Scholar
  51. 51.
    I. A. Kunin, Theory of Elastic Media with Microstructure (Nauka, Moscow, 1975) [in Russian].Google Scholar
  52. 52.
    E. N. Vilchevskaya and A. B. Freidin, “Multiple Appearances of Ellipsoidal Nuclei of a New Phase,” Dokl. Ross. Akad. Nauk 411(6), 770–774 (2006) [Dokl. Phys. (Engl. Transl.) 51 (12), 692–696 (2006)].MathSciNetGoogle Scholar
  53. 53.
    A. B. Freidin and E. N. Vilchevskaya, “Multiple Development of New Phase Inclusions in Elastic Solids,” Int. J. Engng Sci. 47(2), 240–260 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    T. Mura, Micromechanics of Defects in Solids (Kluwer Academic, Dordrecht, 1987).CrossRefGoogle Scholar

Copyright information

© Allerton Press, Inc. 2013

Authors and Affiliations

  • E. N. Vilchevskaya
    • 1
  • R. A. Filippov
    • 1
  • A. B. Freidin
    • 1
  1. 1.Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia

Personalised recommendations