Journal of Elasticity

, Volume 136, Issue 1, pp 55–85 | Cite as

Effective Thermoelasticity of Polymer-Bonded Particle Composites with Imperfect Interfaces and Thermally Expansive Interphases

  • Kane C. Bennett
  • Darby J. LuscherEmail author


A micromechanical model for the thermoelasticity of polymer-bonded composites is presented. The model is particularly aimed at describing materials where the polymeric binder phase undergoes non-negligible thermal expansion affecting the overall thermoelastic response. Constitutive choices for modeling a mixed binder-void interphase layer are proposed, and an associated decomposition of total eigenstrains into classical, elastic imperfection (damage), and binder thermal expansivity parts is examined within the context of imperfect inter-particle interfaces. A novel temperature dependent modified Eshelby tensor is identified, making possible the development of a temperature dependent modified self-consistent homogenization scheme—what we call the M\(\theta\)-SCH model. A method for distinguishing between dispersed and isolated parts of the binder and void phases in the model is also provided, along with a description of particle coating (or interphase) thickness derived from particle morphology and mesoscale effective properties. Although the theory is general, its development is motivated by the need to model anisotropic and highly nonlinear observed thermal expansion behavior of the polymer bonded explosive PBX 9502, for which model simulations are performed and compared with existing measurements.


Thermoelasticity Coated-inclusion Multiphase-composite Micromechanics Self-consistent homogenization PBX 9502 

Mathematics Subject Classification

74Q15 74A50 74A15 74A60 74F05 74F20 15A72 74B05 74E05 74E30 74E25 74M25 



The authors are grateful to several LANL programs, especially funding support from D. Trujillo, K. Smale, and P. Buntain. This work was performed under the auspices of the U.S. Department of Energy under contract DE-AC52-06NA25396.


  1. 1.
    Aboudi, J.: Damage in composites—modeling of imperfect bonding. Compos. Sci. Technol. 28(2), 103–128 (1987) Google Scholar
  2. 2.
    Bedrov, D., Borodin, O., Smith, G.D., Sewell, T.D., Dattelbaum, D.M., Stevens, L.L.: A molecular dynamics simulation study of crystalline 1,3,5-triamino-2,4,6-trinitobenzene as a function of pressure and temperature. J. Chem. Phys. 131, 224 (2009) Google Scholar
  3. 3.
    Benjamin, A.S., Ahart, M., Gramsch, S.A., Stevens, L.L., Orler, E.B., Dattelbaum, D.M., Hemley, R.J.: Acoustic properties of Kel F-800 copolymer up to 85 GPa. J. Chem. Phys. 137(1), 014 (2012) Google Scholar
  4. 4.
    Bennett, K.C., Borja, R.I.: Hyper-elastoplastic/damage modeling of rock with application to porous limestone. Int. J. Solids Struct. 143, 218–231 (2018). Google Scholar
  5. 5.
    Bennett, K.C., Regueiro, R.A., Borja, R.I.: Finite strain elastoplasticity considering the Eshelby stress for materials undergoing plastic volume change. Int. J. Plast. 77, 214–245 (2016) Google Scholar
  6. 6.
    Bennett, K.C., Luscher, D.J., Buechler, M.A., Yeager, J.D.: A micromechanical framework and modified self-consistent homogenization scheme for the thermoelasticity of porous bonded-particle assemblies. Int. J. Solids Struct. 139–140, 224–237 (2018). Google Scholar
  7. 7.
    Benveniste, Y.: The effective mechanical behaviour of composite materials with imperfect contact between the constituents. Mech. Mater. 4(2), 197–208 (1985) MathSciNetGoogle Scholar
  8. 8.
    Benveniste, Y., Dvorak, G.J., Chen, T.: Stress fields in composites with coated inclusions. Mech. Mater. 7(4), 305–317 (1989) Google Scholar
  9. 9.
    Benveniste, Y., Dvorak, G.J., Chen, T.: On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media. J. Mech. Phys. Solids 39(7), 927–946 (1991) ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Bonfoh, N., Hounkpati, V., Sabar, H.: New micromechanical approach of the coated inclusion problem: Exact solution and applications. Comput. Mater. Sci. 62, 175–183 (2012) Google Scholar
  11. 11.
    Borja, R.I.: On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int. J. Solids Struct. 43(6), 1764–1786 (2006) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Borja, R.I., Choo, J.: Cam-Clay plasticity, Part VIII: A constitutive framework for porous materials with evolving internal structure. Comput. Methods Appl. Mech. Eng. 309, 653–679 (2016) ADSMathSciNetGoogle Scholar
  13. 13.
    Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media. Editions Technip, Paris (1987). Translation of: Acoustique des milieux poreux Google Scholar
  14. 14.
    Brown, E.N., Rae, P.J., Gray, G.T.: The influence of temperature and strain rate on the tensile and compressive constitutive response of four fluoropolymers. J. Phys. IV 134, 935–940 (2006) Google Scholar
  15. 15.
    Buechler, M.A., Miller, N.A., Luscher, D.J., Schwarz, R.B., Thompson, D.: Modeling the effects of texture on thermal expansion in pressed PBX 9502 components. In: ASME International Mechanical Engineering Congress and Exposition, vol. 9: Mechanics of Solids, Structures and Fluids. ASME, New York (2016) Google Scholar
  16. 16.
    Cady, H.H.: Growth and defects of explosives crystals. In: MRS Proceedings, vol. 296, p. 243. Cambridge University Press, Cambridge (1992) Google Scholar
  17. 17.
    Capolungo, L., Benkassem, S., Cherkaoui, M., Qu, J.: Self-consistent scale transition with imperfect interfaces: Application to nanocrystalline materials. Acta Mater. 56(7), 1546–1554 (2008) Google Scholar
  18. 18.
    Castañeda, P.P.: The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39(1), 45–71 (1991) ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Castañeda, P.P.: Stationary variational estimates for the effective response and field fluctuations in nonlinear composites. J. Mech. Phys. Solids 96, 660–682 (2016) MathSciNetGoogle Scholar
  20. 20.
    Chang, C.S., Bennett, K.C.: Micromechanical modeling for the deformation of sand with noncoaxiality between the stress and material axes. J. Eng. Mech. 143(1), C4015001 (2015) Google Scholar
  21. 21.
    Cherkaoui, M., Muller, D., Sabar, H., Berveiller, M.: Thermoelastic behavior of composites with coated reinforcements: a micromechanical approach and applications. Comput. Mater. Sci. 5(1), 45–52 (1996). Computational Modelling of the Mechanical Behaviour of Materials Google Scholar
  22. 22.
    Cherkaoui, M., Sabar, H., Berveiller, M.: Elastic behavior of composites with coated inclusions: micromechanical approach and applications. Compos. Sci. Technol. 56(7), 877–882 (1996) Google Scholar
  23. 23.
    Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in 3 phase sphere and cylinder models. J. Mech. Phys. Solids 27(4), 315–330 (1979) ADSzbMATHGoogle Scholar
  24. 24.
    Cunningham, B., Andreski, H., Weese, R., Turner, H., Lauderbach, L.: Thermal expansion measurements on samples cored from hemispherical pressings. Tech. Rep. (2005) Google Scholar
  25. 25.
    Dinzart, F., Sabar, H.: Homogenization of the viscoelastic heterogeneous materials with multi-coated reinforcements: an internal variables formulation. Arch. Appl. Mech. 84(5), 715–730 (2014) ADSzbMATHGoogle Scholar
  26. 26.
    Dinzart, F., Sabar, H., Berbenni, S.: Homogenization of multi-phase composites based on a revisited formulation of the multi-coated inclusion problem. Int. J. Eng. Sci. 100, 136–151 (2016) MathSciNetzbMATHGoogle Scholar
  27. 27.
    Dumont, S., Lebon, F., Raffa, M.L., Rizzoni, R., Welemane, H.: Multiscale Modeling of Imperfect Interfaces and Applications, pp. 81–122. Springer, Cham (2016) Google Scholar
  28. 28.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc., Math. Phys. Eng. Sci. 241, 376–396 (1957) ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Gao, Z.J.: A circular inclusion with imperfect interface: Eshelby’s tensor and related problems. J. Appl. Mech. 62(4), 860–866 (1995). International Mechanical Engineering Congress and Exhibition/Winter Annual Meeting of the ASME, San Francisco, CA, Nov. 12–17, 1995 ADSzbMATHGoogle Scholar
  30. 30.
    Gao, S.L., Mäder, E.: Characterisation of interphase nanoscale property variations in glass fibre reinforced polypropylene and epoxy resin composites. Composites, Part A, Appl. Sci. Manuf. 33(4), 559–576 (2002) Google Scholar
  31. 31.
    Gavazzi, A.C., Lagoudas, D.C.: On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites. Comput. Mech. 7(1), 13–19 (1990) Google Scholar
  32. 32.
    Gorham, J.M., Woodcock, J.W., Scott, K.C.: Challenges, strategies and opportunities for measuring carbon nanotubes within a polymer composite by X-ray photoelectron spectroscopy. NIST Special Publication 1200-10 (2015) Google Scholar
  33. 33.
    Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press, London (1954) zbMATHGoogle Scholar
  34. 34.
    Hashin, Z.: Analysis of composite materials—a survey. J. Appl. Mech. 50(3), 481–505 (1983) ADSzbMATHGoogle Scholar
  35. 35.
    Hashin, Z.: Thermoelastic properties of particulate composites with imperfect interface. J. Mech. Phys. Solids 39(6), 745–762 (1991) ADSMathSciNetGoogle Scholar
  36. 36.
    Herve, E., Zaoui, A.: \(n\)-Layered inclusion-based micromechanical modelling. J. Mech. Phys. Solids 13(4), 213–222 (1993) zbMATHGoogle Scholar
  37. 37.
    Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13(4), 213–222 (1965) ADSMathSciNetGoogle Scholar
  38. 38.
    Hill, R.: The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15(2), 79–95 (1967) ADSMathSciNetGoogle Scholar
  39. 39.
    Hill, R.: Interfacial operators in the mechanics of composite media. J. Mech. Phys. Solids 31(4), 347–357 (1983) ADSMathSciNetzbMATHGoogle Scholar
  40. 40.
    Hori, M., Nemat-Nasser, S.: Double-inclusion model and overall moduli of multi-phase composites. Mech. Mater. 14(3), 189–206 (1993) Google Scholar
  41. 41.
    Hu, G.K., Weng, G.J.: The connections between the double-inclusion model and the Ponte Castaneda-Willis, Mori-Tanaka, and Kuster-Toksoz models. Mech. Mater. 32(8), 495–503 (2000) Google Scholar
  42. 42.
    Huang, Y., Hu, K.X., Wei, X., Chandra, A.: A generalized self-consistent mechanics method for composite materials with multiphase inclusions. J. Mech. Phys. Solids 42(3), 491–504 (1994) ADSzbMATHGoogle Scholar
  43. 43.
    Kim, J.K., Sham, M.L., Wu, J.: Nanoscale characterisation of interphase in silane treated glass fibre composites. Composites, Part A, Appl. Sci. Manuf. 32(5), 607–618 (2001) Google Scholar
  44. 44.
    Kolb, J.R., Rizzo, H.F.: Growth of 1,3,5-triamino-2,4,6-trinitobenzene (TATB): I. Anisotropic thermal-expansion. Propellants Explos. Pyrotech. 4, 10–16 (1979) Google Scholar
  45. 45.
    Laws, N.: On interfacial discontinuities in elastic composites. J. Elast. 5(3), 227–235 (1975) zbMATHGoogle Scholar
  46. 46.
    Lebensohn, R.A., Tomé, C.N., Maudlin, P.J.: A self-consistent formulation for the prediction of the anisotropic behavior of viscoplastic polycrystals with voids. J. Mech. Phys. Solids 52(2), 249–278 (2004) ADSMathSciNetzbMATHGoogle Scholar
  47. 47.
    Li, J.Y.: On micromechanics approximation for the effective thermoelastic moduli of multi-phase composite materials. Mech. Mater. 31(2), 149–159 (1999) Google Scholar
  48. 48.
    Lipinski, P., Barhdadi, E.H., Cherkaoui, M.: Micromechanical modelling of an arbitrary ellipsoidal multi-coated inclusion. Philos. Mag. 86(10), 1305–1326 (2006) ADSGoogle Scholar
  49. 49.
    Luscher, D.J., Buechler, M.A., Miller, N.A.: Self-consistent modeling of the influence of texture on thermal expansion in polycrystalline TATB. Model. Simul. Mater. Sci. Eng. 22(7), 075008 (2014) ADSGoogle Scholar
  50. 50.
    March, A.: Mathematical theory on regulation according to the particle shape and affine deformation. Z. Kristallogr. 81, 285–297 (1932) Google Scholar
  51. 51.
    Mavko, G., Mukerji, T., Dvorkin, J.: Rock Physics Handbook—Tools for Seismic Analysis in Porous Media. Cambridge University Press, Cambridge (2003) Google Scholar
  52. 52.
    Meidani, M., Chang, C.S., Deng, Y.: On active and inactive voids and a compression model for granular soils. Eng. Geol. 222, 156–167 (2017) Google Scholar
  53. 53.
    Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21(5), 571–574 (1973) Google Scholar
  54. 54.
    Mouden, M.E., Cherkaoui, M., Molinari, A., Berveiller, M.: The overall elastic response of materials containing coated inclusions in a periodic array. Int. J. Eng. Sci. 36(7), 813–829 (1998) Google Scholar
  55. 55.
    Mura, T.: Micromechanics of Defects in Solids, 2nd edn. Nijhof, Dordrecht (1987) zbMATHGoogle Scholar
  56. 56.
    Nandi, A.K., Kasar, S.M., Thanigaivelan, U., Ghosh, M., Mandal, A.K., Bhattacharyya, S.C.: Synthesis and characterization of ultrafine TATB. J. Energ. Mater. 25(4), 213–231 (2007) Google Scholar
  57. 57.
    Nemat-Nasser, S., Hori, M.: Applied mathematics and mechanics. In: Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland Series in Applied Mathematics and Mechanics, vol. 37, pp. ii–687. North-Holland, Amsterdam (1993) Google Scholar
  58. 58.
    Nemat-Nasser, S., Iwakuma, T., Hejazi, M.: On composites with periodic structure. Mech. Mater. 1(3), 239–267 (1982) Google Scholar
  59. 59.
    Ostoja-Starzewski, M.: Material spatial randomness: from statistical to representative volume element. Probab. Eng. Mech. 21(2), 112–132 (2006) MathSciNetGoogle Scholar
  60. 60.
    Qu, J.: Eshelby tensor for an elastic inclusion with slightly weakened interface. J. Appl. Mech. 60(4), 1048–1050 (1993) ADSzbMATHGoogle Scholar
  61. 61.
    Qu, J.: The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech. Mater. 14, 269–281 (1993) Google Scholar
  62. 62.
    Qu, J., Cherkaoui, M.: Fundamentals of Micromechanics of Solids. Wiley, New York (2006) Google Scholar
  63. 63.
    Rae, P.: The linear thermal expansion of 11 polymers from approximately \(-100 \mbox{ to } +100~{}^{\circ}\mbox{C}\). Tech. rep., Los Alamos National Laboratory (2015) Google Scholar
  64. 64.
    Salari, M.R., Saeb, S., Willam, K.J., Patchet, S.J., Carrasco, R.C.: A coupled elastoplastic damage model for geomaterials. Comput. Methods Appl. Mech. Eng. 193(27), 2625–2643 (2004) ADSzbMATHGoogle Scholar
  65. 65.
    Schlenker, J.L., Gibbs, G.V., Boisen, M.B.: Strain-tensor components expressed in terms of lattice parameters. Acta Crystallogr. A, Found. Crystallogr. 34(1), 52–54 (1978) ADSGoogle Scholar
  66. 66.
    Schöneich, M., Dinzart, F., Sabar, H., Berbenni, S., Stommel, M.: A coated inclusion-based homogenization scheme for viscoelastic composites with interphases. Mech. Mater. 105, 89–98 (2017) Google Scholar
  67. 67.
    Sidhom, M., Dormieux, L., Lemarchand, E.: Poroelastic properties of a nanoporous granular material with interface effects. J. Nanomech. Micromech. 5(3), 04014001 (2014) Google Scholar
  68. 68.
    Sun, J., Kang, B., Xue, C., Liu, Y., Xia, Y., Liu, X.: Crystal state of 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) undergoing thermal cycling process. J. Energ. Mater. 28, 189–201 (2010) ADSGoogle Scholar
  69. 69.
    Torquato, S., Rintoul, M.D.: Effect of the interface on the properties of composite media. Phys. Rev. Lett. 75, 4067–4070 (1995) ADSGoogle Scholar
  70. 70.
    Walpole, L.J.: A coated inclusion in an elastic medium. Math. Proc. Camb. Philos. Soc. 83, 495 (1978) MathSciNetzbMATHGoogle Scholar
  71. 71.
    Wei, P.J., Huang, Z.P.: Dynamic effective properties of the particle-reinforced composites with the viscoelastic interphase. Int. J. Solids Struct. 41(24), 6993–7007 (2004) zbMATHGoogle Scholar
  72. 72.
    Yeager, J.D., Luscher, D.J., Vogel, S.C., Clausen, B., Brown, D.W.: Neutron diffraction measurements and micromechanical modelling of temperature-dependent variations in TATB lattice parameters. Propellants Explos. Pyrotech. 41, 514–525 (2016) Google Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Fluid Dynamics & Solid Mechanics Group (T-3), Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA

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