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

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Abstract

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.

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References

  1. Aboudi, J.: Damage in composites—modeling of imperfect bonding. Compos. Sci. Technol. 28(2), 103–128 (1987)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.ijsolstr.2018.03.011

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.ijsolstr.2018.02.001

    Article  Google Scholar 

  7. Benveniste, Y.: The effective mechanical behaviour of composite materials with imperfect contact between the constituents. Mech. Mater. 4(2), 197–208 (1985)

    Article  MathSciNet  Google Scholar 

  8. Benveniste, Y., Dvorak, G.J., Chen, T.: Stress fields in composites with coated inclusions. Mech. Mater. 7(4), 305–317 (1989)

    Article  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  13. Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media. Editions Technip, Paris (1987). Translation of: Acoustique des milieux poreux

    Google Scholar 

  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. 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. Cady, H.H.: Growth and defects of explosives crystals. In: MRS Proceedings, vol. 296, p. 243. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  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)

    Article  Google Scholar 

  18. Castañeda, P.P.: The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39(1), 45–71 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  24. Cunningham, B., Andreski, H., Weese, R., Turner, H., Lauderbach, L.: Thermal expansion measurements on samples cored from hemispherical pressings. Tech. Rep. (2005)

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  33. Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press, London (1954)

    MATH  Google Scholar 

  34. Hashin, Z.: Analysis of composite materials—a survey. J. Appl. Mech. 50(3), 481–505 (1983)

    Article  ADS  MATH  Google Scholar 

  35. Hashin, Z.: Thermoelastic properties of particulate composites with imperfect interface. J. Mech. Phys. Solids 39(6), 745–762 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  36. Herve, E., Zaoui, A.: \(n\)-Layered inclusion-based micromechanical modelling. J. Mech. Phys. Solids 13(4), 213–222 (1993)

    MATH  Google Scholar 

  37. Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13(4), 213–222 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  38. Hill, R.: The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15(2), 79–95 (1967)

    Article  ADS  MathSciNet  Google Scholar 

  39. Hill, R.: Interfacial operators in the mechanics of composite media. J. Mech. Phys. Solids 31(4), 347–357 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Hori, M., Nemat-Nasser, S.: Double-inclusion model and overall moduli of multi-phase composites. Mech. Mater. 14(3), 189–206 (1993)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  45. Laws, N.: On interfacial discontinuities in elastic composites. J. Elast. 5(3), 227–235 (1975)

    Article  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Li, J.Y.: On micromechanics approximation for the effective thermoelastic moduli of multi-phase composite materials. Mech. Mater. 31(2), 149–159 (1999)

    Article  Google Scholar 

  48. Lipinski, P., Barhdadi, E.H., Cherkaoui, M.: Micromechanical modelling of an arbitrary ellipsoidal multi-coated inclusion. Philos. Mag. 86(10), 1305–1326 (2006)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  50. March, A.: Mathematical theory on regulation according to the particle shape and affine deformation. Z. Kristallogr. 81, 285–297 (1932)

    Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  55. Mura, T.: Micromechanics of Defects in Solids, 2nd edn. Nijhof, Dordrecht (1987)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  58. Nemat-Nasser, S., Iwakuma, T., Hejazi, M.: On composites with periodic structure. Mech. Mater. 1(3), 239–267 (1982)

    Article  Google Scholar 

  59. Ostoja-Starzewski, M.: Material spatial randomness: from statistical to representative volume element. Probab. Eng. Mech. 21(2), 112–132 (2006)

    Article  MathSciNet  Google Scholar 

  60. Qu, J.: Eshelby tensor for an elastic inclusion with slightly weakened interface. J. Appl. Mech. 60(4), 1048–1050 (1993)

    Article  ADS  MATH  Google Scholar 

  61. Qu, J.: The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech. Mater. 14, 269–281 (1993)

    Article  Google Scholar 

  62. Qu, J., Cherkaoui, M.: Fundamentals of Micromechanics of Solids. Wiley, New York (2006)

    Book  Google Scholar 

  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)

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    Article  Google Scholar 

  67. Sidhom, M., Dormieux, L., Lemarchand, E.: Poroelastic properties of a nanoporous granular material with interface effects. J. Nanomech. Micromech. 5(3), 04014001 (2014)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  69. Torquato, S., Rintoul, M.D.: Effect of the interface on the properties of composite media. Phys. Rev. Lett. 75, 4067–4070 (1995)

    Article  ADS  Google Scholar 

  70. Walpole, L.J.: A coated inclusion in an elastic medium. Math. Proc. Camb. Philos. Soc. 83, 495 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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Acknowledgements

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.

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  1. Fluid Dynamics & Solid Mechanics Group (T-3), Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Kane C. Bennett & Darby J. Luscher

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Bennett, K.C., Luscher, D.J. Effective Thermoelasticity of Polymer-Bonded Particle Composites with Imperfect Interfaces and Thermally Expansive Interphases. J Elast 136, 55–85 (2019). https://doi.org/10.1007/s10659-018-9688-z

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