Abstract
A maximum entropy-based stochastic micromechanical framework considering the inter-particle interaction effect is proposed to characterize the probabilistic behavior of the effective properties of two-phase composite materials. Based on our previous work, the deterministic micromechanical model of the two-phase composites is derived by introducing the strain concentration tensors considering the inter-particle interaction effect. By modeling the volume fractions and properties of constituents as stochastic, we extend the deterministic framework to stochastics, to incorporate the inherent randomness of effective properties among different specimens. A distribution-free method is employed to get the unbiased probability density function based on the maximum entropy principle. Further, the normalization procedures are utilized to make the probability density functions more stable. Numerical examples including limited experimental validations, comparisons with existing micromechanical models, commonly used probability density functions and the direct Monte Carlo simulations indicate that the proposed models provide an accurate and computationally efficient framework in characterizing the effective properties of two-phase composites.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Torquato S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, Berlin (2001)
Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 241, 376–396 (1957)
Eshelby J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 252, 561–569 (1959)
Eshelby J.D.: Elastic inclusions and inhomogeneities. Prog. Solid Mech. 2, 89–140 (1961)
Hashin Z., Shtrikman S.: On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335–342 (1962)
Hashin Z., Shtrikman S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10, 343–352 (1962)
Hashin Z., Shtrikman S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140 (1963)
Torquato S.: Random heterogeneous media: microstructure and improved bounds on effective properties. Appl. Mech. Rev. 44, 37–76 (1991)
Willis J.: On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids 39, 73–86 (1991)
Beran M., Molyneux J.: Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Quart. Appl. Math. 24, 107–118 (1966)
Willis J.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185–202 (1977)
Hill R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222 (1965)
Roscoe R.: Isotropic composites with elastic or viscoelastic phases: general bounds for the moduli and solutions for special geometries. Rheol. Acta 12, 404–411 (1973)
Mori T., Tanaka K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973)
Benveniste Y.: A new approach to the application of Mori–Tanaka’s theory in composite materials. Mech. Mater. 6, 147–157 (1987)
Christensen R., Lo K.: Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330 (1979)
Sheng P.: Effective-medium theory of sedimentary rocks. Phys. Rev. B 41, 4507–4512 (1990)
Sheng P., Callegari A.: Differential effective medium theory of sedimentary rocks. Appl. Phys. Lett. 44, 738–740 (1984)
Nguyen N., Giraud A., Grgic D.: A composite sphere assemblage model for porous oolitic rocks. Int. J. Rock Mech. Min. Sci. 48, 909–921 (2011)
Li G., Zhao Y., Pang S.S.: Four-phase sphere modeling of effective bulk modulus of concrete. Cem. Concr. Res. 29, 839–845 (1999)
Wang H., Li Q.: Prediction of elastic modulus and Poisson’s ratio for unsaturated concrete. Int. J. Solids Struct. 44, 1370–1379 (2007)
Yaman I., Aktan H., Hearn N.: Active and non-active porosity in concrete part II: evaluation of existing models. Mater. Struct. 35, 110–116 (2002)
Zhu H.H., Chen Q., Yan Z.G., Ju J.W., Zhou S.: Micromechanical models for saturated concrete repaired by electrochemical deposition method. Mater. Struct. 47, 1067–1082 (2014)
Yan Z.G., Chen Q., Zhu H.H., Ju J.W., Zhou S., Jiang Z.W.: A multiphase micromechanical model for unsaturated concrete repaired by electrochemical deposition method. Int. J. Solids Struct. 50, 3875–3885 (2013)
Yang Q.S., Tao X., Yang H.: A stepping scheme for predicting effective properties of the multi-inclusion composites. Int. J. Eng. Sci. 45, 997–1006 (2007)
Garboczi E., Berryman J.: Elastic moduli of a material containing composite inclusions: Effective medium theory and finite element computations. Mech. Mater. 33, 455–470 (2001)
Chen H.S., Acrivos A.: The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Int. J. Solids Struct. 14, 349–364 (1978)
Ju J., Chen T.M.: Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mech. 103, 103–121 (1994)
Ju J., Chen T.: Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mech. 103, 123–144 (1994)
Ju J., Sun L.: A novel formulation for the exterior-point Eshelby’s tensor of an ellipsoidal inclusion. J. Appl. Mech. 66, 570–574 (1999)
Ju J., Zhang X.: Micromechanics and effective transverse elastic moduli of composites with randomly located aligned circular fibers. Int. J. Solids Struct. 35, 941–960 (1998)
Ju J., Sun L.: Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal inhomogeneities. Part I: micromechanics-based formulation. Int. J. Solids Struct. 38, 183–201 (2001)
Sun L., Ju J.: Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal inhomogeneities. Part II: applications. Int. J. Solids Struct. 38, 203–225 (2001)
Sun L., Ju J.: Elastoplastic modeling of metal matrix composites containing randomly located and oriented spheroidal particles. J. Appl. Mech. 71, 774–785 (2004)
Ju J., Yanase K.: Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions. Acta Mech. 215, 135–153 (2010)
Ju J., Yanase K.: Micromechanical effective elastic moduli of continuous fiber-reinforced composites with near-field fiber interactions. Acta Mech. 216, 87–103 (2011)
Ju J., Yanase K.: Size-dependent probabilistic micromechanical damage mechanics for particle-reinforced metal matrix composites. Int. J. Damage Mech. 20, 1021–1048 (2011)
Yanase K., Ju J.W.: Effective elastic moduli of spherical particle reinforced composites containing imperfect interfaces. Int. J. Damage Mech. 21, 97–127 (2012)
Ferrante F., Graham-Brady L.: Stochastic simulation of non-Gaussian/non-stationary properties in a functionally graded plate. Comput. Methods Appl. Mech. Eng. 194, 1675–1692 (2005)
Banchs R.E., Klie H., Rodriguez A., Thomas S.G., Wheeler M.F.: A neural stochastic multiscale optimization framework for sensor-based parameter estimation. Integr. Comput. Aided Eng. 14, 213–223 (2007)
Biswal B., Øren P.-E., Held R., Bakke S., Hilfer R.: Stochastic multiscale model for carbonate rocks. Phys. Rev. E 75, 1–5 (2007)
Chakraborty A., Rahman S.: Stochastic multiscale models for fracture analysis of functionally graded materials. Eng. Fract. Mech. 75, 2062–2086 (2008)
Chakraborty A., Rahman S.: A parametric study on probabilistic fracture of functionally graded composites by a concurrent multiscale method. Probab. Eng. Mech. 24, 438–451 (2009)
Ganapathysubramanian B., Zabaras N.: A stochastic multiscale framework for modeling flow through random heterogeneous porous media. J. Comput. Phys. 228, 591–618 (2009)
Liu, W.K., Siad, L., Tian, R., Lee, S., Lee, D., Yin, X., Yin, X., Yin, X.: Complexity science of multiscale materials via stochastic computations. Int. J. Numer. Methods Eng. 80, 932–978 (2009)
Rahman S.: Multi-scale fracture of random heterogeneous materials. Ships Offshore Struct. 4, 261–274 (2009)
Yin X.L., Lee S., Chen W., Liu W.K., Horstemeyer M.F.: Efficient random field uncertainty propagation in design using multiscale analysis. J. Mech. Des. 131, 1–10 (2009)
Ferrante F.J., Brady L.L.G., Acton K., Arwade S.R.: An overview of micromechanics based techniques for the analysis of microstructural randomness in functionally graded materials. AIP Conf. Proc. 973, 190–195 (2008)
Rahman S., Chakraborty A.: A stochastic micromechanical model for elastic properties of functionally graded materials. Mech. Mater. 39, 548–563 (2007)
Xu X., Graham-Brady L.: A stochastic computational method for evaluation of global and local behavior of random elastic media. Comput. Methods Appl. Mech. Eng. 194, 4362–4385 (2005)
Xu X., Chen X.: Stochastic homogenization of random elastic multi-phase composites and size quantification of representative volume element. Mech. Mater. 41, 174–186 (2009)
Chen, Q., Zhu, H.H., Ju, J.W., Guo, F., Wang, L.B., Yan, Z.G., Deng, T., Zhou, S.: A stochastic micromechanical model for multiphase composite containing spherical inhomogeneities. Acta Mech. doi:10.1007/s00707-014-1278-y
Li X.J., Chen X.Q., Zhu H.H.: Reliability analysis of shield lining sections using Spreadsheet method. Chin. J. Geotech. Eng. 9, 1642–1649 (2013)
Jaynes E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)
Zhu H.H., Zuo Y.L., Li X.J., Deng J., Zhuang X.Y.: Estimation of the fracture diameter distributions using the maximum entropy principle. Int. J. Rock Mech. Min. Sci. 72, 127–137 (2014)
Li X.J., Zuo Y.L., Zhuang X.Y., Zhu H.H.: Estimation of fracture trace length distributions using probability weighted moments and L-moments. Eng. Geol. 168, 69–85 (2014)
Qu J., Cherkaoui M.: Fundamentals of Micromechanics of Solids. Wiley, New York (2006)
Mura T.: Micromechanics of Defects in Solids. Kluwer, Dordrecht (1987)
Er G.K.: A method for multi-parameter PDF estimation of random variables. Struct. Saf. 20, 25–36 (1998)
Smith J.C.: Experimental values for the elastic constants of a particulate-filled glassy polymer. J. Res. NBS 80, 45–49 (1976)
Walsh J.B., Brace W.E., England A.W.: Effect of porosity on compressibility of glass. J. Am. Ceram. Soc. 48, 605–608 (1965)
Weng G.: Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions. Int. J. Eng. Sci. 22, 845–856 (1984)
Weng G.: The theoretical connection between Mori–Tanaka’s theory and the Hashin–Shtrikman–Walpole bounds. Int. J. Eng. Sci. 28, 1111–1120 (1990)
Parameswaran V., Shukla A.: Processing and characterization of a model functionally gradient material. J. Mater. Sci. 35, 21–29 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, H.H., Chen, Q., Ju, J.W. et al. Maximum entropy-based stochastic micromechanical model for a two-phase composite considering the inter-particle interaction effect. Acta Mech 226, 3069–3084 (2015). https://doi.org/10.1007/s00707-015-1375-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-015-1375-6