The aim of this research is to develop the mechanisms of calculation of stress and strain fields statistical characteristics in components of heterogeneous solid media in dependence on microstructural parameters.Multiscale hierarchy of heterogeneous (composite) materials is investigated using the Repre- sentative Volume Element (RVE) concept when parameters of larger scale models are measured or calculated on a smaller scale. Fields of stress, strain and displacements are presented in the form of statistically homogeneous coordinate functions.Analytical expressions for statistical characteristics of structural fields, such as mean values and dispersions, are formed using solution of stochastic boundary value problems and contain structural multipoint correlation functions. The order of the required correlation functions is determined by the solution of stochastic boundary value problem. The boundary value problem is being reduced to inte- gral-differential equation in fluctuations of displacements. The second approximation of the problem solution is obtained. To determine relation between deformation in components and macroscopic de- formation, the iteration procedure was organized.New analytical expressions and numerical results for statistical characteristics of stress and strain fields in components of elastoplastic composite materials are derived with the second approxima- tion of the boundary value problem and correlation functions up to fifth order. 3D models of representa- tive volume of material microstructure with polydisperse ellipsoidal inclusions are synthesized; the mul- tipoint correlation functions up to fifth order have been obtained for them. Numerical results are obtained for porous composites with polydisperse ellipsoidal inclusions in simple shear state of strain for micro- structures with various inclusions volume fraction.

About the authors

M A Tashkinov

Perm National Research Polytechnic University

29, Komsomolsky av., 614990, Perm, Russian Federation


  1. Buryachenko V. Micromechanics of heterogeneous materials. - New York: Springer, 2007. - 686 p.
  2. Torquato S. Random heterogenous materials, microstructure and mac-roscopic properties. - Springer, 2001. - 701 p.
  3. Determination of the size of the representative volume element for random composites: statistical and numerical approach / T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin // International Journal of Solids and Structures. - 2003. - Vol. 40. - P. 3647-3679.
  4. Liu K.C., Ghoshal A. Validity of random microstructures simulation in fiber-reinforced composite materials // Composites Part B: Engi-neering. - 2014. - Vol. 57. - P. 56-70.
  5. Tashkinov M.A., Wildemann V.E., Mikhailova N.V. Method of suc-cessive approximations in stochastic elastic boundary value problem for structurally heterogenous materials // Computational Materials Science. - 2012. - Vol. 52. - P 101-106. doi: 10.1016/j.commatsci.2011.04.025
  6. Хорошун Л.П. Методы случайных функций в задачах о макро-скопических свойствах микронеоднородных сред // Прикл. меха-ника. - 1978. - Т. 14. - Вып. 2. - С. 3-17.
  7. Волков С.Д., Ставров В.П. Статистическая механика композит-ных материалов. - Минск: Изд-во Белорус. гос. ун-та, 1978. - 208 с.
  8. Вильдеман В.Э., Соколкин Ю.В., Ташкинов А.А. Механика не-упругого деформирования и разрушения композиционных материалов. - М.: Наука, 1997. - 288 с.
  9. Saheli G., Garmestani H., Adams B.L. Microstructure design of a two phase composite using two-point correlation functions // Journal of Computer-Aided Materials Design. - 2004. - Vol. 11. - P. 103-115.
  10. Шермергор Т.Д. Теория упругости микронеоднородных сред. - М.: Наука, 1976. - 400 с.
  11. Михайлова Н.В., Ташкинов А.А. Упругопластическое деформи-рование дисперсных композитов с разреженной случайной структурой // Механика композиционных материалов и конструкций. - 2010. - Т. 16, № 4. - С. 469-482.
  12. Паньков А.А. Статистическая механика пьезокомпозитов. - Пермь: Изд-во Перм. гос. техн. ун-та, 2009. - 480 с.
  13. Соколкин Ю.В., Волкова Т.А. Расчет распределения деформаций и напряжений в зернистых композитах с учетом реальных мо-ментных функций свойств микроструктуры // Механика компо-зиционных материалов и конструкций. - 1998. - Т. 4, № 3. - С. 70-85.
  14. Соколкин Ю.В., Ташкинов А.А. Механика деформирования и разрушения структурно-неоднородных тел. - М.: Наука, 1984. - 116 с.
  15. Jiao Y., Stillinger F.H., Torquato S. Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and appli-cations // Physical Review. -2008. - Vol. 77. - No. 3. - Р. 031135. doi: 10.1103/PhysRevE.77.031135
  16. Kaminski M.M. Computational mechanics of composite materials. - Springer, 2005. - 433 p.
  17. Multiscale methods for composites: a review / P. Kanouté, D.P. Boso, J.L. Chaboche, B.A. Schrefler // Arch. Comput. Methods. Eng. - 2009. - Vol. 16. - Р. 31-75.
  18. Ильиных А.В., Радионова М.В., Вильдеман В.Э. Компьютерный синтез и статистический анализ распределения структурных ха-рактеристик зернистых композиционных материалов // Механика композиционных материалов и конструкций. - 2010. - Т. 16, № 2. - С. 251-265.
  19. Rasool A., Böhm H.J. Effects of particle shape on the macroscopic and microscopic linear behaviors of particle reinforced composites // International Journal of Engineering Science. - 2012. - Vol. 58. - Р. 21-34.
  20. Hori M., Kubo J. Analysis of probabilistic distribution and range of average stress in each phase of heterogeneous materials // J. Mech. Phys. Solids. - 1998. - Vol. 46. - Р. 537-556.
  21. Tashkinov M.A., Vildeman V.E., Mikhailova N.V. Method of succes-sive approximations in a stochastic boundary-value problem in the elasticity theory of structurally heterogeneous media // Composites: Mechanics, Computations, Applications. - 2011. - Vol. 2. - No. 1. - Р. 21-37.
  22. Silberschmidt V.V. Account for random microstructure in multiscale models // Multiscale Modeling and Simulation of Composite Materials and Structures. Eds. Y.W. Kwon, D.H. Allen and R. Talreja. - New York: Springer, 2008. - Р. 1-35.
  23. Lotwick H.W. Simulations on some spatial hard core models, and the complete packing problem // J. Statist. Comp. Simul. - 1982. - Vol. 15. - Р. 295-314.



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