Abstract
The basic approaches used in mathematical models and general methods for solution of the equations of the mechanics of stochastic composites are generalized. They can be reduced to the stochastic equations of the theory of elasticity of a structurally inhomogeneous medium, to the equations of the theory of effective elastic moduli, to the equations of the theory of elastic mixtures, or to more general equations of the fourth order. The solution of the stochastic equations of the elastic theory for an arbitrary domain involves substantial mathematical difficulties and may be implemented only rather approximately. The construction of the equations of the theory of effective moduli is associated with the problem on the effective moduli of a stochastically inhomogeneous medium, which can be solved by the perturbation method, by the method of moments, or by the method of conditional moments. The latter method is most appropriate. It permits one to determine the effective moduli in a two-point approximation and nonlinear deformation properties. In the structure of equations, the theory of elastic mixtures is more general than the theory of effective moduli; however, since the state equations have not been strictly substantiated and the constants have not been correctly determined, theoretically or experimentally, this theory cannot be used for systematic designing composite structures. A new model of the nonuniform deformation of composites is more promising. It is constructed by performing strict mathematical transformations and averaging the output stochastic equations, all the constants being determined. In the zero approximation, the equations of the theory of effective moduli follow from this model, and, in the first approximation, fourth-order equations, which are more general than those of the theory of mixtures, follow from it
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
N. S. Bakhvalov and G. P. Panasenko, Averaging of Processes in Periodic Media [in Russian], Nauka, Moscow (1984).
M. J. Beran, “Application of statistical theories to determine the thermal, electric, and magnetic properties of inhomogeneous materials,” in: G. P. Sendeckyi (ed.), The Mechanics of Composite Materials, Vol. 2 of the series L. J. Broutman and R. H. Krock (eds.), Composite Materials, Acad. Press, New York–London (1974).
V. V. Bolotin, Statistical Methods in Structural Mechanics [in Russian], Izd. Inostr. Lit. Stroit., Moscow (1965).
V. V. Bolotin and V. N. Moskalenko, “Determination of the elastic constants of a microinhomogeneous medium,” Zh. Prikl. Mekh. Tekhn. Fiz., No. 1, 66–72 (1968).
G. A. Van Fo Fy, The Theory of Reinforced Materials [in Russian], Naukova Dumka, Kiev (1971).
G. A. Vanin, The Micromechanics of Composites [in Russian], Naukova Dumka, Kiev (1985).
S. D. Volkov and V. Ya. Dolgikh, “Structural stresses in reinforced plates,” Mekh. Polimer., No. 3, 413–420 (1967).
S. D. Volkov and V. P. Stavrov, The Statistical Mechanics of Composites [in Russian], Izd. Belorusskogo Univ., Minsk (1978).
M. I. Gai, L. I. Manevich, and V. G. Oshmyan, “Percolation effects in mechanical systems,” Dokl. Akad. Nauk SSSR, 276, No. 6, 1389–1391 (1984).
V. T. Golovchan, Physicomechanical Anisotropy of Composites [in Russian], Naukova Dumka, Kiev (1987).
V. T. Golovchan, A. N. Guz', Yu. V. Kokhanenko, and V. I. Kushch, The Statics of Materials, Vol. 1 of the 12-volume series The Mechanics of Composites [in Russian], Naukova Dumka, Kiev (1983).
A. N. Guz', “The theory of composites with initial stresses,” in: The Mechanics of Deformable Bodies and Structures [in Russian], Nauka, Moscow (1975), pp. 140–148.
A. N. Guz', “Determination of the reduced elastic constants of laminated composite materials with initial stresses,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 3, 216–219 (1975).
W. D. Kingery, Property Measurements at High Temperatures. Factors Affecting and Methods of Measuring Material Properties at Temperatures above 1400 °C (2500 °F), Wiley, New York, Chapman and Hall, London (1960).
M. A. Krivoglaz and A. S. Cherevko, “The elastic moduli of a solid mixture,” Fiz. Metal. Metalloved., No. 2, 161–165 (1959).
R. M. Christensen, Introduction to the Mechanics of Composites [Russian translation], Mir, Moscow (1982).
I. M. Lifshits and L. N. Rozentsveig, “The theory of the elastic properties of polycrystals,” Zh. Eksp. Teor. Fiz., No. 11, 967–980 (1946).
V. A. Lomakin, “The deformation of microinhomogeneous elastic bodies,” Prikl. Mat. Mekh., No. 5, 888–893 (1965).
V. A. Lomakin, “Influence of microheterogeneities in the structure of materials on their mechanical properties,” in: Reliability Problems in Structural Mechanics [in Russian], Vilnius (1968), pp. 107–112.
V. A. Lomakin, Statistical Problems of the Mechanics of Deformable Solids [in Russian], Nauka, Moscow (1970).
A. K. Malmeister, V. P. Tamuzh, and T. A. Teters, Resistance of Polymeric and Composite Materials [in Russian], Zinatne, Riga (1980).
R. I. Nigmatulin, Fundamentals of the Mechanics of Heterogeneous Media [in Russian], Nauka, Moscow (1978).
W. Nowacki, The Theory of Elasticity [Russian translation], Mir, Moscow (1975).
N. J. Pagano, “The exact moduli of anisotropic laminated composites,” in: G. P. Sendeckyi (ed.), The Mechanics of Composite Materials, Vol. 2 of the series L. J. Broutman and R. H. Krock (eds.), Composite Materials, Acad. Press, New York–London (1974).
V. S. Pugachev, Introduction to Probability Theory [in Russian], Nauka, Moscow (1968).
Kh. A. Rakhmatulin, “Fundamentals of the gas dynamics of interpenetrating motions of compressed media,” Prikl. Mat. Mekh., 20, No. 2, 184–195 (1956).
Kh. A. Rakhmatulin, Ya. U. Saatov, I. G. Filippov, and T. U. Artykov, Waves in Two-Component Media [in Russian], FAN, Tashkent (1974).
Ya. Ya. Rushchitskii, “Determination of the physical constants of the theory of elastic mixtures using experimental dispersion curves,” Prikl. Mekh., 15, No. 6, 26–32 (1979).
Ya. Ya. Rushchitskii, Elements of the Theory of Mixtures [in Russian], Naukova Dumka, Kiev (1991).
G. N. Savin and L. P. Khoroshun, “On the elastic constants of stochastically reinforced materials,” in: Continuum Mechanics and Related Analysis Problems [in Russian], Nauka, Moscow (1972), pp. 437–444.
G. P. Sendeckyi, “The elastic properties of composites,” in: G. P. Sendeckyi (ed.), The Mechanics of Composite Materials, Vol. 2 of the series L. J. Broutman and R. H. Krock (eds.), Composite Materials, Acad. Press, New York–London (1974).
V. P. Tamuzh and V. S. Kuksenko, The Fracture Micromechanics of Polymeric Materials [in Russian], Zinatne, Riga (1978).
I. G. Filippov, “The dynamic theory of relative flows of multicomponent media,” Prikl. Mekh., 7, No. 10, 92–99 (1971).
A. G. Fokin and T. D. Shermergor, “Calculation of the effective elastic moduli of composites with allowance for many-particle interactions,” Zh. Prikl. Mekh. Tekhn. Fiz., No. 1, 51–57 (1969).
A. G. Fokin and T. D. Shermergor, “The elastic moduli of texturized materials,” Inzh. Zh. Mekh. Tverd. Tela, No. 1, 129–133 (1967).
A. G. Fokin and T. D. Shermergor, “The elastic and rheological characteristics of fibrous composites, “ Izv. Akad. Nauk SSR, Mekh. Tverd. Tela, No. 6, 123–129 (1973).
L. P. Khoroshun, “Stress–strain dependences in laminated media,” Prikl. Mekh., 2, No. 2, 14–19 (1966).
L. P. Khoroshun, “Some issues of the correlation theory of structurally inhomogeneous elastic bodies,” Mekh. Polimer., No. 3, 365-371 (1966).
L. P. Khoroshun, “The thermoelastic properties of stochastically reinforced media,” Prikl. Mekh., 2, No. 8, 99–106 (1966).
L. P. Khoroshun, “The theory of isotropic deformation of elastic bodies with random inhomogeneities,” Prikl. Mekh., 3, No. 9, 12–19 (1967).
L. P. Khoroshun, “The statistical theory of deformation of unidirectional fibrous materials,” Prikl. Mekh., 4, No. 7, 8–15 (1968).
L. P. Khoroshun, “The statistical theory of deformation and strength of porous bodies,” in: Reliability Problems in Structural Mechanics [in Russian], Vilnius (1968), pp. 122–127.
L. P. Khoroshun, “The elastic properties of materials reinforced with unidirectional short fibers,” Prikl. Mekh., 8, No. 12, 86–92 (1972).
L. P. Khoroshun, “Prediction of the thermoelastic properties of materials strengthened with unidirectional discrete fibers,” Prikl. Mekh., 10, No. 12, 23–30 (1974).
L. P. Khoroshun, “The theory of thermal conduction of biphase solids,” Prikl. Mekh., 12, No. 7, 24–32 (1976).
L. P. Khoroshun, “The theory of saturated porous media,” Prikl. Mekh., 12, No. 12, 35–41 (1976).
L. P. Khoroshun, “The theory of interpenetrating elastic mixtures,” Prikl. Mekh., 13, No. 10, 124–132 (1977).
L. P. Khoroshun, “Methods of the theory of random functions in problems on the macroscopic properties of microinhomogeneous media,” Prikl. Mekh., 14, No. 2, 3–17 (1978).
L. P. Khoroshun, “Determination of the interpenetration coefficient for biphase elastic bodies,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 541–544 (1979).
L. P. Khoroshun, “The generalized theory of elastic mixtures,” Prikl. Mekh., 16, No. 4, 20–28 (1980).
L. P. Khoroshun, “The force of interphase interaction in the theory of elastic mixtures,” Prikl. Mekh., 18, No. 5, 23–29 (1982).
L. P. Khoroshun, “The Helmholtz theorem in the theory of elastic mixtures,” Prikl. Mekh., 20, No. 7, 115–118 (1984).
L. P. Khoroshun, “Refined models of deformation of composites,” Mekh. Kompos. Mater., No. 5, 798–804 (1984).
L. P. Khoroshun, “The concept of a mixture in constructing the theory of laminated plates and shells,” Prikl. Mekh., 21, No. 4, 110–118 (1985).
L. P. Khoroshun, “The method of conditional moments in the mechanics of composites,” Prikl. Mekh., 23, No. 10, 100–108 (1987).
L. P. Khoroshun, “Fundamentals of the thermomechanics of saturated porous media,” Prikl. Mekh., 24, No. 4, 3–13 (1988).
L. P. Khoroshun, “A new mathematical model of the nonuniform deformation of composites,” Mekh. Kompos. Mater., 31, No. 3, 310–318 (1995).
L. P. Khoroshun and Yu. A. Vetsalo, “The theory of effective properties of ideally plastic composites,” Prikl. Mekh., 23, No. 1, 86–90 (1987).
L. P. Khoroshun, V. P. Georgievskii, and E. N. Shikula, “Prediction of the nonlinear deformation properties of metal-composites,” Prikl. Mekh., 25, No. 9, 45–51 (1989).
L. P. Khoroshun and P. V. Leshchenko, “The theory of effective properties of piezoelectric composites,” Mekh. Kompos. Mater., No. 3, 419–425 (1986).
L. P. Khoroshun, P. V. Leshchenko, and L. V. Nazarenko, “Prediction of the thermoelastic properties of laminated and fibrous composites with orthotropic components,” Prikl. Mekh., 24, No. 3, 15–22 (1988).
L. P. Khoroshun, P. V. Leshchenko, and L. V. Nazarenko, “The effective thermoelastic constants of discrete-fibrous composites with anisotropic components,” Prikl. Mekh., 24, No. 10, 21–28 (1988).
L. P. Khoroshun and P. V. Leshchenko, “The effective electroelastic properties of polycrystals,” Prikl. Mekh., 30, No. 4, 74–81 (1994).
L. P. Khoroshun, B. P. Maslov, and P. G. Shishkin, “The reduced thermoelastic characteristics of a plate with inclusions,” Prikl. Mekh., 12, No. 6, 34–40 (1976).
L. P. Khoroshun and B. P. Maslov, “The effective elastic moduli for bodies weakened by oriented cracks,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 10, 924–927 (1976).
L. P. Khoroshun and B. P. Maslov, “The effective characteristics of materials spatially reinforced by short fibers,” Mekh. Kompos. Mater., No. 1, 3–9 (1979).
L. P. Khoroshun and B. P. Maslov, “Fundamentals of the theory of predicting the electromechanical properties of inhomogeneous piezoelectric materials,” Prikl. Mekh., 15, No. 8, 3–7 (1979).
L. P. Khoroshun, B. P. Maslov, E. N. Shikula, and L. V. Nazarenko, The Statistical Mechanics and Effective Properties of Materials [in Russian], Vol. 3 of the 12-volume series The Mechanics of Composites, Naukova Dumka, Kiev (1993).
L. P. Khoroshun, A. Kh. Melikbekyan, and V. M. Pinchuk, “Prediction of the properties of disoriented fibrous composites,” Prikl. Mekh., 12, No. 12, 13–19 (1976).
L. P. Khoroshun, A. Kh. Melikbekyan, and P. G. Shishkin, “The thermoelastic constants of fiberglass of oblique-angled winding,” Prikl. Mekh., 15, No. 1, 13–18 (1979).
L. P. Khoroshun and L. V. Nazarenko, “The thermoelasticity of orthotropic composites with ellipsoidal inclusions,” Prikl. Mekh., 26, No. 9, 3–10 (1990).
L. P. Khoroshun and Yu. N. Podil'chuk, “Representation of the solutions of the static equations for two-component mixtures,” Prikl. Mekh., 17, No. 4, 16–23 (1981).
L. P. Khoroshun and N. S. Soltanov, “Construction of the thermoelastic equations for a two-temperature continuum,” Prikl. Mekh., 13, No. 5, 112–115 (1977).
L. P. Khoroshun and N. S. Soltanov, “The theory of coupled thermoelasticity of two-component mixtures,” Prikl. Mekh., 17, No. 6, 21–28 (1981).
L. P. Khoroshun and N. S. Soltanov, “The thermostressed state of a two-temperature half-space with periodically varied surface temperature,” Prikl. Mekh., 1, No. 11, 30–34 (1982).
L. P. Khoroshun and N. S. Soltanov, The Thermoelasticity of Two-Component Mixtures [in Russian], Naukova Dumka, Kiev (1984).
L. P. Khoroshun and V. Z. Tydnyuk, “The dynamic problem for biphase elastic bodies,” Prikl. Mekh., 15, No. 2, 35–40 (1979).
L. P. Khoroshun and S. N. Shevchenko, “The efficiency of reinforcement with discrete unidirectional fibers,” Prikl. Mekh., 11, No. 9, 104–108 (1975).
L. P. Khoroshun and E. N. Shikula, “The thermoelastic properties of spatially reinforced materials,” Prikl. Mekh., 27, No. 1, 15–24 (1991).
L. P. Khoroshun and E. N. Shikula, “The nonlinear deformation properties of granular metal-composites,” Prikl. Mekh., 27, No. 5, 32–37 (1991).
L. P. Khoroshun and A. S. Shcherbakov, The Strength and Deformability of Gypsum Fiber Concrete [in Russian], Naukova Dumka, Kiev (1979).
T. D. Shermergor, The Theory of Elasticity of Microinhomogeneous Media [in Russian], Nauka, Moscow (1977).
V. I. Shklovskii and A. L. Éfros, “The flow theory and conductivity of strongly inhomogeneous media,” Usp. Fiz. Nauk, 117, No. 3, 401–437 (1975).
N. A. Shul'ga, Fundamentals of the Mechanics of Laminated Media with a Periodic Structure [in Russian], Naukova Dumka, Kiev (1981).
A. M. Yaglom, “Introduction to the theory of stationary random functions,” Usp. Mat. Nauk, 7, No. 5, 3–168 (1952).
R. J. Atkin and R. E. Crain, “Continuum theory of mixtures: basic and historical development,” Quart. J. Mech. Appl. Math., 29, No. 2, 209–244 (1976).
A. Betford and M. Stern, “On wave propagation in fiber-reinforced materials,” Trans. ASME, J. Appl. Mech., 37, No. 4, 1190–1192 (1970).
A. Bedford and M. Stern, “Toward a diffusing continuum theory of composite materials,” Trans. ASME, J. Appl. Mech., 8, No. 1, 8–14 (1971).
A. Bedford and M. Stern, “A multi-continuum theory for composite elastic materials,” Acta Mech., 14, No. 1, 85–102 (1972).
A. Bedford and D. S. Drumheller, “On a generalized effective stiffness theory,” Trans. ASME, J. Appl. Mech., 41, No. 1, 305–307 (1974).
A. Bedford, D. S. Drumheller, and H. J. Sutherland, “On modeling the dynamics of composite materials,” in: S. Nemat-Nasser (ed.), Mechanics Today, No. 3, 1–54 (1976).
M. J. Beran, Statistical Continuum Theories, Interscience Publishers, New York (1968).
M. J. Beran and J. Molyneux, “Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media,” Quart. Appl. Math., 24, No. 2, 107–118 (1966).
B. Budiansky, “On the elastic moduli of some heterogeneous materials,” J. Mech. Phys. Solids, 13, No. 4, 223–227 (1965).
D. S. Drumheller and A. Bedford, “On a continuum theory for a laminated medium,” Trans. ASME, J. Appl. Mech., 40, No. 2, 527–532 (1973).
A. E. Green and T. R. Steel, “Constitutive equations for interacting continua,” Int. J. Eng. Sci., 4, No. 4, 483–500 (1966).
Z. Hashin, “Theory of mechanical behavior of heterogeneous media,” Appl. Mech. Revs., 17, No. 1, 1–10 (1964).
Z. Hashin and S. Shtrikman, “On some variational principles in anisotropic and nonhomogeneous elasticity,” J. Mech. Phys. Solids, 10, No. 4, 335–342 (1962).
Z. Hashin and S. Shtrikman, “A variational approach to the theory of the elastic behavior of multiphase materials,” J. Mech. Phys. Solids, 11, No. 2, 127–135 (1963).
R. Hill, “A self-consistent mechanics of composite materials,” J. Mech. Phys. Solids, 13, No. 4, 213–222 (1965).
R. Hill, “Interfacial operators in the mechanics of composite media,” J. Mech. Phys. Solids, 31, No. 4, 347–357 (1983).
M. Hori and F. Jonesawa, “Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials,” J. Math. Phys., 16, No. 9, 1772–1775 (1975).
O. Ishal and L. Cohen, “Elastic properties of filled and porous epoxy composites,” Int. J. Mech. Sci., 19, 539–546 (1967).
L. P. Khoroshun and E. N. Shikula, “Nonlinear straining of porous composite materials,” Int. Appl. Mech., 30, No. 6, 983–988 (1994).
L. P. Khoroshun, “Principles of the micromechanics of material damage. 1. Short-term damage,” Int. Appl. Mech., 34, No. 4, 1035–1041 (1998).
L. P. Khoroshun and E. N. Shikula, “Influence of the spread of the strength of components on the deformation of a laminar composite with microfailures,” Int. Appl. Mech., 33, No. 3, 679–684 (1997).
L. P. Khoroshun and E. N. Shikula, “Effect of the spread of strength characteristics of the binder on the deformation of laminar-fibrous composite materials,” Int. Appl. Mech., 33, No. 7, 39–45 (1997).
E. Kroner, “Elastic moduli of perfectly disordered composite material,” J. Mech. and Phys. Solids, 15, No. 5, 319–324 (1967).
J. J. McCoy, “Macroscopic response of continua with random microstructures,” in: Mechanics Today, Vol. 6, Pergamon Press Inc., (1981), pp. 1–40.
H. D. McNiven and Y. Mengi, “A mathematical model for the linear dynamic behavior of two-phase periodic materials,” Int. J. Solids Struct., 15, No. 4, 271–280 (1979).
H. D. McNiven and Y. Mengi, “A mixture theory for elastic laminated composites,” Int. J. Solids Struct., 15, No. 4, 281–302 (1979).
A. N. Nayfeh and G. A. Curtman, “A continuum approach to the propagation of shear waves in laminated wave guides,” Trans ASME, J. Appl. Mech., 4, No. 1, 106–110 (1974).
A. N. Nayfeh and E. A. Nassar, “A simulation of the influence of bonding materials on the dynamic behavior of laminated composites,” Trans. ASME, J. Appl. Mech., 45, No. 10, 822–828 (1978).
R. W. Ogden, “On the overall moduli of nonlinear elastic composite materials,” J. Mech. Phys. Solids, 22, No. 6, 541–554 (1974).
A. Reuss, “Berechnung der Fliessgreze von Mischkristallen auf Grund der Plastizitats-bedingung für Einkristalle,” Z. Angew. Math. Mech., 9, No. 1, 49–58 (1929).
T. R. Steel, “Applications of a theory of interacting continua,” Quart. J. Mech. Appl. Math., 20, No. 1, 57–72 (1967).
T. R. Steel, “Determination of the constitutive coefficients for a mixture of two solids,” Int. J. Solids Struct., 4, No. 12, 1149–1160 (1968).
M. Stern and A. Bedford, “Wave propagation in elastic laminates using a multicontmuum theory,” Acta Mech., 15, No. 1, 21–38 (1972).
C. T. Sun, J. D. Achenbach, and G. Herrmann, “Continuum theory for a laminated medium,” Trans. ASME, J. Appl. Mech., 35, No. 3, 467–475 (1968).
H. J. Sutherland, “Dispersion of acoustic waves by an aluminia-epoxy mixture,” J. Compos. Mat., 13, No. 1, 35–47 (1979).
T. R. Tiersten and M. Jahanmir, “A theory of composites modeled as interpenetrating solid continuum,” Arch. Ration. Mech. Anal., 54, No. 2, 153–163 (1977).
W. Voight, Lehrbuch der Kristallphysik, Teubner, Berlin (1928).
L. J. Walpole, “On bounds for the overall elastic moduli of inhomogeneous systems. I,” J. Mech. Phys. Solids, 14, No. 3, 151–162 (1966).
L. J. Walpole, “On bounds for the overall elastic moduli of inhomogeneous systems. II,” J. Mech. Phys. Solids, 14, No. 5, 289–301 (1966).
J. R. Willis, “The overall elastic response of composite materials,” Trans. ASME, Ser. E, 50, No. 46, 1202–1209 (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Khoroshun, L.P. Mathematical Models and Methods of the Mechanics of Stochastic Composites. International Applied Mechanics 36, 1284–1316 (2000). https://doi.org/10.1023/A:1009482032355
Issue Date:
DOI: https://doi.org/10.1023/A:1009482032355