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Extension of the microstructural theory of two-phase mixtures to composite materials

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International Applied Mechanics Aims and scope

Abstract

Studies in the microstructural theory of two-phase elastic, viscoelastic, and piezoelastic mixtures are analyzed with reference to composite materials. It is confirmed that an integral part of the studies in this subject area has been carried out at the S.P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine.

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References

  1. A. A. Abduganiev,Development of a Microstructural Theory of Porous Media Saturated with a Liquid and Analysis of Wave Propagation [in Russian], Ph.D. Thesis, Inst. Mekh. Akad. Nauk Ukr. SSR (1990).

  2. V. O. Bilyi,Plane Wave Triplets in Two-Phase Media [in Ukrainian], Ph.D. Thesis, Kievskii Univ. im. T. G. Shevchenko (1997).

  3. V. V. Bolotin, “The bending of multilayered plates,”Izv. Akad. Nauk SSSR, Mekh. Mash., No. 3, 65–72 (1963)

    Google Scholar 

  4. G. A. Van FoFy, “Elastic waves in reinforced materials,” in:Composite Fibrous Materials [Russian translation], Naukova Dumka, Kiev (1970), pp. 127–137.

    Google Scholar 

  5. G. A. Vanin,The Micromechanics of Composite Materials [in Russian], Naukova Dumka, Kiev (1985).

    Google Scholar 

  6. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov,The Theory of Waves [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  7. A. N. Guz',The Stability of Three-Dimensional Deformable Bodies [in Russian], Naukova Dumka, Kiev (1971).

    Google Scholar 

  8. A. N. Guz',The Stability of Elastic Bodies Subject to Finite Strains [in Russian], Naukova Dumka, Kiev (1973).

    Google Scholar 

  9. L. K. Zarembo and V. A. Krasil'nikov,Introduction to Nonlinear Acoustics [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  10. L. K. Zarembo and V. I. Timoshenko,Nonlinear Acoustics [in Russian], MGU, Moscow (1984).

    Google Scholar 

  11. G. M. Zaslavskii and R. Z. Sagdeev,Introduction to Nonlinear Physics: From a Pendulum to Turbulence and Chaos [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  12. V. A. Krasil'nikov and V. V. Krylov,Introduction to Physical Acoustics [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  13. B. P. Maslov, Ya. Ya. Rushchitskii, and A. P. Kovalenko, “Complete sets of physical constants for the nonlinear microstructural theory of two-phase elastic, mixtures evaluated for some structural granulated composites,”Prikl. Mekh.,32, No. 11, 41–49 (1996).

    Google Scholar 

  14. A. N. Guz (ed.),The Mechanics of Composites and Structural Elements [in Russian], in three vols., Naukova Dumka, Kiev (1982).

    Google Scholar 

  15. A. N. Guz' (ed.),The Mechanics of Composites [in Russian], in 12 vols., Naukova Dumka, Kiev (1993–2000).

    Google Scholar 

  16. R. I. Nigmatulin,The Fundamentals of the Mechanics of Heterogeneous Media [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  17. R. I. Nigmatulin,The Dynamics of Multiphase Media [in Russian], in two parts, Nauka, Moscow (1987).

    Google Scholar 

  18. B. E. Pobedrya and V. I. Gorbachev, “Static problems of elastic composites,”Vest. Mosk. Univ., Ser. Mat. Mekh., No. 5, 101–110 (1977).

    Google Scholar 

  19. B. E. Pobedrya, “The theory of viscoelasticity of composites” in:Mechanics of Composites [in Russian], Vol. 3 (1979), pp. 414–423.

  20. B. E. Pobedrya,Numerical Methods in the Theory of Elasticity and Plasticity [in Russian], Izd. Mosk. Univ. (1981).

  21. M. I. Rabinovich and D. I. Trubetskov,Introduction to the Theory of Vibrations and Waves [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  22. Kh. A. Rakhmatulin, Ya. U. Saatov, I. G. Filippov, and T. U. Artykov,Waves in Two-Component Media [in Russian], Fan, Tashkent (1974).

    Google Scholar 

  23. Ya. Ya. Rushchitskii, “General solutions in the theory of a mixture of two elastic media,” in:Abstracts of Papers Read at the 4th Polish-Soviet Symp. on Nonclassical Problems of Mechanics [in Russian], Warsaw (1973), p. 21.

  24. Ya. Ya. Rushchitskii, “The use of the Papkovich-Neiber solution in the theory of a mixture of two solids,”Prikl. Mekh. 9, No. 9, 17–23 (1973).

    Google Scholar 

  25. Ya. Ya. Rushchitskii, “A plane problem of the theory of a mixture of two solids,”Prikl. Mekh.,10, No. 2, 52–61 (1974).

    Google Scholar 

  26. Ya. Ya. Rushchitskii, “Torsional waves in a cylinder made from a composite material,”Prikl. Mekh.,12, No. 5, 18–23 (1976).

    Google Scholar 

  27. Ya. Ya. Rushchitskii, “One case of wave propagation in mixtures of elastic materials,”Prikl. Mekh.,14, No. 1, 25–33 (1978).

    Google Scholar 

  28. Ya. Ya. Rushchitskii, “Laws of wave propagation in mixtures of elastic materials,” in:Abstracts of Papers Read at the 6th Polish-Soviet Symp. on Nonclassical Problems of Mechanics [in Russian], Warsaw (1978), p. 18.

  29. Ya. Ya. Rushchitskii, “Using experimental dispersion curves to determine the physical constants of the theory of a mixture of elastic materials,”Prikl. Mekh.,15, No. 6, 26–32 (1979).

    Google Scholar 

  30. Ya. Ya. Rushchitskii, “Analysis of the boundary effect in a linear mixture of elastic media,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 4, 52–55 (1980).

    Google Scholar 

  31. Ya. Ya. Rushchitskii, “Applicability of the linear theory of mixtures,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 6, 58–62 (1980).

    Google Scholar 

  32. Ya. Ya. Rushchitskii, “General representations for the solutions of the equations of the linear theory of a mixture of elastic media,”Prikl. Mekh.,16, No. 7, 3–11 (1980).

    Google Scholar 

  33. Ya. Ya. Rushchitskii, “Variation principles of the linearized theory of mixtures,”Prikl. Mekh.,16, No. 10, 134–136 (1980).

    Google Scholar 

  34. Ya. Ya. Rushchitskii, “Phase velocities, group velocities, and velocities of movement of the energy of a plane wave in a mixture of elastic materials,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 8, 50–52 (1981).

    Google Scholar 

  35. Ya. Ya. Rushchitskii, “The torsion of a direct beam made from a mixture of elastic materials,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 9, 51–54 (1981).

    Google Scholar 

  36. Ya. Ya. Rushchitskii, “Physical constants in the theory of mixtures,”Prikh. Mekh.,17, No. 11 134–136 (1981).

    Google Scholar 

  37. Ya. Ya. Rushchitskii, “The stability of the equilibrium state in the theory of mixtures,”Prikl. Mekh.,17, No. 12, 51–58 (1981).

    Google Scholar 

  38. Ya. Ya. Rushchitskii, “Propagation of plane viscoelastic waves in a mixture of elastic materials,”Prikl. Mekh.,18, No. 1, 93–99 (1982).

    Google Scholar 

  39. Ya. Ya. Rushchitskii, “Study of the basic problems of the mechanics of a mixture of elastic media,”Author's Abstract and Thesis of Doctoral Dissertation [in Russian], Inst. Mekh. Akad. Nauk Ukrainy, Kiev, (1982).

    Google Scholar 

  40. Ya. Ya. Rushchitskii, “Inherent instability in mixtures,”Prikl. Mekh.,18, No. 12, 48–53 (1982).

    Google Scholar 

  41. Ya. Ya. Rushchitskii, “Description of wave phenomena in materials with a microstructure using the theory of mixtures”, in:Abstracts of Papers Read at the 5th National Congress on Theoretical and Applied Mechanics [in Russian], Varna (1985), p. 93.

  42. Ya. Ya. Rushchitskii, “A new microstructural model of the deformation of materials”Trans. Lavrentiev's Readings on Mathematics, Mechanics, and Physics [in Russian], Naukova Dumka, Kiev (1985), pp. 184–186.

    Google Scholar 

  43. Ya. Ya. Rushchitskii, “Phenomenological description of interaction in the linear theory of a mixture of elastic media,”Prikl. Mekh.,22, No. 5, 90–95 (1986).

    Google Scholar 

  44. Ya. Ya. Rushchitskii, “The validity of a mixture model in the mechanics of composites,” in:Abstracts of Papers Read at the 6th All-Union Congress on Theoretical and Applied Mechanics [in Russian], Tashkent (1986), p. 54.

  45. Ya. Ya. Rushchitskii, “Stability analysis in the mechanics of composites using the theory of mixtures,” in:Abstracts of Papers Read at the All-Union Symp. on Stability in the Mechanics of a Deformable Solid [in Russian], Kalinin-Tver' (1986), pp. 195–196.

  46. Ya. Ya. Rushchitskii, “Microstructural theories in the mechanics of composites,” in:Abstracts of Papers Read at the 7th Polish-Soviet Symp. on Nonclassical Problems of Mechanics [in Russian], Warsaw-Yablonna (1987), p. 33.

  47. Ya. Ya. Rushchitskii, “Microstructural theories in the mechanics of composites,” in:Advances in the Machine Industry: Composition Materials. Int. Center Sci. Eng. Inf., A. A. Blagonravov Inst. Mech. Eng., MTsNTI, Moscow (1987), pp. 83–98.

    Google Scholar 

  48. Ya. Ya. Rushchitskii, “A nonlinear wave in a two-phase material,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 1, 49–52 (1990).

    Google Scholar 

  49. Ya. Ya. Rushchitskii, “A plane nonlinear wave in a two-phase material,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 2, 45–47 (1990).

    Google Scholar 

  50. Ya. Ya. Rushchitskii, “Element of the Theory of Mixtures [in Russian], Naukova Dumka, Kiev (1991).

    Google Scholar 

  51. Ya. Ya. Rushchitskii, “Interaction of elastic waves in a two-phase material,”Prikl. Mekh.,28, No. 5, 13–21 (1992).

    Google Scholar 

  52. Ya. Ya. Rushchitskii, “A Continual Model of a Mixture in the Mechanics of Laminates, Chapter 9 of the monographThe Dynamics and Stability of Laminated Composites [in Russan], Naukova Dumka, Kiev (1993), pp. 228–263.

    Google Scholar 

  53. Ya. Ya. Rushchitskii, “Propagation of Long Waves in Composites (a Continual Theory of the First Order), Chapter 10, Vol. 2 of the 12-volume seriesMechanics of Composites [in Russian], Naukova Dumka, Kiev (1993–2002), pp. 377–405.

    Google Scholar 

  54. Ya. Ya. Rushchitskii, “Interaction of compression and shear waves in a composite material with nonlinearly elastic components in the microstructure,”Prikl. Mekh.,29, No. 4, 18–26 (1993).

    Google Scholar 

  55. Ya. Ya. Rushchitskii, “Riemann's simple waves in hyperelastic bodies,”Dop. Akad. Nauk Ukrainy, No. 10, 56–58 (1993).

    Google Scholar 

  56. Ya. Ya. Rushchitskii, “Nonlinear plane waves in an orthotropic body,”Dop. Akad. Nauk Ukrainy, No. 12, 60–62 (1993).

    Google Scholar 

  57. Ya. Ya. Rushchitskii, “The resonance interaction of nonlinear waves in two-phase mixtures,”Prikl. Mekh.,30, No. 5, 32–41 (1994).

    Google Scholar 

  58. Ya. Ya. Rushchitskii, “Simple waves in quadratically nonlinear hyperelastic materials,”Prikl. Mekh.,30, No. 8, 35–41 (1994).

    Google Scholar 

  59. Ya. Ya. Rushchitskii, “Analysis of wave interaction in two-phase materials within the framework of the microstructural theory of mixtures,” in:Abstracts of Papers Read at the Int. Sci. Colloq. on Nonlinear Dynamics of Solids with a Microstructure [in Russian], Nizhnyi Novgorod (1995), pp. 21–22.

  60. Ya. Ya. Rushchitskii, “A model of a solid mixture for granulated piezoelectric materials,” in:Abstracts of Papers read at the 4th Int. Conf. on the Mechanics of Inhomogeneous Structures [in Ukrainian], L'vov-Ternopol' (1995), p. 89.

  61. Ya. Ya. Rushchitskii, “Waves and the direct piezoelectric effect in two-phase piezoelectrics,”Prikl. Mekh.,31, No. 10, 42–48 (1995).

    Google Scholar 

  62. Ya. Ya. Rushchitskii, “Three-wave interaction and generation of the second harmonic in single-phase and two-phase hyperelastic media,”Prikl. Mekh.,32, No. 7, 38–45 (1996).

    Google Scholar 

  63. Ya. Ya. Rushchitskii, “Simple waves in solid mixtures,”Prikl. Mekh.,32, No. 9, 31–39 (1996).

    Google Scholar 

  64. Ya. Ya. Rushchitskii, “Simple waves,”Dop. Akad. Nauk Ukrainy, No. 12, 55–60 (1996).

    Google Scholar 

  65. Ya. Ya. Rushchitskii, “Nonlinear waves in solid mixtures (review),”Prikl. Mekh.,33, No. 1, 3–39 (1997).

    Google Scholar 

  66. Ya. Ya. Rushchitskii, “The interaction of simple waves in solid mixtures,”Dop. NAN Ukrainy, No. 3, 64–70 (1999).

    Google Scholar 

  67. Ya. Ya. Rushchitskii, “Unconventional ordered pairs of thermodynamic parameters: from Podstrigach's theory of diffusion elasticity to the Bedford-Drumheller theory of mixtures,”Mat. Metody Fiz.-Mekh. Polya, L'vov,41, No. 3, 117–120 (1998–1999).

    Google Scholar 

  68. Ya. Ya. Rushchitskii and A. A. Abduganiev, “A theory of mixtures based on the Bedford-Drumheller model for a saturated porous medium,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 12, 38–39 (1989).

    Google Scholar 

  69. Ya. Ya. Rushchitskii and A. A. Abduganiev, “Two unconventional concepts of the mechanics of a multiphase mixture,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 11, 61–64 (1992).

    Google Scholar 

  70. Ya. Ya. Rushchitskii and A. A. Abduganiev, “A new class of linearized problems depending on the parameters of the initial state,”Prikl. Mekh.,30, No. 7, 9–16 (1994).

    Google Scholar 

  71. Ya. Ya. Rushchitskii and V. A. Belyi, “Existence of one type of combination dissipation of sound by sound in a two-phase solid,”Dop. NAN Ukrainy, No. 10, 58–63 (1994).

    Google Scholar 

  72. Ya. Ya. Rushchitskii and V. A. Belyi, “The decomposing instability of triplets in a hyperelastic two-phase medium,”Dop. NAN Ukrainy, No. 11, 57–63 (1996).

    Google Scholar 

  73. Ya. Ya. Rushchitskii, A. P. Kovalenko, and E. V. Savel'eva, “Self-generation of lateral waves in hyperelastic media,”Prikl. Mekh.,32, No. 5, 30–38 (1996).

    Google Scholar 

  74. Ya. Ya. Rushchitskii, A. P. Kovalenko, and S. I. Tsurpal, “The distortion of a simple bell-shaped wave propagating in a two-phase material,”Dop. NAN Ukrainy, No. 8, 63–66 (1996).

    Google Scholar 

  75. Ya. Ya. Rushchitskii, A. P. Kovalenko, V. A. Belyi, and S. I. Tsurpal, “The distortion of a simple longitudinal wave in an inhomogeneous solid,”Prikl. Mekh.,33, No. 9, 15–23 (1997).

    Google Scholar 

  76. Ya. Ya. Rushchitskii and I. A. Ostrakov, “Generation of new harmonics of nonlinear elastic waves in a composite material,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 10, 63–66 (1991).

    Google Scholar 

  77. Ya. Ya. Rushchitskii and I. A. Ostrakov, “The distortion of a plane harmonic wave propagating in a composite material,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 11, 51–54 (1991).

    Google Scholar 

  78. Ya. Ya. Rushchitskii and S. O. Rushchitskaya, “Spherical waves in the theory of mixtures,”Prikl. Mekh.,12, No. 6, 63–69 (1976).

    Google Scholar 

  79. Ya. Ya. Rushchitskii and S. O. Rushchitskaya, “Features of solutions in the theory of mixtures,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 6, 531–534 (1976).

    Google Scholar 

  80. Ya. Ya. Rushchitskii and S. O. Rushchitskaya, “The generalized solution of one differential problem of mechanics,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 7, 632–636 (1974).

    Google Scholar 

  81. Ya. Ya. Rushchitskii and S. O. Rushchitskaya, “The Sommerfeld conditions in the theory of mixtures,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 8, 730–733 (1976).

    Google Scholar 

  82. Ya. Ya. Rushchitskii and S. O. Rushchitskaya, “Saint Venant's principle in the theory of a mixture of elastic media,”Prikl. Mekh.,13, No. 6, 12–17 (1977).

    Google Scholar 

  83. Ya. Ya. Rushchitskii and S. O. Rushchitskaya, “Description of features of the wave phenomena in composite materials,”Mekh. Komp. Mater., No. 5, 894–900 (1996).

    Google Scholar 

  84. Ya. Ya. Rushchitskii and E. V. Savel'eva, “The evolution of a harmonic wave propagating in a two-phase material,”Prikl. Mekh.,28, No. 9, 42–46 (1992).

    Google Scholar 

  85. Ya. Ya. Rushchitskii and I. M. Khotenko, “Linear waves in two-phase piezoelectric materials,”Dop. NAN Ukrainy, No. 3, 41–43 (1995).

    Google Scholar 

  86. Ya. Ya. Rushchitskii and B. M. Khotenko, “Creation of a quadratically nonlinear theory of wave propagation in two-phase piezoelectric powders,”Dop. NAN Ukrainy, No. 9, 61–67 (1998).

    Google Scholar 

  87. Ya. Ya. Rushchitskii and B. M. Khotenko, “Nonlinear plane harmonic waves in two-phase piezoelectric powders,”Dop. NAN Ukrainy, No. 10, 60–66 (1998).

    Google Scholar 

  88. Ya. Ya. Rushchitskii and S. I. Tsurpal, “Generation of a new simple wave by a longitudinal simple wave in a hyperelastic two-phase medium,”Dop. NAN Ukrainy, No. 7, 56–60 (1996).

    Google Scholar 

  89. Ya. Ya. Rushchitskii and S. B. Tsurpal,Waves in Materials with a Microstructure [in Russian], Inst, Mekh. NANU, Kiev (1998).

    Google Scholar 

  90. Ya. Ya. Rushchitskii and B. B. Érgashev, “Calculation of the shear-interaction factor for isotropic structural materials filled with microspheres,”Prikl. Mekh.,23, No. 9, 52–58 (1987).

    Google Scholar 

  91. Ya. Ya. Rushchitskii and B. B. Érgashev, “The theory of effective moduli of composite materials filled with hollow microspheres,”Prikl. Mekh.,23, No. 11, 17–21 (1987).

    Google Scholar 

  92. Ya. Ya. Rushchitskii and B. B. Érgashev, “Theoretical analysis of nonstationary effects in a massive specimen made from fibrous composite and subject to a short-term harmonic impulse,”Prikl. Mekh.,23, No. 12, 81–86 (1987).

    Google Scholar 

  93. Ya. Ya. Rushchitskii and B. B. Érgashev, “The vibrations of a hollow cylinder made from a composite and subject to impulse internal pressure,”Prikl. Mekh.,25, No. 7, 30–37 (1989).

    Google Scholar 

  94. Ya. Ya. Rushchitskii and B. B. Érgashev, “The shear-interaction coefficient for granulated composites filled with solid microspheres,”Prikl. Mekh.,25, No. 9, 126–128 (1989).

    Google Scholar 

  95. Ya. Ya. Rushchitskii and B. B. Érgashev, “The relations for calculation of the physical constants of the isotropic theory of mixtures,”Prikl. Mekh.,25, No. 11, 62–66 (1989).

    Google Scholar 

  96. Ya. Ya. Rushchitskii and B. B. Érgashev, “Some features of the natural vibrations of a mixture,”Prikl. Mekh.,26, No. 1, 16–22 (1990).

    Google Scholar 

  97. Ya. U. Saatov,Plane Problems of the Mechanics of Elastoporous Media [in Russian], Fan, Tashkent (1975).

    Google Scholar 

  98. E. V. Savel'eva,The Evolution of Plane Waves in Multiphase Media [in Ukrainian], Author's Abstract of Ph.D. Thesis, Inst. Mekh. NANU, Kiev (1997).

    Google Scholar 

  99. G. N. Savin and Ya. Ya. Rushchitskii, “Plane waves in composites,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 9, 821–824 (1974).

    Google Scholar 

  100. G. N. Savin and Ya. Ya. Rushchitskii, “The theory of wave propagation in a mixture of elastic materials,” in:Abstracts of Papers Read at the 4th All-Union Congress on Theoretical and Applied Mechanics [in Russian], Kiev (1976), p. 107.

  101. G. N. Savin and Ya. Ya. Rushchitskii, “Surface waves in the theory of a mixture of elastic materials,”Abstracts of Papers Read at the 4th All-Union Symp. on Propagation of Elastic and Elastoplastic Waves [in Russian], Bishkek (1976), pp. 118–119.

  102. L. I. Sedov,Continuum Mechanics [in Russian], Vol. 2, Nauka, Moscow (1970).

    MATH  Google Scholar 

  103. I. G. Filippov, “The dynamic theory of the relative flow of multicomponent media,”Prikl. Mekh.,7, No. 10, 92–99 (1971).

    Google Scholar 

  104. I. G. Filippov and V. G. Cheban,Nonsteady Motions of Continuous Compressible Media [in Russian], Shtiintsa, Kishinev (1973).

    Google Scholar 

  105. Z. M. Khotenko,Waves in Solid Nonlinearly Elastic Piezoelectric Mixtures [in Ukrainian], Author's Abstract of Ph.D. Thesis, Inst. Gidromekh. NANU, Kiev (1999).

    Google Scholar 

  106. S. I. Tsurpal,Single Simple Plane Bell-Shaped Waves in Composites [in Ukrainian], Author's Abstract of Ph.D. Thesis, Inst. Mekh. NANU, Kiev (1996).

    Google Scholar 

  107. B. B. Érgashev,Nonstationary Waves in Isotropic Composites with a Microstructure [in Russian], Author's Abstract of Ph.D. Thesis, Inst. Mekh. Akad. Nauk Ukr. SSR, Kiev (1988).

    Google Scholar 

  108. J. D. Achenbach, “Generalized continuum theories for directionally reinforced solids,”Arch. Mech.,35, No. 4, 257–278 (1976).

    Google Scholar 

  109. J. D. Achenbach and J. Herrmann, “Dispersion of free harmonic waves in filler armed composites,”Rocket Tech. Cosmon., No. 10, 8–12 (1968).

    Google Scholar 

  110. R. J. Atkin, P. Chadvick, and T. R. Stell, “Uniqueness theorems for linearized theories of interacting continua,”Mathematika,14, No. 1, 27–37 (1967).

    Google Scholar 

  111. R. J. Atkin and R. E. Crain, “Continuum theory of mixtures: basic theory and historical developments,”Quart. J. Mech. Appl. Math.,29, No. 2, 209–244 (1976).

    MATH  ADS  Google Scholar 

  112. M. Aron, “On existence and uniqueness of solutions in the linear theory of mixtures of two elastic solids,”Arch. Mech.,26, No. 4, 717–725 (1974).

    MATH  Google Scholar 

  113. A. Bedford and M. Stern, “On wave propagation in fiber-reinforced materials,”Trans. ASME, J. Appl. Mech. 37, No. 4, 1190–1192 (1970).

    Google Scholar 

  114. A. Bedford and M. Stern, “Toward a diffusing continuum theory of composite materials,”Trans. ASME, J. Appl Mech.,38, No. 1, 8–14 (1971).

    Google Scholar 

  115. A. Bedford and M. Stern, “A multi-continuum theory for composite elastic materials,”Acta Mechanica,14, No. 1, 85–102 (1972).

    Google Scholar 

  116. A. Bedford, D. S. Drumheller, and H. J. Sutherland, “On modeling the dynamics of composite materials,” in:Nemat-Nasser S. Mechanics Today, Vol. 3, Pergamon Press, New York (1976), pp. 1–54.

    Google Scholar 

  117. A. Bedford and D. S. Drumheller, “Theories of immiscible and structured mixtures,”Int. J. Eng. Sci.,21, No. 8, 863–960 (1983).

    MATH  Google Scholar 

  118. M. Ben-Amoz, “On wave propagation in laminated composites.—I. Propagation parallel to the laminate,”Int. J. Eng Sci., No. 1, 43–56 (1975).

    Google Scholar 

  119. M. Ben-Amoz, “On wave propagation in laminated composites.—II. Propagation normal to the laminate,”Int. J. Eng. Sci., No. 1, 57–67 (1975).

    Google Scholar 

  120. V. V. Bolotin, “Vibration of layered elastic plates,”Proceedings of Vibration Problems,4, No. 4, 331–346 (1963).

    Google Scholar 

  121. P. M. Bowen,Mixtures and EM Field Theories (A. C. Eringen, editor), Vol. 3, in:Continuum Physics, Academic Press, New York (1976), pp. 1–127.

    Google Scholar 

  122. P. M. Bowen, “Toward a thermodynamics and mechanics of mixtures,”Arch. Rat. Mech. Anal.,24, No. 5, 370–403 (1967).

    MATH  Google Scholar 

  123. R. M. Christensen,Mechanics of Composite Materials, John Wiley and Sons, New York (1979).

    Google Scholar 

  124. L. J. Broutman and R. H. Krock, (eds.),Composite Materials, in 8 vols., Academic Press, New York (1974).

    Google Scholar 

  125. D. S. Drumheller and H. J. Sutherland, “A lattice model for stress wave propagation in composite materials,”Trans. ASME, J. Appl. Mech.,40, No. 1, 157–164 (1973).

    Google Scholar 

  126. D. S. Drumheller and A. Bedford, “Wave propagation in elastic laminates using a second order microstructure theory,”Int. J. Solids Struct., No. 10, 61–76 (1974).

    Google Scholar 

  127. A. E. Green and T. R. Steel, “Constitutive equations for interacting continua,”Int. J. Eng. Sci.,4, No. 4, 483–500 (1966).

    Google Scholar 

  128. A. N. Guz,Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Springer-Verlag, Berlin (1999).

    MATH  Google Scholar 

  129. A. N. Guz' and J. J. Rushchitskii, “Main Ukrainian historical and modern sources on mechanics (Focus on the S. P. Timoshenko Institute of Mechanics),”Appl. Mech. Reviews, No. 3, 199–208 (1998).

    Google Scholar 

  130. A. C. Eringen and E. S. Suhubi, “Nonlinear theory of simple microelastic solids,”Int. J. Eng., Sci.,2, No. 2, 189–203 (1964).

    MATH  Google Scholar 

  131. A. C. Eringen, “Theory of micromorphic materials with memory,”Int. J. Eng. Sci.,10, No. 7, 623–641 (1972).

    Google Scholar 

  132. A. C. Eringen, “Point charge, infra-red dispersion and conduction in nonlocal piezoelasticity,” in:Proc. IUTAM-IUPAP Symposium on the Mechanical Behavior of Electromagnetic Solid Continua (G. A. Vaugin, editor) (Paris, 1983), North-Holland, Amsterdam (1984), pp. 187–196.

    Google Scholar 

  133. G. A. Hegemier, “On a theory of interacting continua for wave propagation in composites,” in:Dynamics of Composite Materials (E. H. Lee, editor), ASME, New York (1972), pp. 70–121.

    Google Scholar 

  134. G. A. Hegemier and A. N. Nayfeh, “A continuum theory for wave propagation in laminated composites.—Case I: propagation normal to the laminates,”Trans. ASME, J. Appl. Mech.,40, No. 2, 503–510 (1973).

    Google Scholar 

  135. G. A. Hegemier and T. C. Bache, “A continuum theory for wave propagation in laminated composites.—Case II: propagation parallel to the laminates,”J. Elasticity,3, No. 2, 125–140 (1973).

    Google Scholar 

  136. G. A. Hegemier and T. C. Bache, “A general continuum theory with microstructure for wave propagation in elastic laminated composites,”Trans. ASME, J. Appl. Mech.,41, No. 1, 101–105 (1974).

    MATH  Google Scholar 

  137. G. Lubin (ed.),Handbook on Experimental Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey (1987).

    Google Scholar 

  138. A. S. Kobayashi (ed.),Handbook on Experimental Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey (1987)

    Google Scholar 

  139. G. Herrmann, R. K. Kaul, and T. G. Delph, “On continuum modeling of the dynamic behavior of layered composites,”Arch. Mech.,28, No. 3, 405–421 (1978).

    Google Scholar 

  140. R. J. Knops and T. R. Stell, “Uniqueness in the linear theory of a mixture of two elastic solids,”Int. J. Eng. Sci.,7, No. 6, 571–577 (1969).

    MATH  Google Scholar 

  141. B. Lempriere, “On practicability of analysing waves in composites by the theory of mixtures,” in: Report No. LNSC-6-78-69-21, Lockheed Palo Alto Research Laboratory (1969), pp. 76–90.

  142. B. Lempriere, “The practicability of analyzing waves in composites by the theory of mixtures,” in:Abstracts of Papers Read at the Colloq. on Dynamic Behavior of Composite Materials, Univ. of California, San Diego (1969), pp. 84–85.

  143. J. Lighthill,Waves in Fluids, Cambridge University Press, Cambridge-London (1978).

    MATH  Google Scholar 

  144. G. A. Maugin and A. C. Eringen, “On the equations of the electrodynamics of deformable bodies of finite extent,”J. Mech.,16, No. 1, 101–147 (1977).

    MATH  Google Scholar 

  145. G. A. Maugin, “A continuum approach to magnon-phonon couplings—I. II,”Int. J. Eng. Sci.,17, No. 11, 1073–1091, 1093–1108 (1979).

    MATH  Google Scholar 

  146. G. A. Maugin,Nonlinear Electromechanical Effects and Applications, Academic Press, New York (1985).

    Google Scholar 

  147. G. A. Maugin,Continuum Mechanics of Electromagnetic Solids, North Holland, Amsterdam (1988).

    MATH  Google Scholar 

  148. G. A. Maugin, B. Collet, and J. Pouget,Nonlinear Wave Propagation in Coupled Electromechanical Systems, in:Nonlinear Wave Propagation (T. W. Wright, editor), ASME Publications, New York (1986).

    Google Scholar 

  149. 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).

    MATH  Google Scholar 

  150. H. D. McNiven and Y. Mengi, “A mixture theory for elastic laminated composites,”Int. J. Solids Struct.,15, No. 4, 281–302 (1979).

    Google Scholar 

  151. H. D. McNiven and Y. Mengi, “Propagation of transient waves in elastic laminated composites,”Int. J. Solids Struct.,15, No. 4, 303–318 (1979).

    Google Scholar 

  152. R. D. Mindlin, “Microstructure in linear elasticity,”Arch. Rat. Mech. Analysis. 16, No. 1, 51–78 (1964).

    MATH  ADS  Google Scholar 

  153. A. M. Nayfeh, “Time-harmonic waves propagating normal to the layers of multilayered periodic media,”Trans. ASME, J. Appl. Mech.,41, No. 1, 92–105 (1974).

    MATH  Google Scholar 

  154. A. N. Nayfeh and G. A. Gurtman, “A continuum approach to the propagation of shear waves in laminated wave guides,”Trans. ASME, J. Appl. Mech.,41 No. 1, 106–110 (1974).

    MATH  Google Scholar 

  155. J. Pouget, “Electro-acoustic echoes in piezoelectric powders,” in:Proc. IUTAM-IUPAP Symp. on the Mechanical Behavior of Electromagnetic Solid Continuums (G. A. Vaugin, editor), (Paris, 1983), North-Holland, Amsterdam (1984), pp. 177–184.

    Google Scholar 

  156. B. Riemann,Über die Fortpflanzung ebener Luftwellen von endlichen Schwingungsweite. Abhandlungen der Königischen Gesellschaft zu Göttingen, 1860, Bd. VIII, in:Bernhards Riemann's gesammelte mathematische Werke und wissenschaftlicher Nachlass, 2-te Auflage, Leipzig, Teubner Verlag (1892), pp. 157–179, in: B. Riemann,Gesammelte mathematische Werke, wissenschaflicher Nachlass und Nachtrage, Collected Papers, Springer-Verlag/Teubner Verlaggesellschaft, Berlin-Leipzig (1990).

    Google Scholar 

  157. J. J. Ruszczycky, “O rozwazaniu typu Papkowicz-Neibera w zagadnieniach mechaniki osrodkow wieloskladnikowych”,Rozprawy Inzynierskie—Eng. Trans.,22, No. 4, 577–585 (1974).

    Google Scholar 

  158. J. J. Rushchitsky, “The microstructural theory of mixtures”, in:Annotations of Papers Read at IUTAM/ICM Symp. on Loading, Failure, and Fracture of Anisotropic Solids, Antony Sawczuk Memoriam, Grenoble, France (1987), p. 35.

  159. J. J. Rushchitsky, “Microstructural theory of the mixture: conception, main mechanical effects, features of elastic and viscoelastic waves, practicability to composite materials”,Proc. Intern. Congress on Appl. Mechanics, Vol. 2, Beijing, China (1989), pp. 42–46.

  160. J. J. Rushchitsky, “Nonlinear interaction of elastic waves in composite materials,” in:Abstracts of Lectures Read at the Yugoslav Conf. on Nonlinear Deterministic and Stochastic Processes in Dynamical Systems with Applications, University of Nis, Yugoslavia, Nis (1991), p. 57.

  161. J. J. Rushchitsky, “Nonlinear elastic waves in solids with inner microstructure, new effects and features,” in:Annotations of Lectures Read at the 15th Int. Symp. on Vibrations in Physical Systems, Poznan (1992), p. 140.

  162. J. J. Rushchitsky, “Nonlinear elastic waves in isotropic multi-phase materials (composites): interaction, second harmonics generation, new complex waves, evolution, distortion, blow up, triplets”, inAbstracts of Papers Read at the IUTAM Symp. on Nonlinear Waves in Solids, Victoria, British Columbia, Canada (1993), p. 43.

  163. J. J. Rushchitsky, “Analytical investigation of wave triplets in a multi-phase medium”, in:Abstracts of Papers Read at the 1st European and 13th Intern. Conf. on Nonlinear Vibrations, Harburg, Germany (1993), p. 138.

  164. J. J. Rushchitsky, “The combinational scattering of sound on sound in multi-phase solids”, in:Annotations of Lectures Read at the 15th Intern. Symp. on Vibrations in Physical Systems, Poznan, Poland (1994), p. 124.

  165. J. J. Rushchitsky, “Simple plane waves in hyperelastic solids”, in:Abstract Book 2nd European Solid Mechanics Conference, Section:Vibration, Waves and Impacts, S. I, School of Architecture of the University of Genoa, Genoa, Italy (1994).

  166. J. J. Rushchitsky, “Lax-Nelson problem on the direct piezoelectric effect in two-phase materials”, in:Abstracts of Papers Read at the 3rd Intern. Congress on Industrial and Applied Mathematics (ICIAM 95), Hamburg, Germany (1995), p. 465.

  167. J. J. Rushchitsky, “Interaction of elastic waves in an isotropic multi-phase materials”,J. Theor. Appl. Mech., Bulgarian Academy of Sciences, No. 3-4, 82–93 (1994–1995).

    Google Scholar 

  168. J. J. Rushchitsky, “Wave triplets in two-phase composites,”Mech. Composite Mater. No. 5, 660–670 (1995).

    Google Scholar 

  169. J. J. Rushchitsky, “Lax-Nelson problem on the direct piezoelectric effect in two-phase materials”, in:Zeitschrift für Angewandte Mathematik und Mechanik, Special issue for lectures the ICIAM/GAMM 95,76, Supplement 2 (1996), pp. 663–664.

  170. J. J. Rushchitsky, “New dispersive simple wave (new partial solution) in nonlinear solid mixtures”, in:Abstracts of Papers Read at the 2nd European Conf. on Nonlinear Vibrations, Prague, Czech Republic (1996), p. 413.

  171. J. J. Rushchitsky, “Phenomena of a break-down instability and parametric amplification of waves in two-phase materials,” in:Abstracts of Papers Read at the 31st Solid Mechanics Conf. SolMech'96, Olsztyn-Mierki, Poland (1996), pp. 211–212.

  172. J. J. Rushchitsky, “New properties of transverse waves in quadratic nonlinear elastic materials (nonclassic results in classic theory),” in:Annotations of Lectures Read at the 15th Int. Symp. on Vibrations in Physical Systems, Poznan, Poland (1996), pp. 228–229.

  173. J. J. Rushchitsky, “Notes on simple wave energy”,Technicka Mechanika—Engineering Mechanics,4, No. 6, 327–334 (1997).

    Google Scholar 

  174. J. J. Rushchitsky, “New solitary waves and their interaction in solid mixtures (granular composites)”,Zeitschrift für Angewandte Mathematik und Mechanik, Special issue for lectures the GAMM 97,78, Supplement 1 (1998), pp. 626–627.

    Google Scholar 

  175. J. J. Rushchitsky, “New dispersive simple wave (new partial solution) in nonlinear solid mixtures”, in:Proc. 2nd European Nonlinear Oscillations Conference,3, Prague, Czech Republic (1997), pp. 141–144.

  176. J. J. Rushchitsky, “About energy propagation velocity of classic, dispersive, and simple waves in materials”, in:Annotation of Lectures Read at the 18th Symp. on Vibrations in Physical Systems, Poznan, Poland (1998), pp. 302–303.

  177. J. J. Rushchitsky, “Nonlinear shear piezoelastic waves in piezoelectric powders”, in:Abstracts of Papers Read at the GAMM 98 Jahrestagung, Bremen, Germany (1998), pp. 117.

  178. J. J. Rushchitsky, “Interaction of solitary waves in two-phase materials”,Int. Appl. Mech.,34, No. 11, 1134–1139 (1998).

    Google Scholar 

  179. J. J. Rushchitsky, “On classification of elastic waves”,Int. Appl. Mech.,35, No. 11, pp. 1104–1110 (1999).

    Google Scholar 

  180. J. J. Rushchitsky, “Interaction of waves in solid mixtures”,Appl. Mech. Rev.,52, No. 2, 35–74 (1999).

    Google Scholar 

  181. J. J. Rushchitsky, “Nonlinear interaction of elastic solitary waves in materials”, in:Abstracts of Papers Read at the 3rd European Conf. on Nonlinear Oscillations (August 8–12), Denmark, Copenhagen (Lyngby) (1999), pp. 36–37.

  182. J. J. Rushchitsky, “Interaction of solitary and harmonic waves in materials with the microstructure”, in:Abstracts of Papers Read at the 27th Int. Conf.-School on Analysis and Synthesis of Nonlinear Mechanical Oscillating Systems (September, 1–10) St-Petersburg, Russia (1999).

  183. J. J. Rushchitsky, “About the transition of the continuous wave into the shock wave during its propagation in a composite material”, in:Proc. EUROMECH 400 Colloq. on Impact and Damage Tolerance Modeling of Composite Materials and Structures (September, 27–29), London, United Kingdom (1999), pp. 154–157.

  184. J. J. Rushchitsky, and I. M. Khotenko, “Nonlinear plane waves in piezoelectric powders”,Annotations of Lectures Read at the 18th Symp. on Vibrations in Physical Systems, Poznan, Poland (1998), pp. 230–231.

  185. J. J. Rushchitsky and I. M. Khotenko, “On nonlinear waves in two-phase piezopowders”,Int. Appl. Mech.,34, No. 8, 750–755 (1998).

    Google Scholar 

  186. J. J. Rushchitsky and S. O. Rushchitska, “An interaction of simple solitary waves with profiles in the form of Chebyshov-Hermite, Mathieu, and Whittaker functions in media with the microstructure”,Proc. Int. Symp. TRECOP'98 on Trends in Continuum Physics (D. Maruszewski, W. Muschik, and A. Radowicz, editors), World Scientific, Singapore-London (1999), pp. 392–300.

    Google Scholar 

  187. J. J. Rushchitsky and S. O. Rushchitska, “On simple waves with profiles in the form of some special functions—Chebyshov-Hermite, Mathieu, and Whittaker—in two-phase media”, in:Proc. Mark Krein Int. Conf. on Series-Operator Theory, Advances and Applications, Vol. 117,Differential Operators and Related Topics (R. Kaashoek, E. Adamian, M. Gorbachuk, and E. Popov, editors), Kluwer, Amsterdam (2000), pp. 313–322.

    Google Scholar 

  188. I. N. Sneddon and D. S. Berry,The Classical Theory of Elasticity, Vol. 6,Flugge Encyclopedia of Physics, Springer-Verlag, Berlin (1958).

    Google Scholar 

  189. T. R. Steel, “Applications of a theory of interacting continua”,Q. J. Mech. Appl. Math.,20, No. 1, 57–72 (1967).

    MATH  Google Scholar 

  190. T. R. Steel, “Linearized theory of plane strain of mixture of two solids”,Int. J. Eng. Sci. 20, No. 10, 775–790 (1967).

    Google Scholar 

  191. T. R. Steel, “Determination of the constitutive coefficients for a mixture of two solids”,Int. J. Solids Struct.,4, No. 12, 1149–1160 (1968).

    MATH  ADS  Google Scholar 

  192. H. J. Sutherland and H. H. Calvit, “A dynamic investigation of fiber-reinforced viscoelastic materials. Experimental and theoretical analysis of pulse propagation in glass- and nylon-reinforced urethane filaments”,Exp. Mech., No. 8, 304–310 (1974).

    Google Scholar 

  193. M. Zh. Svanadze, “On existence of eigenfrequencies in the theory of two-component elastic mixtures”,Q. J. Mech. Appl. Math.,51, No. 3, 427–437 (1998).

    MATH  Google Scholar 

  194. T. R. Tiersten and M. Jahanmir, “A theory of composites modeled as interpenetrating solid continua”,Arch. Rat. Mech. Anal.,54, No. 2, 153–163 (1977).

    Google Scholar 

  195. C. Truesdell,A First Course in Rational Continuum Mechanics, The John Hopkins University, Baltimore (1972).

    Google Scholar 

  196. C. Truesdell and W. Noll,The Nonlinear Field Theories of Mechanics, Flugge's Handbuch der Physik, Band III/3, Springer Verlag, Berlin (1965).

    Google Scholar 

  197. J. Whitham,Linear and Nonlinear Waves, Wiley Interscience, New York (1974).

    MATH  Google Scholar 

  198. C. H. Yew and P. N. Jogi, “A study of wave motion in fiber-reinforced medium”,Int. J. Solids Struct.,12, No. 6, 693–703 (1976).

    Google Scholar 

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S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 33–64, May, 2000.

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Rushchitskii, Y.Y. Extension of the microstructural theory of two-phase mixtures to composite materials. Int Appl Mech 36, 586–614 (2000). https://doi.org/10.1007/BF02682074

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