Skip to main content
SpringerLink
Log in

From the Chemical Potential Tensor and Concentration Tensor to Nonlocal Continuum Theories

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

The nontraditional thermodynamic pair (the chemical potential tensor and the concentration tensor) was introduced in the pioneering studies by Pidstryhach (also spelled as Podstrigach). Eliminating the chemical potential tensor and the concentration tensor from the constitutive equations for the stress tensor, Pidstryhach obtained the space-time-nonlocal equation for the stress tensor. We discuss the development of Pidstryhach’ scientific ideas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1955).

    MATH  Google Scholar 

  2. N. O. Virchenko and V. Ya. Rybak, Foundations of Fractional Integro-Differentiation [in Ukrainian], Zadruha, Kyiv (2007).

    Google Scholar 

  3. M. A. Grinfeld, Methods of Continuum Mechanics in the Theory of Phase Transformations [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  4. M. A. Grinfeld, “On the two types of heterogeneous phase equilibria,” Dokl. Akad. Nauk SSSR, 258, No. 3, 567–569 (1981).

    Google Scholar 

  5. M. A. Grinfeld, “Stability of heterogeneous equilibrium in systems containing solid elastic phases,” Dokl. Akad. Nauk SSSR, 265, No. 4, 836–840 (1982).

    MathSciNet  Google Scholar 

  6. O. Hrytsyna and V. Kondrat, Thermomechanics of Condensed Systems with Regard for the Local Displacements of Masses. I. Foundations of Theory [in Ukrainian], Rastr-7, Lviv (2017).

  7. M. A. Guzev, “Structure of the chemical potential tensor for a two-phase elastic medium under dynamic conditions,” Zh. Fiz. Khim., 79, No. 9, 1639–1643 (2005); English translation: Russ. J. Phys. Chem. A, 79, No. 9, 1451–1454 (2005).

  8. M. A. Guzev, “Chemical potential tensor for a two-phase continuous medium model,” Prikl. Mekh. Tekh. Fiz., No. 3, 12–22 (2005); English translation: J. Appl. Mech. Tech. Phys., 46, No. 3, 315–323 (2005).

  9. M. A. Guzev, “Conditions at the interphase boundary in a nonlinear elastic material in the dynamic case,” Dokl. Akad, Nauk SSSR, 416, No. 6, 763–765 (2007); English translation: Dokl. Phys., 52, No. 10, 571–573 (2007).

  10. V. I. Kondaurov, “The Clausius–Clapeyron equations for phase transitions of the first kind in a thermoelastic material,” Prikl. Mat. Mekh., 68, No. 1, 73–90 (2004); English translation: J. Appl. Math. Mech., 68, No. 1, 65–79 (2004).

  11. S. T. Konobeevsky, “On the theory of phase transitions. II. Diffusion in solid solutions under the influence of the distribution of stresses,” Zh. Éksper. Teor. Fiz., 13, No. 6, 200–214 (1943).

    Google Scholar 

  12. A. M. Kosevich and A. S. Kovalev, “Averaged equations of equilibrium and motion of an elastic medium with point defects,” Fiz. Tverd. Tela, 13, No. 1, 218–224 (1971).

    Google Scholar 

  13. I. A. Kunin, Theory of Elastic Bodies with Microstructure. Nonlocal Theory of Elasticity [in Russian], Nauka, Moscow (1975).

  14. É. S. Makarov, “A version of construction and some plane problems of the theory of chemoplasticity,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 13, No. 5, 70–72 (1989).

    Google Scholar 

  15. É. S. Makarov, “Application of the theory of chemoplasticity to the analysis of limit states of corrosive pipes,” Fiz.-Khim. Mekh. Mater., 25, No. 2, 115–117 (1989).

    Google Scholar 

  16. É. S. Makarov and I. E. Agureev, “Introduction to the theory of chemoplasticity,” in: A. L. Tolokonnikov (editor), Problems of Pure and Applied Mathematics [in Russian], Priokskoe Knizh. Izd., Tula (1988), pp. 196–212.

    Google Scholar 

  17. É. S. Makarov and I. E. Agureev, “Application of the theory of chemoplasticity to analysis of pressure treatment processes,” Izv. Vyssh. Uchebn. Zaved., Mashinostr., No. 3, 110–114 (1988).

    Google Scholar 

  18. Ya. S. Pidstryhach, “Differential equations of the diffusion strain theory of a solid,” Dop. Akad. Nauk Ukr. RSR, No. 3, 336–339 (1963).

  19. Ya. S. Pidstryhach, “Differential equations of the thermodiffusion problem in a deformable solid,” Dop. Akad. Nauk Ukr. RSR, No. 2, 169–172 (1961).

    Google Scholar 

  20. Ya. S. Pidstryhach, “On one generalization of a theoretical model of solid,” Dop. Akad. Nauk Ukr. RSR, No. 8, 1015–1017 (1963).

    Google Scholar 

  21. Ya. S. Pidstryhach and V. S. Pavlyna, “General relations of the thermodynamics of solid solutions,” Ukr. Fiz. Zh., 6, No. 5, 655–663 (1961).

    Google Scholar 

  22. Yu. Z. Povstenko, “A circular rotational dislocation loop in a nonlocally elastic medium,” Mat. Met. Fiz.-Mekh. Polya, 38, 95–98 (1995); English translation: J. Math. Sci., 81, No. 6, 3080–3083 (1996).

  23. Yu. Z. Povstenko, “The mathematical theory of defects in a Cosserat continuum,” Mat. Met. Fiz.-Mekh. Polya, Issue 27, 34–40 (1988); English translation: J. Soviet Math., 62, No. 1, 2524–2530 (1992).

  24. Yu. Z. Povstenko, “Nonlocal and gradient theories of elasticity and their application to the description of imperfections in solids,” Mat. Met. Fiz.-Mekh. Polya, 46, No. 2, 136–146 (2003).

    MathSciNet  MATH  Google Scholar 

  25. Yu. Z. Povstenko, “Description of surface phenomena in elastically polarized solids,” Zh. Prikl. Mekh. Tech. Fiz., No. 1, 117–121 (1983); English translation: J. Appl. Mech. Tech. Phys., 24, No. 1, 102–105 (1983).

  26. Yu. Z. Povstenko, “Straight dislocations, disclinations, and concentrated forces in a nonlocally elastic medium,” Mat. Met. Fiz.-Mekh. Polya, 44, No. 1, 124–129 (2001).

    MATH  Google Scholar 

  27. Yu. Z. Povstenko, “The diffusion equation for the concentration tensor,” Mat. Met. Fiz.-Mekh. Polya, Issue 33, 36–39 (1991); English translation: J. Soviet Math., 65, No. 5, 1841–1844 (1993).

  28. Yu. Z. Povstenko, “Tensor thermodynamic functions for deformable solids,” Mat. Met. Fiz.-Mekh. Polya, Issue 18, 41–43 (1983).

  29. Yu. Z. Povstenko, “Thermodynamics of the processes of diffusion and heat conduction in a Cosserat continuum,” in: A. I. Lopushanskaya (editor), Thermodynamics of Irreversible Processes [in Russian], Nauka, Moscow (1992), pp. 150–156.

    Google Scholar 

  30. Yu. Z. Povstenko, “Point defect in a nonlocal elastic medium,” Mat. Met. Fiz.-Mekh. Polya, 41, No. 3, 85–89 (1998); English translation: J. Math. Sci., 104, No. 5, 1501–1505 (2001).

  31. Yu. Z. Povstenko and O. A. Matkovskii, “Screw dislocation in a nonlocally elastic medium with moment stresses,” Dop. Nats. Akad. Nauk Ukr., No. 10, 57–60 (1995).

    Google Scholar 

  32. Yu. Z. Povstenko and O. A. Matkovskii, “Boundary dislocation in a nonlocally elastic medium with moment stresses,” Mat. Met. Fiz.-Mekh. Polya, 40, No. 3, 98–102 (1997); English translation: J. Math. Sci., 96, No. 1, 2883–2886 (1999).

  33. Ya. S. Podstrigach, “Diffusion theory of deformation of an isotropic continuum,” in: Problems of the Mechanics of Real Solids [in Russian], Issue 2 (1964), pp. 71–99.

    Google Scholar 

  34. Ya. S. Podstrigach, “Diffusion theory of the inelasticity of metals,” Zh. Prikl. Mekh. Tech. Fiz., No. 2, 67–72 (1965); English translation: J. Appl. Mech. Tech. Phys., 6, No. 2, 56–60 (1965).

  35. Ya. S. Podstrigach, “On a nonlocal theory of solid body deformation,” Prikl. Mekh., 3, No. 2, 71–76. (1967); English translation: Soviet Appl. Mech., 3, No. 2, 44–46 (1967).

  36. Ya. S. Podstrigach and V. S. Pavlina, “Differential equations of thermodynamic processes in n -component solid solutions,” Fiz.-Khim. Mekh. Mater., 1, No. 4, 383–389 (1965); English translation: Soviet Mater. Sci., 1, No. 4, 259–264 (1966).

  37. Ya. S. Podstrigach and Yu. Z. Povstenko, Introduction to the Mechanics of Surface Phenomena in Deformable Solids [in Russian], Naukova Dumka, Kiev (1985).

  38. A. I. Rusanov, “The development of the fundamental concepts of surface thermodynamics,” Kolloid. Zh., 74, No. 2, 148–166 (2012); English translation: Colloid J., 74, No. 2, 136–153 (2012).

  39. Ya. Ya. Rushchyts’kyi, “Nontraditional ordered pairs of thermodynamic parameters: from the Pidstryhach theory of diffusion elasticity to the Bedford–Drumheller theory of mixtures,” Mat. Met. Fiz.-Mekh. Polya, 41, No. 3, 117–120 (1998); English translation: J. Math. Sci., 104, No. 5, 1538–1541 (2001).

  40. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Newark, etc. (1993).

    MATH  Google Scholar 

  41. O. V. Temnov, “Dia- and paraelastic polarization in solids,” Issled. Teor. Uprug. Plast., No. 10, 83–102 (1974).

    Google Scholar 

  42. L. M. Truskinovskii, “Equilibrium phase interfaces,” Dokl. Akad. Nauk SSSR, 265, No. 2, 306–310 (1982); English translation: Sov. Phys. Dokl., 27, No. 7, 551–552 (1982).

  43. L. M. Truskinovskii, “On the chemical potential tensor,” Geokhimiya, No. 12, 1730–1744 (1983).

    Google Scholar 

  44. N. S. Fastov, “On the thermodynamics of irreversible processes in elastically deformable bodies,” Probl. Metalloved. Fiz. Met., No. 5, 550–576 (1958).

    Google Scholar 

  45. R. P. Araujo and D. L. S. McElwain, “A mixture theory for the genesis of residual stresses in growing tissues I: A general formulation,” SIAM J. Appl. Math., 65, No. 4, 1261–1284 (2005).

    MathSciNet  MATH  Google Scholar 

  46. N. Ari and A. C. Eringen, “Nonlocal stress field at Griffith crack,” Cryst. Lattice Defects Amorph. Mater., 10, 937–945 (1983).

    Google Scholar 

  47. M. A. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., 27, No. 3, 240–253 (1956).

    MathSciNet  MATH  Google Scholar 

  48. R. M. Bowen, “The thermochemistry of reacting mixture of elastic materials with diffusion,” Arch. Rational Mech. Anal., 34, No. 2, 97–127 (1969).

    MathSciNet  MATH  Google Scholar 

  49. R. M. Bowen, “Theory of mixtures,” in: A. C. Eringen (editor), Continuum Physics, Vol. 3: Mixtures and EM Field Theories, Academic Press, New York (1976), pp. 1–127.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  51. R. M. Bowen and J. C. Wiese, “Diffusion in mixtures of elastic materials,” Int. J. Eng. Sci., 7, No. 7, 689–722 (1969).

    MATH  Google Scholar 

  52. F. Buratti, Y. Huo, and I. Müller, “Eshelby tensor as a tensor of free enthalpy,” J. Elast., 72, No. 1-3, 31–42 (2003).

    MathSciNet  MATH  Google Scholar 

  53. M. Ciarletta and S. Chiriţă, “Some non-standard problems related with the mathematical model of thermoelasticity with microtemperatures,” J. Therm. Stresses, 36, No. 6, 517–536 (2013).

    Google Scholar 

  54. J.-M.-C. Duhamel, “Second mémoire sur les phénomènes thermo-mécanique,” J. Ecole Polytech., 15, 1–57 (1837).

    Google Scholar 

  55. D. G. B. Edelen, “Nonlocal field theories,” in: A. C. Eringen (editor), Continuum Physics, Vol. 4: Polar and Nonlocal Theories, Academic Press, New York (1976), pp. 75–204.

    Google Scholar 

  56. A. C. Eringen, “Edge dislocation in nonlocal elasticity,” Int. J. Eng. Sci., 15, No. 3, 177–183 (1977).

    MathSciNet  MATH  Google Scholar 

  57. A. C. Eringen, “Line crack subject to shear,” Int. J. Fracture., 14, No. 4, 367–379 (1978).

    MathSciNet  Google Scholar 

  58. A. C. Eringen, “Linear theory of nonlocal elasticity and dispersion of plane waves,” Int. J. Eng. Sci., 10, No. 5, 425–435 (1972).

    MathSciNet  MATH  Google Scholar 

  59. A. C. Eringen, Nonlocal Continuum Field Theories, Springer, New York (2002).

    MATH  Google Scholar 

  60. A. C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” J. Appl. Phys., 54, No. 9, 4703–4710 (1983).

    Google Scholar 

  61. A. C. Eringen, “Screw dislocation in nonlocal elasticity,” J. Phys. D: Appl. Phys., 10, No. 5, 671–678 (1977).

    Google Scholar 

  62. A. C. Eringen, “Theory of micropolar elasticity,” in: H. Liebowitz (editor), Fracture, Vol. 2, Academic Press, New York (1968), pp. 621–729.

    Google Scholar 

  63. A. C. Eringen, “Theory of nonlocal thermoelasticity,” Int. J. Eng. Sci., 12, No. 12, 1063–1077 (1974).

    MathSciNet  MATH  Google Scholar 

  64. A. C. Eringen, “Vistas of nonlocal continuum physics,” Int. J. Eng. Sci., 30, No. 10, 1551–1565 (1992).

    MathSciNet  MATH  Google Scholar 

  65. J. D. Eshelby, “The continuum theory of lattice defects,” in: F. Seitz and D. Turnbull (editors), Solid State Physics, Vol. 3, Academic Press, New York (1956), pp. 79–144.

    Google Scholar 

  66. A. Fick, “Über Diffusion,” Ann. Phys., 94, 59–86 (1855).

    Google Scholar 

  67. J. B. J. Fourier, Théorie Analytique de la Chaleur, Firmin Didot, Paris (1822).

    MATH  Google Scholar 

  68. R. Gorenflo, Yu. Luchko, and F. Mainardi, “Analytical properties and applications of the Wright function,” Fract. Calc. Appl. Anal., 2, No. 4, 383–414 (1999).

    MathSciNet  MATH  Google Scholar 

  69. R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in: A. Carpinteri and F. Mainardi (editors), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien (1997), pp. 223–276.

    MATH  Google Scholar 

  70. W. S. Gorsky, “Theorie der elastischen Nachwirkung in ungeordneten Misch–Kristallen (elastische Nachwirkung zweiter Art),” Phys. Z. Sowjetunion., 8, No. 1, 457–471 (1935).

    Google Scholar 

  71. A. E. Green and P. M. Naghdi, “On thermal effects in the theory of shells,” Proc. Roy. Soc. London. Ser. A., 365, No. 1721, 161–190 (1979).

    MathSciNet  MATH  Google Scholar 

  72. M. A. Grinfel’d, “On heterogeneous equilibrium of nonlinear elastic phases and chemical potential tensors,” Int. J. Eng. Sci., 19, No. 7, 1031–1039 (1981).

    MATH  Google Scholar 

  73. M. A. Grinfeld, Thermodynamic Methods in the Theory of Heterogeneous Systems, Longman, Harlow (1991).

    Google Scholar 

  74. R. A. Grot, “Thermodynamics of a continuum with microstructure,” Int. J. Eng. Sci., 7, No. 8, 801–814 (1969).

    MATH  Google Scholar 

  75. M. E. Gurtin and A. C. Pipkin, “A general theory of heat conduction with finite wave speeds,” Arch. Ration. Mech. Anal., 31, No. 2, 113–126 (1968).

    MathSciNet  MATH  Google Scholar 

  76. R. Herrmann, Fractional Calculus: An Introduction to Physicists, World Scientific, Singapore (2014).

    MATH  Google Scholar 

  77. D. Ieşan, “Thermoelasticity of bodies with microstructure and microtemperatures,” Int. J. Solids Struct., 44, No. 25-26, 8648–8662 (2007).

    MATH  Google Scholar 

  78. D. Ieşan and R. Quintanilla, “On the theory of thermoelasticity with microtemperatures,” J. Therm. Stresses, 23, No. 3, 199–215 (2000).

    MathSciNet  Google Scholar 

  79. D. Ieşan and R. Quintanilla, “On thermoelastic bodies with inner structure and microtemperatures,” J. Math. Anal. Appl., 354, No. 1, 12–23 (2009).

    MathSciNet  MATH  Google Scholar 

  80. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).

    MATH  Google Scholar 

  81. E. Kröner, “Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen,” Arch. Ration. Mech. Anal., 4, No. 1, 273–334 (1959).

    MathSciNet  MATH  Google Scholar 

  82. E. Kröner, “Elasticity theory of materials with long-rang cohesive forces,” Int. J. Solids Struct., 3, No. 5., 731–742 (1967).

    MATH  Google Scholar 

  83. I. A. Kunin, Theory of Elastic Media with Microstructure I. One-Dimensional Models, Springer, Berlin (1982).

    MATH  Google Scholar 

  84. I. A. Kunin, Theory of Elastic Media with Microstructure II. Three-Dimensional Models, Springer, Berlin (1983).

    MATH  Google Scholar 

  85. I. S. Liu, “On interface equilibrium and inclusion problem,” Continuum Mech. Thermodyn., 4, No. 3, 177–186 (1992).

    MathSciNet  MATH  Google Scholar 

  86. R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publ., Connecticut (2006).

    Google Scholar 

  87. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London (2010).

    MATH  Google Scholar 

  88. F. Neumann, Vorlesungen über die Theorie des Elasticität des festen Körper und des Lichtäthers, Teubner, Leipzig (1895).

    MATH  Google Scholar 

  89. R. R. Nigmatullin, “To the theoretical explanation of the 'universal response',” Phys. Stat. Sol. (b), 123, No. 2, 739–745 (1984).

    Google Scholar 

  90. R. R. Nigmatullin, “On the theory of relaxation for systems with 'remnant' memory,” Phys. Stat. Sol. (b), 124, No. 1, 389–393 (1984).

    Google Scholar 

  91. W. Nowacki, “Dynamical problems of thermodiffusion in solids I, II, III,” Bull. Acad. Polon. Sci. Sér. Sci. Tech., 23, 55–64, 129–135, 257–266 (1974).

  92. W. Nowacki, “Dynamic problems of thermodiffusion in elastic solids,” Proc. Vibr. Probl., 15, No. 2, 105–128 (1974).

    MathSciNet  MATH  Google Scholar 

  93. P. Podio-Guidugli, “Configurational balances via variational arguments,” Interfac. Free Bound., 3, No. 2, 223–232 (2001).

    MathSciNet  MATH  Google Scholar 

  94. I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).

    MATH  Google Scholar 

  95. Y. Z. Povstenko, “Axisymmetric ring loading in a nonlocal elastic space,” Int. J. Eng. Sci., 39, No. 3, 285–302 (2001).

    Google Scholar 

  96. Y. Z. Povstenko, “Circular dislocation loops in non-local elasticity,” J. Phys. D: Appl. Phys., 28, No. 1, 105–111 (1995).

    Google Scholar 

  97. Y. Z. Povstenko, “Fractional Cattaneo-type equations and generalized thermoelasticity,” J. Therm. Stresses, 34, No. 2, 97–114 (2011).

    Google Scholar 

  98. Y. Z. Povstenko, “Fractional heat conduction equation and associated thermal stresses,” J. Therm. Stresses, 28, No. 1, 83–102 (2005).

    Google Scholar 

  99. Y. Z. Povstenko, “Imperfections in nonlocal elasticity,” J. Phys. (Paris), 8, No. 8, 309–316 (1998).

    Google Scholar 

  100. Y. Z. Povstenko, “Non-local equations in mathematics and physics. Theory of non-local elasticity,” Prace Nauk. Wyższ. Szkoły Pedagog. Częstochow., Matematyka, 5, 89–96 (1997).

    Google Scholar 

  101. Y. Z. Povstenko, “Straight disclinations in nonlocal elasticity,” Int. J. Eng. Sci., 33, No. 4, 575–582 (1995).

    MATH  Google Scholar 

  102. Y. Z. Povstenko, “Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation,” Int. J. Eng. Sci., 43, No. 11-12, 977–991 (2005).

    MathSciNet  MATH  Google Scholar 

  103. Y. Z. Povstenko, “The use of differential equations of nonlocal elasticity for description of crystal imperfections,” Z. Angew. Math. Mech., 75, Suppl. 5, 407–408 (1996).

    MATH  Google Scholar 

  104. Y. Z. Povstenko, “Thermoelasticity which uses fractional heat conduction equation,” Mat. Met. Fiz.-Mekh. Polya 51, No. 2, 239– 246 (2008); English translation: J. Math. Sci., 162, No. 2, 296–305 (2009).

  105. Y. Z. Povstenko, “Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation,” Int. J. Solids Struct., 44, No. 7-8, 2324–2348 (2007).

    MathSciNet  MATH  Google Scholar 

  106. Y. Z. Povstenko and I. Kubik, “Concentrated ring loading in a nonlocal elastic medium,” Int. J. Eng. Sci., 43, No. 5-6, 457–471 (2005).

    MathSciNet  MATH  Google Scholar 

  107. Y. Z. Povstenko and O. A. Matkovskii, “Circular disclination loops in nonlocal elasticity,” Int. J. Solids Struct., 37, No. 44, 6419–6432 (2000).

    MATH  Google Scholar 

  108. Y. Povstenko, Fractional Thermoelasticity, Springer, New York (2015).

    MATH  Google Scholar 

  109. Y. Povstenko, Generalized theory of diffusive stresses associated with the time-fractional diffusion equation and nonlocal constitutive equations for the stress tensor, Comput. Math. Appl., 78, No. 6, 1819–1825 (2019), https://doi.org/10.1016/j.camwa.2016.02.034.

    Article  MathSciNet  Google Scholar 

  110. Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, Birkhäuser, New York (2015).

    MATH  Google Scholar 

  111. Y. Povstenko, “Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder,” Fract. Calcul. Appl. Anal., 14, No. 3, 418–435 (2011).

    MathSciNet  MATH  Google Scholar 

  112. Y. Povstenko, “Theories of thermal stresses based on space-time-fractional telegraph equations,” Comput. Math. Appl., 64, No. 10, 3321–3328 (2012).

    MathSciNet  MATH  Google Scholar 

  113. Y. Povstenko, “Theory of thermoelasticity based on the space-time-fractional heat conduction equation,” Phys. Scr. T, 136, 014017–1–6 (2009).

  114. Y. Povstenko, “Thermoelasticity based on fractional heat conduction equation,” in: F. Ziegler, R. Heuer, and C. Adam (editors), Proc. 6th Internat. Congr. on Thermal Stresses (May 26–29, 2005, Vienna, Austria), Vol. 2, Vienna University of Technology, Vienna (2005), pp. 501–504.

    Google Scholar 

  115. M. B. Rubin, “A uniqueness theorem for thermoelastic shells with generalized boundary conditions,” Quart. Appl. Math., 44, No. 3, 431–440 (1986).

    MathSciNet  MATH  Google Scholar 

  116. A. I. Rusanov, “Advances in thermodynamics of solid surfaces,” Pure Appl. Chem., 61, No. 11, 1945–1948 (1989).

    Google Scholar 

  117. A. I. Rusanov, “Problems on surface thermodynamics,” Pure Appl. Chem., 64, No. 1, 111–124 (1992).

    Google Scholar 

  118. A. I. Rusanov, “Thermodynamics of solid surfaces,” Surface Sci. Rep., 23, No. 6-8, 173–247 (1996).

    Google Scholar 

  119. A. I. Rusanov, “Surface thermodynamics revisited,” Surface Sci. Rep., 58, No. 5-8, 111–239 (2005).

    Google Scholar 

  120. A. I. Rusanov, A. K. Shchekin, and D. V. Tatyanenko, “Grand potential in thermodynamics of solid bodies and surfaces,” J. Chem. Phys., 131, No. 16., 161104 (2009).

    Google Scholar 

  121. J. Rushchitsky, Theory of Waves in Materials, Ventus Publ. ApS, Copenhagen (2011).

    Google Scholar 

  122. I. Samohýl, “Thermodynamics of mixtures of reacting and nonreacting fluids with heat conduction, diffusion, and viscosity,” Int. J. Non-Lin. Mech., 32, No. 2, 241–257 (1997).

    MathSciNet  MATH  Google Scholar 

  123. I. Samohýl, “Thermodynamics of nonreacting mixtures of any symmetry with heat conduction, diffusion, and viscosity,” Int. J. Non-Lin. Mech., 32, No. 2, 235–240 (1997).

    MathSciNet  MATH  Google Scholar 

  124. I. Samohýl and W. Pabst, “Phase equilibrium in non-fluids and non-fluid mixtures,” Int. J. Non-Lin. Mech., 39, No. 2, 247–263 (2004).

    MATH  Google Scholar 

  125. I. Samohýl and W. Pabst, “The Eshelby relation in mixtures,” Int. J. Non-Lin. Mech., 32, No. 2, 227–233 (1997).

    MATH  Google Scholar 

  126. A. Scalia and M. Svanadze, “On the representations of solutions of the theory of thermoelasticity with microtemperatures,” J. Therm. Stresses, 29, No. 9, 849–863 (2006).

    MathSciNet  Google Scholar 

  127. A. Scalia and M. Svanadze, “Potential method in the linear theory of thermoelasticity with microtemperatures,” J. Therm. Stresses, 32, No. 10, 1024–1042 (2009).

    Google Scholar 

  128. A. Scalia, M. Svanadze, and R. Tracinà, “Basic theorems on the equilibrium theory of thermoelasticity with microtemperatures,” J. Therm. Stresses, 33, No. 8, 721–753 (2010).

    Google Scholar 

  129. I. Shimizu, “Nonhydrostatic and nonequilibrium thermodynamics of deformable materials,” J. Geophys. Res., 97, No. B4, 4587–4597 (1992).

    Google Scholar 

  130. B. Stuke, “Allgemeine Rahmengleichungen der Kontinuumsdynamik,” Phys. Lett., 21, No. 6, 649–650 (1966).

    Google Scholar 

  131. B. Stuke, “Tensorielle chemische Potential: eine notwendige Erweiterung der Gibbs’schen Thermodynamik,” Z. Naturforsch. A, 30, No. 11, 1433–1440 (1975).

    Google Scholar 

  132. M. Svanadze, “Fundamental solutions of the equations of the theory of thermoelasticity with microtemperatures,” J. Therm. Stresses, 27, No. 2, 151–170 (2004).

    MathSciNet  Google Scholar 

  133. C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York (1969).

    MATH  Google Scholar 

  134. V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Heidelberg (2013).

    MATH  Google Scholar 

  135. C. Wei, “A theoretical framework for modeling the chemomechanical behavior of unsaturated soils,” Vadose Zone J., 13, No. 9, 1–21 (2014).

    Google Scholar 

  136. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York (2003).

    Google Scholar 

  137. C. Woźniak, “Thermoelasticity of bodies with microstructure,” Arch. Mech. Stos., 19, No. 3, 335–365 (1967).

    MATH  Google Scholar 

  138. C. Woźniak, “Thermoelasticity of nonsimple oriented materials,” Int. J. Eng. Sci., 5, No. 8, 605–612 (1967).

    Google Scholar 

  139. J. Wyrwał, A. Marynowicz, and J. Świrska, “On tensorial forms of thermodynamic potentials in mixtures theory,” Int. J. Solids Struct., 46, No. 11-12, 2293–2297 (2009).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, J. Długosz University in Częstochowa, Częstochowa, Poland

    Y. Z. Povstenko

Authors
  1. Y. Z. Povstenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. Z. Povstenko.

Additional information

Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 61, No. 1, pp. 71–85, January–March, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Povstenko, Y.Z. From the Chemical Potential Tensor and Concentration Tensor to Nonlocal Continuum Theories. J Math Sci 249, 389–403 (2020). https://doi.org/10.1007/s10958-020-04949-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-04949-0

Keywords

Search

Navigation