, Volume 51, Issue 1, pp 211–221 | Cite as

Solution of some boundary value thermoelasticity problems for a rectangular parallelepiped taking into account micro-thermal effects



A three-dimensional system of differential equations is considered that describes a thermoelastic equilibrium of homogeneous isotropic elastic materials, microelements of which, in addition to classical displacements and thermal fields, are also characterized by microtemperatures. In the Cartesian system of coordinates the general solution of this system of equations is constructed using harmonic and metaharmonic functions. Some boundary value micro-thermoelasticity problems are stated for the rectangular parallelepiped. An analytical solution of this class of boundary value problems is constructed using the above-mentioned general solution. When the coefficients characterizing microthermal effects are zero, the obtained solutions lead to the solutions of corresponding classical boundary value thermoelasticity problems, the majority of which have been solved for the first time. It should be noted that the aim of the given work is to construct an effective (analytical) solution for a class of boundary vale problems rather than to investigate the validity or applicability of the involved theory.


Micro-thermoelasticity Rectangular parallelepiped General solution Boundary value problems 



The designated project has been fulfilled by a financial support of Shota Rustaveli National Science Foundation (Grant SRNSF/AR/91/5-109/11). Any idea in this publication is possessed by the author and may not represent the opinion of Shota Rustaveli National Science Foundation itself.


  1. 1.
    Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solids, I and II. Int J Eng Sci 2:189–203, 389–403Google Scholar
  2. 2.
    Mindlin RD (1964) Microstructure in linear elasticity. Arch Ration Mech Anal 16:51–77MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Green AE, Rivlin RS (1964) Multipolar continuum mechanics. Arch Ration Mech Anal 17:113–147MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Eringen AC, Kafadar CB (1976) Polar field theories. In: Eringen AC (ed) Continuum physics, vol 4. Academic Press, New YorkGoogle Scholar
  5. 5.
    Grot R (1969) Thermodynamics of a continuum with microstructure. Int J Eng Sci 7:801–814MATHCrossRefGoogle Scholar
  6. 6.
    Iesan D, Quintanilla R (2000) On a theory thermoelasticity with microtemperatures. J Therm Stress 23:199–215MathSciNetCrossRefGoogle Scholar
  7. 7.
    Svanadze M (2004) Fundamental solutions of the equations of the theory of thermoelasticity with microtemperatures. J Therm Stress 27(2):151–170MathSciNetCrossRefGoogle Scholar
  8. 8.
    Scalia A, Svanadze M (2006) On the representations of solutions of the theory of thermoelasticity with microtemperatures. J Therm Stress 29(9):849–863MathSciNetCrossRefGoogle Scholar
  9. 9.
    Scalia A, Svanadze M (2009) Potential method in the linear theory of thermoelasticity with microtemperatures. J Therm Stress 32(10):1024–1042CrossRefGoogle Scholar
  10. 10.
    Scalia A, Svanadze M, Tracinà R (2010) Basic theorems in the equilibrium theory of thermoelasticity with microtemperatures. J Therm Stress 33(8):721–753CrossRefGoogle Scholar
  11. 11.
    Khomasuridze NG (1998) Thermoelastic equilibrium of bodies in generalized cylindrical coordinates. Georgian Math J 5(6):521–544MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Khomasuridze NG (2007) On the symmetry principle in continuum mechanics. J Appl Math Mech 71(1):20–29Google Scholar

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.TbilisiGeorgia

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