Skip to main content
SpringerLink
Log in

Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The present paper is concerned with the investigation of disturbances in a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion, when a moving source is acting along one of the co-ordinate axis on the boundary of the medium. Eigen value approach is applied to study the disturbance in Laplace-Fourier transform domain for a two dimensional problem. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using a numerical technique for the inversion of Laplace transform based on Fourier expansion techniques. These expressions are calculated numerically for a copper like material and depicted graphically. As special cases, the results in generalized thermoelastic and elastic media are obtained. Effect of presence of diffusion is analyzed theoretically and numerically.

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

Abbreviations

λ, µ :

Lame’s constants

ρ :

density of the medium

σ ij :

components of stress tensor

e ij :

components of strain tensor

u i :

components of displacement vector

C E :

specific heat at constant strain

t :

time

T :

absolute temperature

T 0 :

reference temperature chosen so that \(\tfrac{{\left| {T - T_0 } \right|}}{{T_0 }} \ll 1\)

Θ:

T − T 0

K :

thermal conductivity

e kk :

dilatation

δ ij :

Kronecker delta

P :

chemical potential per unit mass

C :

mass concentration

D :

thermodiffusion constant

τ 0 :

thermal relaxation time

τ :

diffusion relaxation time

a :

measure of thermodiffusion effect

b :

measure of diffusive effects

β 1 :

= (3λ + 2µ)α t

β 2 :

= (3λ + 2µ)α c

α t :

coefficient of linear thermal expansion

α c :

coefficient of linear diffusion expansion

F 0 :

intensity of the applied mechanical load

u :

displacement vector

ϕ :

scalar potential

ψ :

vector potential

δ(.):

Dirac delta function

References

  1. Biot M. Thermoelasticity and irreversible thermo-dynamics[J]. Journal of Applied Physics, 1956, 27:240–253.

    Article  MATH  MathSciNet  Google Scholar 

  2. Lord H, Shulman Y. A Generalized thermodynamical theory of thermoelasticity[J]. J Mech Phys Solids, 1967, 15:299–309.

    Article  MATH  Google Scholar 

  3. Dhaliwal R, Sherief H. Generalized thermoelasticity for an isotropic media[J]. Quart Appl Math, 1980, 33:1–8.

    Article  MathSciNet  Google Scholar 

  4. Sherief H. Fundamental solution of the generalized thermoelastic problem for small time[J]. J Thermal Stresses, 1986, 9:151–164.

    Article  Google Scholar 

  5. Sherief H, Anwar M. Problem in generalized thermoelasticity[J]. J Thermal Stresses, 1986, 9:165–181.

    Article  Google Scholar 

  6. Sherief H, Ezzat M. Solution of the generalized problem of thermoelasticity in the form of series of functions[J]. J Thermal Stresses, 1994, 17:75–95.

    Article  MathSciNet  Google Scholar 

  7. Sherief H, Anwar M. State space approach to two-dimensional generalized thermoelasticity problems[J]. J Thermal Stresses, 1994, 17:567–590.

    Article  MathSciNet  Google Scholar 

  8. Sinha A N, Sinha S B. Reflection of thermoelastic waves at a solid half-space with thermal relaxation[J]. J Phys Earth, 1974, 22:237–244.

    Google Scholar 

  9. Abd-Alla A N, Al-Dawy A A. The reflection phenomena of SV waves in a generalized thermoelastic medium[J]. Int J Math Sci, 2000, 23:529–546.

    Article  MATH  MathSciNet  Google Scholar 

  10. Sharma J N, Kumar V, Chand D. Reflection of generalized thermoelastic waves from the boundary of a half-space[J]. J Thermal Stresses, 2003, 26:925–942.

    Article  Google Scholar 

  11. Oriani R A. Thermomigration in solid metals[J]. J Phys Chem Solids, 1969, 30:339–351.

    Article  Google Scholar 

  12. Fryxel R E, Aitken E A. High temperature studies of urania in a thermal gradient[J]. J Nucl Mater, 1969, 30:50–56.

    Article  Google Scholar 

  13. Nowacki W. Dynamic problems of thermoelastic diffusion in solids[J]. I Bulletin de l’Academic Polonaise des Sciences Serie des Sciences Techniques, 1974, 22:55–64.

    MathSciNet  Google Scholar 

  14. Nowacki W. Dynamic problems of thermoelastic diffusion in solids[J]. II Bulletin de l’Academic Polonaise des Sciences Serie des Sciences Techniques, 1974, 22:129–135.

    Google Scholar 

  15. Nowacki W. Dynamic problems of thermoelastic diffusion in solids[J]. III Bulletin de l’Academic Polonaise des Sciences Serie des Sciences Techniques, 1974, 22:266–275.

    Google Scholar 

  16. Nowacki W. Dynamic problems of diffusion in solids[J]. Eng Frac Mech, 1976, 8:261–266.

    Article  Google Scholar 

  17. Sherief H, Hamza F, Saleh H. The theory of generalized thermoelastic diffusion[J]. Int J Eng Sci, 2004, 42:591–608.

    Article  MathSciNet  Google Scholar 

  18. Singh B. Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion[J]. J Earth Syst Sci, 2005, 114:159–168.

    Article  Google Scholar 

  19. Sherief H, Saleh H. A half-space problem in the theory of generalized thermoelastic diffusion[J]. Int J Solid Structures, 2005, 42:4484–4493.

    Article  MATH  Google Scholar 

  20. Aouadi M. Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion[J]. Zeitschrift fr Angewandte Mathematik und Physik (ZAMP), 2006, 57(2):350–366.

    Article  MATH  MathSciNet  Google Scholar 

  21. Aouadi M. A generalized thermoelastic diffusion problem for an infinitely long solid cylinder[J]. Int J Mathematics and Mathematical Sci, 2006, 2006:123–137.

    MathSciNet  Google Scholar 

  22. Verrujit A, Cordova C. Moving loads on an elastic half plane with hysteretic damping[J]. J Appl Mechanics, 2001, 68:915–922.

    Article  Google Scholar 

  23. Honig G, Hirdes U. A method for the numerical inversion of the Laplace transform[J]. J Comp Appl Math, 1984, 10:113–132.

    Article  MATH  MathSciNet  Google Scholar 

  24. Press WH, Teukolsky S A, Vellerlig W T, Flannery B P. Numerical recipes in FORTRAN (2nd edition)[M]. Cambridge: Cambridge University Press, 1986.

    Google Scholar 

  25. Thomas L. Fundamentals of heat transfer[M]. New Jersey: Prentice-Hall Inc Englewood Cliffs, 1980.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar, 125001, Haryana, India

    Sunita Deswal

  2. Department of Mathematics, Govt. College, Nalwa (Hisar), 125037, Haryana, India

    Suman Choudhary

Authors
  1. Sunita Deswal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suman Choudhary
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sunita Deswal.

Additional information

Communicated by CHENG Chang-jun

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deswal, S., Choudhary, S. Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion. Appl. Math. Mech.-Engl. Ed. 29, 207–221 (2008). https://doi.org/10.1007/s10483-008-0208-5

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-008-0208-5

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation