Abstract
The disturbance caused by the application of continuous mechanical source on the free surface of a homogeneous, isotropic elastic half space in the context of the theory of generalized thermoelastic diffusion with one relaxation time parameter is investigated in the Laplace-Fourier transform domain for a two dimensional problem using eigenvalue approach. The integral transforms are inverted by using a numerical technique. The expressions for displacement components, stresses, temperature field, concentration and chemical potential so obtained in the physical domain are computed numerically and illustrated graphically at different times, for copper like material. As a special case the effect of diffusion on various expressions has also been obtained analytically and depicted graphically.
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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 \(\frac{|T-T_{0}|}{T_{0}}\ll 1\)
- Θ :
-
= T−T 0
- K :
-
thermal conductivity
- e kk :
-
dilatation
- δ ij :
-
Kronecker delta
- P :
-
chemical potential per unit mass
- C :
-
non-equilibrium concentration
- C 0 :
-
mass concentration at natural state
- c :
-
= C−C 0
- 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
- H(⋅):
-
Heaviside function
References
Abd-Alla AN, Al-Dawy AA (2000) The reflection phenomena of SV waves in a generalized thermoelastic medium. Int J Math Sci 23:529–546
Biot M (1956) Thermoelasticity and irreversible thermo-dynamics. J Appl Phys 27:240–253
Dhaliwal R, Sherief H (1980) Generalized thermoelasticity for an isotropic media. Q Appl Math 33:1–8
Duhamel J (1837) Some memoire sur les phenomenes thermo-mechanique. J Ec Polytech 15:1–31
Hoing G, Hirdes U (1984) A method for the numerical inversion of the Laplace transform. J Comput Appl Math 10:113–132
Lord H, Shulman Y (1967) A generalized thermodynamical theory of thermoelasticity. J Mech Phys Solids 15:299–309
Neumann F (1885) Vorlesungen Uber die Theorie der Elasticitat. Meyer, Brestau
Nowacki W (1974) Dynamic problems of thermoelastic diffusion in solids. I. Bull Acad Pol Sci Ser Sci Tech 22:55–64
Nowacki W (1974) Dynamic problems of thermoelastic diffusion in solids. II. Bull Acad Pol Sci Ser Sci Tech 22:129–135
Nowacki W (1974) Dynamic problems of thermoelastic diffusion in solids. III. Bull Acad Pol Sci Ser Sci Tech 22:266–275
Nowacki W (1976) Dynamic problems of diffusion in solids. Eng Fract Mech 8:261–266
Olesiak ZS, Pyryev YuA (1995) A coupled Quasi-stationary problem of thermodiffusion for an elastic cylinder. Int J Eng Sci 33:773–780
Press WH, Teukolsky SA, Vellerlig WT, Flannery BP (1986) Numerical recipes in FORTRAN, 2nd edn. Cambridge University Press, Cambridge
Sharma JN, Kumar V, Chand D (2003) Reflection of generalized thermoelastic waves from the boundary of a half-space. J Therm Stress 26:925–942
Sherief H (1986) Fundamental solution of the generalized thermoelastic problem for small time. J Therm Stress 9:151–164
Sherief H, Anwar M (1986) Problem in generalized thermoelasticity. J Therm Stress 9:165–181
Sherief H, Anwar M (1994) State space approach to two-dimensional generalized thermoelasticity problems. J Therm Stress 17:567–590
Sherief H, Ezzat M (1994) Solution of the generalized problem of thermoelasticity in the form of series of functions. J Therm Stress 17:75–95
Sherief H, Saleh H (2005) A half-space problem in the theory of generalized thermoelastic diffusion. Int Solids Struct 42:4484–4493
Sherief H, Hamza F, Saleh H (2004) The theory of generalized thermoelastic diffusion. Int J Eng Sci 42:591–608
Singh B (2005) Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion. J Earth Syst Sci 114:159–168
Sinha AN, Sinha SB (1974) Reflection of thermoelastic waves at a solid half-space with thermal relaxation. J Phys Earth 22:237–244
Thomas L (1980) Fundamentals of heat transfer. Englewood Cliffs, Prentice-Hall
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Choudhary, S., Deswal, S. Mechanical loads on a generalized thermoelastic medium with diffusion. Meccanica 45, 401–413 (2010). https://doi.org/10.1007/s11012-009-9260-9
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DOI: https://doi.org/10.1007/s11012-009-9260-9