Abstract
In this work, the field equations of the linear theory of thermoelasticity have been constructed in the context of a new consideration of Fourier law of heat conduction with time-fractional order and three-phase lag. A uniqueness and reciprocity theorems are proved. One-dimensional application for a half-space of elastic material in the presence of heat sources has been solved using Laplace transform and state space techniques Ezzat (Canad J Phys Rev 86:1241–1250, 2008). According to the numerical results and its graphs, conclusion about the new theory has been established.
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Abbreviations
- λ, μ :
-
Lame’s constants
- ρ :
-
density
- C E :
-
specific heat at constant strain
- t :
-
time
- T :
-
absolute temperature
- Θ:
-
T − T 0
- υ:
-
thermal displacement
- T o :
-
reference temperature chosen so that \({\left|\frac{\theta}{T_o}\right| < <1 }\)
- Q :
-
strength of the heat source
- F i :
-
mass force
- q i :
-
components of heat flux
- σ ij :
-
components of stress tensor
- e ij :
-
components of strain tensor
- u i :
-
components of displacement vector
- k :
-
thermal conductivity
- δ ij :
-
Kronecker delta function
- α T :
-
coefficient of linear thermal expansion
- τ o , τ q , τ υ, τ T :
-
relaxation times
- α :
-
fractional order of the differentiation 0 < α ≤ 1
- γ :
-
= (3λ + 2μ)α T
- \({\varepsilon}\) :
-
= γ /ρ c E , thermal coupling parameter
- \({c_o^2}\) :
-
\({=\frac{(\lambda+2\mu)}{\rho}}\)
- H(.):
-
Heaviside unit step function
- δ(.):
-
Dirac delta function
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Ezzat, M.A., El Karamany, A.S. & Fayik, M.A. Fractional order theory in thermoelastic solid with three-phase lag heat transfer. Arch Appl Mech 82, 557–572 (2012). https://doi.org/10.1007/s00419-011-0572-6
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DOI: https://doi.org/10.1007/s00419-011-0572-6