Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer

Abstract

The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials (FGM) (i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach. The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.

A study on the thermoelasticity of three-phase-lag dipolar materials with voids

“…-the geometric equations (1), where the notation is: At the same time, the surface traction components t i , the surface couple components μ jk , the surface heat flux q, and the equilibrated stress vector λ corresponding to the voids, are defined in the following forms:…”

“…Between the systems of charges S (δ) and their corresponding solutions s (δ) , the next Betti-type relation of reciprocity holds: D ρF (1) i * u (2) i + ρG (1) ij * ϕ (2) ij + ρL (1) * ν (2) t * R (1) * θ (2) -1 T 0 t * q (1) i * β (2) i dV + ∂D t * t (1) i * u (2) i + t * μ (1) ij * ϕ (2) ij + t * λ (1) * ν (2) (2) i * u (1) i + ρG (2) ij * ϕ (1) ij + ρL (2) * ν (1) t * R (2) * θ (1) -1 T 0 t * q (2) i * β (1) i dV + ∂D t * t (2) i * u (1) i + t * μ (2) ij * ϕ (1) ij + t * λ (2) * ν (1)…”

“…μ (1) ijk * χ (2) ijk -F ijkmn ε (1) mn * χ (2) ijk -D mnijk γ (1) mn * χ (2) ijkf ijkm ν (1) ,m * χ (2) ijk c ijk ν (1) * χ (2) ijk + γ ijk θ (1) * χ (2) ijk = μ (2) ijk * χ (1) ijk -F ijkmn ε (2) mn * χ (1) ijk -D mnijk γ (2) mn * χ (1) ijk f ijkm ν (2) ,m * χ (1) ijkc ijk ν (2) * χ (1) ijk + γ ijk θ (2) * χ (1) ijk ,…”

“…where t α , t q and t T are the notations for the thermal displacement gradient, the heat flux, respectively the temperature gradient, in the three-phase-lag heat conduction theory, K ij and K * ij representing the tensor of the thermal conductivity, respectively the tensor of the conductivity rate. The results of applying the three-phase-lag theory to particular materials, a theory that comes from the dual-phase-lag theory, see [2,29,43], have been studied in many papers, for example in [3,5,6,12,13,24,25,28,30,36,38], other studies presenting the advantages of this theory, such as [1,9,39].…”

Fractional thermoelasticity applications for porous asphaltic materials

“…The verification process was done by comparison of model predictions with the previous work and shows good agreement is achieved. Abbas (2014Abbas ( , 2015 solved some problems on fractional order theory of thermoelasticity for a functional graded material. Some applications of fractional calculus to various problems in continuum mechanics are reviewed in the literature (Ezzat et al 2012a, b;2013a, b;2014a, b, c, d;2015).…”