Dual-Phase-Lag Model on Generalized Magneto-Thermoelastic Interaction in a Functionally Graded Material

Abstract

In this work, we consider the problem of magneto-thermoelastic interactions in a functionally graded material (FGM) under dual-phase-lag model in the presence of thermal shock. The generalized thermoelasticity theory with one relaxation time has been employed. The material is assumed to be elastic and functionally graded (FGM) (i.e. material with spatially varying properties). The basic equations have been written in the form of a vectormatrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. Numerical inversion of the transforms is carried out using the Stehfest method. Further, graphs have been drawn to show the effect of the nonhomogeneity parameter, magnetic field, and dual-phase-lag parameters on displacement, temperature, stress, and strain.

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

“…This study deepens and expands the results obtained previously and by modeling the linear theory of the dipolar materials with voids into the three-phase-lag theory we develop this analysis around the support offered by the notion of the volume fraction, which is related to the introduction of an additional kinematic freedom degree. The approach of this theoretical model consists in obtaining results regarding the uniqueness of the solution, different from the classical ones, such as the use of the dissipative inequality instead of the Laplace transform technique, applied for instance in [1,2,5,12,13,23]. Along with the other results obtained throughout this article, they provide a basis for developing the study of these structures in multiple directions, one of which is to consider a specific aspect: that of the isotropic and homogeneous materials.…”

“…Taking into account that I ks = I sk and r i = 1 T 0q i , we see that r is also of the form r = 1 T 0q , from Eq. (70), we deduce the following identity: D ρF (1) i * u (2) i + ρG (1) ij * ϕ (2) ij + ρL (1) * ν (2) t * R (1) * θ (2) t * r (1) i * θ (2) ,i dV + ∂D t * t (1) i * u (2) i + t * μ (1) ij * ϕ (2) ij + t * λ (1) * ν (2) + t * r (1) * θ (2) dA = D ρF (2) i * u (1) i + ρG (2) ij * ϕ (1) ij + ρL (2) * ν (1) t * R (2) * θ (1) t * r (2) i * θ (1) ,i dV + ∂D t * t (2) i * u (1) i + t * μ (2) ij * ϕ (1) ij + t * λ (2) * ν (1) + t * r (2) * θ (1) dA.…”

“…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)…”

“…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].…”

Heating a 2D Thermoelastic Half-Space Induced by Volumetric Absorption of a Laser Radiation

“…The DPL model or the generalized theory with dual-phase-lag is one of the essential generalizations developed by Tzou [10,11], where the energy equation was modified to contain two distinct phase-lags: one symbolizes the temperature gradient, while the other symbolizes the heat flux. Several authors have employed the DPL model to study thermoelastic waves under the effect of different fields [12][13][14][15][16][17][18][19].…”