Many theories describing the heat conduction have been proposed in recent years. This is to propose a theory where the propagation of heat
is demonstrated with a finite propagation speed, in contrast to the classical model using Fourier’s law leading to infinite propagation speed of heat signals. Good reviews are the articles written by
by Chandrasekharaiah [27] or
Hetnarski and Ignaczak [36, 37].
The dual-phase-lag model developed by
Tzou [25, 26] describes the thermal relaxation and thermalization behaviors that
are interwoven in the ultrafast process of heat transport in the electron gas. Two intrinsic delay times, called phase-lags denoted by and
, were introduced to account for the finite times required for the thermal equilibrium
( ) and effective collisions
( ) between electrons and phonons to take place. According to this model, the classical Fourier’s law
\(q(x,t)=-K∇θ(x,t)\)
is replaced by an universal relation between the heat flux vector at a point of the material at the time and the temperature gradient at the same point at the time
\(q(x,t+τ_q)=-K∇θ(x,t+τ_θ)\)
where represents the variance temperature in which is the absolute temperature above the reference temperature and is the thermal conductivity. The phase-lags, and
, are positive values and intrinsic properties of the material. The phase-lag of heat
flux can be interpreted as the time delay
due to the fast transient effect of thermal inertia, while the phase-lag of the temperature
gradient represents the effect of phonon-electron interactions and phonon scattering during ultrafast heat transfer [25].
Later on, Green and Naghdi [15-17] proposed a
completely new thermoelasticity theory by developing an alternative formulation of heat propagation. They incorporated the approach based on the Fourier law (refereed as type-I), the theory without energy dissipation (type-
II), and theory with energy dissipation (type-
III). The type-III model is a more general one. In the theories by Green and Naghdi,
the gradient of thermal displacement, is considered as a new constitutive variable. Also, the
scalar is interpreted as the counterpart in thermal fields of the mechanical displacement in mechanical fields;
it is called the thermal-displacement. The thermal displacement, satisfies
. The heat conduction law for GN-III model [15]
is given by
(3)
Here is a material constant characteristic of the theory and its unit is carried out by the unit of (conductivity/time).
Extending the thermoelastic model introduced by Green–Naghdi [15],
Roychoudhuri [35] proposed a three-phase-lag heat (TPL) conduction theory.
Roychoudhuri [35] has proposed the TPL heat conduction model in which the Fourier’s law of heat conduction
is replaced by an approximation to an improved form with introducing different phase lags for the heat flux vector, temperature gradient and for the thermal displacement gradient. In this model, the generalized constitutive equation for heat conduction proposed to describe the lagging behavior is of the form:
(4)
Roychoudhuri [35] in Eq. (4) introducing the phase lag of thermal displacement gradient
( ),
in addition to the phase lags and of heat flux and temperature gradient. Alternative parabolic and hyperbolic types of heat conduction can
be obtained by applying the Taylor series expansion of Eq. (4)
with respect to time as
(5)
Also in this case, exactly as for the constitutive equation by
Tzou [25] or its extension
Roychoudhuri [35], a new time differential (three-phase-lag) model can
be considered. In this model, the Fourier law of heat conduction
is replaced by an approximation to a modification of the Fourier law with introducing three different phase lags for the heat flux vector, the temperature gradient and the thermal displacement gradient. This model
obtained through the Taylor series expansions of both sides of Eq. (4) and keeping terms up to specific orders in
, , and i.e. the heat equation is
(6)
We have expanded until
order the heat
flux, until
order the thermal displacement gradient and until
order thermodynamic temperature. It is also manifest that
Eqs. (3) and (5) can be understood as a special case
with respect to the new one modeled by Eq. (6). In particular, in [35],
the time differential three-phase-lag heat conduction model is treated, starting from Eq. (3) and
retaining terms up to the second order in and up to the first order in and .
On taking the time-derivative of this equation and using
, we
obtain (7)
The increment of the entropy satisfies the following equations:
where denotes the specific heat at constant strain, represents the stress temperature modulus, in which denotes the thermal expansion
coefficient, , are
Lamé’s constants, is the displacement vector, is the density of the medium and is the power of the source of heat per unit mass.
The energy balance equation is given from (8) and (9) by
Further on differentiation of the energy equation (6)
with respect to time and then elimination
of from Eqs. (7) and (10) leads to the
modified heat transport equation with
three-phase-lag of higher derivative orders:
(11)
The lagging behavior
is closely associated with the possibility of explaining the problems related to the applications on the nanoscale: for instance, in the chemical nanotechnologies, transferring energy or the technique in which the chemical reactions occurring under
extremely fast transient conditions require very specific treatment and cannot
be treated by the classical macroscopic theories. Other areas of interest are
currently under investigation are those relating to various
possible combinations of Taylor series expansion orders for both associates of the constitutive equations.
It is worth to mention that a study upon the constitutive equations of type (6) or (11) made in the article by
Chiriţă et al. [39] shows that for they lead to an unbalanced system and therefore they cannot describe a real situation. For the case of the
modified heat equation (6) reduces to one considered by Green and Naghdi III [15, 17]. The case with and corresponds to the classical Fourier’s law. When
, and
, Eqs. (6) and (11) reduce to the heat conduction equation of Lord and Shulman theory [1]. We can recover the classical theory of thermoelasticity with three phase-lags proposed by
Roychoudhuri [35] when and take
. This model also retrieved thermoelasticity with two phase-lags proposed by [25] when and .
For a homogeneous and isotropic material, the additional governing equations of this model are as given below.
The stress-Strain relations:
(12)
The relationship between the strain and displacement
(13)
The equation of motion
(14)
The above system is a fully hyperbolic system in the sense that both the equations of motion (14) and the equation of heat transport (11) present in the system are of a hyperbolic-type.