Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 21

where T r is a reference temperature. Therefore, the wave-like features will become
significant when: (1) t is high, or (3) t is small. Some typical

situations where hyperbolic heat conduction differs from classical parabolic heat
conduction include those concerned with a localized moving heat source with high
intensity, a rapidly propagating crack tip, shock wave propagation, thermal reso-
nance, interfacial effects between dissimilar materials, laser material processing,
and laser surgery (Chandrasekharaiah 1986, 1998, Joseph and Preziosi 1989, 1990,
Tzou 1992a, 1995a, 1997, Wang 1994, 2000a).
When

τ 0 is large, (2) ∂ T / ∂

τ 0 is finite, the CV constitutive relation (1.25) and the
hyperbolic heat-conduction equation (1.26) become (Joseph and Preziosi 1989)

τ 0 → ∞ but k ef f = k /

∂ q ( r , t )

t = −

k eff ∇ T ( r , t ) , (1.30)

∂

and

2 T
∂

1
α eff

∂ 1

k eff

∂ F

∂ t 2 = Δ

T +

t ,

(1.31)

and c are the density and the specific heat of the material,
respectively. Therefore, when

where α eff = k ef f / ρ c , ρ

τ 0 is very large, a temperature gradient established
at a point of the material results in an instantaneous heat flux rate at that point,
and vice-versa. Eq. (1.31) is a classical wave equation that predicts thermal wave
propagation with speed V CV , like Eq. (1.26). A major difference exists, however,
between Eqs. (1.26) and (1.31): the former allows damping of thermal waves, the
latter does not (Wang and Zhou 2000, 2001).

The Dual-Phase-Lagging Constitutive Relation

It has been confirmed by many experiments that the CV constitutive relation gener-
ates a more accurate prediction than the classical Fourier law. However, some of its
predictions do not agree with experimental results either (Tzou 1995a, 1997, Wang
1994). A thorough study shows that the CV constitutive relation has only taken ac-
count of the fast-transient effects, but not the micro-structural interactions. These
two effects can be reasonably represented by the dual-phase-lag between q and ∇ T ,

a further modification of Eq. (1.28) (Tzou 1995a, 1997),

q ( r , t + τ 0 )= − k ∇ T ( r , t + τ T ) . (1.32)

According to this relation, the temperature gradient at a point r of the material at
time t + τ T corresponds to the heat flux density vector at r at time t + τ 0 . The delay

time τ T is interpreted as being caused by the micro-structural interactions (small-

scale heat transport mechanisms occurring in the micro-scale, or small-scale effects
of heat transport in space) such as phonon-electron interaction or phonon scattering,
and is called the phase-lag of the temperature gradient (Tzou 1995a, 1997). The