Propagation of thermoelastic waves across an interface with consideration of couple stress and second sound

Abstract

The reflection and transmission of thermoelastic waves across an interface between two different couple stress solids are studied based on the thermoelastic Green–Naghdi theory with consideration of second sound. First, some thermodynamic equations of a couple stress elastic solid are formulated and the function of free energy density is postulated. Second, equations of thermal motion and heat conduction of the couple stress elasticity are derived and constitutive relations with thermoelastic coupled effects are obtained. From these equations, four kinds of dispersive waves, namely, thermal-mechanically coupled MT1 wave and MT2 wave, uncoupled SV wave, and an evanescent wave that becomes the surface waves at interface, are derived. Then, the interfacial conditions of couple stress elastic solids with consideration of force stress, couple stress, and thermal effects are used to determine the amplitude ratios of the reflection and transmission waves with respect to the incident wave. The numerical results are validated by consideration of energy conservation.

Keywords

Couple stress thermoelasticity thermal wave coupled wave reflection and transmission

1. Introduction

Different mechanical behaviors have been observed experimentally between micro/nanoscales and large scales [1 –4]. However, classical elastic theory does not predict these differences in mechanical behavior. In order to reflect the micro/nanoscale mechanical behavior and capture the micro-size effects, generalized continuum theories, e.g. the couple stress theory [5,6], the micromorphic theory [7], the micropolar theory [8], microstretch theory [9], and the nonlocal theory [10], were proposed successively. The couple stress theory as one of the generalized continuum theories has received wide attention due to it having fewer material parameters. A couple stress elastic solid differs from a classical elastic solid in that not only the force stress but also the couple stress can be taken into account. It is also different from the micropolar elastic solid. Each mass point has not only the translation motion but also the dependent micro-rotation motion, while the micropolar elastic solid has extra independent micro-rotation motion apart from the translation motion [11 –13]. Toupin [6], Mindlin and Tiersten [5], Koiter [14], and others have contributed to the establishment and development of the elastic theory of couple stress solids. In the linear theory of couple stress solids, there is an additional elastic modulus (called a modulus of bending and twisting). The square root of the ratio of the bending–twisting modulus to the usual shear modulus has the dimension of length. The length $l$ is an important material parameter which carries with it all of the differences between the couple stress solid and the classical elastic solid. Usually, the length $l$ is small in comparison with the dimension of a body and the wavelength normally encountered. However, its influences might become important as dimensions of a body or the wavelength encountered diminishes to the same order of the length $l$ .

The convenience of conversion between the mechanical energy and thermal energy makes the thermo-mechanically coupled theory of everlasting interest. In the conventional coupled thermoelasticity (CTE) based on the parabolic equation of heat transport and the Fourier law of the heat conduction, the thermal signals propagate with infinite speed. However, the generalized thermoelasticity based on the hyperbolic equation of heat transport and the non-Fourier law of heat conduction predicts the existence of the thermal wave (called second sound), which admits a finite propagation speed of thermal signals. Lord and Shulman [15] proposed a generalized thermodynamic theory (L-S theory) which introduces a relaxation time between the thermal flux and the temperature gradient, and consequently the hyperbolic equation of heat transport is derived. Abbas and Othman [16] and Lotfy and Othman [17] studied the propagation of thermoelastic waves in a solid half-space based on L-S theory. Different from the L-S theory, the generalized thermoelasticity proposed by Green and Lindsay [18] (G-L theory) possesses two relaxation times. Agarwal [19] studied the propagation of thermoelastic waves of assigned frequency under the G-L theory. Through comparison with the L-S theory, all the corresponding results of the L-S theory are recovered after introducing some assumptions. Sinha and Elsibai [20] studied the reflection and refraction of thermal elastic waves at an interface of two semi-infinite media by applying G-L theory and considered two kinds of interfaces, i.e. liquid–liquid and liquid–solid. The numerical results were compared with that of L-S theory in order to investigate the effect of the second relaxation time. Singh [21] studied the refection and refraction of plane sound waves at an interface between a liquid half-space and a micropolar generalized thermoelastic solid half-space by use of L-S and G-L theories. The numerical results showed the thermal effect of second thermal relaxation time taken by G-L theory is nonexistence. Green and Naghdi [22 –24] put forward a new theory (G-N theory) by introducing the concept of thermal displacement. In contrast to the classical thermoelasticity characterized by the Fourier law, the heat flux in G-N theory does not involve energy dissipation. A constitutive equation for an entropy flux vector is determined by the same potential function which also determines the stress, and it permits the transmission of heat as thermal waves at finite speed. Othman et al. [25] studied the reﬂection of thermoelastic waves in a semi-inﬁnite elastic solid by using G-N theory. In the context of various linear theories of thermoelasticity, i.e. L-S, G-L, and G-N theories, CTE, and uncoupled thermoelasticity, Sharma et al. [26] investigated the problem of thermoelastic wave reﬂection from insulated and isothermal stress-free interfaces as well as rigidly ﬁxed boundaries of homogeneous isotropic solid half-spaces. A comprehensive comparison was made of the various models of thermoelasticity mentioned above based on the numerical results. Recently, based on the G-N theory, Chakraborty and Singh [27] further studied reﬂection and refraction of a thermoelastic wave at the interface of two thermoelastic solid half-spaces in the presence of normal initial stress. Other related works are those reported by Othman and Abbas [28 –30].

In this paper, the reflection and transmission problems of a thermoelastic coupled wave at an interface of dissimilar solid half-spaces governed by the couple stress thermoelasticity are studied. The coupled governing equation between the displacement field and the temperature field and the constitutive equation with consideration of the couple stress effects and the thermoelastic coupled effects are derived based on the G-N theory. These equations result in two thermo-mechanically coupled P waves, an uncoupled SV wave, and an evanescent wave. By using the interfacial conditions of thermoelastic solids with consideration of force stress, couple stress, and thermal effects, the amplitude ratios of the reflection and transmission waves with respect to the incident wave are determined. The numerical results are validated by consideration of energy conservation, which must be satisfied by the incident waves, the reflection waves, and the transmission waves.

2. Thermodynamically formulas of couple stress elasticity

According to the law of energy conservation in thermodynamics (the first law of thermodynamics),

\dot{K} + \dot{E} = H + W,

(1)

where $K$ is the total kinetic energy, $E$ is total internal energy, $H$ is the total rate of heat supply, and $W$ is the total rate of the work done by external forces. The dots on the letters stand for the derivative of time. The explicit expressions for $\overset{\cdot}{K}, \overset{\cdot}{E},$ H and W in the couple stress elastic solid are written as

\dot{K} = \int_{V} ρ \overset{··}{u} \cdot \dot{u} d V,

(2a)

\overset{\cdot}{E} = \int_{V} ρ \overset{\cdot}{ε} dV,

(2b)

H = - \int_{S} Q \cdot n dS = - \int_{V} \nabla \cdot Q dV,

(2c)

\begin{matrix} W = \int_{S} n \cdot τ \cdot \overset{\cdot}{u} dS + \int_{S} n \cdot μ \cdot \overset{\cdot}{ω} dS = \int_{V} \nabla \cdot (τ \cdot \overset{\cdot}{u}) dV + \int_{V} \nabla \cdot (μ \cdot \overset{\cdot}{ω}) dV \\ = \int_{V} (\nabla \cdot τ \cdot \overset{\cdot}{u} + τ : \nabla \overset{\cdot}{u}) dV + \int_{V} (\nabla \cdot μ \cdot \overset{\cdot}{ω} + μ : \nabla \overset{\cdot}{ω}) dV, \end{matrix}

(2d)

where $ε$ is the internal energy per unit mass, $Q$ is the heat flux vector, $u$ is the displacement vector, and $ω$ is the axial vector of the antisymmetric part of the displacement gradient. $τ (= τ^{S} + τ^{A})$ is the force stress tensor ( $τ^{s}$ is the symmetric part and $τ^{A}$ the antisymmetric part), while $μ (= μ^{0} + μ^{D})$ is the couple stress tensor ( $μ^{0}$ is the spherical part while $μ^{D}$ is the deviatoric part).

Consider that

ω = \frac{1}{2} \nabla \times u,

(3a)

τ^{A} : \nabla \overset{\cdot}{u} = \frac{1}{2} (τ_{•}^{\times} I) \cdot \nabla \times \overset{\cdot}{u} . (I is an unit tensor of two order)

(3b)

Inserting equations (2) and (3) into equation (1) leads to

\begin{matrix} \int_{V} [(ρ \overset{\cdot\cdot}{u} - \nabla \cdot τ) \cdot \overset{\cdot}{u} - \frac{1}{2} (\nabla \cdot μ + τ_{•}^{\times} I) \cdot \nabla \times \overset{\cdot}{u} \\ + (ρ \overset{\cdot}{ε} + \nabla \cdot Q - τ^{S} : \nabla \overset{\cdot}{u} - \frac{1}{2} μ : \nabla \nabla \times \overset{\cdot}{u})] dV = 0 . \end{matrix}

(4)

Equation (4) holds for any arbitrary volume element. Therefore, the following equations are obtained:

\nabla \cdot τ = ρ \overset{\cdot\cdot}{u},

(5a)

\nabla \cdot μ + τ_{•}^{\times} I = 0,

(5b)

ρ \overset{\cdot}{ε} = τ^{S} : \nabla \overset{\cdot}{u} + \frac{1}{2} μ : \nabla \nabla \times \overset{\cdot}{u} - \nabla \cdot Q .

(5c)

Equations (5a) and (5b) are the equilibrium equations of the force stress and the couple stress, respectively, and equation (5c) is the energy equilibrium equation of the interior energy of per unit mass. Consider that

μ^{D} = μ - μ^{0} = μ - \frac{1}{3} μ : II,

(6)

I : \nabla \nabla \times \overset{\cdot}{u} = 0 .

(7)

Inserting equations (6) and (7) into equation (5c) leads to

ρ \overset{\cdot}{ε} = τ^{S} : \nabla \overset{\cdot}{u} + \frac{1}{2} μ^{D} : \nabla \nabla \times \overset{\cdot}{u} - \nabla \cdot Q

(8)

= τ^{S} : \nabla \overset{\cdot}{u} + \frac{1}{2} μ^{D} : \nabla \nabla \times \overset{\cdot}{u} - Φ \cdot \nabla T - T \nabla \cdot Φ,

(9)

where $Q = T Φ$ , $Φ$ is the entropy flux vector, and $T$ is absolute temperature.

In order to establish the boundary condition, let us consider again the energy conservation in a representative volume element in the absence of body force and body couple:

\overset{\cdot}{K} + \overset{\cdot}{E} = \int_{S} (n \cdot τ \cdot \overset{\cdot}{u} + n \cdot μ \cdot \overset{\cdot}{ω} + n \cdot Q) dS .

(10)

Consider that

\begin{matrix} n \cdot μ \cdot \overset{\cdot}{ω} = n \cdot μ \cdot nn \cdot \overset{\cdot}{ω} + n \cdot μ \cdot (I - nn) \cdot \overset{\cdot}{ω} \\ = \frac{1}{2} μ_{nn} n \cdot \nabla \times \overset{\cdot}{u} + n \cdot μ \cdot (I - nn) \cdot \overset{\cdot}{ω}, \end{matrix}

(11)

\frac{1}{2} μ_{nn} n \cdot \nabla \times \overset{\cdot}{u} = \frac{1}{2} n \cdot \nabla \times (μ_{nn} \overset{\cdot}{u}) - \frac{1}{2} n \times \nabla μ_{nn} \cdot \overset{\cdot}{u},

(12)

and, if the surface $S$ is smooth,

\int_{S} n \cdot \nabla \times (μ_{nn} \overset{\cdot}{u}) dS = 0 .

(13)

Equation (10) becomes

\begin{matrix} \overset{\cdot}{K} + \overset{\cdot}{E} = \int_{S} [(n \cdot τ - \frac{1}{2} n \times \nabla μ_{nn}) \cdot \overset{\cdot}{u} + n \cdot μ \cdot (I - nn) \cdot \overset{\cdot}{ω} + n \cdot Q) dS \\ = \int_{S} (p \cdot \overset{\cdot}{u} + m \cdot \overset{\cdot}{ω} + n \cdot Φ T) dS, \end{matrix}

(14)

where

p = (n \cdot τ - \frac{1}{2} n \times \nabla μ_{nn}) = n \cdot τ^{s} + \frac{1}{2} n \times (\nabla \cdot μ^{D} - \nabla μ_{nn}^{D}),

(15)

m = n \cdot μ \cdot (I - nn) .

(16)

In view of equation (14), $(p, u)$ , $(m, ω)$ , and $(n \cdot Φ, T)$ are the work-conjugate pairs. Therefore, the boundary conditions require that one of a pair should be specified on the boundary, i.e.

The displacement vector $u$ is given or the modified traction vector $p$ is given.

The tangent component $m$ of the surface couple vector $n \cdot μ$ is given or the axial vector $ω$ of the antisymmetric part of the displacement gradient tensor $\nabla u$ is given.

The normal component of entropy flux vector, i.e. $n \cdot Φ$ , is given or temperature T is given.

In order to obtain the constitutive relation, define

ψ = ε - T φ,

(17)

where $ψ$ is the Helmholtz free energy per unit mass and $φ$ is entropy density per unit mass. Taking the derivative to equation (17) with respect to time leads to

ρ \overset{\cdot}{ψ} = ρ \overset{\cdot}{ε} - ρ \overset{\cdot}{T} φ - ρ T \overset{\cdot}{φ} .

(18)

Introducing the equilibrium equation of entropy

ρ \overset{\cdot}{φ} = ρ ζ - \nabla \cdot Φ,

(19)

where $ζ$ is internal rate of production of entropy. The external rate of supply of entropy is absent. Inserting equation (19) into equation (18), then inserting into equation (9) leads to

τ^{S} : \overset{\cdot}{ε} + μ^{D} : \overset{\cdot}{χ} - ρ (\overset{\cdot}{ψ} + \overset{\cdot}{T} φ) - ρ T ζ - Φ \cdot \nabla T = 0,

(20)

where $χ = \frac{1}{2} \nabla \nabla \times u$ is the curvature-twist tensor. Equation (20) stands for the energy conservation (in terms of Helmholtz free energy) of the couple stress thermoelastic solid.

Define the free energy per unit mass, $ψ$ , as a function of the strain, $ε_{ij}$ , the curvature-twist, $χ_{ij}$ , the temperature, $T$ , and the thermal displacement gradient, $α_{, i}$ , namely, $ψ = ψ (T, α_{, i}, ε_{ij}, χ_{ij})$ , where $α$ is called the thermal displacement [23], which stands for the average displacement magnitude of the molecular thermal motion, while $T$ represents the average velocity magnitude of the molecular thermal motion, namely, $\overset{\cdot}{α} = T$ . Then, equation (20) can be rewritten as

ρ (\frac{\partial ψ}{\partial T} + φ) \overset{\cdot}{T} + (Φ_{i} + \frac{\partial (ρ ψ)}{\partial α_{, i}}) T_{, i} + (\frac{\partial (ρ ψ)}{\partial ε_{ij}} - τ_{ij}^{s}) {\overset{\cdot}{ε}}_{ij} + (\frac{\partial (ρ ψ)}{\partial χ_{ij}} - μ_{jk}^{d}) {\overset{\cdot}{χ}}_{ij} + ρ T ζ = 0 .

(21)

Choosing $\overset{\cdot}{T}$ , $T_{, i}$ , ${\overset{\cdot}{ε}}_{ij}$ , ${\overset{\cdot}{χ}}_{ij}$ as independent variables, the following equations are obtained:

ρ φ = - \frac{\partial (ρ ψ)}{\partial T}, Φ_{i} = - \frac{\partial (ρ ψ)}{\partial α_{, i}}, τ_{ij}^{s} = \frac{\partial (ρ ψ)}{\partial ε_{ij}}, μ_{ij}^{d} = \frac{\partial (ρ ψ)}{\partial χ_{ij}}, ρ T ζ = 0 .

(22)

The first four equations give the coupled constitutive relations of thermal and mechanical qualities. The last equation, i.e. $ρ T ζ = 0$ , is equivalent to $ζ = 0$ . In other words, the absence of entropy source is equivalent to the absence of internal rate of production of entropy, so that there is no dissipation of energy.

In the present work, a phenomenological simplified version of the free energy density for a centrosymmetric and isotropic material is given as following,

ρ ψ = \frac{1}{2} (λ ε_{ii} ε_{jj} + 2 μ ε_{ij} ε_{ij}) + 2 (η χ_{ij} χ_{ij} + η' χ_{ji} χ_{ij}) - \frac{ρ C_{r} θ^{2}}{2 T_{0}} - ℜ θ ε_{pp} + \frac{κ^{*}}{2 T_{0}} α_{, i} α_{, i},

(23)

where $λ$ and $μ$ are the Lamé constants of the classical elastic solid, $η$ and $η'$ are the new material constants introduced for the couple stress solid, and $C_{r}$ is the specific heat capacity. $ℜ = (3 λ + 2 μ) h$ is the thermoelastic coupling coefficient, $h$ is the linear thermal expansion coefficient, and $κ^{*}$ is a material constant related to the specific thermal properties and hereafter called generalized thermal conduction coefficients. $T_{0}$ is the reference temperature and $θ (= T - T_{0})$ is the temperature change. The corresponding constitutive relations are

τ_{ij}^{s} = \frac{\partial (ρ ψ)}{\partial ε_{ij}} = λ δ_{ij} ε_{pp} + 2 μ ε_{ij} - ℜ θ δ_{ij},

(24a)

μ_{ij}^{D} = \frac{\partial (ρ ψ)}{\partial χ_{ij}} = 4 η χ_{ij} + 4 η' χ_{ji},

(24b)

ρ φ = - \frac{\partial (ρ ψ)}{\partial θ} = \frac{ρ C_{r} θ}{T_{0}} + ℜ ε_{pp},

(24c)

Φ_{i} = - \frac{\partial (ρ ψ)}{\partial α_{, i}} = - \frac{κ^{*}}{T_{0}} α_{, i} .

(24d)

Inserting equations (24c) and (24d) into equation (19) leads to

ρ C_{r} \overset{\cdot\cdot}{α} + T_{0} ℜ \nabla \cdot \overset{\cdot}{u} = κ^{*} \nabla^{2} α .

(25)

Considering that $\overset{\cdot}{α} = T, \overset{\cdot\cdot}{α} = \overset{\cdot}{T} = \overset{\cdot}{θ},$ and $\nabla^{2} T = \nabla^{2} θ,$ equation (25) can be rewritten as

ρ C_{r} \overset{\cdot\cdot}{θ} + T_{0} ℜ \nabla \cdot \overset{\cdot\cdot}{u} = κ^{*} \nabla^{2} θ .

(26)

Equation (26) is the heat transport equation without the energy dissipation. When the thermoelastic couple effect is not considered, namely, the thermoelastic coupling coefficient $ℜ = 0$ , the speed of the thermal wave can be derived from equation (26), i.e. $v_{T} = \sqrt{\frac{κ^{*}}{ρ C_{r}}}$ . The heat conduction equation can be derived from equation (24d):

Q = - κ^{*} \nabla α or \overset{\cdot}{Q} = - κ^{*} \nabla T .

(27a,b)

3. Reflection and transmission at the interface

By using

τ^{A} = - \frac{1}{2} I \times (τ_{•}^{\times} I); \nabla \times \nabla \cdot μ^{0} = 0,

(28a,b)

equations (5a) and (5b) can be incorporated into

\nabla \cdot τ^{s} + \frac{1}{2} \nabla \times \nabla \cdot μ^{D} = ρ \overset{\cdot\cdot}{u} .

(29)

Inserting equations (24a,b) into equation (29) leads to the equation of motion in terms of displacement:

(λ + μ + η \nabla^{2}) \nabla \nabla \cdot u + (μ - η \nabla^{2}) \nabla^{2} u - ℜ \nabla θ = ρ \overset{\cdot\cdot}{u} .

(30)

Taking the divergence and curl operation on both sides of equation (30) leads to

V_{p}^{2} \nabla^{2} \nabla \cdot u - (ℜ / ρ) \nabla^{2} θ = \nabla \cdot \overset{\cdot\cdot}{u},

(31a)

V_{s}^{2} (1 - l^{2} \nabla^{2}) \nabla^{2} \nabla \times u = \nabla \times \overset{\cdot\cdot}{u},

(31b)

where $V_{p} = \sqrt{(λ + 2 μ) / ρ}$ and $V_{s} = \sqrt{μ / ρ}$ are the phase speeds of dilatational and distortional waves, respectively, in the classic elastic solids and $l = \sqrt{η / μ}$ is a microstructure parameter in the couple stress solid.

In order to uncouple the vector equation (30), the application of Helmholtz vector representation theorem,

u (x, y) = u_{x} (x, y) e_{x} + u_{y} (x, y) e_{y} = \nabla δ (x, y) + \nabla \times ϑ (x, y) e_{z},

(32)

leads to

V_{p}^{2} \nabla^{4} δ - (ℜ / ρ) \nabla^{2} θ = \nabla^{2} \overset{\cdot\cdot}{δ},

(33a)

V_{s}^{2} (1 - l^{2} \nabla^{2}) \nabla^{2} ϑ = \overset{\cdot\cdot}{ϑ} .

(33b)

Taking equation (26) into consideration, equations (33a,b) can be rewritten as

(\nabla^{2} + σ_{p}^{2}) (\nabla^{2} + ℑ_{p}^{2}) δ = 0,

(34a)

(\nabla^{2} + σ_{s}^{2}) (\nabla^{2} - τ_{s}^{2}) ϑ = 0,

(34b)

where

\begin{matrix} σ_{p}^{2} = \frac{{\hat{ω}}^{2}}{2 V_{p}^{2}} (\frac{ρ C_{r} V_{p}^{2}}{κ^{*}} + \frac{ℜ^{2} T_{0}}{ρ κ^{*}} + 1 + \sqrt{Δ}) (\hat{ω} is frequency), \\ ℑ_{p}^{2} = \frac{{\hat{ω}}^{2}}{2 V_{p}^{2}} (\frac{ρ C_{r} V_{p}^{2}}{κ^{*}} + \frac{ℜ^{2} T_{0}}{ρ κ^{*}} + 1 - \sqrt{Δ}), \\ σ_{s}^{2} = [{(1 + 4 {\hat{ω}}^{2} l^{2} / V_{s}^{2})}^{1 / 2} - 1] / 2 l^{2}, τ_{s}^{2} = [{(1 + 4 {\hat{ω}}^{2} l^{2} / V_{s}^{2})}^{1 / 2} + 1] / 2 l^{2}, and \\ Δ = {(\frac{ρ C_{r} V_{p}^{2}}{κ^{*}} + \frac{ℜ^{2} T_{0}}{ρ κ^{*}} - 1)}^{2} + \frac{4 ℜ^{2} T_{0}}{ρ κ^{*}} . \end{matrix}

Equation (34) means that there are three traveling waves of wavenumber $σ_{p}$ , $ℑ_{P}$ , and $σ_{s}$ and one evanescent wave with attenuation factor $τ_{s}$ . The two bulk waves of wavenumber $σ_{p}$ and $ℑ_{P}$ are thermo-mechanically coupled longitudinal waves and are named as MT1 wave (first sound) and MT2 wave (second sound) hereafter. The bulk wave of wavenumber $σ_{s}$ is a pure mechanical transverse wave (SV wave). The evanescent wave of wavenumber $τ_{s}$ cannot exist in bulk and exists only in the vicinity of interface and reduces to an S-type surface wave (SS wave) when it propagates along the interface.

It is noted that the thermal effects affect only longitudinal waves, while the couple stress affects only the shear wave. The MT1 wave and MT2 wave are non-dispersive, while the SV wave and SS wave are both dispersive. The phase speeds of the four kinds of waves can be expressed as

v_{MT 1}^{g} = \frac{\hat{ω}}{σ_{p}} = \sqrt{2} V_{p} {(\frac{ρ C_{r} V_{p}^{2}}{κ^{*}} + \frac{ℜ^{2} T_{0}}{ρ κ^{*}} + 1 + \sqrt{Δ})}^{- \frac{1}{2}},

(35a)

v_{MT 2}^{g} = \frac{\hat{ω}}{ℑ_{p}} = \sqrt{2} V_{p} {(\frac{ρ C_{r} V_{p}^{2}}{κ^{*}} + \frac{ℜ^{2} T_{0}}{ρ κ^{*}} + 1 - \sqrt{Δ})}^{- \frac{1}{2}},

(35b)

v_{sv}^{g} = \frac{\hat{ω}}{σ_{s}} = V_{s} (1 + l^{2} σ_{s}^{2})^{\frac{1}{2}},

(35c)

v_{ss}^{g} = \frac{\hat{ω}}{τ_{s}} = V_{s} (l^{2} τ_{s}^{2} - 1)^{\frac{1}{2}} .

(35d)

It is noted that larger values of specific heat capacity $C_{r}$ and thermoelastic coupling coefficient ℜ make the speeds of MT1 and MT2 waves decrease, while a larger value of the generalized thermal conduction coefficient $κ^{*}$ will enhance the speed of MT1 and MT2 waves. It is also noted that the speed of the MT1 wave is much greater than that of the MT2 wave. Moreover, the speed of the SV wave is always greater than that of the SS wave at a given frequency. As the frequency $\hat{ω}$ tends to zero, the speed of the SV wave tends to that of the classic elastic material, while the speed of the SS wave tends to zero, namely, the SS wave will disappear. As the frequency $\hat{ω}$ increases gradually, the speeds of the SV wave and SS wave also increase and the deviation between them tends to zero.

In the reflection and transmission problem, the apparent wavenumber of all waves (incident waves, reflection waves, and transmission waves) should be equal. Therefore, applying the Snell theorem leads to (see Figure 1)

\frac{\hat{ω} \sin ϕ_{1}}{v_{MT 1}^{g 1}} = \frac{\hat{ω} \sin ϕ_{2}}{v_{MT 2}^{g 1}} = \frac{\hat{ω} \sin ϕ_{3}}{v_{s}^{g 1}} = \frac{\hat{ω} \sin ϕ_{4}}{v_{MT 1}^{g 2}} = \frac{\hat{ω} \sin ϕ_{5}}{v_{MT 2}^{g 2}} = \frac{\hat{ω} \sin ϕ_{6}}{v_{s}^{g 2}} = ξ,

(36)

where $ν^{gj}$ represents the phase speeds of various waves and $j = 1$ represents medium 1 while $j = 2$ medium 2; $ξ$ is the apparent wavenumber. Let

β_{MT 1}^{2} = σ_{p}^{2} - ξ^{2}, β_{MT 2}^{2} = ℑ_{p}^{2} - ξ^{2}, β_{s}^{2} = σ_{s}^{2} - ξ^{2}, γ_{s}^{2} = τ_{s}^{2} + ξ^{2} .

Then, the potential functions of various waves can be expressed as

\begin{matrix} δ = A_{1} \exp [i (ξ x + β_{MT 1} y - \hat{ω} t)] + A_{2} \exp [i (ξ x - β_{MT 1} y - \hat{ω} t)] \\ + A_{3} \exp [i (ξ x + β_{MT 2} y - \hat{ω} t)] + A_{4} \exp [i (ξ x - β_{MT 2} y - \hat{ω} t)], \end{matrix}

(37a)

\begin{matrix} ϑ = B_{1} \exp [i (ξ x + β_{s} y - \hat{ω} t)] + B_{2} \exp [i (ξ x - β_{s} y - \hat{ω} t)] \\ + D_{1} \exp [- γ_{s} y + i (ξ x - \hat{ω} t)] + D_{2} \exp [γ_{s} y + i (ξ x - \hat{ω} t)] . \end{matrix}

(37b)

Inserting equation (37) into equation (26) leads to

\begin{matrix} θ = n_{1} {A_{1} \exp [i (ξ x + β_{MT 1} y - \hat{ω} t)] + A_{2} \exp [i (ξ x - β_{MT 1} y - \hat{ω} t)]} \\ + n_{2} {A_{3} \exp [i (ξ x + β_{MT 2} y - \hat{ω} t)] + A_{4} \exp [i (ξ x - β_{MT 2} y - \hat{ω} t)]}, \end{matrix}

(38)

where $n_{1} = \frac{{\hat{ω}}^{2} ℜ T_{0} σ_{p}^{2}}{{\hat{ω}}^{2} ρ C_{r} - κ^{*} σ_{p}^{2}}$ and $n_{2} = \frac{{\hat{ω}}^{2} ℜ T_{0} ℑ_{p}^{2}}{{\hat{ω}}^{2} ρ C_{r} - κ^{*} ℑ_{p}^{2}}$ .

Inserting equation (37) into equation (25) leads to

\begin{matrix} α = m_{1} {A_{1} \exp [i (ξ x + β_{MT 1} y - \hat{ω} t)] + A_{2} \exp [i (ξ x - β_{MT 1} y - \hat{ω} t)]} \\ + m_{2} {A_{3} \exp [i (ξ x + β_{MT 2} y - \hat{ω} t)] + A_{4} \exp [i (ξ x - β_{MT 2} y - \hat{ω} t)]}, \end{matrix}

(39)

where $m_{1} = \frac{i \hat{ω} ℜ T_{0} σ_{p}^{2}}{{\hat{ω}}^{2} ρ C_{r} - κ^{*} σ_{p}^{2}}$ and $m_{2} = \frac{i \hat{ω} ℜ T_{0} ℑ_{p}^{2}}{{\hat{ω}}^{2} ρ C_{r} - κ^{*} ℑ_{p}^{2}}$ .

Consider that an incident coupled MT1 wave or SV wave from medium 1 propagates obliquely toward the interface (see Figure 1). When it impinges the interface, reflection waves in medium 1 and transmission waves in medium 2 are created.

Incident waves:

δ^{(0)} = A_{0} \exp [i (ξ x - β_{MT 1}^{(1)} y - \hat{ω} t)],

(40a)

ϑ^{(0)} = B_{0} \exp [i (ξ x - β_{s 1} y - \hat{ω} t)],

(40b)

θ^{(0)} = n_{1}^{(1)} A_{0} \exp [i (ξ x - β_{MT 1}^{(1)} y - \hat{ω} t)],

(40c)

where $n_{1}^{(1)} = \frac{{\hat{ω}}^{2} ℜ_{1} T_{01} σ_{p 1}^{2}}{{\hat{ω}}^{2} ρ_{1} C_{r 1} - κ_{1}^{*} σ_{p 1}^{2}}$ .

Reflection waves:

δ^{(1)} = A_{1} \exp [i (ξ x + β_{MT 1}^{(1)} y - \hat{ω} t)] + A_{3} \exp [i (ξ x + β_{MT 2}^{(1)} y - \hat{ω} t)],

(41a)

ϑ^{(1)} = B_{1} \exp [i (ξ x + β_{s 1} y - \hat{ω} t)] + D_{1} \exp [- γ_{s 1} y + i (ξ x - \hat{ω} t)],

(41b)

θ^{(1)} = n_{1}^{(1)} A_{1} \exp [i (ξ x + β_{MT 1}^{(1)} y - \hat{ω} t)] + n_{2}^{(1)} A_{3} \exp [i (ξ x + β_{MT 2}^{(1)} y - \hat{ω} t)],

(41c)

where $n_{1}^{(1)} = \frac{{\hat{ω}}^{2} ℜ_{1} T_{01} σ_{p 1}^{2}}{{\hat{ω}}^{2} ρ_{1} C_{r 1} - κ_{1}^{*} σ_{p 1}^{2}}$ and $n_{2}^{(1)} = \frac{{\hat{ω}}^{2} ℜ_{1} T_{01} ℑ_{p 1}^{2}}{{\hat{ω}}^{2} ρ_{1} C_{r 1} - κ_{1}^{*} ℑ_{p 1}^{2}}$ .

Transmission waves:

δ^{(2)} = A_{2} \exp [i (ξ x - β_{MT 1}^{(2)} y - \hat{ω} t)] + A_{4} \exp [i (ξ x - β_{MT 1}^{(2)} y - \hat{ω} t)],

(42a)

ϑ^{(2)} = B_{2} \exp [i (ξ x - β_{s 2} y - \hat{ω} t)] + D_{2} \exp [+ γ_{s 2} y + i (ξ x - \hat{ω} t)],

(42b)

θ^{(2)} = n_{1}^{(2)} A_{2} \exp [i (ξ x - β_{MT 1}^{(2)} y - \hat{ω} t)] + n_{2}^{(2)} A_{4} \exp [i (ξ x - β_{MT 2}^{(2)} y - \hat{ω} t)],

(42c)

where $n_{1}^{(2)} = \frac{{\hat{ω}}^{2} ℜ_{2} T_{02} σ_{p 2}^{2}}{{\hat{ω}}^{2} ρ_{2} C_{r 2} - κ_{2}^{*} σ_{p 2}^{2}}$ and $n_{2}^{(2)} = \frac{{\hat{ω}}^{2} ℜ_{2} T_{02} ℑ_{p 2}^{2}}{{\hat{ω}}^{2} ρ_{2} C_{r 2} - κ_{2}^{*} ℑ_{p 2}^{2}}$ .

Let $(A_{1}, A_{3}, B_{1}, D_{1})$ represent the amplitude of reflection waves, ( $A_{2}, A_{4}, B_{2}, D_{2}$ ) represent the amplitude of transmission waves, and $(A_{0}, B_{0})$ represent the amplitude of incident waves, MT1 wave, and SV wave, respectively. $A_{1} / A_{0}$ , $A_{3} / A_{0}$ , $B_{1} / A_{0}$ , and $D_{1} / A_{0}$ are the amplitude ratios of various reflection waves (coupled MT1 wave, coupled MT2 wave, SV wave, and the S-type surface wave) with respect to the incident wave. Similarly, $A_{2} / A_{0}$ , $A_{4} / A_{0}$ , $B_{2} / A_{0}$ , and $D_{2} / A_{0}$ are the amplitude ratios of various transmission waves with respect to the incident wave. These amplitude ratios are determined by the interface conditions between two thermoelastic solids with the couple stress considered. For the thermodynamically ideal interface, eight mechanical and thermal quantities are continuous across the interface. However, if the interface is designed elaborately, some extreme cases are also possible. Here, four kinds of extreme cases are also considered. The explicit expressions for these interface conditions can be written as follows:

The perfect interface

{(u_{i}^{(1)} - u_{i}^{(2)}) |}_{y = 0} = 0, {(p_{i}^{(1)} - p_{i}^{(2)}) |}_{y = 0} = 0, (i = x, y)

(43a,b)

{(ω_{z}^{(1)} - ω_{z}^{(2)}) |}_{y = 0} = 0, {(m_{z}^{(1)} - m_{z}^{(2)}) |}_{y = 0} = 0,

(43c,d)

{(θ^{(1)} - θ^{(2)}) |}_{y = 0} = 0, {(Q_{y}^{(1)} - Q_{y}^{(2)}) |}_{y = 0} = 0,

(43e,f)

where

$p_{x} = τ_{y x}^{s} + \frac{1}{2} (μ_{x z, x}^{D} + μ_{y z, y}^{D}), p_{y} = τ_{y y}^{s} - ℜ θ, ω_{z} = \frac{1}{2} (u_{y, x} - u_{x, y}), m_{z} = μ_{y z}^{D}, Q_{i} = - κ^{*} α_{, i} .$

Interface of heat insulation

{(u_{i}^{(1)} - u_{i}^{(2)}) |}_{y = 0} = 0, {(p_{i}^{(1)} - p_{i}^{(2)}) |}_{y = 0} = 0, (i = x, y)

(44a,b)

{(ω_{z}^{(1)} - ω_{z}^{(2)}) |}_{y = 0} = 0, {(m_{z}^{(1)} - m_{z}^{(2)}) |}_{y = 0} = 0,

(44c,d)

{Q_{y}^{(1)} |}_{y = 0} = {Q_{y}^{(2)} |}_{y = 0} = 0 .

(44e,f)

Interface of constant temperature

{(u_{i}^{(1)} - u_{i}^{(2)}) |}_{y = 0} = 0, {(p_{i}^{(1)} - p_{i}^{(2)}) |}_{y = 0} = 0, (i = x, y)

(45a,b)

{(ω_{z}^{(1)} - ω_{z}^{(2)}) |}_{y = 0} = 0, {(m_{z}^{(1)} - m_{z}^{(2)}) |}_{y = 0} = 0,

(45c,d)

{θ^{(1)} |}_{y = 0} = {θ^{(2)} |}_{y = 0} = 0 .

(45e,f)

Interface without micromoment

{(u_{i}^{(1)} - u_{i}^{(2)}) |}_{y = 0} = 0, {(p_{i}^{(1)} - p_{i}^{(2)}) |}_{y = 0} = 0, (i = x, y)

(46a,b)

{{m_{z}^{(1)} |}_{y = 0} = m_{z}^{(2)} |}_{y = 0} = 0,

(46c,d)

{(θ^{(1)} - θ^{(2)}) |}_{y = 0} = 0, {(Q_{y}^{(1)} - Q_{y}^{(2)}) |}_{y = 0} = 0 .

(46e,f)

Interface without microrotation

{(u_{i}^{(1)} - u_{i}^{(2)}) |}_{y = 0} = 0, {(p_{i}^{(1)} - p_{i}^{(2)}) |}_{y = 0} = 0, (i = x, y)

(47a,b)

{{ω_{z}^{(1)} |}_{y = 0} = ω_{z}^{(2)} |}_{y = 0} = 0,

(47c,d)

{(θ^{(1)} - θ^{(2)}) |}_{y = 0} = 0, {(Q_{y}^{(1)} - Q_{y}^{(2)}) |}_{y = 0} = 0 .

(47e,f)

Equations (43)–(47) can be written in the matrix form

AX = B + C,

(48)

where $X = {[A_{1} / A_{1} A_{0} A_{0}, A_{3} / A_{3} A_{0} A_{0}, B_{1} / B_{1} A_{0} A_{0}, D_{1} / D_{1} A_{0} A_{0}, A_{2} / A_{2} A_{0} A_{0}, A_{4} / A_{4} A_{0} A_{0}, B_{2} / B_{2} A_{0} A_{0}, D_{2} / D_{2} A_{0} A_{0}]}^{T}$ . The explicit expressions for the matrixes $A$ , $B$ , and $C$ are listed in the Appendix A . $B$ is related with the incident coupled MT1 wave and $C$ with the incident SV wave. The reflection and transmission amplitude ratios $X$ can be obtained by solving equation (48).

4. Energy flux and energy flux conservation

Energy flux density of a coupled wave along the propagation direction $n$ can be obtained by

q (n, t) = - p \cdot \overset{\cdot}{u} - m \cdot \overset{\cdot}{ω} + Q (n, t),

(49)

where the first term represents the work ratio done by the force stress and the second term represents the work ratio done by the couple stress. The third term represents the thermal flux along the propagation direction. Due to the time dependence of the energy flux, the average energy flux over one period, i.e. $\bar{q} (n) = \frac{\hat{ω}}{2 π} \int_{0}^{\frac{2 π}{\hat{ω}}} q (n, t) dt$ , is more interesting and can be calculated for various waves.

The transverse waves are independent of the thermal wave. Therefore, the average energy fluxes of the transverse waves can be calculated by

{\bar{q}}_{s}^{(0)} (n_{s}^{(0)}) = \frac{1}{2} \hat{ω} σ_{s 1}^{3} μ_{1} (1 + l_{1}^{2} σ_{s 1}^{2}) B_{0} B_{0}^{*},

(50a)

{\bar{q}}_{s}^{(j)} (n_{s}^{(j)}) = \frac{1}{2} \hat{ω} σ_{sj}^{3} μ_{j} (1 + l_{j}^{2} σ_{sj}^{2}) B_{j} B_{j}^{*},

(50b)

where equation (50a) is the average energy flux of an incident SV wave. Equation (50b) is the average energy flux of reflection waves (for $j = 1$ ) and transmission waves (for $j = 2$ ), respectively.

The dilatational waves are coupled with the thermal wave (second sound). The average energy fluxes of the coupled waves can be calculated by

{\bar{q}}_{MT 1}^{(0)} (n_{MT 1}^{(0)}) = \frac{1}{2} \hat{ω} σ_{p 1}^{3} (λ_{1} + 2 μ_{1}) A_{0} A_{0}^{*} + \hat{ω} σ_{p 1} ℜ_{1} n_{1}^{(1)} A_{0} A_{0}^{*},

(51a)

{\bar{q}}_{MT 1}^{(j)} (n_{MT 1}^{(j)}) = \frac{1}{2} \hat{ω} σ_{pj}^{3} (λ_{j} + 2 μ_{j}) A_{j} A_{j}^{*} + \hat{ω} σ_{pj} ℜ_{j} n_{j}^{(1)} A_{j} A_{j}^{*},

(51b)

{\bar{q}}_{MT 2}^{(j)} (n_{MT 2}^{(j)}) = \frac{1}{2} \hat{ω} ℑ_{pj}^{3} (λ_{j} + 2 μ_{j}) A_{j + 2} A_{j + 2}^{*} + \hat{ω} ℑ_{Tj} ℜ_{j} n_{j}^{(2)} A_{j + 2} A_{j + 2}^{*},

(51c)

where equation (51a) is the average energy flux of an incident MT1 wave. Equation (51b) is the average energy flux of reflection (for $j = 1$ ) and transmission (for $j = 2$ ) MT1 waves. Similarly, equation (51c) is the average energy flux of reflection and transmission MT2 waves.

For the SS surface waves, the energy flux density on the wave front decreases gradually with the increase of $z$ . Here, a unit area on the wavefront is taken to be $l_{x} \times l_{y} = γ_{s} \times 1 / 1 γ_{s} γ_{s}$ near the surface, which is so chosen to make $γ_{s}$ disappear in the expression of energy flux. The average energy flux of the S type surface wave can be calculated by

{\bar{q}}_{ss}^{(j)} (n) = \frac{1}{2} M μ \hat{ω} ξ (3 τ_{sj}^{2} + 4 ξ^{2} + 2 l^{2} τ_{sj}^{4}) (D_{j} D_{j}^{*}),

(52)

where $M = \frac{1 - \exp (- 2)}{2}$ ; $j = 1$ and $j = 2$ stand for the reflection and transmission waves.

Define the reflection and transmission coefficients as the energy flux ratio (along the propagation directions) of various reflection waves and transmission waves with respect to incident waves, namely, ${\bar{q}}_{MT 1}^{(j)} / {\bar{q}}_{MT 1}^{(j)} {\bar{q}}_{MT 1}^{(0)} {\bar{q}}_{MT 1}^{(0)}$ , ${\bar{q}}_{MT 2}^{(j)} / {\bar{q}}_{MT 2}^{(j)} {\bar{q}}_{MT 1}^{(0)} {\bar{q}}_{MT 1}^{(0)}$ , ${\bar{q}}_{s}^{(j)} / {\bar{q}}_{s}^{(j)} {\bar{q}}_{MT 1}^{(0)} {\bar{q}}_{MT 1}^{(0)}$ , and ${\bar{q}}_{ss}^{(j)} / {\bar{q}}_{ss}^{(j)} {\bar{q}}_{MT 1}^{(0)} {\bar{q}}_{MT 1}^{(0)}$ in the case of an incident MT1 wave. In the case of an incident SV wave, the denominator should be replaced by ${\bar{q}}_{s}^{(0)}$ . Energy conservation requires

E = \frac{{\bar{q}}_{MT 1}^{(1)} \cos ϕ_{1} + {\bar{q}}_{MT 2}^{(1)} \cos ϕ_{2} + {\bar{q}}_{s}^{(1)} \cos ϕ_{3} + {\bar{q}}_{MT 1}^{(2)} \cos ϕ_{4} + {\bar{q}}_{MT 2}^{(2)} \cos ϕ_{5} + {\bar{q}}_{s}^{(2)} \cos ϕ_{6}}{{\bar{q}}^{(0)} \cos ϕ} = 1 .

(53)

This means that the energy flux of an incident wave (input energy flux) through a unit area at the interface is equal to the energy flux of reflection and transmission waves (output energy flux) through same area at the interface. Equation (53) is used to validate the numerical results in the next section.

5. Numerical examples and discussions

It is known that the reflection and transmission coefficients are dependent upon the material constants of two couple stress solids $(v_{i}, μ_{i}, ρ_{i}, l_{i}, C_{ri}, κ_{i}^{*}, ℜ_{i}, T_{0 i})$ ( $v_{1}$ and $v_{2}$ are the Poisson ratios) and the parameter of incident wave $(A_{0}, \hat{ω}, ϕ)$ , namely,

X = f (v_{1}, μ_{1}, ρ_{1}, l_{1}, C_{r 1}, κ_{1}^{*}, ℜ_{1}, T_{0 i}, v_{2}, μ_{2}, ρ_{2}, l_{2}, C_{r 2}, κ_{2}^{*}, ℜ_{2}, T_{02}, \hat{ω}, ϕ) .

(54)

Choosing $(ρ_{1}, l_{1}, μ_{1}, T_{01})$ as the basic physical quantities, then the non-dimensional form of equation (54) is

X = f (v_{1}, C_{1}, K_{1}, R_{1}, \bar{v}, \bar{μ}, \bar{ρ}, \bar{l}, {\bar{C}}_{r}, {\bar{κ}}^{*}, \bar{ℜ}, \bar{T}, \bar{ω}, ϕ),

(55)

where $C_{1} = \frac{ρ_{1} T_{01} C_{r 1}}{μ_{1}}$ , $K_{1} = \frac{ρ_{1} T_{01} κ_{1}^{*}}{μ_{1}^{2}}$ , $R_{1} = \frac{T_{01} ℜ_{1}}{μ_{1}}$ , $\bar{l} = \frac{l_{2}}{l_{1}}$ , $\bar{μ} = \frac{μ_{2}}{μ_{1}}$ , $\bar{v} = \frac{v_{2}}{v_{1}}$ , $\bar{ρ} = \frac{ρ_{2}}{ρ_{1}}$ ,

ℜ = \frac{ℜ_{2}}{ℜ_{1}}, {\bar{κ}}^{*} = \frac{κ_{2}^{*}}{κ_{1}^{*}}, {\bar{C}}_{r} = \frac{C_{r 2}}{C_{r 1}}, \bar{T} = \frac{T_{02}}{T_{01}}, \bar{ω} = \frac{{\hat{ω}}^{2} l_{1}^{2}}{V_{s 1}^{2}} .

In the numerical example, the following normalized quantities are given, namely, $C_{1} = 1.4, K_{1} = 1, R_{1} = 2, \bar{μ} = 1 / 3, \bar{v} = 1, \bar{ρ} = 1, \bar{ℜ} = 1, {\bar{κ}}^{*} = 0.1, {\bar{C}}_{r} = 0.5, and \bar{T} = 1 .$

First, let us investigate the influences of the various thermal interfaces on the reflection and transmission coefficients. Figure 2 shows the reflection and transmission coefficients of various waves in the case of an incident MT1 wave. It is found that only the reflection and transmission MT2 waves are sensitive to the thermal interface conditions. Other reflection and transmission waves are not sensitive to the thermal interface conditions. Consider that the mechanical wave dominates in the coupled MT1 wave while the thermal wave (second sound) dominates in the coupled MT2 wave. Why the MT2 wave is sensitive to the thermal interface condition is understandable. Figure 3 shows the reflection and transmission coefficients of various waves in the case of an incident SV wave. A similar phenomenon is observed, namely, only the reflection and transmission MT2 waves are sensitive to the thermal interface conditions.

Apart from the thermal interface conditions, the microstructure interface conditions are still interesting. Figure 4 shows the reflection and transmission coefficients in the case of an incident MT1 wave for different microstructure interfaces. It is found that the reflection and transmission waves of shear type, no matter whether it is a bulk wave or surface wave, are more sensitive to the microstructure interface conditions than the reflection and transmission waves of dilatational type. Especially, the surface waves of shear type are evidently dependent upon the microstructure interface conditions. It is known that the surface wave of shear type does not exist in the classic elastic solids and exists only in the microstructure solid with couple stress considered [31]. Therefore, the microstructure interface conditions concerning the couple stress and microrotation have strong influences upon the surface waves of shear type. Figure 5 shows the reflection and transmission coefficients in the case of an incident SV wave. Different from the case of an incident MT1 wave, the microstructure interface conditions have evident influences on the reflection and transmission waves of all type, namely, the reflection and transmission waves of dilatational type are also sensitive to the microstructure interface conditions.

In the couple stress solid, the microstructure parameter $l$ plays an important role. The couple stress effects disappear as $l \to 0$ . For the reflection and transmission problem at an interface of two different couple stress solids, how does the microstructure parameter ratio $\bar{l} = l_{2} / l_{1}$ influence the reflection and transmission waves? In order to give an insight on the microstructure parameter ratio, the reflection and transmission coefficients in the case of an incident MT1 wave for different microstructure parameter ratio $\bar{l}$ are calculated and shown in Figure 6. It is observed that the reflection and transmission MT1 waves and reflection SV and SS waves increase, while the reflection and transmission MT2 waves and transmission SV and SS waves decrease when the microstructure parameter ratio $\bar{l}$ increases. However, the influences of the microstructure parameter ratio $\bar{l}$ are somewhat different in the case of an incident SV wave. Figure 7 shows the reflection and transmission coefficients in the case of an incident SV wave. It is observed that the reflection/transmission MT1 waves and reflection MT2 wave increase, while the reflection/transmission SS wave and transmission MT2 wave decrease when the microstructure parameter ratio $\bar{l}$ increases. The reflection and transmission SV waves are not sensitive to the microstructure parameter ratio $\bar{l}$ .

The thermoelastic waves due to the thermoelastic coupling effects are also evidently dependent on the thermal parameters of two solids. Figure 8 shows the influences of three thermal parameters on the reflection MT2 wave in the case of an incident MT1 wave. It is observed that the increase of the non-dimensional thermoelastic coupling coefficient $R_{1}$ and the general heat conduction coefficient $K_{1}$ makes the reflection MT2 wave increase, while the increase of non-dimensional specific heat capacity $C_{1}$ makes the reflection MT2 wave decrease. This phenomenon is understandable by considering the explicit expression of the coupling coefficient $n_{2}^{(1)}$ (in equation (41c)) between the temperature field $θ (x, y, t)$ and the dilatational displacement potential $δ (x, y, t)$ . Obviously, the increase of the thermoelastic coupling coefficient $ℜ_{1}$ and the general heat conduction coefficient $κ_{1}^{*}$ makes the coupling coefficient $n_{2}^{(1)}$ of the reflection MT2 wave increase, while the increase of the specific heat capacity $C_{r 1}$ makes the coupling coefficient $n_{2}^{(1)}$ decrease. In fact, the mechanical counterpart of the generalized heat conduction coefficient $κ_{1}^{*}$ is the elastic bulk modulus, while the counterpart of the specific heat capacity $C_{r 1}$ is the mass density. This perception is crucial to understand the propagation behavior of the thermoelastic waves.

In order to validate the numerical results obtained in this section, the energy fluxes carried by various waves are estimated and the energy conservation between the incident waves and the reflection waves as well as the transmission waves is checked. Figure 9 shows the energy conservation index $E$ in equation (53) in the case of incident MT1 wave and SV wave for various thermal interface conditions and the microstructure interface conditions. It is observed that the energy conservation is satisfied approximately with the largest error being less than 5% at total range of incident angle.

6. Conclusions

In the present work, not only the microstructure effects but also the thermal wave (second sound) effects upon the propagation behavior of elastic waves across an interface are considered. Based on the theoretical analysis and the numerical simulation, the following conclusions can be drawn:

The existence of couple stress makes the SV wave not only dispersive but also creates one evanescent wave which does not exist in the classic elastic solid. However, the dilatational wave is not affected by the couple stress.

The thermoelastic coupling effects create two dilatational waves which are coupled. In one of them the mechanical energy dominates while the thermal energy dominates in the other. However, the thermoelastic coupling effect does not affect the shear waves.

The thermoelastic theory used in the present work admits the existence of the “second sound,” namely, the wave-type thermal signal propagates with a finite speed. The first sound and second sound are always coupled together, namely, the displacement field and the temperature field are in proportion to each other with a proportionality coefficient which is dependent upon frequency.

The propagation behavior of the reflection and transmission waves across an interface is strongly dependent upon the interface conditions. Only the reflection and transmission MT2 waves are sensitive to the thermal interface conditions. However, the micromechanical interface conditions affect all types of reflection and transmission waves. In particular, the shear-type waves are very sensitive to the micromechanical interface conditions.

The microstructure characteristic parameter and the thermal parameters of two solids on both sides of the interface also affect the reflection and transmission waves. The MT2 wave is mainly sensitive to the thermal parameter, while the microstructure parameter mainly affects the shear-type waves.

Laser ultrasonic detection is a new non-contact detection technique. The generation laser has a short pulse (from nanoseconds to femtoseconds) and high peak power. The generation mechanism of the ultrasonic wave is either thermoelastic effects or ablation effects. In the thermoelastic regime, the ultrasound is generated by the sudden thermal expansion due to the heating of a tiny surface of the material by the laser pulse. If the laser power is sufficient to heat the surface above the material melting point, some material is evaporated (typically some nanometers) and ultrasound is generated by the recoil effect of the expanding material evaporated. The frequency of the generated ultrasound is partially determined by the frequency of the laser pulses. The present work helps to understand some physical phenomena arising in the laser ultrasonic detection.

Footnotes

Appendix A

The explicit expressions of matrix $A = {(a_{ij})}_{8 \times 8}$ , $B = {(b_{i})}_{8 \times 1}$ , and $C = {(c_{i})}_{8 \times 1}$ in equation (43) for the perfect interface are (only the non-zero components are listed):

a_{11} = a_{12} = ξ, a_{13} = β_{s 1}, a_{14} = γ_{s 1} i, a_{15} = a_{16} = - ξ, a_{17} = β_{s 2}, a_{18} = γ_{s 2} i, a_{21} = β_{MT 1}^{(1)},

a_{22} = β_{MT 2}^{(1)}, a_{23} = a_{24} = - ξ, a_{25} = β_{MT 1}^{(2)}, a_{26} = β_{MT 2}^{(2)}, a_{27} = a_{28} = ξ, a_{31} = - 2 μ_{1} ξ β_{MT 1}^{(1)},

a_{32} = - 2 μ_{1} ξ β_{MT 2}^{(1)}, a_{33} = (ξ^{2} - β_{s 1}^{2}) - l_{1}^{2} σ_{s 1}^{4}, a_{34} = (ξ^{2} + γ_{s 1}^{2}) - l_{1}^{2} σ_{s 1}^{4}, a_{35} = - 2 μ_{2} ξ β_{MT 1}^{(2)},

a_{36} = - 2 μ_{2} ξ β_{MT 2}^{(2)}, a_{37} = - [(ξ^{2} - β_{s 2}^{2}) - l_{2}^{2} σ_{s 2}^{4}], a_{38} = - [(ξ^{2} + γ_{s 2}^{2}) - l_{2}^{2} τ_{s 2}^{4}],

a_{41} = - λ_{1} σ_{p 1}^{2} - 2 μ_{1} {(β_{MT 1}^{(1)})}^{2} - ℜ_{1} n_{1}^{(1)}, a_{42} = - λ_{1} ℑ_{p 1}^{2} - 2 μ_{1} {(β_{MT 2}^{(1)})}^{2} - ℜ_{1} n_{2}^{(1)}, a_{43} = 2 μ_{1} ξ β_{s 1},

a_{44} = 2 μ_{1} ξ γ_{s 1} i, a_{45} = λ_{2} σ_{p 2}^{2} + 2 μ_{2} {(β_{MT 1}^{(2)})}^{2} + ℜ_{2} n_{1}^{(2)}, a_{46} = λ_{2} ℑ_{p 2}^{2} + 2 μ_{2} {(β_{MT 2}^{(2)})}^{2} + ℜ_{2} n_{2}^{(2)},

a_{47} = 2 μ_{2} ξ β_{s 2}, a_{48} = 2 μ_{2} ξ γ_{s 2} i, a_{53} = - σ_{s 1}^{2}, a_{54} = τ_{s 1}^{2}, a_{57} = σ_{s 2}^{2}, a_{58} = - τ_{s 2}^{2},

a_{63} = - η_{1} σ_{s 1}^{2} β_{s 1} i, a_{64} = - η_{1} τ_{s 1}^{2} γ_{s 1}, a_{67} = - η_{2} σ_{s 2}^{2} β_{s 2} i, a_{68} = - η_{2} τ_{s 2}^{2} γ_{s 2}, a_{71} = n_{1}^{(1)},

a_{72} = n_{2}^{(1)}, a_{75} = - n_{1}^{(2)}, a_{76} = - n_{2}^{(2)}, a_{81} = - κ_{1}^{*} i β_{MT 1}^{(1)} m_{1}^{(1)}, a_{82} = - κ_{1}^{*} i β_{MT 2}^{(1)} m_{2}^{(1)},

a_{85} = - κ_{2}^{*} i β_{MT 1}^{(1)} m_{1}^{(2)}, a_{86} = - κ_{2}^{*} i β_{MT 2}^{(2)} m_{2}^{(2)},

b_{11} = - ξ, b_{21} = β_{MT 1}^{(1)}, b_{31} = - 2 μ_{1} ξ β_{MT 1}^{(1)}, b_{41} = λ_{1} σ_{p 1}^{2} + 2 μ_{1} {(β_{MT 1}^{(1)})}^{2} + ℜ_{1} n_{1}^{(1)}, b_{71} = - n_{1}^{(1)},

b_{81} = - κ_{1}^{*} i β_{MT 1}^{(1)} m_{1}^{(1)},

c_{11} = β_{s 1}, c_{21} = ξ, c_{31} = - [(ξ^{2} - β_{s 1}^{2}) - l_{1}^{2} σ_{s 1}^{4}], a_{41} = 2 μ_{1} ξ β_{s 1}, a_{51} = σ_{s 1}^{2}, c_{61} = - η_{1} σ_{s 1}^{2} β_{s 1} i .

For the other interface conditions, only the seventh and eighth or fifth and sixth rows of the matrixes need to be modified.

Funding

This work was supported by Fundamental Research Funds for the Central Universities (grant number FRF-BR-15-026A), the National Natural Science Foundation of China (grant number 10972029), Heilongjiang provincial education department basic business special project (grant number 135109232), and the Heilongjiang Natural Science Fund (grant number B2015019).

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