### Mathematical formulation

Consider the Cartesian coordinate system with the direction of *z*-axis normal to the mid-plane of a composite heat-conducting laminate spanned by *x-* and *y*-axes. Outer surfaces of the laminate are at *z* = ± *d*/2. Thermoelastic Lamb waves propagate in an arbitrary direction *θ*, which is defined in the counterclockwise relative to the *x*-axis.

Constitutive equations for the linear thermoelastic anisotropic materials in the context of generalized thermoelasticity are stress-strain-temperature relations

$$ {\sigma}_{ij}={c}_{ij kl}{\varepsilon}_{kl}-{\beta}_{ij} T $$

(1)

where *β*
_{
ij
} = *c*
_{
ijkl
}
*α*
_{
kl
}, *i, j, k, l* = 1, 2, 3.

Each layer of the composite laminate with an arbitrary orientation in the global coordinate system (*x*, *y*, *z*) is considered as a monoclinic material having *x*-*y* as a plane of symmetry. The stress-strain and temperature relations for monoclinic material therefore take the following matrix form:

$$ \left[\begin{array}{c}\hfill {\sigma}_{xx}\hfill \\ {}\hfill {\sigma}_{yy}\hfill \\ {}\hfill {\sigma}_{zz}\hfill \\ {}\hfill {\sigma}_{yz}\hfill \\ {}\hfill {\sigma}_{xz}\hfill \\ {}\hfill {\sigma}_{xy}\hfill \end{array}\right]=\left[\begin{array}{cccccc}\hfill {c}_{11}\hfill & \hfill {c}_{12}\hfill & \hfill {c}_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {c}_{16}\hfill \\ {}\hfill {c}_{12}\hfill & \hfill {c}_{22}\hfill & \hfill {c}_{23}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {c}_{26}\hfill \\ {}\hfill {c}_{13}\hfill & \hfill {c}_{23}\hfill & \hfill {c}_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {c}_{36}\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {c}_{44}\hfill & \hfill {c}_{45}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {c}_{45}\hfill & \hfill {c}_{55}\hfill & \hfill 0\hfill \\ {}\hfill {c}_{16}\hfill & \hfill {c}_{26}\hfill & \hfill {c}_{36}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {c}_{66}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\varepsilon}_{xx}\hfill \\ {}\hfill {\varepsilon}_{yy}\hfill \\ {}\hfill {\varepsilon}_{zz}\hfill \\ {}\hfill {\varepsilon}_{yz}\hfill \\ {}\hfill {\varepsilon}_{xz}\hfill \\ {}\hfill {\varepsilon}_{xy}\hfill \end{array}\right]-\left[\begin{array}{c}\hfill {\beta}_{xx}\hfill \\ {}\hfill {\beta}_{yy}\hfill \\ {}\hfill {\beta}_{zz}\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill {\beta}_{xy}\hfill \end{array}\right] T $$

(2)

where, since *σ*
_{
ij
}, *ε*
_{
kl
}, *α*
_{
kl
} and *C*
_{
ijkl
} are tensors, and we are conducting analysis in the global (*x*, *y*, *z*) co-ordinates, any orthogonal transformation of the primed to the non-primed co-ordinates, i.e., (*x*′, *y*′, *z*′) to (*x*, *y*, *z*) transforms according to (3).

Generally, when the global coordinate system (*x*, *y*, *z*) does not coincide with the principal material coordinate system (*x*′, *y*′, *z*′) but makes an angle *ϕ* with the *z*-axis, the stiffness matrix *c*
_{
ijkl
} (*c*
_{
ij
}) system can be obtained from the stiffness matrix \( {c}_{ij kl}^{\prime }\ \left({c}_{ij}^{\prime}\right) \) system by using transformations

$$ \begin{array}{l}{c}_{11}={c}_{11}^{\prime }{u}^4+{c}_{22}^{\prime }{v}^4+2\left({c}_{12}^{\prime }+2{c}_{66}^{\prime}\right){u}^2{v}^2,\\ {}{c}_{12}=\left({c}_{11}^{\prime }+{c}_{22}^{\prime }-4{c}_{66}^{\prime}\right){u}^2{v}^2+{c}_{12}^{\prime}\left({u}^4+{v}^4\right),\\ {}{c}_{13}={c}_{13}^{\prime }{u}^2+{c}_{23}^{\prime }{v}^2,{c}_{16}=\left({c}_{11}^{\prime }-{c}_{12}^{\prime }-2{c}_{66}^{\prime}\right){u}^3 v+\left({c}_{12}^{\prime }-{c}_{22}^{\prime }+2{c}_{66}^{\prime}\right){v}^3 u,\\ {}{c}_{22}={c}_{11}^{\prime }{v}^4+{c}_{22}^{\prime }{u}^4+2\left({c}_{12}^{\prime }+2{c}_{66}^{\prime}\right){u}^2{v}^2,{c}_{23}={c}_{23}^{\prime }{u}^2+{c}_{13}^{\prime }{v}^2,\\ {}{c}_{26}=\left({c}_{11}^{\prime }-{c}_{12}^{\prime }-2{c}_{66}^{\prime}\right) u{v}^3+\left({c}_{12}^{\prime }-{c}_{22}^{\prime }+2{c}_{66}^{\prime}\right) v{u}^3,{c}_{33}={c}_{33}^{\prime },\\ {}{c}_{36}=\left({c}_{23}^{\prime }-{c}_{13}^{\prime}\right) u v,{c}_{45}=\left({c}_{44}^{\prime }-{c}_{55}^{\prime}\right) u v,{c}_{44}={c}_{44}^{\prime }{u}^2+{c}_{55}^{\prime }{v}^2,{c}_{55}={c}_{55}^{\prime }{u}^2+{c}_{44}^{\prime }{v}^2,\\ {}{c}_{66}=\left({c}_{11}^{\prime }+{c}_{22}^{\prime }-2{c}_{12}^{\prime }-2{c}_{66}^{\prime}\right){u}^2{v}^2+{c}_{16}^{\prime}\left({u}^4+{v}^4\right),{\alpha}_{12}=\left({\alpha}_{33}^{\prime }-{\alpha}_{11}^{\prime}\right) u v,\\ {}{\beta}_{xx}={\beta}_{xx}^{\prime }{u}^2+{\beta}_{zz}^{\prime }{v}^2,{\beta}_{yy}={\beta}_{xx}^{\prime }{v}^2+{\beta}_{zz}^{\prime }{u}^2,{\beta}_{xy}=\left({\beta}_{zz}^{\prime }-{\beta}_{xx}^{\prime}\right) u v,{\beta}_{zz}={\beta}_{zz}^{\prime}\end{array} $$

(3)

where *u* = cos *φ* and *v* = sin *φ*.

The linear strain-displacement relations are

$$ \begin{array}{l}\ {\varepsilon}_{xy}=\frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v\ }{\partial x}\right),{\varepsilon}_{yz}=\frac{1}{2}\left(\frac{\partial v\ }{\partial z}+\frac{\partial w\ }{\partial y}\right),\\ {}{\varepsilon}_{zx}=\frac{1}{2}\left(\frac{\partial u\ }{\partial z}+\frac{\partial w\ }{\partial x}\right),{\varepsilon}_{xx}=\frac{\partial u}{\partial x},\\ {}{\varepsilon}_{yy}=\frac{\partial v\ }{\partial y},{\varepsilon}_{zz}=\frac{\partial w\ }{\partial z}\end{array} $$

(4)

The summation convention is implied; *u*, *v* and *w* are the displacements in the *x*, *y*, and *z* directions and *σ*
_{
ij
} and *ε*
_{
ij
} are the stress and strain tensors, respectively; *β*
_{
ij
} is the thermal modulus; *α*
_{
ij
} is the thermal expansion tensor; *T* is the temperature; and the fourth-order tensor of the elasticity *C*
_{
ijkl
} satisfies the (Green) symmetry conditions:

*c*
_{
ijkl
} = *c*
_{
klij
} = *c*
_{
ijlk
} = *c*
_{
jikl
}, *ε*
_{
ij
} = *ε*
_{
ji
} and *α*
_{
ij
} = *α*
_{
ji
}, *β*
_{
ij
} = *β*
_{
ji
}, *K*
_{
ij
} = *K*
_{
ji
}

The equations of motion and energy are given by

$$ \left[{c}_{i j kl}\frac{\partial^2{u}_k}{\partial {x}_j\partial {x}_l}-{\beta}_{i j}\frac{\partial T}{\partial {x}_j}\right]=\rho \frac{\partial^2{u}_i}{\partial {t}^2} $$

(5)

$$ {K}_{i j}\frac{\partial^2 T}{\partial {x}_i\partial {x}_j}-\rho {C}_e\left(\frac{\partial T}{\partial t}+{\tau}_0\frac{\partial^2 T}{\partial^2 t}\right)={T}_0{\beta}_{i j}\left(\frac{\partial }{\partial t}+{\tau}_0\frac{\partial^2}{\partial^2 t}\right)\frac{\partial {u}_i}{\partial {x}_j} $$

(6)

where *ρ* is the density, *t* is the time, *K*
_{
ij
} are the thermal conductivities, *C*
_{
e
} and τ_{0} are respectively the specific heat at constant strain, and thermal relaxation time.

### Analysis

For transient waves propagating in the *x-y* plane of the plate in a direction of angle *ϕ* with respect to the *x* axis, the displacements have the form

$$ \left({u}_1,{u}_2,{u}_3, T\right)=\left( U(z), V(z), W(z),\varTheta (z)\right){e}^{i\left[\left({\ell}_x\ x+{\ell}_y y\right)-\omega t\right]}\times i=\sqrt{-1} $$

(7)

where **k** = [*ℓ*
_{
x
}, *ℓ*
_{
y
}]^{T} is the wave vector, and its magnitude,

$$ \left|\mathbf{k}\right|= k=\sqrt{{\ell^2}_x+{\ell^2}_y}=\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{${c}_p$}\right., $$

(8)

is the wave number, *ω* is the angular frequency, and *c*
_{
p
} is the phase velocity. Here, **k** points in the direction of propagation. Substituting Equation 7 into Equations 2
*via*
4, we obtain

$$ \begin{array}{l}{\sigma}_{x x}=\left[{c}_{11}{\ell}_x U+{c}_{12}{\ell}_y V- i{c}_{13}{W}^{\prime }+{c}_{16}\left({\ell}_y U+{\ell}_x V\right)+ i{\beta}_{x x} T\right] E\\ {}{\sigma}_{y y}=\left[{c}_{12}{\ell}_x U+{c}_{22}{\ell}_y V- i{c}_{23}{W}^{\prime }+{c}_{26}\left({\ell}_y U+{\ell}_x V\right)+ i{\ell}_y T\right] E\\ {}{\sigma}_{z z}=\left[{c}_{13}{\ell}_x U+{c}_{23}{\ell}_y V- i{c}_{33}{W}^{\prime }+{c}_{36}\left({\ell}_y U+{\ell}_x V\right)+ i{\beta}_z T\right] E\end{array} $$

$$ \begin{array}{l}{\sigma}_{y z}=\left[{c}_{44}\left({V}^{\prime }+ i{\ell}_y W\right)+{c}_{45}\left({U}^{\prime }+ i{\ell}_x W\right)\right] E\\ {}{\sigma}_{x z}=\left[{c}_{45}\left({V}^{\prime }+ i{\ell}_y W\right)+{c}_{55}\left({U}^{\prime }+ i{\ell}_x W\right)\right] E\\ {}{\sigma}_{x y}=\Big[{c}_{16}{\ell}_x U+{c}_{26}{\ell}_y V- i{c}_{36}{W}^{\prime}\\ {}\kern1.75em +{c}_{66}\left({\ell}_y U+{\ell}_x V\right)+ i{\beta}_{x y} T\Big] E\end{array} $$

(9a)

where \( E= i{e}^{i\left[\left({\ell}_x\ x+{\ell}_y y\right)-\omega t\right]} \)

The thermal gradient

$$ \frac{\partial T}{\partial z}={\varTheta}^{\prime }(z){e}^{i\left[\left({\ell}_x\ x+{\ell}_y y\right)-\omega t\right]} $$

(9b)

where the prime indicates the derivative with respect to *z*. Substituting Equations 9 into the equation of motion and heat conduction Equations 5 and 6 for monoclinic material, gives

$$ \begin{array}{l}-{c}_{55}{U}^{{\prime\prime} }-{c}_{45}{V}^{{\prime\prime} }+\left({c}_{11}{\ell}_x^2+2{c}_{16}{\ell}_x{\ell}_y+{c}_{66}{\ell}_y^2-\rho {\omega}^2\right) U\\ {}+\left[{c}_{16}{\ell}_x^2+\left({c}_{12}+{c}_{66}\right){\ell}_x{\ell}_y+{c}_{26}{\ell}_y^2\right] V\\ {}- i\left[\left({c}_{13}+{c}_{55}\right){\ell}_x+\left({c}_{36}+{c}_{45}\right){\ell}_y\right)\Big]{W}^{\prime}\\ {}- i\left[{\beta}_x{\ell}_x+{\beta}_{x y}{\ell}_y\right]\Theta =0\end{array} $$

(10a)

$$ \begin{array}{l}-{c}_{45}{U}^{{\prime\prime} }-{c}_{44}{V}^{{\prime\prime} }+\left[{c}_{16}{\ell}_x^2+\left({c}_{12}+{c}_{66}\right){\ell}_x{\ell}_y+{c}_{26}{\ell}_y^2\right] U\\ {}+\left({c}_{66}{\ell}_x^2+2{c}_{26}{\ell}_x{\ell}_y+{c}_{22}{\ell}_y^2-\rho {\omega}^2\right) V\\ {}- i\left[\left({c}_{36}+{c}_{45}\right){\ell}_x+\left({c}_{23}+{c}_{44}\right){\ell}_y\right]{W}^{\prime}\\ {}- i\left({\beta}_{x y}{\ell}_x+{\beta}_y{\ell}_y\right]\Theta =0\end{array} $$

(10b)

$$ \begin{array}{l}- i\left[\left({c}_{13}+{c}_{55}\right){\ell}_x+\left({c}_{36}+{c}_{45}\right){\ell}_y\right]{U}^{\prime}\\ {}- i\left[\left({c}_{36}+{c}_{45}\right){\ell}_x+\left({c}_{23}+{c}_{44}\right){\ell}_y\right]{V}^{\prime }-{c}_{33}{W}^{{\prime\prime}}\\ {}+\left[{c}_{55}{\ell}_x^2+2{c}_{45}{\ell}_x{\ell}_y+{c}_{44}{\ell}_y^2-\rho {\omega}^2\right] W-{\beta}_z{\Theta}^{\prime }=0\\ {}-{k}_{33}{\Theta}^{{\prime\prime} }+\left[{k}_{11}{\ell}_x^2+2{k}_{12}{\ell}_x{\ell}_y+{c}_{22}{\ell}_y^2-\rho {c}_p^2{C}_e\tau \right)\Theta \\ {}-{T}_0 ik{c}_p^2\tau \Big\{\left({\beta}_x{\ell}_x+{\beta}_{x y}{\ell}_y\right) U\\ {}-\left({\beta}_{x y}{\ell}_x+{\beta}_y{\ell}_y\right) V+ i{\beta}_z{W}^{\prime }=0\end{array} $$

(10c)

### Waves in a composite lamina

In an off-axis lamina, the solutions of Equations 10 can be simply separated into symmetric and antisymmetric wave modes, which make the analytical representation particularly simple:

$$ \begin{array}{l}\left({U}_s,{V}_s,{W}_s,{\varTheta}_s\right)=\left({A}_s \cos \alpha z,{B}_s \cos \alpha z,{C}_s \sin \alpha z,{D}_s \cos \alpha z\ \right)\\ {}\left({U}_a,{V}_a,{W}_a,{\varTheta}_a\right)=\left({A}_a \sin \alpha z,{B}_a \sin \alpha z,{C}_a \cos \alpha z,{D}_a \sin \alpha z\right)\end{array} $$

(11a, b)

Substituting these expressions into the equation of motion and heat conduction equation is an unknown variable *α* to be determined later; moreover, the subscripts *s* and *a* represent symmetric and antisymmetric modes, respectively. First, substituting the symmetric mode into the equations of motion, Equation 10 may be expressed in a matrix form

$$ \left[\begin{array}{cccc}\hfill {\varGamma}_{11}-\rho {c}_p^2\hfill & \hfill {\varGamma}_{12}\hfill & \hfill {\varGamma}_{13}\hfill & \hfill {\varGamma}_{14}\hfill \\ {}\hfill {\varGamma}_{12}\hfill & \hfill {\varGamma}_{22}-\rho {c}_p^2\hfill & \hfill {\varGamma}_{23}\hfill & \hfill {\varGamma}_{24}\hfill \\ {}\hfill {\overline{\varGamma}}_{13}\hfill & \hfill {\overline{\varGamma}}_{23}\hfill & \hfill {\varGamma}_{33}-\rho {c}_p^2\hfill & \hfill {\varGamma}_{34}\hfill \\ {}\hfill {\overline{\varGamma}}_{41}\hfill & \hfill {\overline{\varGamma}}_{42}\hfill & \hfill {\varGamma}_{43}\hfill & \hfill {\varGamma}_{44}-\frac{\rho {\omega}^2{C}_e{T}_0\tau}{T_0{\omega}^2\tau}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {A}_s\hfill \\ {}\hfill {B}_s\hfill \\ {}\hfill {C}_s\hfill \\ {}\hfill {D}_s\hfill \end{array}\right]=0 $$

(12)

where the bar indicates complex conjugate. The elements in the above matrix defined by (**Γ** − *ρω*
^{2}
**Ι**) are as follows:

$$ \begin{array}{l}{\varGamma}_{11}={c}_{11}{\ell}_x^2+2{c}_{16}{\ell}_x{\ell}_y+{c}_{66}{\ell}_y^2+{c}_{55}{\alpha}^2,\\ {}{\varGamma}_{12}={c}_{16}{\ell}_x^2+\left({c}_{12}+{c}_{66}\right){\ell}_x{\ell}_y+{c}_{26}{\ell}_y^2+{c}_{45}{\alpha}^2,\\ {}{\varGamma}_{14}=- i\left[{\beta}_{x x}{\ell}_x+{\beta}_{x y}{\ell}_y\right],\\ {}{\varGamma}_{22}={c}_{66}{\ell}_x^2+2{c}_{26}{\ell}_x{\ell}_y+{c}_{22}{\ell}_y^2+{c}_{44}{\alpha}^2,\\ {}{\varGamma}_{23}=- i\left[\left({c}_{36}+{c}_{45}\right){l}_x+\left({c}_{23}+{c}_{44}\right){l}_y\right]\alpha, \\ {}{\varGamma}_{24}=- i\left[{\beta}_{x y}{\ell}_x+{\beta}_{y y}{\ell}_y\right],\\ {}{\varGamma}_{31}= i\left[\left({c}_{13}+{c}_{55}\right){\ell}_x+\left({c}_{36}+{c}_{45}\right){\ell}_y\right]\alpha ={\overline{\varGamma}}_{13},\\ {}{\varGamma}_{32}= i\left[\left({c}_{36}+{c}_{45}\right){\ell}_x+\left({c}_{23}+{c}_{44}\right){\ell}_y\right]\alpha ={\overline{\varGamma}}_{23},\\ {}{\varGamma}_{33}={c}_{55}{\ell}_x^2+2{c}_{45}{\ell}_x{\ell}_y+{c}_{44}{\ell}_y^2+{c}_{33}{\alpha}^2,\\ {}{\varGamma}_{34}={\beta}_z\alpha, \\ {}{\varGamma}_{41}= i\left({\beta}_x{\ell}_x+{\beta}_{x y}{\ell}_y\right)={\overline{\varGamma}}_{14},{\varGamma}_{41}= i\left({\beta}_{x x}{\ell}_x+{\beta}_{x y}{\ell}_y\right)={\overline{\varGamma}}_{14},\\ {}{\varGamma}_{42}= i\left({\beta}_{x y}{\ell}_x+{\beta}_{y y}{\ell}_y\right)={\overline{\varGamma}}_{24},\\ {}{\varGamma}_{43}={\beta}_{z z}\alpha ={\varGamma}_{34},\\ {}{\varGamma}_{44}=-\left[{k}_{11}{\ell}_x^2+2{k}_{12}{\ell}_x{\ell}_y+{k}_{22}{\ell}_y^2+{k}_{33}{\alpha}^2\right]/{T}_0{\omega}^2\tau, \tau ={\tau}_0+\raisebox{1ex}{$ i$}\!\left/ \!\raisebox{-1ex}{$\omega $}\right.\kern0.5em \end{array} $$

(13)

For nontrivial solutions of *A*
_{
s
}, *B*
_{
s
}, *C*
_{
s
} and *D*
_{
s
} in Equation 12, the determinant of the 4 × 4 matrix vanishes, which leads to the following eight-degree polynomial in terms of *α*:

$$ \begin{array}{l}{P}_0{\alpha}^8+\left({P}_1+\varepsilon {C}_0\right){\alpha}^6+\left({P}_2+\varepsilon {C}_1\right){\alpha}^4\\ {}+\left({P}_3+\varepsilon {C}_2\right){\alpha}^2+\left({P}_4+\varepsilon {C}_3\right)=0\end{array} $$

(14)

Since Equation 14 is a biquadratic equation in *α*
^{2}, it has four roots for *α*
^{2} (*α*
_{
j,} *j* = 1, 2, 3, 4).

Hence, the eight roots for *α* can be arranged in four pairs as

$$ {\alpha}_{j+1}=-{\alpha}_j,\kern0.75em \left( j=1,3,5,7\right) $$

(15)

In Equation 14, *ε* is a coupling constant, if coupling constant is zero, then Equation 14 reduces to

$$ {P}_0{\upalpha}^8+{P}_1{\upalpha}^6+{P}_2{\upalpha}^4+{P}_3{\upalpha}^2+{P}_4=0, $$

$$ {P}_0=\left({c}_{45}^2{c}_{33}-{c}_{33}{c}_{44}{c}_{55}\right){k}_{33} $$

$$ \begin{array}{l}{P}_1=\left({c}_{33}{c}_{44}{c}_{55}-{c}_{45}^2{c}_{33}\right){F}_4\\ {}\kern1em +\left\{\begin{array}{l}\left({c}_{45}^2-{c}_{44}{c}_{55}\right){F}_3-{c}_{33}{c}_{55}{F}_2-{c}_{33}{c}_{44}{F}_1+\left({c}_{55}{F}_{23}-{c}_{45}{F}_{13}\right){F}_{32}\\ {}+\left({c}_{44}{F}_{13}-{c}_{45}{F}_{23}\right){F}_{31}+2{c}_{33}{c}_{45}{F}_{12}\end{array}\right\}{k}_{33}\kern1.25em \end{array} $$

$$ \begin{array}{l}{P}_2=\left\{\begin{array}{l}\left({c}_{44}{c}_{55}-{c}_{45}^2\right){F}_3+{c}_{33}{c}_{55}{F}_2+{c}_{33}{c}_{44}{F}_1-2{c}_{33}{c}_{45}{F}_{12}\\ {}-\left({c}_{44}{F}_{13}-{c}_{45}{F}_{23}\right){F}_{31}-\left({c}_{55}{F}_{23}-{c}_{45}{F}_{13}\right){F}_{32}\end{array}\right\}{F}_4\\ {}\kern1.25em +\left\{\begin{array}{l}\left({F}_{32}{F}_{23}-{c}_{44}{F}_3-{c}_{33}{F}_2\right){F}_1+\left({F}_{31}{F}_{13}-{c}_{55}{F}_3\right){F}_2\\ {}+2{c}_{45}{F}_{12}{F}_3-\left({F}_{13}{F}_{32}+{F}_{23}{F}_{31}\right){F}_{12}+{c}_{33}{F}_{12}^2\end{array}\right\}{k}_{33}\end{array} $$

$$ \begin{array}{l}{P}_3=\left\{\begin{array}{l}\left({c}_{55}{F}_2+{c}_{44}{F}_1-2{c}_{45}{F}_{12}\right){F}_3+\left({c}_{33}{F}_1-{F}_{31}{F}_{13}\right){F}_2\\ {}+\left({F}_{13}{F}_{12}-{F}_1{F}_{23}\right){F}_{32}+{F}_{12}{F}_{23}{F}_{31}-{c}_{33}{F}_{12}^2\end{array}\right\}{F}_4\\ {}\kern1.75em -\left({F}_1{F}_2-{F}_{12}^2\right){F}_3{k}_{33}\end{array} $$

$$ {P}_4=\left({F}_1{F}_2-{F}_{12}^2\right){F}_3{F}_4 $$

$$ {C}_0=\left({c}_{45}^2-{c}_{44}{c}_{55}\right){F}_{34}{F}_{43} $$

$$ \begin{array}{l}{C}_1=\left({c}_{33}{c}_{55}{F}_{42}-{c}_{33}{c}_{45}{F}_{41}+{c}_{45}{F}_{31}{F}_{43}-{c}_{55}{F}_{32}{F}_{43}\right)\left(-{F}_{24}\right)\\ {}\kern2em +\left(2{c}_{45}{F}_{12}-{c}_{55}{F}_2-{c}_{44}{F}_1\right){F}_{34}{F}_{43}+\left({c}_{44}{F}_{14}{F}_{31}-{c}_{45}{F}_{14}{F}_{32}\right){F}_{43}\\ {}\kern1.75em +\left({c}_{44}{F}_{13}-{c}_{45}{F}_{23}\right){F}_{34}{F}_{41}-{c}_{33}{c}_{44}{F}_{14}{F}_{41}\\ {}\kern1.75em +\left({c}_{55}{F}_{23}-{c}_{45}{F}_{13}\right){F}_{34}{F}_{42}+{c}_{33}{c}_{45}{F}_{14}{F}_{42}\end{array} $$

$$ \begin{array}{l}{C}_2=\left(\begin{array}{l}\left({c}_{33}{F}_{42}-{F}_{43}{F}_{32}\right){F}_1+\left({c}_{55}{F}_{42}-{c}_{45}{F}_{41}\right){F}_3\\ {}+\left({F}_{13}{F}_{32}-{c}_{33}{F}_{12}\right){F}_{41}+\left({F}_{12}{F}_{43}-{F}_{13}{F}_{42}\right){F}_{31}\end{array}\right)\left(-{F}_{24}\right)\\ {}\kern1.75em +\left({F}_{23}{F}_{42}-{F}_2{F}_{43}\right){F}_{34}{F}_1\\ {}\kern1.75em +\left({F}_{31}{F}_{14}{F}_{43}+{F}_{13}{F}_{41}{F}_{34}-{c}_{33}{F}_{14}{F}_{41}\right){F}_2\\ {}\kern1.75em +\left({F}_{23}{F}_{32}-{c}_{44}{F}_3\right){F}_{14}{F}_{41}\\ {}\kern1.5em +\left({F}_{12}{F}_{34}-{F}_{14}{F}_{32}\right){F}_{43}{F}_{12}\\ {}\kern1.75em -\left({F}_{13}{F}_{42}+{F}_{23}{F}_{41}\right){F}_{12}{F}_{34}\\ {}\kern1.5em +\left({c}_{45}{F}_3+{c}_{33}{F}_{12}-{F}_{31}{F}_{23}\right){F}_{14}{F}_{42}\end{array} $$

$$ {C}_3={F}_3\left({F}_{41}{F}_{12}-{F}_1{F}_{42}\right){F}_{24}-{F}_{14}{F}_3\left({F}_{41}{F}_2-{F}_{42}{F}_{12}\right) $$

$$ \begin{array}{l}{F}_{12}={c}_{16}{\ell}_x^2+\left({c}_{12}+{c}_{66}\right){\ell}_x{\ell}_y+{c}_{26}{\ell}_y^2,\\ {}{F}_{13}=- i\left(\left({c}_{13}+{c}_{55}\right){\ell}_x+\left({c}_{36}+{c}_{45}\right){\ell}_y\right)\\ {}{F}_{14}=- i\left({\beta}_{x x}{\ell}_x+{\beta}_{x y}{\ell}_y\right),\\ {}{F}_{24}=- i\left({\beta}_{x y}{\ell}_x+{\beta}_{y y}{\ell}_y\right)\\ {}{F}_{23}=- i\left(\left({c}_{36}+{c}_{45}\right){\ell}_x+\left({c}_{23}+{c}_{44}\right){\ell}_y\right)\end{array} $$

$$ \begin{array}{l}{F}_1={c}_{11}{\ell}_x^2+2{c}_{16}{\ell}_x{\ell}_y+{c}_{66}{\ell}_y^2,\\ {}{F}_2={c}_{66}{\ell}_x^2+2{c}_{26}{\ell}_x{\ell}_y+{c}_{22}{\ell}_y^2\\ {}{F}_3={c}_{55}{\ell}_x^2+2{c}_{45}{c}_{16}{\ell}_x{\ell}_y+{c}_{44}{\ell}_y^2,\\ {}{F}_4=-\left({K}_{11}{\ell}_x^2+2{K}_{12}{\ell}_x{\ell}_y+{K}_{22}{\ell}_y^2\right)\end{array} $$

$$ \begin{array}{l}{F}_{21}={c}_{16}{\ell}_x^2+\left({c}_{12}+{c}_{66}\right){\ell}_x{\ell}_y+{c}_{26}{\ell}_y^2={F}_{12},\\ {}{F}_{31}= i\left(\left({c}_{13}+{c}_{55}\right){\ell}_x+\left({c}_{36}+{c}_{45}\right){\ell}_y\right)\\ {}{F}_{32}= i\left(\left({c}_{36}+{c}_{45}\right){\ell}_x+\left({c}_{23}+{c}_{44}\right){\ell}_y\right),\kern0.5em \\ {}{F}_{11}= i\left({\beta}_{x x}{\ell}_x+{\beta}_{x y}{\ell}_y\right),\\ {}{F}_{42}= i\left({\beta}_{x y}{\ell}_x+{\beta}_{y y}{\ell}_y\right),{F}_{34}={F}_{43}={\beta}_{zz}.\end{array} $$

Here *ε* is a coupling parameter, and for each *ξ*
_{
j
}, in symmetric modes, *A*
_{
s
}, *B*
_{
s
}, *C*
_{
s
}, and *D*
_{
s
} can be expressed in terms of *A*
_{
s
}
*via* Equation 12 as

$$ {B}_s={\Delta}_1/\Delta = R{A}_s,{C}_s={\Delta}_2/\Delta = iS{A}_s,{D}_s={\Delta}_3/\Delta =\Omega {A}_s $$

(16)

Where

$$ \begin{array}{l}\Delta =\left({c}_{33}{c}_{44}{F}_{14}-{c}_{33}{c}_{45}{F}_{24}+\left({c}_{45}{F}_{23}-{c}_{44}{F}_{13}\right){F}_{34}\right){\alpha}^4\\ {}\kern1.75em +\Big(\left({c}_{33}{F}_2+{c}_{44}{F}_3-{F}_{23}{F}_{32}\right){F}_{14}+\left({F}_{12}{F}_{23}-{F}_2{F}_{13}\right){F}_{34}\\ {}\kern1.25em +\left({F}_{13}{F}_{32}-{c}_{45}{F}_3-{c}_{33}{F}_{12}\right){F}_{24}\Big){\alpha}^2+\left({F}_{14}{F}_2-{F}_{24}{F}_{12}\right){F}_3\end{array} $$

$$ \begin{array}{l}{\Delta}_1=\left({c}_{33}{c}_{55}{F}_{24}-{c}_{33}{c}_{45}{F}_{14}+\left({c}_{45}{F}_{13}-{c}_{55}{F}_{23}\right){F}_{34}\right){\alpha}^4\\ {}\kern1.75em +\Big(\left({c}_{33}{F}_1+{c}_{55}{F}_3-{F}_{13}{F}_{31}\right){F}_{24}+\left({F}_{12}{F}_{13}-{F}_1{F}_{23}\right){F}_{34}\\ {}\kern1.5em +\left({F}_{13}{F}_{23}-{c}_{45}{F}_3-{c}_{33}{F}_{12}\right){F}_{14}\Big){\alpha}^2+\left({F}_{24}{F}_1-{F}_{14}{F}_{12}\right){F}_3\end{array} $$

$$ \begin{array}{l}{\Delta}_2=\alpha \left\{\begin{array}{l}\left({c}_{44}{c}_{55}-{c}_{45}^2\right){F}_{34}{\alpha}^4\\ {}+\left(\begin{array}{l}\left({c}_{45}{F}_{14}-{c}_{55}{F}_{24}\right){F}_{32}+\left({c}_{45}{F}_{24}-{c}_{44}{F}_{14}\right){F}_{31}\\ {}+\left({c}_{44}{F}_1+{c}_{55}{F}_2-2{c}_{45}{F}_{12}\right){F}_{34}\end{array}\right){\alpha}^2\end{array}\right\}\\ {}\kern1.75em +\left({F}_{12}{F}_{24}-{F}_2{F}_{14}\right){F}_{31}+\left({F}_{12}{F}_{14}-{F}_1{F}_{24}\right){F}_{32}\\ {}\kern1.75em +\left({F}_1{F}_2-{F}_{{}_{12}}^2\right){F}_{34}\end{array} $$

$$ \begin{array}{l}{\Delta}_3=\left({c}_{45}^2{c}_{33}-{c}_{33}{c}_{44}{c}_{55}\right){\alpha}^6\\ {}\kern1.25em +\left(\begin{array}{l}\left({c}_{45}^2-{c}_{44}{c}_{55}\right){F}_3+\left({c}_{44}{F}_{13}-{c}_{45}{F}_{23}\right){F}_{31}\\ {}+\left({c}_{55}{F}_{23}-{c}_{45}{F}_{13}\right){F}_{32}+{c}_{33}\left(2{F}_{12}{c}_{45}-{c}_{55}{F}_2-{c}_{44}{F}_1\right)\end{array}\right){\alpha}^4\\ {}\kern1.9em +\Big(\left({c}_{33}{F}_{12}-{F}_{23}{F}_{31}+2{c}_{45}{F}_3-{F}_{13}{F}_{32}\right){F}_{12}+{F}_2{F}_{13}{F}_{31}\\ {}\kern1.5em +{F}_1{F}_{23}{F}_{32}-{c}_{33}{F}_1{F}_2-{c}_{55}{F}_2{F}_3-{c}_{44}{F}_1{F}_3\Big){\alpha}^2+{F}_{{}_{12}}^2{F}_3-{F}_1{F}_2{F}_3\end{array} $$

and similarly for antisymmetric modes, *B*
_{
a
} = *RA*
_{
a
}, *C*
_{
a
} = − *iSA*
_{
a
} and *D*
_{
a
} = *ΩA*
_{
a
}. With the above equations, the polarization displacement vectors and temperature are determined from the three roots. Consequently, the general solution of Equation 11 is

$$ \begin{array}{l}\left({U}_s,{V}_s,{W}_s,{\varTheta}_s\right)={\displaystyle \sum_{j=1}^4{A}_{s j}\left\{ \cos {\alpha}_j z,{R}_j \cos {\alpha}_j z, i{S}_j \sin {\alpha}_j z,{\Omega}_j \cos {\alpha}_j z\right\}}\\ {}\left({U}_a,{V}_a,{W}_a,{\varTheta}_a\right)={\displaystyle \sum_{j=1}^4{A}_{a j}\left\{ \sin {\alpha}_j z,{R}_j \sin {\alpha}_j z,- i{S}_j \cos {\alpha}_j z,{\Omega}_j \sin {\alpha}_j z\right\}}\end{array} $$

(17a,b)

Substituting Equation 17 into Equation 9, the expression of \( {\sigma}_{zz},{\sigma}_{yz},{\sigma}_{xz},\mathrm{and}\ \frac{\partial T}{\partial \mathrm{z}} \) with traction and thermal gradient-free boundary conditions on the top and bottom surfaces \( z=\pm \raisebox{1ex}{$ h$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \), then \( {\sigma}_{zz},{\sigma}_{yz},{\sigma}_{xz},\mathrm{and}\ \frac{\partial T}{\partial \mathrm{z}} \) may be expressed for the symmetric and antisymmetric modes respectively as

$$ \begin{array}{l}{\left.\left({\sigma}_{z z},{\sigma}_{yz},{\sigma}_{xz},\frac{\partial T}{\partial \mathrm{z}}\right)\right|}_{z= h/2}={\displaystyle \sum_{j=1}^4\left\{\begin{array}{l}{N}_{1 j}\left(\begin{array}{c}\hfill \cos {\alpha}_j z\hfill \\ {}\hfill \sin {\alpha}_j z\hfill \end{array}\right),{N}_{2 j}\left(\begin{array}{c}\hfill \sin {\alpha}_j z\hfill \\ {}\hfill \cos {\alpha}_j z\hfill \end{array}\right),\\ {}{N}_{3 j}\left(\begin{array}{c}\hfill \sin {\alpha}_j z\hfill \\ {}\hfill \cos {\alpha}_j z\hfill \end{array}\right),{N}_{4 j}\left(\begin{array}{c}\hfill \sin {\alpha}_j z\hfill \\ {}\hfill \cos {\alpha}_j z\hfill \end{array}\right)\end{array}\right\}{A}_j}\\ {}\end{array} $$

(18)

$$ {\mathrm{N}}_{1 j}={c}_{13}{\ell}_x+{c}_{23}{\ell}_y{R}_j+{c}_{33}{\alpha}_j{S}_j+{c}_{36}\left({\ell}_y+{\ell}_x{R}_j\right)+ i{\beta}_z{\Omega}_j $$

$$ {\mathrm{N}}_{2 j}={c}_{44}\left({\alpha}_j{R}_j+{\ell}_y{S}_j\right)+{c}_{45}\left({\alpha}_j+ i{\ell}_x{S}_j\right) $$

$$ {\mathrm{N}}_{3 j}={c}_{45}\left({\ell}_j{R}_j+{\alpha}_y{S}_j\right)+{c}_{55}\left({\alpha}_j+{\ell}_x{S}_j\right) $$

$$ {\mathrm{N}}_{4 j}=-{\alpha}_j{\Omega}_j $$

(19)

The existence of a nontrivial solution of Equation 18 leads to closed-form dispersion relations as

$$ {N}_{11}{G}_1 \tan \left({\alpha}_1 h/2+\chi \right)+{N}_{12}{G}_3 \tan \left({\alpha}_2 h/2+\chi \right)+{N}_{13}{G}_5 \tan \left({\alpha}_3 h/2+\chi \right)+{N}_{14}{G}_7 \tan \left({\alpha}_4 h/2+\chi \right)=0 $$

(20)

$$ {G}_1=\left|\begin{array}{ccc}\hfill {N}_{22}\hfill & \hfill {N}_{23}\hfill & \hfill {N}_{24}\hfill \\ {}\hfill {N}_{32}\hfill & \hfill {N}_{33}\hfill & \hfill {N}_{34}\hfill \\ {}\hfill {N}_{42}\hfill & \hfill {N}_{43}\hfill & \hfill {N}_{44}\hfill \end{array}\right|,{G}_2=\left|\begin{array}{ccc}\hfill {N}_{21}\hfill & \hfill {N}_{23}\hfill & \hfill {N}_{24}\hfill \\ {}\hfill {N}_{31}\hfill & \hfill {N}_{33}\hfill & \hfill {N}_{34}\hfill \\ {}\hfill {N}_{41}\hfill & \hfill {N}_{43}\hfill & \hfill {N}_{44}\hfill \end{array}\right|,{G}_3=\left|\begin{array}{ccc}\hfill {N}_{21}\hfill & \hfill {N}_{22}\hfill & \hfill {N}_{24}\hfill \\ {}\hfill {N}_{31}\hfill & \hfill {N}_{32}\hfill & \hfill {N}_{34}\hfill \\ {}\hfill {N}_{41}\hfill & \hfill {N}_{42}\hfill & \hfill {N}_{44}\hfill \end{array}\right|,{G}_4=\left|\begin{array}{ccc}\hfill {N}_{21}\hfill & \hfill {N}_{22}\hfill & \hfill {N}_{23}\hfill \\ {}\hfill {N}_{31}\hfill & \hfill {N}_{32}\hfill & \hfill {N}_{33}\hfill \\ {}\hfill {N}_{41}\hfill & \hfill {N}_{42}\hfill & \hfill {N}_{43}\hfill \end{array}\right| $$

(21)

where *χ* = 0 and *χ* = *π*/2 represent antisymmetric and symmetric Lamb wave modes, respectively. Equation 20 is a transcendental equation implicitly relating *ω* to **k**. For a fixed *θ*, a numerical iterative root-finding method is employed to compute the admissible *ω* for a range of **k** values, leading to dispersion relations of Lamb wave modes in the direction of propagation. Furthermore, in general, the frequency *ω* of each mode is the single-valued function of **k**.

### Waves in a composite laminate

In formulating thermoelastic Lamb waves in a heat-conducting laminate, the interfaces between layers are assumed to be perfectly bonded. The displacement and temperature components of each layer in the *z* axis Equation 11 needs to be modified in exponential forms to accommodate the inhomogeneity of the multilayered laminates.

$$ U= A{e}^{i\alpha z}, V= B{e}^{i\alpha z}, W=- iC{e}^{i\alpha z} $$

(22)

$$ {\Theta}_s= D{e}^{i\alpha z} $$

(23)

Substituting these expressions into the equations of motion, Equation 10 may be rearranged in a matrix form.

For nontrivial solutions of *A*, *B*, *C* and *D* in Equation 12, the determinant of the 4 × 4 matrix vanishes,

$$ \left[\begin{array}{cccc}\hfill {\Gamma}_{11}-\rho {c}_p^2\hfill & \hfill {\Gamma}_{12}\hfill & \hfill {\Gamma}_{13}\hfill & \hfill {\Gamma}_{14}\hfill \\ {}\hfill {\Gamma}_{12}\hfill & \hfill {\Gamma}_{22}-\rho {c}_p^2\hfill & \hfill {\Gamma}_{23}\hfill & \hfill {\Gamma}_{24}\hfill \\ {}\hfill {\overline{\Gamma}}_{13}\hfill & \hfill {\overline{\Gamma}}_{23}\hfill & \hfill {\Gamma}_{33}-\rho {c}_p^2\hfill & \hfill {\Gamma}_{34}\hfill \\ {}\hfill {\overline{\Gamma}}_{41}\hfill & \hfill {\overline{\Gamma}}_{42}\hfill & \hfill {\Gamma}_{43}\hfill & \hfill {\Gamma}_{44}-\frac{\rho {\omega}^2{C}_e{T}_0\tau}{T_0{\omega}^2\tau}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A\hfill \\ {}\hfill B\hfill \\ {}\hfill C\hfill \\ {}\hfill D\hfill \end{array}\right]=0 $$

(24)

giving the following eight-degree polynomial in terms of *α*, which is a biquadratic equation, it has four roots for *α*
^{2} (*α*
_{
j,} *j* = 1, 2, 3, 4), and these eight roots for *α* can be arranged in four pairs as

$$ {\alpha}_{j+1}=-{\alpha}_j,\kern0.75em \left( j=1,3,5,7\right) $$

(25)

For each *α*
_{
j
}, *B*, *C* and *D* can be expressed in terms of *A via* Equation 23 as *B* = *RA*, *C* = − *SA* and *D* = *ΩA*.

Thus, the general solution of Equations 22 and 23 in each lamina is

$$ \left( U, V, W, T\right)= E{\displaystyle \sum_{j=1}^8\left({A}_j,{R}_j{A}_j,{S}_j{A}_j,{\varOmega}_j{A}_j\right){e}^{ik{\alpha}_j z}} $$

(26)

The inter-laminar stress and thermal gradient components, *σ*
_{
zz
}, *σ*
_{
yz
}, *σ*
_{
xz
} and \( \frac{\partial T}{\partial \mathrm{z}} \) in each lamina may be expressed as

$$ \left({\sigma}_{z z},{\sigma}_{y z},{\sigma}_{x z},\frac{\partial T}{\partial \mathrm{z}}\right)\left|{}_{z=\pm h/2}= ik{e}^{i\left[\right(\left[\left({\ell}_x\ x+{\ell}_y y\right)-\omega t\right]}{\displaystyle \sum_{j=1}^8\Big({N}_{1 j}},{N}_{2 j},{N}_{3 j},{N}_{4 j}\right){A}_j{e}^{i k{\alpha}_j z} $$

(27)

Generally, there are two methods, namely transfer matrix method and assemble matrix method, for obtaining the dispersion relations in laminates. Although the procedures of these two methods seem different, they are identical in principle by both satisfying traction-free boundary conditions on the outer surfaces of the laminate and continuity of interface conditions between two adjacent laminas in a different manner. Both methods can calculate dispersion curves in a general laminate with an arbitrary stacking sequence. Using Equations 22 and 23, it may be observed that symmetric and antisymmetric wave modes in general laminates cannot be decoupled. However, in designing the composite structures, symmetric laminates are practically used. A robust method is proposed to separate the two types of wave modes by imposing boundary conditions at both top and mid-plane surface. Traction-free boundary conditions on the top surface of the laminate are given by

$$ \left({\sigma}_{z z},{\sigma}_{yz},{\sigma}_{xz},\frac{\partial T}{\partial \mathrm{z}}\right)\Big|{}_{z= h/2}=0 $$

(28)

Because of the symmetric geometry and symmetric material property of the laminate, only half of the laminate needs to be considered and then the following conditions on the stress and displacement components at the mid-plane for symmetric modes are imposed

$$ \left( w,{\sigma}_{yz},{\sigma}_{xz},\frac{\partial T}{\partial \mathrm{z}}\right)\Big|{}_{z=0}=0 $$

(29)

Likewise, the boundary conditions of antisymmetric modes at the mid-plane are

$$ \left( u, v,{\sigma}_{z z}, T\right)\Big|{}_{z=0}=0 $$

(30)

By imposing displacement and stress continuity conditions along the interfaces of half lay-up of an *N*-layered laminate, a total of 4*N* equations are constructed if the assemble matrix method is used. Then set the determinant of the 4*N* equations to zero, and numerically solve the resulting transcendental equation for the dispersion relations of Lamb waves in symmetric laminates.