Accurate modeling of PZT-induced Lamb wave propagation in structures by using a novel spectral finite element method

Consider a PZT-bonded base plate that refers to the global coordinate system (x, y, z) located on the mid-plane of the base plate, as shown in figure 1. If the excitation frequency is not too high, the first-order shear deformation theory (FSDT), which takes the effect of the lateral contraction in the z-direction into account [4, 20], can be adopted in modeling Lamb wave propagation in plate-like structures. Theoretical analysis demonstrates that the simple plate theory can be used when the product of the wave frequency and the plate thickness is within 1.0 MHz mm, although the Lamb wave structure can be fairly complicated. Details on this issue can be found in the appendix.

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According to the first-order shear deformation theory, the displacement fields for the base plate are given by

$$\begin{eqnarray}&&\left\{ \begin{array}{lllllllllllllll} {{u}_{b}}\left( x,y,z,t \right)={{u}_{b0}}\left( x,y,t \right)+z{{\theta }_{b}}\left( x,y,t \right)={{u}_{S}}+{{u}_{A}} \\ {{v}_{b}}\left( x,y,z,t \right)={{v}_{b0}}\left( x,y,t \right)+z{{\phi }_{b}}\left( x,y,t \right)={{v}_{S}}+{{v}_{A}} \\ {{w}_{b}}\left( x,y,z,t \right)=z{{\psi }_{b}}\left( x,y,t \right)+{{w}_{b0}}\left( x,y,t \right)={{w}_{S}}+{{w}_{A}} \\ \end{array} \right.,\end{eqnarray} \tag{ 1 }$$

where the subscripts A and S represent the anti-symmetric and symmetric modes, ${{u}_{b0}}\left( x,y,t \right)$, ${{v}_{b0}}\left( x,y,t \right)$ and ${{w}_{b0}}\left( x,y,t \right)$ are the in-plane and out-of-plane displacements on the mid-plane of the base plate, ${{\theta }_{b}}\left( x,y,t \right)$ and ${{\phi }_{b}}\left( x,y,t \right)$ are the rotations of the cross section normal to the mid-plane of the base plate, ${{\psi }_{b}}\left( x,y,t \right)$ is the lateral contraction and t is time, respectively. Note that ${{u}_{b0}}$, ${{v}_{b0}}$ and ${{\psi }_{b}}$ are related to the symmetric modes, and ${{w}_{b0}}$, ${{\theta }_{b}}$ and ${{\phi }_{b}}$ are related to the anti-symmetric modes.

To employ the simple way shown in [19] to avoid the inherent thickness locking in the simple plate theory, the strain fields of the base plate are also decomposed into anti-symmetric and symmetric modes, namely

$$\begin{eqnarray}\left\{ \begin{array}{lllllllllllllll} {{\varepsilon }_{x}} & = & \frac{\partial {{u}_{b0}}}{\partial x}+z\frac{\partial {{\theta }_{b}}}{\partial x}=\varepsilon _{x}^{S}+\varepsilon _{x}^{A}, \\ {{\gamma }_{xy}} & = & \left( \frac{\partial {{u}_{b0}}}{\partial y}+\frac{\partial {{v}_{b0}}}{\partial x} \right)+z\left( \frac{\partial {{\theta }_{b}}}{\partial y}+\frac{\partial {{\phi }_{b}}}{\partial x} \right)=\gamma _{xy}^{S}+\gamma _{xy}^{A}{\mkern 1mu} , \\ {{\varepsilon }_{y}} & = & \frac{\partial {{v}_{b0}}}{\partial y}+z\frac{\partial {{\phi }_{b}}}{\partial y}=\varepsilon _{y}^{S}+\varepsilon _{y}^{A}, \\ {{\gamma }_{yz}} & = & z\frac{\partial {{\psi }_{b}}}{\partial y}+\left( {{\phi }_{b}}+\frac{\partial {{w}_{b0}}}{\partial y} \right)=\gamma _{yz}^{S}+\gamma _{yz}^{A}{\mkern 1mu} , \\ {{\varepsilon }_{z}} & = & {{\psi }_{b}}+0=\varepsilon _{z}^{S}+\varepsilon _{z}^{A}, \\ {{\gamma }_{xz}} & = & z\frac{\partial {{\psi }_{b}}}{\partial x}+\left( {{\theta }_{b}}+\frac{\partial {{w}_{b0}}}{\partial x} \right)=\gamma _{xz}^{S}+\gamma _{xz}^{A}{\mkern 1mu} . \\ \end{array} \right.\end{eqnarray} \tag{ 2 }$$

For the PZT layer, the displacement fields are given by

$$\begin{eqnarray}&&\left\{ \begin{array}{lllllllllllllll} {{u}_{p}}\left( x,y,z,t \right)={{u}_{p0}}\left( x,y,{{h}_{1}},t \right)+\left( z-{{h}_{1}} \right){{\theta }_{p}}\left( x,y,t \right){\mkern 1mu} , \\ {{v}_{p}}\left( x,y,z,t \right)={{v}_{p0}}\left( x,y,{{h}_{1}},t \right)+\left( z-{{h}_{1}} \right){{\phi }_{p}}\left( x,y,t \right){\mkern 1mu} , \\ {{w}_{p}}\left( x,y,z,t \right)={{w}_{p0}}\left( x,y,{{h}_{1}},t \right)+\left( z-{{h}_{1}} \right){{\psi }_{p}}\left( x,y,t \right){\mkern 1mu} . \\ \end{array} \right.\end{eqnarray} \tag{ 3 }$$

where ${{u}_{p0}}\left( x,y,{{h}_{1}},t \right)$, ${{v}_{p0}}(x,y,{{h}_{1}},t)$ and ${{w}_{p0}}(x,y,{{h}_{1}},t)$ are the in-plane and out-of-plane displacements on the mid-plane of the PZT layer, ${{\theta }_{p}}(x,y,t)$ and ${{\phi }_{p}}\left( x,y,t \right)$ are the rotations of the cross section normal to the mid-plane of the PZT layer, ${{\psi }_{p}}\left( x,y,t \right)$ is the lateral contraction and ${{h}_{1}}$ is the distance between the mid-planes of the base plate and the PZT layer, shown in figure 1.

Since the PZT layer is perfectly bonded on the base plate, the displacements at the interface should be compatible. After some manipulations, one has

$$\begin{eqnarray}\left\{ \begin{array}{ccccccccccccccc} {{u}_{p}}\left( x,y,z,t \right){\mkern 1mu} =\ \;{{u}_{b0}}\left( x,y,t \right)+\frac{{{h}_{b}}}{2}{{\theta }_{b}}\left( x,y,t \right) & {} \\ {} & +\bar{z}{{\theta }_{p}}\left( x,y,t \right) \\ {{v}_{p}}\left( x,y,z,t \right){\mkern 1mu} =\ \;{{v}_{b0}}\left( x,y,t \right)+\frac{{{h}_{b}}}{2}{{\phi }_{b}}\left( x,y,t \right) & {} \\ {} & +\bar{z}{{\phi }_{p}}\left( x,y,t \right) \\ {{w}_{p}}\left( x,y,z,t \right)\,=\ \;{{w}_{b0}}\left( x,y,t \right)+\frac{{{h}_{b}}}{2}{{\psi }_{b}}\left( x,y,t \right) & {} \\ {} & +\bar{z}{{\psi }_{p}}\left( x,y,t \right), \\ \end{array} \right.\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&\bar{z}=z-\frac{{{h}_{b}}}{2},\end{eqnarray} \tag{ 5 }$$

and the strains in the PZT layer can be given by

$$\begin{eqnarray}\left\{ \begin{array}{lllllllllllllll} {{\varepsilon }_{x}} & = & \frac{\partial {{u}_{b0}}}{\partial x}+\frac{{{h}_{b}}}{2}\frac{\partial {{\theta }_{b}}}{\partial x}+\bar{z}\frac{\partial {{\theta }_{p}}}{\partial x}, \\ {{\gamma }_{xy}} & = & \frac{\partial {{u}_{b0}}}{\partial y}+\frac{\partial {{v}_{b0}}}{\partial x}+\frac{{{h}_{b}}}{2}\left( \frac{\partial {{\theta }_{b}}}{\partial y}+\frac{\partial {{\phi }_{b}}}{\partial x} \right) \\ {} & {} & +\bar{z}\left( \frac{\partial {{\theta }_{p}}}{\partial y}+\frac{\partial {{\phi }_{p}}}{\partial x} \right){\mkern 1mu} , \\ {{\varepsilon }_{y}} & = & \frac{\partial {{v}_{b0}}}{\partial y}+\frac{{{h}_{b}}}{2}\frac{\partial {{\phi }_{b}}}{\partial y}+\bar{z}\frac{\partial {{\phi }_{p}}}{\partial y}, \\ {{\gamma }_{yz}} & = & {{\phi }_{p}}+\frac{\partial {{w}_{b0}}}{\partial y}+\frac{{{h}_{b}}}{2}\frac{\partial {{\psi }_{b}}}{\partial y}+\bar{z}\frac{\partial {{\psi }_{p}}}{\partial y}{\mkern 1mu} , \\ {{\varepsilon }_{z}} & = & {{\psi }_{p}}, \\ {{\gamma }_{xz}} & = & {{\theta }_{p}}+\frac{\partial {{w}_{b0}}}{\partial x}+\frac{{{h}_{b}}}{2}\frac{\partial {{\psi }_{b}}}{\partial x}+\bar{z}\frac{\partial {{\psi }_{p}}}{\partial x}{\mkern 1mu} . \\ \end{array} \right.\end{eqnarray} \tag{ 6 }$$

An assumption is made that the electric potential for the PZT layer is distributed linearly through the thickness [21, 22]. The electric voltage is equal to zero on the lower surface of the PZT layer and is equal to V on the upper surface of the PZT layer. Therefore, the electric voltage ${{V}_{p}}$ in the PZT layer can be calculated by

$$\begin{eqnarray}&&{{V}_{p}}=\frac{V}{{{h}_{p}}}\left( z-\frac{{{h}_{b}}}{2} \right){\mkern 1mu} .\end{eqnarray} \tag{ 7 }$$

The resulting electric field ${{E}_{z}}$ is

$$\begin{eqnarray}&&{{E}_{z}}=-\frac{\partial {{V}_{p}}}{\partial z}=-\frac{V}{{{h}_{p}}}=-[{{{\bf B}}_{V}}]V,\end{eqnarray} \tag{ 8 }$$

where ${{h}_{p}}$ is the thickness of the PZT layer, and $[{{{\bf B}}_{V}}]=[1/{{h}_{p}}]$.

Equation (2) expresses three-dimensional (3D) strain fields; however, the 3D constitutive equations should not be used directly. Otherwise, thickness locking would occur if no remedial measure was used since the displacement shown in equation (1) varies only linearly through the plate thickness.

It is known that anti-symmetric modes are primarily flexural waves and are more likely to be influenced by thickness locking [16]. Equation (2) shows that $\varepsilon _{z}^{A}=0$. However, $\varepsilon _{z}^{A}=-\nu \varepsilon _{x}^{A}-\nu \varepsilon _{y}^{A}$ due to Poisson's effect. In other words, $\varepsilon _{z}^{A}\ne 0$ for nonzero $\varepsilon _{x}^{A}$ or $\varepsilon _{y}^{A}$. This contradiction, known as thickness locking, can be avoided if a plane stress state is assumed [19]. In this way, $\varepsilon _{z}^{A}$ is not involved in equation (9) to calculate the stress components. Alternatively, it is observed that the anti-symmetric displacement components, shown in equation (1), are exactly the same as the ones in the well-known Mindlin plate theory. Therefore, the plane stress state as adopted in the Mindlin plate theory should also be employed for the anti-symmetric mode.

In summary, the 3D constitutive equation is only used for the symmetric strains, and a different constitutive equation is used for the anti-symmetric strains [19]. In detail, the stress-strain equations are given by

$$\begin{eqnarray}&&\{{\boldsymbol{ \sigma }} \}={{\{{\boldsymbol{ \sigma }} \}}^{S}}+{{\{{\boldsymbol{ \sigma }} \}}^{A}}=[{\bf C}]{{\{{\boldsymbol{ \varepsilon }} \}}^{S}}+[\tilde{{\bf C}}]{{\{{\boldsymbol{ \varepsilon }} \}}^{A}},\end{eqnarray} \tag{ 9 }$$

where $\{{\boldsymbol{ \sigma }} \}$ and $\{{\boldsymbol{ \varepsilon }} \}$ are the 3D stress and strain vectors, the superscripts A and S represent the anti-symmetric and symmetric modes, $[{\bf C}]$ and $[\tilde{{\bf C}}]$ are given by

$$\begin{eqnarray}[{\bf C}]=\left[ \begin{array}{ccccccccccccccc} {{C}_{11}} & {{C}_{12}} & {{C}_{13}} & 0 & 0 & 0 \\ {{C}_{12}} & {{C}_{22}} & {{C}_{23}} & 0 & 0 & 0 \\ {{C}_{13}} & {{C}_{23}} & {{C}_{33}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {{C}_{44}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{C}_{55}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{C}_{66}} \\ \end{array} \right]\end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray}[\tilde{{\bf C}}]=\left[ \begin{array}{ccccccccccccccc} {{\tilde{C}}_{11}} & {{\tilde{C}}_{12}} & 0 & 0 & 0 & 0 \\ {{\tilde{C}}_{12}} & {{\tilde{C}}_{22}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\kappa }^{2}}{{C}_{44}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\kappa }^{2}}{{C}_{55}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{C}_{66}} \\ \end{array} \right],\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{C}_{11}}={{C}_{22}}={{C}_{33}}=\frac{E\left( 1-\nu \right)}{\left( 1+\nu \right)\left( 1-2\nu \right)} \\ {{C}_{12}}={{C}_{13}}={{C}_{23}}=\frac{E\nu }{\left( 1+\nu \right)\left( 1-2\nu \right)} \\ {{C}_{44}}={{C}_{55}}={{C}_{66}}=\frac{E}{2\left( 1+\nu \right)} \\ \end{array}\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\tilde{C}}_{11}}={{C}_{11}}-\frac{C_{13}^{2}}{{{C}_{33}}}=\frac{E}{1-{{\nu }^{2}}}\;\ \;\ \;\left( \lt {{C}_{11}} \right) \\ {{\tilde{C}}_{12}}={{C}_{12}}-\frac{{{C}_{13}}{{C}_{23}}}{{{C}_{33}}}=\frac{E\nu }{1-{{\nu }^{2}}}\;\ \;\ \;\left( \lt {{C}_{12}} \right) \\ {{\tilde{C}}_{22}}={{C}_{22}}-\frac{C_{23}^{2}}{{{C}_{33}}}=\frac{E}{1-{{\nu }^{2}}}\;\ \;\ \;\left( \lt {{C}_{22}} \right){\mkern 1mu} . \\ \end{array}\end{eqnarray} \tag{ 13 }$$

Equation (11) is obtained based on the assumption of the plane stress state and ${{\kappa }^{2}}$ is the shear correction factor as adopted in the Mindlin plate theory. In equations (12) and (13), E and v are the elasticity modulus and Poisson's ratio, respectively.

h_p is much smaller than h_b . However, h_p /L_p is much larger than h_b /L_b , where L_p and L_b are the smaller in-plane dimension of the PZT layer and the base plate. Besides, the PZT layer is usually bonded on the upper surface of the base plate; thus, pure anti-symmetric modes will not exist practically. Therefore, for the PZT layer, the 3D constitutive equations are used, namely

$$\begin{eqnarray}&&\left\{ \begin{array}{lllllllllllllll} \{{\boldsymbol{ \sigma }} \}=[{{{\bf C}}^{E}}]\{{\boldsymbol{ \varepsilon }} \}-{{[{\bf e}]}^{T}}\{{\bf E}\} \\ \{{\bf D}\}=[{\bf e}]\{{\boldsymbol{ \varepsilon }} \}+[{{{\boldsymbol{ \varepsilon }} }^{S}}]\{{\bf E}\} \\ \end{array} \right.,\end{eqnarray} \tag{ 14 }$$

where $[{\bf e}]$ and $[{{{\boldsymbol{ \varepsilon }} }^{S}}]$ are matrices of the piezoelectric constant and the dielectric constant measured at zero strain and {D} and {E} are vectors of the electric displacement and electric field, respectively.

In developing a 2D spectral finite element without PZT layers, each node of the element owns six displacement degrees of freedom ${{q}_{k}}\left( k=1,2,\ldots ,6 \right)$, i.e., ${{u}_{b0}},{{v}_{b0}},{{w}_{b0}},{{\theta }_{b}},{{\phi }_{b}}$ and ${{\psi }_{b}}$. Thus, the total number of DOFs is the same as the hybrid spectral element in [17] if the node distribution in the $\xi -\eta$ plane is the same. For a 2D SFE with one PZT layer, each node of the element owns nine displacement degrees of freedom ${{q}_{k}}\left( k=1,2,\ldots ,9 \right)$, i.e., ${{u}_{b0}},{{v}_{b0}},{{w}_{b0}},{{\theta }_{b}},{{\phi }_{b}},{{\psi }_{b}},{{\theta }_{p}},{{\phi }_{p}}$ and ${{\psi }_{p}}$.

In the proposed 2D spectral finite element with a PZT layer, each PZT element shares only one conjunct voltage DOF V, no matter how many SFEs are used to model the PZT element. In other words, each PZT element (actuator or sensor) has only one voltage DOF. This is quite different from the existing methods [2, 6, 8] in that each node has one voltage DOF, and the weighted average of all of the voltage DOFs is taken as the voltage applied on the PZT element. It is obvious that the proposed approach is beneficial to the CPU time in numerical simulations, especially when the PZTs are modeled by more than one spectral finite element, or the total number of voltage DOFs is large.

It is known that if the extended Chebyshev points are used as the element nodes, a larger time step can be used in the numerical integration by the central difference method [23]. The non-dimensional node position in [−1, 1] can be explicitly computed by

$$\begin{eqnarray}\begin{array}{} {{\xi }_{k}} & = & -\frac{{\rm cos} \left[ \left( 2k-1 \right)\pi /\left( 2N \right) \right]}{{\rm cos} \left[ \pi /\left( 2N \right) \right]} \\ {} & {} & \quad \ \;k=1,2,\ldots ,N;\xi \in \left[ -1,1 \right] \\ \end{array}\end{eqnarray} \tag{ 15 }$$

For illustration, a 6 × 6 node 2D SFE, shown in figure 2, is presented. The element contains a PZT layer, and each node has nine DOFs.

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The nine displacements within the element can be expressed as

$$\begin{eqnarray}&&{{\{{{q}_{k}}\}}^{T}}=[{\bf N}]\{{\bf q}\}=\sum \limits_{i=1}^{6} \sum \limits_{j=1}^{6} \left[ {{N}_{ij}}\left( \xi ,\eta \right) \right]{{\left\{ {{q}_{k}}\left( {{\xi }_{i}},{{\eta }_{j}} \right) \right\}}^{T}},\end{eqnarray} \tag{ 16 }$$

where $[{\bf N}]$ is the shape function matrix, ${{N}_{ij}}\left( \xi ,\eta \right)={{L}_{i}}\left( \xi \right){{L}_{j}}\left( \eta \right)\left( i,j=1,2,\ldots ,6 \right)$ are shape functions, ${{L}_{i}}(\xi )$ and ${{L}_{j}}(\eta )$ are Lagrange interpolation functions and {q} is the vector of the nodal DOFs, respectively.

Element strain fields can be obtained by substituting equation (16) into equations (2) and (6). One has

$$\begin{eqnarray*}\begin{array}{lllllllllllllll} {{\{{\boldsymbol{ \varepsilon }} \}}_{b}} & = & {{\{{\boldsymbol{ \varepsilon }} \}}^{A}}+{{\{{\boldsymbol{ \varepsilon }} \}}^{S}}=([{\bf B}_{b}^{A}]+[{\bf B}_{b}^{S}])\{{\bf q}\} \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {} & = & \sum \limits_{i=1}^{6} \sum \limits_{j=1}^{6} \left( \left[ B_{bij}^{A}(\xi ,\eta ) \right]+\left[ B_{bij}^{S}(\xi ,\eta ) \right] \right) \\ {} & {} & \ \;\times {{\left\{ {{q}_{k}}({{\xi }_{i}},{{\eta }_{j}}) \right\}}^{T}} \\ {{\{{\boldsymbol{ \varepsilon }} \}}_{p}} & = & [{{{\bf B}}_{p}}]\{{\bf q}\}=\sum \limits_{i=1}^{6} \sum \limits_{j=1}^{6} \left[ {{B}_{pij}}(\xi ,\eta ) \right] \\ {} & {} & \ \;\times {{\left\{ {{q}_{k}}({{\xi }_{i}},{{\eta }_{j}}) \right\}}^{T}} \\ \end{array}\end{eqnarray} \tag{ 17 }$$

Based on the virtual work principle or on Hamilton's principle [2, 6], a set of equations of motion for each element can be derived and written in a matrix form as

$$\begin{eqnarray}&&\left\{ \begin{array}{lllllllllllllll} [{{{\bf M}}^{e}}]\{{\bf \ddot{q}}\}+[{\bf K}_{uu}^{e}]\{{\bf q}\}+[{\bf K}_{uV}^{e}]V=\{{{{\bf F}}^{e}}\} \\ [{\bf K}_{Vu}^{e}]\{{\bf q}\}+[{\bf K}_{VV}^{e}]V={{Q}^{e}}{\mkern 1mu} , \\ \end{array} \right.\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \begin{array}{lllllllllllllll} [{{{\bf M}}^{e}}] & = & {{[{{{\bf M}}^{e}}]}_{b}}+{{[{{{\bf M}}^{e}}]}_{p}} \\ {} & = & \int \limits_{-\frac{{{h}_{b}}}{2}}^{\frac{{{h}_{b}}}{2}} \int \limits_{-1}^{1} \int \limits_{-1}^{1} {{[{{{\bf L}}_{b}}]}^{T}}{{\rho }_{b}}[{{{\bf L}}_{b}}]\left| J \right|d\xi d\eta dz \\ {} & {} & +\int \limits_{\frac{{{h}_{b}}}{2}}^{\frac{{{h}_{b}}}{2}+{{h}_{p}}} \int \limits_{-1}^{1} \int \limits_{-1}^{1} {{[{{{\bf L}}_{p}}]}^{T}}{{\rho }_{p}}[{{{\bf L}}_{p}}]\left| J \right|d\xi d\eta dz \\ \end{array} \\ \begin{array}{lllllllllllllll} [{\bf K}_{uu}^{e}] & = & {{[{\bf K}_{uu}^{e}]}_{b}}+{{[{\bf K}_{uu}^{e}]}_{p}} \\ {} & = & [{\bf K}_{uu}^{e}]_{b}^{A}+[{\bf K}_{uu}^{e}]_{b}^{S}+{{[{\bf K}_{uu}^{e}]}_{p}} \\ {} & = & \int \limits_{-\frac{{{h}_{b}}}{2}}^{\frac{{{h}_{b}}}{2}} \int \limits_{-1}^{1} \int \limits_{-1}^{1} {{[{\bf B}_{b}^{A}]}^{T}}[\tilde{{\bf C}}][{\bf B}_{b}^{A}]\left| J \right|d\xi d\eta dz \\ {} & {} & +\int \limits_{-\frac{{{h}_{b}}}{2}}^{\frac{{{h}_{b}}}{2}} \int \limits_{-1}^{1} \int \limits_{-1}^{1} {{[{\bf B}_{b}^{S}]}^{T}}[{\bf C}][{\bf B}_{b}^{S}]\left| J \right|d\xi d\eta dz \\ {} & {} & +\int \limits_{\frac{{{h}_{b}}}{2}}^{\frac{{{h}_{b}}}{2}+{{h}_{p}}} \int \limits_{-1}^{1} \int \limits_{-1}^{1} {{[{{{\bf B}}_{p}}]}^{T}}[{{{\bf C}}^{E}}][{{{\bf B}}_{p}}]\left| J \right|d\xi d\eta dz \\ \end{array} \\ \begin{array}{lllllllllllllll} [{\bf K}_{uV}^{e}] & = & {{[{\bf K}_{Vu}^{e}]}^{T}} \\ {} & = & \int \limits_{\frac{{{h}_{b}}}{2}}^{\frac{{{h}_{b}}}{2}+{{h}_{p}}} \int \limits_{-1}^{1} \int \limits_{-1}^{1} {{[{{{\bf B}}_{p}}]}^{T}}{{[{\bf e}]}^{T}}[{{{\bf B}}_{V}}]\left| J \right|d\xi d\eta dz \\ [{\bf K}_{VV}^{e}] & = & -\int \limits_{\frac{{{h}_{b}}}{2}}^{\frac{{{h}_{b}}}{2}+{{h}_{p}}} \int \limits_{-1}^{1} \int \limits_{-1}^{1} {{[{{{\bf B}}_{V}}]}^{T}}[{{{\bf \epsilon } }^{S}}][{{{\bf B}}_{V}}]\left| J \right|d\xi d\eta dz{\mkern 1mu} , \\ \end{array} \\ \end{array}\end{eqnarray} \tag{ 19 }$$

$\left| J \right|$ is the determinant of the Jacobian matrix, and $\{{{{\bf F}}^{e}}\}$ and ${{Q}^{e}}$ are the mechanical force vector due to the applied load and the applied electrical charge on the surface of the PZT layer, respectively.

Note that [B_V ] is given by equation (8), each PZT element (actuator or sensor) has only one voltage DOF and $\int {{\Omega }_{b}} {{[{\bf B}_{b}^{A}]}^{T}}[{\bf C}][{\bf B}_{b}^{S}]d\Omega =\int {{\Omega }_{b}} {{[{\bf B}_{b}^{S}]}^{T}}[\tilde{{\bf C}}][{\bf B}_{b}^{A}]d\Omega =0$, where ${{\Omega }_{b}}$ is the domain of integration defined in the base plate. The Gauss quadrature is used for obtaining the element matrices in equation (19). The diagonal mass matrix can be obtained by the method of row summation before performing the numerical integration [23].

For the base plate model without PZTs, simply remove the terms associated with the PZT layer. Thus, one has

$$\begin{eqnarray}&&{{[{{{\bf M}}^{e}}]}_{{\rm b}}}\{{\bf \ddot{q}}\}+{{[{\bf K}_{uu}^{e}]}_{b}}\{{\bf q}\}=\{{{{\bf F}}^{e}}\}\end{eqnarray} \tag{ 20 }$$

The well-known explicit time-integration scheme, called the central difference method, is adopted to solve equation (18) or equation (20).

To model irregular shapes, the method of the sub-parametric element is adopted. The order of geometric functions is relatively lower than the order of displacement functions. The geometric functions are defined by

$$\begin{eqnarray}&&\left\{ \begin{array}{ccccccccccccccc} x=x\left( \xi ,\eta \right)=\sum \limits_{i=1}^{M} {{f}_{i}}\left( \xi ,\eta \right){{x}_{i}} \\ y=y\left( \xi ,\eta \right)=\sum \limits_{i=1}^{M} {{f}_{i}}\left( \xi ,\eta \right){{y}_{i}} \\ \end{array} \right.\quad \left( -1\leqslant \xi ,\eta \leqslant 1 \right),\end{eqnarray} \tag{ 21 }$$

where x and y are the space variables defined on the physical domain of the element, and $\xi ,\eta$ are the variables defined on the bi-unit square domain, shown in figure 2. ${{x}_{i}},{{y}_{i}}$ are coordinates of nodes on the element boundaries and M is the total number of boundary nodes; thus, ${{f}_{i}}\left( \xi ,\eta \right)$ are the shape functions for serendipity type finite elements with only boundary nodes.

The first order derivatives of shape functions ${{N}_{ij}}\left( \xi ,\eta \right)$ with respect to the space variables x and y can be computed by

$$\begin{eqnarray}&&\left\{ \begin{array}{ccccccccccccccc} \frac{\partial {{N}_{ij}}}{\partial x} \\ \frac{\partial {{N}_{ij}}}{\partial y} \\ \end{array} \right\}={{\left[ J\left( \xi ,\eta \right) \right]}^{-1}}\left\{ \begin{array}{ccccccccccccccc} \frac{\partial {{N}_{ij}}}{\partial \xi } \\ \frac{\partial {{N}_{ij}}}{\partial \eta } \\ \end{array} \right\}\end{eqnarray} \tag{ 22 }$$

With equation (21) and the given $\left( {{x}_{i}},{{y}_{i}} \right)$, the Jacobian matrix can be computed by

$$\begin{eqnarray}\left[ J\left( \xi ,\eta \right) \right]=\left[ \begin{array}{ccccccccccccccc} \frac{\partial x}{\partial \xi } & \frac{\partial y}{\partial \xi } \\ \frac{\partial x}{\partial \eta } & \frac{\partial y}{\partial \eta } \\ \end{array} \right]\end{eqnarray} \tag{ 23 }$$

In order to verify the validity of the proposed 2D SFE established by the simple plate theory, two models with free boundary conditions are considered: (1) a base plate without PZTs and (2) a PZT bonded on the top of the base plate. For the convenience of analysis, assume that the two models shown in figure 3 are in the state of plane strain, similar to the models employed in [6, 16]. To be more specific, both models have a sufficiently large dimension in the y direction.

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In figure 3(a), points A and B are on the mid-plane of the base plate, a force is actuated at point A' and the displacement signal is the output at point B. In figure 3(b), a pitch-catch method using a pair of piezoelectric actuators (PZT1) and sensors (PZT2) is introduced.

The base plate has a total length of L = 2L₁ + L₂ and has the thickness of h_b . The base plate material is aluminum, and the material properties are E = 70.0 GPa, ν = 0.3 and ρ_b  = 2700 kg m⁻³. The material properties of PZTs are listed in table 1.

Table 1.  Material properties of PZTs.

Properties	Value	Properties	Value
$C_{11}^{E}$	147.0 GPa	${{e}_{31}}$	−3.09 C m−2
$C_{12}^{E}$	105.0 GPa	${{e}_{33}}$	16.0 C m−2
$C_{13}^{E}$	93.7 GPa	${{e}_{15}}$	11.6 C m−2
$C_{33}^{E}$	113.0 GPa	$\varepsilon _{11,r}^{S}$	1130
$C_{44}^{E}=C_{55}^{E}$	23.0 GPa	$\varepsilon _{33,r}^{S}$	914
$C_{66}^{E}$	21.2 GPa	${{\rho }_{p}}$	7700.0 kg m−3

In numerical simulations, a frequency of the force or the voltage input signal is chosen to guarantee that only the lowest symmetric S0 mode and the lowest anti-symmetric A0 mode are excited. In the present investigation, h_b  = 1.02 mm and h_p  = 0.25 mm; therefore, both input signals have a center frequency of 100 kHz and a time duration of 50 μs, as shown in figure 4.

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During numerical simulations, it is possible that both the A0 and S0 modes are received simultaneously at point B or by sensor PZT2 since the speeds of the A0 and S0 modes are quite different, and the reflected S0 wave from the edge of the base plate may propagate to point B or sensor PZT2 when the slower A0 wave reaches there. This makes it difficult to recognize or distinguish different Lamb wave modes and makes the calculated velocities inaccurate, or it can even cause them to make no sense. Hence, in order to completely separate the two Lamb wave modes during the simulations, the distance between point A and point B or between PZT1 and PZT2 should be carefully selected.

For illustration, take the base plate model without PZTs as an example. Three propagation paths are considered, namely path #1: S0 mode propagates from point A directly to point B; path #2: A0 mode propagates from point A directly to point B; and path #3: S0 mode propagates from point A to the left end of the base plate, and then the reflected S0 mode propagates to point B. To distinguish the three Lamb wave modes from the signal picked up at point B, the waves propagating along these three paths should reach point B at different times or satisfy the following relations:

$$\begin{eqnarray}&&\left\{ \begin{array}{lllllllllllllll} t_{p1}^{S0}\lt t_{p2}^{A0} \\ t_{p2}^{A0}\lt t_{p3}^{S0}, \\ \end{array} \right.\end{eqnarray} \tag{ 24 }$$

where the subscripts p1 – p3 represent path #1 – path #3, respectively.

Taking the wave packet width of the S0 and A0 modes into consideration and assuming that they have the same width as the input signal, equation (24) should be modified by

$$\begin{eqnarray}&&\left\{ \begin{array}{lllllllllllllll} \frac{{{L}_{2}}}{V_{g}^{S0}}+\frac{\Delta t}{2}\lt \frac{{{L}_{2}}}{V_{g}^{A0}}-\frac{\Delta t}{2} \\ \frac{{{L}_{2}}}{V_{g}^{A0}}+\frac{\Delta t}{2}\lt \frac{2{{L}_{1}}+{{L}_{2}}}{V_{g}^{S0}}-\frac{\Delta t}{2}, \\ \end{array} \right.\end{eqnarray} \tag{ 25 }$$

where $V_{g}^{S0},V_{g}^{A0}$ represent the analytical group velocities of the S0 and A0 modes, and $\Delta t$ is the wave packet width of the input signal.

In the present investigation, the geometric dimensions are chosen according to equation (25). They are L₁ = 406.4 mm and L₂ = 177.8 mm and are different from the ones used in [6]. If the dimensions in [6] are used, then the speed of the A0 mode cannot be accurately calculated by the so-called A0 signal that was picked up by PZT2 since it actually contains both A0 and reflected S0 waves.

Consider first the model of the base plate without PZTs. The plate is modeled by 312 proposed SFEs. For verification, the same model is also analyzed by using ANSYS. In ANSYS, four rectangular plane strain elements (2D plane182) are used in the thickness direction; thus, the finite element model has a total of 14 976 elements. Figure 5 shows the applied load format used in ANSYS and in the SFEM. In ANSYS, a force F_x , shown in figure 5(a), is applied at point A' on the upper surface of the plate. The same force F_x and the resulting bending moment M_y  = F_x h_b/2 are applied at point A' in the SFEM, shown in figure 5(b). The F_x applied at point A excites the S0 mode, and M_y excites the A0 mode. For simplicity and for convenience in discussions, symbols SFE_σz and SFE_3D stand for the proposed 2D SFE and for the one based on the 3D constitutive equations, without taking any remedial measure for the thickness locking, respectively.

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More precisely, the only difference between SFE_3D and SFE_σz is the way in which to compute the stress components. SFE_σz directly uses equation (9) to compute the stress components. SFE_3D also uses equation (9) to compute the stress components but with some modifications, i.e., replacing matrix $[\tilde{{\bf C}}]$ in equation (9) by matrix [C]. All of the other mathematic formulations for the two approaches are exactly the same.

Figures 6(a) and (b) compare the components of displacement ${{u}_{b}}$ and ${{w}_{b}}$ at point B obtained by SFE_σz with those obtained by ANSYS. From equation (1), it is known that the ${{u}_{b}}$ and ${{w}_{b}}$ at point B (z = 0) are the S0 and A0 modes. The S0 wave, excited by force F_x , propagates along the right and left directions of point A simultaneously. The S0 wave propagating along the right direction reaches point B, which is the first wave package shown in figure 6(a). The S0 waves propagating along the left and right directions are reflected successively when they reach the left and right ends of the plate. The two reflected waves meet at point B at the same time, which is the second wave package, shown in figure 6. This explains why the amplitude of the second wave package is twice as high as that of the first wave package, shown in figure 6(a). The good agreement between the results obtained by SFE_σz and those obtained by ANSYS demonstrates that the proposed 2D SFE is capable of simulating the propagation of Lamb waves in plates. It is also seen that the A0 mode with a relatively slower group/phase velocity exhibits more dispersive characteristics compared to the S0 mode.

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In the simulation, the time step is chosen as 3.0 × 10⁻⁸ s for the SFEM and 7.5 × 10⁻⁸ s for ANSYS. Table 2 shows the comparison of the computational resources. It is seen that the SFEM can get similar accurate results with a fewer number of DOFs and much less CPU time; thus, the computational efficiency is greatly increased.

The results obtained by using the SFE_3D are shown in figure 7. It is seen that the predictions of the S0 mode, shown in figure 7(a), agree well with the ANSYS data. However, a significant difference exists between the predictions of the A0 mode shown in figure 7(b) and the ANSYS data. From figures 6 and 7, one may conclude that thickness locking only exists in the anti-symmetric modes, and remedial measures are necessary to avoid the thickness locking.

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Table 3 summarizes the group velocity that was obtained by numerical simulations and compared with theoretical solutions. To extract the accurate wave packet time of flight (TOF) from the signal, a time–frequency technique called the wavelet transform (WT) is used. Since the propagation distance is exactly known, the group velocity can be easily obtained. From table 3, it is seen that SFE_σz shows extremely high accuracy for the case studied and is even higher than the data obtained by ANSYS with fine meshes. It is seen that the results of SFE_3D are not satisfactory for the A0 mode. Due to thickness locking, the error reaches 4.54% for this simple case. The error will be increased with the increase of the center frequency [19]. This implies that remedial measures should be taken to avoid thickness locking, which exists inherently in the simple plate theory.

In practice, the input signal is usually induced by PZT actuators, and the output signals are picked up by PZT sensors. In this section, a model of the base plate with two surface-mounted PZTs, shown in figure 3(b), is investigated. The input voltage signal with an amplitude of 50 V is applied on the top surface of PZT1. The output voltage signal is measured by PZT2. The plate is modeled by 312 proposed SFEs, including four SFEs with PZTs. For verification, the same model is also analyzed using ANSYS. In Contrast to the previous case, besides 14 976 rectangular plane strain elements (2D plane 182), 24 additional rectangular plane strain elements (2D plane13) for each PZT transducer are used to model the two PZT elements. The mesh configurations for the 2D SFEM and the ANSYS models are schematically shown in figure 8.

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To verify the usage of one voltage degree for each PZT element, the problem is analyzed by using the 2D SFE with different numbers of voltage degrees: one with only one voltage degree for the entire PZT element, the other with one voltage degree at each node. Although the magnitude of voltage at each node is quite different, the weighted sum of the voltage on the PZT element is the same as the one obtained by the 2D SFE with only one voltage degree for the entire PZT element. Due to space limitations, the detailed comparisons are omitted, and only the results obtained by the proposed SFEs are presented.

The simulation results, together with the data obtained by ANSYS, are shown in figure 9. The first and third waves are the S0 mode, and the middle one is the A0 mode. A similar phenomenon to the displacements shown in figures 6 and 7 is observed. The data obtained by the proposed SFEs, shown in figure 9(a), agree well with the ones by ANSYS with very fine meshes; thus, the proposed SFEs with and without PZTs are verified. Due to the thickness locking, however, the results obtained by the SFE_3D, shown in figure 9(b), do not agree well with the ANSYS data for the anti-symmetric mode. It is demonstrated again that remedial measures should be taken to avoid thickness locking in formulating a 2D spectral plate element based on the simple plate theory.

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Table 4 compares the group velocities of the A0 and S0 modes with three different methods. Again, the accuracy of the proposed SFE is the highest. Due to the presence of PZT elements, the error appears slightly larger than the base plate without PZTs, shown in table 3. It is reported in [20] that the PZT layer tends to decrease the group velocities of all the wave modes. Due to thickness locking, the error of the group velocity of the A0 mode predicted by SFE_3D is much larger than the other two.

During the Lamb wave propagation, voltages will also be induced by the actuator PZT1, even after the input signal has finished. Figure 10 shows the output signal picked up by PZT1. With the group velocity of the S0 mode, listed in table 4, it is easy to distinguish the four Lamb wave modes. The first wave package with the largest voltage amplitude is excited by the input voltage signal, and the other three wave packages with much smaller amplitudes are all S0 modes, generated by the reflected Lamb waves coming from PZT2 and the left and right edges of the base plate, respectively.

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To demonstrate the capability of the proposed 2D SFE for 3D modeling of piezo-elasto dynamics for PZTs with circular shapes, a pitch-catch model in [17] is analyzed by using the SFE_σz. A 355.6 mm × 508.0 mm × 1.0 mm thick aluminum plate is instrumented with a pair of 6.35 mm radius, 0.25 mm thick PZTs on the upper surface of the plate, as shown in figure 11. PZT1 acts as an actuator, while PZT2 as a sensor. The entire plate is modeled by 17 920 proposed 2D SFEs, including eight SFEs with PZTs. For comparisons, the excitation signal and the material properties of the base plate and the PZTs are the same as the ones listed in [17].

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Snapshots of the y-displacement of the induced waves on the top surface of the plate at four different time instances are shown in figure 12. The excitation frequency is 100 kHz. It can be clearly seen that only the S0 mode and the A0 mode are excited. The S0 mode has a large wavelength and a fast velocity, while the A0 mode has a small wavelength and a low velocity. When the Lamb waves encounter the free boundaries of the plate, they are reflected back into the plate.

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Figure 13 presents the comparison of the voltage results at PZT2 simulated by the proposed 2D SFEs to the experimental data directly cited from [17]. The excitation frequency is 100 kHz. It is seen that the simulated results correlate well with the experimental data in terms of the waveform. A similar correlation between the experimental data and the hybrid spectral element [17] is observed. Since one hybrid spectral element was used in the thickness direction, the total number of DOFs is the same as in the present model, if the same number of elements is used.

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When the excitation frequency increases to 400 kHz, the simulated results by the proposed 2D SFEs are shown in figure 14. The agreement with the experimental data is not as good as the agreement shown in figure 13. Disagreements on the shape of the wave package, especially for the S0 mode, are clearly seen. A similar mismatch to the experimental data also exists for the results obtained by the hybrid spectral element in [17]. It is pointed out by Ha and Chang [17] that the mismatch is attributable to the interface adhesive layer since the effects of the adhesive layer are not considered in the simulations. In contrast to the case of 100 kHz, two hybrid spectral elements were used in the thickness direction to obtain reasonably accurate results. Since the present model uses one element in the thickness direction, the total number of DOFs is much smaller than the hybrid spectral element model in [17] for the case considered.

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