Efficient Layerwise Time-Domain Spectral Finite Element for Guided Wave Propagation Analysis of Multi-layered Panels

Abstract

Guided wave-based structural health monitoring techniques require accurate and fast simulation tools for high-frequency wave propagation in the laminate structures. This article develops an accurate and computationally efficient time-domain spectral finite element (SFE) for wave propagation analysis of laminated composite and sandwich beam and panel-type structures based on the efficient layerwise zigzag theory. It considers the axial displacement to follow a global third-order variation with a layerwise linear variation across the thickness. The independent variables are reduced to only three by imposing the interfacial continuity of transverse shear stress and shear traction-free conditions at the top and bottom surfaces. Regardless of the number of layers in the laminate, the element has only four degrees of freedom (DOFs) per node $u_0$, $w_0$, $\frac{dw_0}{dx}$, and $\psi _0$. The deflection $w_0$ is interpolated using the C$^1$-continuous Lobatto basis function, whereas $u_0$ and $\psi _0$ employ the C$^0$-continuous Lobatto basis shape functions. A thorough numerical study is accomplished to validate and evaluate the proposed element’s accuracy and efficiency for free vibration and Lamb wave propagation analysis of laminated composite and sandwich panels. The study reveals that the developed element is superior to its conventional counterpart and other existing 1D elements with a similar number of DOFs.

1 Introduction

Composite laminates are essential in constructing lightweight structures such as aircraft, wind turbine blades, sports, and medical equipment [1]. However, their heterogeneity causes complexity in their failure modes, requiring reliable online monitoring to prevent catastrophic failure. It has sparked a lot of study into developing reliable structural health monitoring (SHM) approaches for real-time damage identification in composite structures. Lamb waves provide a viable method of detecting damage in thin-walled constructions. [14]. Lamb wave-based SHM techniques require accurate and fast simulation tools for high-frequency wave propagation in the laminated structures.

The conventional finite element method (FEM) is a preferred technique for solving boundary value problems in complicated geometries. However, for the diagnostic Lamb waves (50 kHz–350 kHz), the conventional FEM is known to exhibit numerical dispersion and dissipation errors [8]. For circumventing this problem, various nonconventional FEMs have been reported.

One of the nonconventional FEM is the spectral element method (SEM). These frequency-domain spectral elements use a 1D beam or 2D plate models based on a variety of laminate theories, including the classical lamination theory (CLT) [12], the first-order shear deformation theory (FSDT) [11], and the advanced third-order zigzag theory [10]. However, the effective numerical assessment of the inverse Fourier transforms for multi-degree of freedom systems is computationally hard and prone to wrap-around error and instability.

The time-domain spectral finite element method (TDSFEM) is perhaps the most widely used special-purpose FEM for guided wave propagation problems. It is favored because of its simplicity and the fact that it retains the benefits of conventional FEM while also eliminating its drawbacks by using high-order orthogonal polynomials as shape functions. Till recently, all time-domain SFEs for composite structures employed laminate theories that require only C$^0$-continuity for the displacement interpolation functions [7, 13]. However, Jain and Kapuria [3] have presented a TDSFE for wave propagation analysis of isotropic beams based on the refined third-order shear deformation theory (TOT) that requires C$^1$-continuous interpolation functions for deflection [6]. The advanced efficient layerwise zigzag theory (ZIGT) has been developed, which incorporates the layerwise variation of in-plane displacement. Still, it has the same number of primary variables as the TOT. This theory has shown to be efficient and accurate for the static, dynamic, and buckling analysis of composite structures  [4, 5], and no TDSFE has been reported based on this theory.

This article presents TDSFE based on the ZIGT for wave propagation analysis of laminated composite and sandwich beam- and infinite panel-type structures. The axial displacement $u_0$, deflection $w_0$, slope $\frac{\hbox {d}w_0}{\hbox {d}x}$ and shear rotation $\psi _0$ are four DOFs per node in an element. The DOFs $w_0$ and $\frac{\hbox {d}w_0}{\hbox {d}x}$ are interpolated using the C$^1$-continuous Lobatto spectral interpolation functions [6], whereas C$^0$-continuous Lobatto basis functions for $u_0$ and $\psi _0$. Hamilton’s principle is used for deriving the governing equations of motion for the discretized system. A comprehensive numerical study is conducted to validate and assess the developed element’s accuracy and efficiency for free vibration and guided wave propagation response of composite and soft-core sandwich beams and panels under narrowband tone burst. The validation and assessment are done by comparing with the results available in the literature based on other laminate theories, analytical ZIGT solutions, and 2D elasticity-based FE solutions obtained using commercial software ABAQUS. Further, the spectral FE is compared with that of the conventional FE based on the same laminate theory.

2 Methodology

Consider a laminated composite beam with a total thickness of h and the number of layers L. According to the efficient layerwise zigzag theory (ZIGT), the axial displacement u is assumed to have a global third-order variation throughout the thickness with layerwise linear variation, whereas the transverse displacement w is constant across the laminate thickness. Considering that the displacement field is independent along the width (y) direction under both plane stress and plane strain assumptions, the displacement field is given as

$$\begin{aligned} u(x,z,t)= & {} u_k(x,t) - zw_{{0,x}}+z \psi _k(x,t) +z^2\xi (x,t)+z^3\eta (x,t) \nonumber \\ w(x,z,t)= & {} w_0(x,t) \end{aligned}$$

(1)

where $w_0$ is deflection at $z=0$, which is considered as the reference plane. $u_k$ and $\psi _k$ are the translation and shear rotation variables for the k$^{\hbox {th}}$ layer. The quadratic and cubic variations in z in the above equation are considered by $\xi$ and $\eta$. The partial differentiation with respect to a coordinate, such as x, is indicated by a subscript comma followed by that coordinate. The total displacement variables are $2L+3$, which can be represented in terms of only three primary variables by imposing the $2(L-1)$ continuity conditions of transverse shear stress $\tau _{zx}$ and in-plane displacement u at each layer interface, and two conditions of zero transverse shear traction at the top and bottom surfaces. After elimination, it yields

$$\begin{aligned} u(x,z,t)=u_0(x,t)-zw_{0_{,x}}(x,t) + R^k(z) \psi _0(x,t) \end{aligned}$$

(2)

where $u_0$ and $\psi _0$ are axial displacement and shear rotation at the reference plane, and $R^k(z)$ is a layerwise function of z for the $k^{\hbox {th}}$-layer, which depends on the material properties and stacking sequence of layers in the laminated beam.

Hamilton’s principle for laminated beams/panels subjected to a normal transverse loading of $q_z(x)$ per unit area reduces to

$$\begin{aligned}&\int _x[\langle \rho ^k \ddot{u}\delta u+\rho ^k \ddot{w}\delta w +\sigma _x\delta \varepsilon _x+\tau _{zx}\delta \gamma _{zx}\rangle +b q_z(x)\delta w_0(x,t)]\,\hbox {d}x \nonumber \\&\qquad \qquad -\langle \sigma _x\delta u+\tau _{zx}\delta w_0\rangle \big |_x= 0, \qquad \forall \, \delta u_0, \delta w_0, \delta \psi _0 \end{aligned}$$

(3)

where the notation $\langle \ldots \rangle =\sum _{k=1}^{L}\int _{z^+_{k-1}}^{z^-_k}(\ldots )b\,dz$ indicates integration across the entire laminate thickness, $\rho _k$ is the material mass density of the $k^{\hbox {th}}$ layer, and b is the beam width (=1 for infinite panels).

For the spectral finite element formulation, $u_0$ and $\psi _0$ are interpolated using the C$^0$-continuous Lobatto interpolation functions $\mathbf {N}$ based on the continuity criteria, whereas $w_0$ is interpolated using C$^1$-continuous Lobatto polynomial interpolation functions $\bar{\mathbf {N}}$. The details of these functions can be found in [3]. Thus, the displacement variables can be interpolated as

$$\begin{aligned} u_0=\mathbf {N} \boldsymbol{u_0^e},\qquad \psi _0=\mathbf {N}\boldsymbol{\psi _0^e}, \qquad w_0= \bar{\mathbf {N}}\boldsymbol{w_0^e} \end{aligned}$$

(4)

with

$$\begin{aligned} \begin{array}{ll} \boldsymbol{u_0^e}=\left[ \begin{array}{llll} u_{0_{1}} &{} u_{0_{2}} &{} ... &{} u_{0_{n}}\\ \end{array} \right] ^{\hbox {T}}, \quad \boldsymbol{\psi _0^e}=\left[ \begin{array}{llll} \psi _{0_{1}} &{} \psi _{0_{2}} &{} ... &{} \psi _{0_{n}}\\ \end{array}\right] ^{\hbox {T}} \\ \boldsymbol{w_0^e} =\left[ \begin{array}{lllllll} w_{0_1} &{} \theta _1 &{} w_{0_2} &{} \theta _2 &{} ... &{} w_{0_n} &{} \theta _n \\ \end{array}\right] ^{\hbox {T}} \end{array} \end{aligned}$$

(5)

After substituting above interpolation in variational Eq. (3), we find the element mass matrix ${\textbf {M}}^{\boldsymbol{e}}$, stiffness matrix ${\textbf {K}}^{\boldsymbol{e}}$ and load vector ${\textbf {P}}^{\boldsymbol{e}}$ as

$$\begin{aligned} \mathbf {M}^{\boldsymbol{e}}=\int _0^l {\hat{\mathbf {N}}^{\hbox {T}}} {\tilde{\mathbf {I}}} {\hat{\mathbf {N}}}\,\hbox {d}x, \quad \mathbf {K}^{\boldsymbol{e}}=\int _0^l {\hat{\mathbf {B}}^{\hbox {T}}} {\bar{\mathbf {D}}} {\hat{\mathbf {B}}}\,\hbox {d}x, \quad \mathbf {P}^{\boldsymbol{e}}=\int _0^l {\hat{\mathbf {N}}^{\hbox {T}}} \boldsymbol{f_{u}}\,\hbox {d}x. \end{aligned}$$

(6)

where $\bar{\mathbf {D}}$ and $\tilde{\mathbf {I}}$ are the stiffness coefficient matrix and inertia matrix, respectively, which depends on the material properties and lay-up of the beam with

$$\begin{aligned} \begin{array}{ll} \hat{\mathbf {N}}=\left[ \begin{array}{ccc} \mathbf {N}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}-\bar{\mathbf {N}}_{,x}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {N}\\ \mathbf {0}&{}\bar{\mathbf {N}}&{}\mathbf {0}\\ \end{array} \right] , \qquad \hat{\mathbf {B}}=\left[ \begin{array}{ccc} \mathbf {N}_{,x}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}-\bar{\mathbf {N}}_{,xx}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {N}_{,x}\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {N}\\ \end{array} \right] .\end{array} \end{aligned}$$

(7)

We generate the system of equations by adding the contributions of all the finite elements and using the typical assembly technique of the element matrices as

$$\begin{aligned} \mathbf {M} \boldsymbol{\ddot{\mathbf {U}}}+ {\mathbf {C}} \boldsymbol{\dot{\mathbf {U}}}+ {\mathbf {K}} {\mathbf {U}} ={\mathbf {P}},\end{aligned}$$

(8)

where ${\mathbf {M}}$ and ${\mathbf {K}}$ denote the system mass and stiffness matrices, whereas ${\mathbf {U}}$ and ${\mathbf {P}}$ are the global displacement and load vector, respectively. A Rayleigh damping term $\mathbf {C} \dot{\mathbf {U}}$ has been incorporated in Eq. (8) to account for any viscous damping. The damping matrix $\mathbf {C}=\alpha _1 \mathbf {M} + \alpha _2\mathbf {K}$, where $\alpha _1$ and $\alpha _2$ are the mass and stiffness damping coefficients, respectively.

3 Numerical Results and Discussion

3.1 Free Vibration Analysis

Firstly, consider the free vibration analysis of laminated composite and sandwich beams to validate the developed SFE formulation based on the zigzag theory (ZIGT). The same problem has been analyzed earlier in [4] using an analytical solution based on the ZIGT. The composite beam is made of four layers of graphite-epoxy (Table 1) having a stacking sequence of [0$^\circ$/90$^\circ$/90$^\circ$/0$^\circ$] with equal thickness, whereas the three-layer sandwich beam has graphite-epoxy faces and soft-core (Table 1) with 0.1h/0.8h/0.1h thickness.

Table 1. Material properties

The converged solution for non-dimensional natural frequencies of the first six modes is obtained using only five eight-node spectral finite elements based on the ZIGT. Kapuria et al. [4] have presented the same results using the analytical ZIGT and 2D elasticity-based exact solutions. The results for the composite and sandwich beams are presented in Table 2 and 3, respectively. It is seen that the present solution is in excellent agreement with the analytical ZIGT and the 2D exact solutions.

Table 2. Non-dimensional natural frequencies $\bar{\omega }_n = \omega _n (a^2/h) \sqrt{\rho /Y_2}$ of the first 6 modes of the simply-supported composite beam with [0$^\circ$/90$^\circ$/90$^\circ$/0$^\circ$] lay-up

Table 3. Non-dimensional natural frequencies $\bar{\omega }_n = \omega _n (a^2/h) \sqrt{\rho /Y_2}$ of the first 6 modes of the simply-supported sandwich beam

3.2 Lamb Wave Propagation

For assessing the performance of the present TDSFE, the propagation of Lamb waves in a composite panel generated by a piezoelectric wafer transducer attached to the panel’s top surface is examined. The composite panel is made of a four-layer CFRPT800S/3900-2 laminate [0$^\circ$/90$^\circ$]$_S$ whose unidirectional lamina material properties are listed in Table 1. The panel is of 1 m length and unit width, whereas the thickness of each layer is 0.5 mm. The stress transfer at the ends of a piezoelectric wafer patch actuator of length a is considered to occur under the electric potential excitation following the pin-force model [2]. It is represented by the pair of forces $\pm F(t)$ at edges of the actuator on the panel surface, as shown in Fig. 1(a). The piezoelectric wafer patch is 6.35 mm in length. The excitation force F(t) is a narrowband Hann window modulated sinusoidal tone burst signal, $F(t) = \frac{F_0}{2}\left[ 1-\cos \frac{2\pi (t-t_0)}{T_H} \right] \sin (2 \pi f (t-t_0))$, where f is the central frequency of the modulated signal, and $T_H$ is the Hann window width which is obtained from the number of the cycles ($N_B$) as $T_H=N_B/f$. For the present example, $F_0=1$ N, $t_0 = 0\mu s$, $f=100$ kHz and $N_B=5$.

Fig. 1.

Configuration of panel considered for wave propagation analysis showing (a) pin-force model actuation at transducer edges B$'$ and C$'$ in the 2D model and (b) equivalent force system in 1D models. S denotes the sensing location.

A 2D elasticity-based FE solution is developed using the commercial FE analysis software ABAQUS, employing 2000 and 4 eight-node quadrilateral plane strain elements (CPE8R) across length and thickness directions, respectively, to ascertain the accuracy of the present solution. The response is measured at point S on the panel’s top surface at a distance of 152.4 mm from the actuator’s center. In the 1D model, the forces ($\pm F$) at the top surface are replaced by an equivalent force system with the same forces ($\pm F$) and associated moments () at positions B and C (Fig. 1(b)) on the panel’s mid-plane. However, for the present ZIGT, we need to apply also higher-order bending moments $\pm P (= F(t) \, R^L(h/2))$ at B and C. The analysis is performed for two values of the excitation frequency, $f=50$ kHz and 100 kHz. For converged solutions, the present model requires 30 and 60 eight-node elements for 50 kHz and 100 kHz excitation frequency, respectively. To guarantee that the mesh in the actuator area BC is fine enough, we mesh the remaining three parts AB, CS, and SD so that each section has nearly equal length elements.

We also use the FSDT to solve this problem to compare the relative performances of different 1D theories. In ABAQUS, the FSDT-based eight-node shell element (S8R) with a fine mesh of 2000 (length) $\times$ 8 (width) for a plate of the same length (1 m) and 10 mm width. The shear correction factors are assumed to be 5/6. For imposing the plane strain requirement, displacement DOFs of all nodes in the width direction (y) and rotation around the x-axis were constrained. The Newmark-$\beta$ time-marching scheme was used in both 1D and 2D models, with a time step of $\varDelta$T = $2\times 10^{-7}$ for 50 kHz excitation and $1\times 10^{-7}$ for 100 kHz excitation, respectively.

Figure 2 compares the time history output of the in-plane displacement u obtained from all the above 1D and 2D models. The comparison reveals that the present ZIGT SFEM solution is in excellent agreement with the 2D FE solution for both S$_0$ and A$_0$ modes waveform. The FSDT solution, on the other hand, underestimates the peak amplitude and overestimates the ToA for the A$_0$ mode for all the excitation frequencies and yields a grossly erroneous response.

Fig. 2.

In-plane displacement-time history response based on ZIGT SFE, FSDT FE and 2D FE at location S on the top surface of the composite panel under 5-cycle tone burst signal actuation

3.3 Comparison of SFE with Conventional FE

For the Lamb wave propagation problem mentioned above, we compare the performance of the developed SFE to that of the conventional FE based on the ZIGT, which can be developed by the two-node beam/strip element employing C$^1$-continuous Hermite interpolation for $w_0$ and linear Lagrange interpolation for $u_0$ and $\psi _0$. The comparison is conducted for two values of the 5-cycle modulated tone burst signal’s excitation frequency f = 50 kHz and 100 kHz, evaluated with a time step of $\varDelta T$ = $5 \times 10^{-7}$ s and $2 \times 10^{-7}$ s, respectively. For assessment, the ZIGT FE solution with a fine mesh of 6000 two-node elements has been obtained, which serves as the reference solution. The time-history response for the in-plane displacement u at the sensor position S, computed using eight-node spectral ZIGT elements, is compared to the reference solution in Fig. 3. For 50 kHz and 100 kHz excitations, the converged SFE solutions needed 30 and 60 elements, respectively. For each excitation, the present solution closely resembles the reference solution. In the same figure, conventional FE solutions are plotted based on ZIGT with an equivalent number of DOFs. The latter solution for the A$_0$ mode shows significant error in terms of peak amplitude and ToA.

Fig. 3.

In-plane displacement-time history response at location S on the top surface of the composite panel under 5-cycle tone burst signal actuation

3.4 Lamb Wave Propagation Under Transverse Excitation

For further validation and assessment of the proposed element, we consider the Lamb wave propagation of sandwich beam, which has been earlier analyzed by Nanda et al. [10] using frequency domain SFE based on the ZIGT. The sandwich beam comprises three layers of face and core whose material properties are shown in Table 1. We have considered the case of a beam (width b = 10 mm) with a span to thickness ratio a/h = 10 and the core to face thickness ratio $h_c/h_f$ = 8. A transverse point force F(t) is applied at the tip of the cantilever beam. The excitation force F(t) is a narrowband Hann window modulated sinusoidal tone burst signal with an amplitude of $F_0$ = 1N, f = 80 kHz, and $t_0$ = 125 $\upmu$s.

The present SFE yields the converged solution of the transverse velocity $\dot{w}_0$ at the mid-span of the beam using only 20 eight-node elements. Nanda et al. [10] presented the same response using frequency-domain SFE and used the inverse Fourier transform to get a time-domain response. For comparison, we also obtained the same response using conventional FE with an equivalent number of DOFs. In all the above models, we use the Newmark-$\beta$ time integration scheme with a time step of $\varDelta T$= 1 $\upmu$s. Figure 4 represents the comparison of the above solutions. It is seen that the present solution matches the reference solution [10] very well. In contrast, the conventional FE with equivalent DOFs shows significant errors in the peak amplitude and ToA for both A$_0$- and S$_0$-mode waveforms. In addition, the latter results show spurious oscillations at the end, which are well-known errors of the conventional FEM.

Fig. 4.

Transverse velocity-time history response at the mid-span the beam under 5-cycle tone burst signal actuation of 80 kHz excitation frequency

4 Conclusions

A TDSFE based on the ZIGT has been developed to analyze guided wave propagation in laminated composite and sandwich beam/panel-type structures. The element interpolates $w_0$ and $\frac{\hbox {d}w_0}{\hbox {d}x}$ using the C$^1$-continuous Lobatto spectral interpolation functions established in [6], while $u_0$ and $\psi _0$ are interpolated using the C$^0$-continuous Lobatto basis functions. The developed element for the wave propagation response of laminated composite and sandwich beams and panels under narrowband tone burst has been demonstrated in a rigorous numerical investigation, leading to the following conclusions:

• 1. The ZIGT-based SFE yields excellent accuracy for the natural frequencies with respect to the 2D elasticity-based exact solutions.

• 2. For Lamb wave propagation problems in composite panels under narrowband modulated tone burst excitations, all 1D theories predict the S$_0$ mode response with the same accuracy. Their estimations for the A$_0$ mode, on the other hand, diverge significantly. The present ZIGT SFEM solutions for the A$_0$ mode closely match the 2D elasticity-based FE solutions. The FSDT solution predicts major errors in the peak amplitude and the ToA.

• 3. The present TDSFE gives accurate results for soft-core sandwich beams in comparison with the frequency domain SFE.

• 4. In terms of accuracy, the current SFE outperforms the conventional FE with equal DOFs for straight-crested Lamb wave propagation problems.

In summary, the proposed TDSFE based on the ZIGT is an excellent tool for handling wave propagation problems in laminated composite and sandwich beams and panels than the conventional ZIGT FE. It predicts more accurately than existing laminate theories-based 1D elements. The element will be useful when using model-based SHM approaches for laminated composite and sandwich structures that demand fast and accurate wave propagation response computation.