Effect of carbon nanotube waviness on smart damping of geometrically nonlinear vibrations of fuzzy-fiber reinforced composite plates

Abstract

In this article, we present a finite element model for the three-dimensional analysis of smart constrained layer damping of geometrically nonlinear vibrations of laminated fuzzy-fiber reinforced composite plates. The three-dimensional fractional derivative constitutive relation is implemented for the viscoelastic layer. The constraining layer of the smart constrained layer damping treatment is composed of the vertically/obliquely reinforced 1–3 piezoelectric composites. The von Kármán–type nonlinear strain–displacement relations are used to incorporate the geometric nonlinearity in the model. The main aim of this article is to numerically investigate the effect of carbon nanotube waviness on the nonlinear smart damping. Several thin laminated substrate fuzzy-fiber reinforced composite plates with straight carbon nanotubes and wavy carbon nanotubes with different waviness in different planes are considered with various boundary conditions and stacking sequences to numerically compute their effect on smart damping. The performance of the obliquely reinforced 1–3 piezoelectric composites is discussed and the efficacy of the present smart finite element model in terms of active control authority is also presented.

Keywords

Carbon nanotube smart damping nonlinear vibrations fractional derivative model

1. Introduction

Since the discovery in 1991 (Iijima, 1991), carbon nanotubes (CNTs) have generated remarkable research activities due to their unprecedented material properties. With the superlative thermo-mechanical properties, CNTs have emerged as the new class of reinforcing phase for developing novel composites (Harris, 2004; Jin et al., 1998; Lau and Hui, 2002; Thostenson and Chou, 2003) with improved thermal properties and mechanical strength. In many of the investigated novel composites, polymer matrix is reinforced with dispersed and aligned CNTs as the consolidation temperature required for polymer is comparatively lower than that required for ceramics and metals (Bakshi et al., 2010; Odegard et al., 2003; Paiva et al., 2004). However, using CNTs as the primary reinforcements posed certain challenges like difficulty in producing long CNTs, proper dispersion to avoid clustering of CNTs and aligning CNTs to obtain increased reinforcing effect (García et al., 2008b). On the contrary, using fibers with radially grown CNTs on its surface as reinforcement in the polymer matrix is more practical to minimize the aforementioned challenges. CNTs can be grown on carbon fibers using chemical vapor deposition (CVD) and embedding them in the polymer matrix results in a multiscale composite as there is a huge difference in length scale of CNTs and the carbon fiber (Thostenson et al., 2002). Experimental results reported by Mathur et al. (2008) reveal significant increase in the mechanical properties with the increase in the amount of deposited CNTs on the surface of the carbon fiber. García et al. (2008a, 2008b) demonstrated the fabrication and multifunctional properties of a hybrid and advanced laminated composite where aligned CNTs are radially grown in situ on the alumina fibers and the advanced fibers are embedded into the polymeric matrix. The resulting advanced fibers are called a “fuzzy fiber” (García et al., 2008b) and the hybrid laminate with improved mechanical properties is termed as fuzzy-fiber reinforced composite (FFRC). Soliman et al. (2012) investigated the effect of multi-walled carbon nanotubes (MWCNTs) on the tension and in-plane shear behaviors of carbon fiber-reinforced polymer (FRP) composites. Their investigation reported that using 1.5 wt% functionalized MWCNTs, the in-plane failure strain, ultimate strength, and toughness are improved by 39%, 51%, and 121%, respectively. More recently, Kundalwal and Ray (2011, 2013) have developed micromechanical models for FFRC with radially grown straight CNTs (Kundalwal and Ray, 2011) and periodically wavy CNTs (Kundalwal and Ray, 2013) to predict the elastic properties of the same. The predicted results indicate improvement in the elastic properties of the FFRC with the wavy CNTs over those with the straight CNTs. Boroujeni et al. (2014) implemented a relatively low temperature synthesis technique to grow CNTs over the carbon fibers to experimentally obtain the enhanced mechanical properties of FRP composites. The results show improvement of the on-axis tensile strength and ductility of the hybrid FRPs by 11% and 35%, respectively, and 16% improvement on the off-axis stiffness. Bian and Zhao (2015) established a temperature-dependent and thickness-independent model to compute the elastic properties of single-walled carbon nanotubes (SWCNTs) under a thermal environment and found that Young’s moduli of armchair and zigzag nanotubes decrease slightly when the environmental temperature increases, whereas for a given temperature, Young’s moduli of both armchair and zigzag nanotubes increase with increasing tube diameters. The theoretical and experimental results confirm increase in strength-to-weight ratio of the FFRCs over the composites without CNTs making them more attractive and better candidate for structural and space applications.

Unnecessary vibration of dynamically loaded structures may lead to fatigue failure of the same. The vibration level can be controlled by imposing external damping into the flexible host structure by bonding a viscoelastic layer to it where the viscoelastic layer is called as the free-layer or “unconstrained” treatment (Stanway et al., 2003). Another attractive approach is to exploit the direct and the converse piezoelectric effects inherently present in the piezoelectric materials and use them as distributed actuators and sensors, respectively. However, instead of attaching the piezoelectric layer directly to the host structure, it is advantageous to use a viscoelastic layer in between such that the viscoelastic layer becomes constrained between the host structure and the piezoelectric layer which leads to smart constrained layer damping (SCLD) treatment. The flexural vibration control by the constrained layer damping treatment is due to the dissipation of energy in the constrained viscoelastic layer. As the dissipation of energy is mostly due to the transverse shear deformations of the constrained layer, it can be increased by activating the constraining piezoelectric layer of the SCLD treatment using appropriate control voltage. If the piezoelectric layer is left inactive, the standard passive constrained layer damping (PCLD) is achieved from SCLD treatment. Extensive review reveals that piezoelectric materials have been widely used by several researchers (Balamurugan and Narayanan, 2002; Baz and Ro, 1995; Lentze et al., 2007; Ray and Baz, 2001; Trindade, 2011; Yi et al., 1998; Zhang et al., 2015) for active control of vibrations of flexible lightweight structures.

Recently, piezoelectric composite (PZC) materials are being used as the distributed smart materials in which piezoelectric fibers are the reinforcement materials in epoxy matrix. Since the PZCs possess better electro-mechanical properties compared to the monolithic piezoelectric materials, their usage is also increasing. Among the different types of PZC materials studied by the researchers, the vertically and the obliquely reinforced 1–3 PZC materials are commercially available (Gentilman et al., 1996). Initially, Smith and Auld (1991) proposed a physical model of the 1–3 PZC to control the vibrations along the thickness directions. Later, Ray and Pradhan (2007) modified the existing model of Smith and Auld (1991) to obtain all the electroelastic properties of the vertically/obliquely reinforced 1–3 PZC. Application of the 1–3 PZCs for smart damping of linear and geometrically nonlinear vibrations of laminated composite, functionally graded, and magneto-electroelastic plates and shells can be found in Datta and Ray (2018b), Kattimani and Ray (2018), Panda and Ray (2008), Ray and Pradhan (2007), and Sahoo and Ray (2018). For the time-domain analysis, the constrained viscoelastic layer of the SCLD treatment is modeled by the Golla–Hughes–McTavish (GHM) (Golla and Hughes, 1985; McTavish and Hughes, 1993) method. McTavish and Hughes (1993) extended the Golla–Hughes model (Golla and Hughes, 1985) to formulate the GHM model modeling the material modulus as a series of mini-oscillators. In the GHM model, the finite element model (FEM) derived in the Laplace domain needs to be retransformed to obtain the finite element (FE) governing equations in time-domain. Auxiliary dissipative coordinates are introduced to account for the energy dissipation of the viscoelastic material. Many other methods are available in the open literature (Bagley and Torvik, 1983a, 1983b, 1985; Lesieutre and Bianchini, 1995; Lesieutre and Mingori, 1990) for modeling linear viscoelastic material such as the augmenting thermodynamic fields (ATF) (Lesieutre and Mingori, 1990), the anelastic displacement field (ADF) (Lesieutre and Bianchini, 1995), and the fractional derivative model (FDM) (Bagley and Torvik, 1983a, 1983b, 1985). Bagley and Torvik (1983a) used molecular theories to predict the macroscopic behavior of certain viscoelastic media and showed that the predicted results of these molecular theories are equivalent to constitutive relationships expressed in terms of the fractional calculus. Later using the fractional calculus approach, they proposed an effective time-domain model of the viscoelastic material named FDM (Bagley and Torvik, 1983b, 1985). Modeling of viscoelastic material in time domain for implementing different modern control strategies is a critical issue especially in case of smart damping of nonlinear vibrations of structures involving large computational effort. The use of additional dissipative coordinates to model the viscoelastic material using the GHM, ATF, or ADF method results in increase in the overall generalized degrees of freedom and the model becomes computationally costly. The FDM of viscoelastic material does not increase the overall generalized degrees of freedom and thus is an efficient method for modeling viscoelastic layer in time-domain (Datta and Ray, 2016b). Galucio et al. (2004, 2005) presented an FEM for transient dynamic analysis and SCLD of sandwich beams using one-dimensional fractional derivative constitutive equations for viscoelastic layer. Furthermore, the three-dimensional form of fractional derivative constitutive equations for modeling the constrained viscoelastic layer has been introduced by the authors for the SCLD treatment (Datta and Ray, 2015, 2018a). Significant interest has been grown in static, dynamic, and buckling analyses of CNT reinforced composite structures (Alibeigloo, 2014; Arani et al., 2011; Fan and Wang, 2015; Formica et al., 2010; Heshmati et al., 2015; Shen and Xiang, 2014; Zhu et al., 2012) and active control of the same (Mirghaffari and Rahmani, 2017; Song et al., 2016a, 2016b) in last few years. Recently, Kundalwal et al. (2013) and Kundalwal and Ray (2016) have developed the linear FE model for active control of laminated FFRC plates using 1–3 PZCs. However, the effect of CNT waviness on the control authority of the SCLD patch for passive and active control of geometrically nonlinear vibrations of thin FFRC plates has not yet been investigated and here the authors intend to address this issue for developing much more lighter and superior smart structures.

The present work is concerned with developing the nonlinear FE model of FFRC plates to study the effect of CNT waviness on the smart damping treatment provided by the SCLD patch. The von Kármán strain–displacement relations are considered to incorporate the geometric nonlinearity in the FE model. Several equivalent single layer (ESL) theories are available in the literature (Datta and Ray, 2016a; Reddy, 2004) for analyzing composite structures based on FE model. However, for the present analyses, the first-order shear deformation theory (FSDT) with layerwise effect for the base structure, viscoelastic layer, and the 1–3 PZC layer has been used to model the axial displacements in all the layers of the overall plate, whereas quadratic form of displacement field has been used for the transverse displacement. Several thin laminated substrate FFRC plates with straight CNTs and wavy CNTs with different waviness in different planes are considered with various boundary conditions and stacking sequences to numerically compute their effect on smart damping. The performance of the obliquely reinforced 1–3 PZCs is discussed and the efficacy of the present smart FE model in terms of active control authority is also presented.

2. Basic equations for nonlinear plate analysis

The schematic diagram of the laminae made of FFRC with straight and wavy CNTs is presented in Figure 1(a) and (b), respectively. Also, the cross-sectional views of carbon fiber with radially grown wavy CNTs in the 1–3 and 2–3 planes are illustrated in Figure 2(a) and 2(b), respectively. Figure 3 depicts a rectangular substrate FFRC laminated plate and a rectangular patch is attached at the top surface to achieve smart damping. The patch of the SCLD treatment is composed of the constraining layer of the vertically/obliquely reinforced 1–3 PZC material and the constrained viscoelastic layer with thicknesses $h_{p}$ and $h_{v}$ , respectively, while $h$ denotes the thickness of the substrate FFRC laminated plate having $N$ number of unidirectional fuzzy-fiber reinforced laminae. The laminate coordinate system (xyz) is located on the mid-surface of the substrate laminate representing the reference plane and the lines $x = 0, a$ and $y = 0, b$ are along the boundaries of the same. The thickness coordinates (z) of the top and the bottom surfaces of any kth (k = 1, 2, 3, …, N + 2) layer of the overall plate are denoted by $h_{k + 1}$ and $h_{k}$ , respectively. On the contrary, $θ_{k}$ is the fiber orientation angle in any layer of the substrate plate (k = 1, 2, 3, …, N) in the plane (xy) of the lamina with respect to laminate coordinate system. As mentioned earlier, the orientation of the reinforcing piezoelectric fibers in the 1–3 PZC material may be vertical or oblique in $xz$ or yz planes making an angle ψ with respect to vertical z axis. The displacement field for axial deformations of the overall plate is based on the FSDT. However, a layerwise modification is incorporated to accommodate the drastic change in material properties along the thickness direction from the substrate to the viscoelastic layer and from the viscoelastic layer to the 1–3 PZC layer. The kinematics of the deformation is displayed in Figure 4 where the variables $u_{0}$ and $v_{0}$ are the generalized translational displacements of a point (x, y) on the reference plane (z = 0) along x and y directions, respectively. $θ_{x}$ , $ϕ_{x}$ , and $γ_{x}$ denote the generalized rotations of the normal to the middle planes of the substrate plate, the viscoelastic layer, and the 1–3 PZC layer, respectively, in the xz plane and similar rotations in the yz plane are denoted by $θ_{y}$ , $ϕ_{y}$ , and $γ_{y}$ , respectively. Thus, the axial displacements $u (x, y, z, t)$ and $v (x, y, z, t)$ of a point in any layer of the overall plate along the x and y directions, respectively, can be mathematically expressed as follows

\begin{matrix} u (x, y, z, t) = u_{0} (x, y, t) + {\begin{matrix} z θ_{x} (x, y, t), \forall (- \frac{h}{2} \leq z \leq \frac{h}{2}) \\ \frac{h}{2} θ_{x} (x, y, t) + (z - \frac{h}{2}) ϕ_{x} (x, y, t), \forall (\frac{h}{2} \leq z \leq h_{N + 2}), \\ \frac{h}{2} θ_{x} (x, y, t) + h_{v} ϕ_{x} (x, y, t) + (z - h_{N + 2}) γ_{x} (x, y, t), \forall (h_{N + 2} \leq z \leq h_{N + 3}) . \end{matrix} \\ v (x, y, z, t) = v_{0} (x, y, t) + {\begin{matrix} z θ_{y} (x, y, t), \forall (- \frac{h}{2} \leq z \leq \frac{h}{2}), \\ \frac{h}{2} θ_{y} (x, y, t) + (z - \frac{h}{2}) ϕ_{y} (x, y, t), \forall (\frac{h}{2} \leq z \leq h_{N + 2}), \\ \frac{h}{2} θ_{y} (x, y, t) + h_{v} ϕ_{y} (x, y, t) + (z - h_{N + 2}) γ_{y} (x, y, t), \forall (h_{N + 2} \leq z \leq h_{N + 3}) . \end{matrix} \end{matrix}

(1)

The displacement field is chosen in such a way that the interface continuity of displacements between the substrate and the viscoelastic layer and between the viscoelastic layer and the 1–3 PZC layer is maintained. It may be noted that the magnitude of the transverse piezoelectric coefficient of the 1–3 PZC layer is much greater than that of the in-plane piezoelectric coefficients (Ray and Pradhan, 2007). Thus, control of the flexural vibrations of the substrate plate can be obtained properly only if the transverse normal strain in the overall plate is considered in the present model. The transverse displacement $w (x, y, z, t)$ at any point in the overall structure may be assumed to have quadratic variation with respect to $z$ across their thicknesses and can be expressed as follows

\begin{matrix} w (x, y, z, t) = w_{0} (x, y, t) + z θ_{z} (x, y, t) + z^{2} ϕ_{z} (x, y, t), \\ \forall (- \frac{h}{2} \leq z \leq h_{N + 3}) \end{matrix}

(2)

where $w_{0}$ refers to the transverse displacement at any point on the reference plane, $θ_{z}$ and $ϕ_{z}$ are the generalized displacements representing the gradient and the second-order derivative of the transverse displacement in the overall structure with respect to the thickness coordinate (z). Such quadratic variation of transverse displacement ensures parabolic distribution of transverse shear stress across the thickness of the overall structure. For introducing geometric nonlinearity in the model, von Kármán–type strain–displacement relations are considered as follows

\begin{matrix} ε_{x} = \frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w_{0}}{\partial x})}^{2}, ε_{y} = \frac{\partial v}{\partial y} + \frac{1}{2} {(\frac{\partial w_{0}}{\partial y})}^{2}, \\ ε_{z} = \frac{\partial w}{\partial z}, γ_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \frac{\partial w_{0}}{\partial x} \frac{\partial w_{0}}{\partial y}, \\ γ_{xz} = \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}, and γ_{yz} = \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} \end{matrix}

(3)

For condensing the generalized rotational degrees of freedom in a straightforward manner such that the model becomes computationally efficient during time-domain analysis, the generalized translational and rotational vectors are grouped as follows

{d_{t}} = {[\begin{matrix} u_{0} & v_{0} & w_{0} \end{matrix}]}^{T} and {d_{r}} = {[\begin{matrix} {Θ_{r}} & {Φ_{r}} {Γ_{r}} \end{matrix}]}^{T}

(4)

where ${Θ_{r}} = [\begin{matrix} θ_{x} & θ_{y} & θ_{z} \end{matrix}]$ , ${Φ_{r}} = [\begin{matrix} ϕ_{x} & ϕ_{y} & ϕ_{z} \end{matrix}]$ , and ${Γ_{r}} = [\begin{matrix} γ_{x} & γ_{y} \end{matrix}]$ .

In order to implement the selective integration rule for computing the stiffness matrices corresponding to transverse shear deformations, the state of strain at any point in the overall system is divided into the bending strain vector ${ε_{b}^{k}}$ and the out of plane shear strain vector ${ε_{s}^{k}}$ as follows

\begin{matrix} {ε_{b}^{k}} = {\begin{matrix} ε_{x}^{k} & ε_{y}^{k} & γ_{xy}^{k} & ε_{z}^{k} \end{matrix}}^{T} and {ε_{s}^{k}} = {\begin{matrix} γ_{xz}^{k} & γ_{yz}^{k} \end{matrix}}^{T}, \\ k = 1, 2, 3, \dots, N + 2 \end{matrix}

(5)

where $ε_{x}^{k}$ , $ε_{y}^{k}$ , and $ε_{z}^{k}$ are the normal strains along x, y, and z directions, respectively; $γ_{xy}^{k}$ is the in-plane shear strain; and $γ_{xz}^{k}$ , $γ_{yz}^{k}$ are the transverse shear strains at any point in the kth layer. Using equations (1) and (2) in equation (3), the bending and shear strain vectors are expressed in terms of the generalized displacement vectors. Thus, the bending strain vectors in the substrate plate, the constrained viscoelastic layer, and the active constraining layer can be expressed as follows

\begin{matrix} {ε_{b}^{k}} = {ε_{bt}} + [Z_{1}] {ε_{br}} + {ε_{bnl}}, k = 1, 2, 3, \dots, N, \\ {ε_{b}^{N + 1}} = {ε_{bt}} + [Z_{3}] {ε_{br}} + {ε_{bnl}} and \\ {ε_{b}^{N + 2}} = {ε_{bt}} + [Z_{5}] {ε_{br}} + {ε_{bnl}} \end{matrix}

(6)

It may be noted that the layer numbers N + 1 and N + 2 represent the viscoelastic layer and the piezoelectric layer, respectively. Similarly, the transverse shear strain in the substrate plate, the constrained viscoelastic layer, and the active constraining layer are given by

\begin{matrix} {ε_{s}^{k}} = {ε_{st}} + [Z_{2}] {ε_{sr}}, k = 1, 2, 3, \dots, N, \\ {ε_{s}^{N + 1}} = {ε_{st}} + [Z_{4}] {ε_{sr}} and \\ {ε_{s}^{N + 2}} = {ε_{st}} + [Z_{6}] {ε_{sr}} \end{matrix}

(7)

The forms of the matrices $[Z_{β}]$ (β = 1, 2, …, 6) appearing in equations (6) and (7) have been elucidated in Appendix 1, while the generalized strain vectors are given by

\begin{matrix} {ε_{bt}} = [\nabla_{bt}] {d_{t}}, {ε_{st}} = [\nabla_{st}] {d_{t}}, \\ {ε_{br}} = {\begin{matrix} [\nabla_{br 1}] {Θ_{r}}^{T} \\ [\nabla_{br 1}] {Φ_{r}}^{T} \\ [\nabla_{br 2}] {Γ_{r}}^{T} \end{matrix}}, {ε_{sr}} = {\begin{matrix} [\nabla_{sr 1}] {Θ_{r}}^{T} \\ [\nabla_{sr 1}] {Φ_{r}}^{T} \\ I_{2} {Γ_{r}}^{T} \end{matrix}} and \\ {ε_{bnl}} = \frac{1}{2} {[\begin{matrix} {(\frac{\partial w_{0}}{\partial x})}^{2} & {(\frac{\partial w_{0}}{\partial y})}^{2} & 2 \frac{\partial w_{0}}{\partial x} \frac{\partial w_{0}}{\partial y} & 0 \end{matrix}]}^{T} \end{matrix}

(8)

The matrices $[\nabla_{bt}]$ , $[\nabla_{st}]$ , $[\nabla_{br 1}]$ , $[\nabla_{br 2}]$ , and $[\nabla_{sr 1}]$ contain the partial derivative operators and explicitly shown in Appendix 2 and $I_{2}$ is 2 × 2 identity matrix. Similar to the strain vectors given by equation (5), the state of stresses at any point in the overall plate are described by the following bending and transverse shear stress vectors

{σ_{b}^{k}} = {[\begin{matrix} σ_{x}^{k} & σ_{y}^{k} & σ_{xy}^{k} & σ_{z}^{k} \end{matrix}]}^{T} and {σ_{s}^{k}} = {[\begin{matrix} σ_{xz}^{k} & σ_{yz}^{k} \end{matrix}]}^{T}

(9)

where $σ_{x}^{k}$ , $σ_{y}^{k}$ , and $σ_{z}^{k}$ are the normal stresses along x, y, and z directions, respectively; $σ_{xy}^{k}$ is the in-plane shear stress; and $σ_{xz}^{k}$ , $σ_{yz}^{k}$ are the transverse shear stresses at any point in the kth layer.

2.1. Constitutive relation for the FFRC material

As the laminated FFRC plate is made of several fuzzy-fiber reinforced laminae (Figure 3) with their principal material axes arbitrarily oriented with respect to the laminate coordinate system $(xyz)$ , the constitutive equations of each layer are transformed to the laminate coordinate system (Reddy, 2004). Thus, the stress–strain relations for the kth off-axis lamina with respect to the laminate coordinate system can be arranged in the context of the present formulation as follows

\begin{matrix} {σ_{b}^{k}} = [{\bar{C}}_{b}^{k}] {ε_{b}^{k}} and {σ_{s}^{k}} = [{\bar{C}}_{s}^{k}] {ε_{s}^{k}}, \\ k = 1, 2, 3, \dots, N \end{matrix}

(10)

where $[{\bar{C}}_{b}^{k}]$ and $[{\bar{C}}_{s}^{k}]$ are the transformed elastic coefficient matrices with respect to the laminate coordinate system. The elastic coefficient matrices $[{\bar{C}}_{b}^{k}]$ and $[{\bar{C}}_{s}^{k}]$ are explicitly presented in Appendix 3.

2.2. FDM of the viscoelastic material

The three-dimensional FDM is implemented for the viscoelastic layer and is derived using the four-parameter one-dimensional fractional derivative constitutive relation. The detailed mathematical formulation for the three-dimensional FDM is presented elsewhere (Datta and Ray, 2015, 2018a). In the context of present formulation, considering the form presented in equations (5) and (9) for the strains and stresses, respectively, the three-dimensional FDM for any $(n + 1) th$ time step is considered as follows (Datta and Ray, 2015)

\begin{matrix} {σ_{b}^{N + 1}} + τ^{α} \frac{d^{α}}{d t^{α}} {σ_{b}^{N + 1}} = [C^{N + 1}] {ε_{b}^{N + 1}} \\ + τ^{α} \frac{E_{\infty}}{E_{0}} [C^{N + 1}] \frac{d^{α}}{d t^{α}} {ε_{b}^{N + 1}}, \\ {σ_{s}^{N + 1}} + τ^{α} \frac{d^{α}}{d t^{α}} {σ_{s}^{N + 1}} = G_{0} {ε_{s}^{N + 1}} + τ^{α} G_{\infty} \frac{d^{α}}{d t^{α}} {ε_{s}^{N + 1}} \end{matrix}

(11)

where $G_{0}$ and $G_{\propto}$ are the relaxed and non-relaxed shear moduli, respectively, and $(G_{\infty} / G_{0}) = (E_{\infty} / E_{0})$ , with $E_{0}$ and $E_{\propto}$ being the relaxed and non-relaxed elastic moduli, $τ$ is the relaxation time and $α$ is the fractional order of the time derivative $(0 < α < 1)$ . $[C^{N + 1}]$ is the elastic coefficient matrix of the viscoelastic layer and has a form similar to that of isotropic material. At this juncture, an internal variable (Galucio et al., 2004) can be introduced in the form of two anelastic strain functions as follows

\begin{matrix} {σ_{b}^{N + 1}} = \frac{E_{\infty}}{E_{0}} [C^{N + 1}] ({ε_{b}^{N + 1}} - {{\bar{ε}}_{b}^{N + 1}}) \\ {σ_{s}^{N + 1}} = G_{\infty} ({ε_{s}^{N + 1}} - {{\bar{ε}}_{s}^{N + 1}}) \end{matrix}

(12)

where the anelastic bending strain vector ${{\bar{ε}}_{b}^{N + 1}}$ and the anelastic transverse shear strain vector ${{\bar{ε}}_{s}^{N + 1}}$ are given by

\begin{matrix} {{\bar{ε}}_{b}^{N + 1}} = {[\begin{matrix} {\bar{ε}}_{x}^{N + 1} & {\bar{ε}}_{y}^{N + 1} & {\bar{γ}}_{xy}^{N + 1} & {\bar{ε}}_{z}^{N + 1} \end{matrix}]}^{T} and \\ {{\bar{ε}}_{s}^{N + 1}} = {[\begin{matrix} {\bar{γ}}_{xz}^{N + 1} & {\bar{γ}}_{yz}^{N + 1} \end{matrix}]}^{T} \end{matrix}

(13)

Substituting equation (12) into equation (11), the following equations are obtained which contain only one fractional derivative term

\begin{matrix} {{\bar{ε}}_{b}^{N + 1}} + τ^{α} \frac{d^{α}}{d t^{α}} {{\bar{ε}}_{b}^{N + 1}} = \frac{E_{\infty} - E_{0}}{E_{\infty}} {ε_{b}^{N + 1}}, \\ {{\bar{ε}}_{s}^{N + 1}} + τ^{α} \frac{d^{α}}{d t^{α}} {{\bar{ε}}_{s}^{N + 1}} = \frac{E_{\infty} - E_{0}}{E_{\infty}} {ε_{s}^{N + 1}} \end{matrix}

(14)

Different definitions of fractional operator are available in the literature, out of which the best known are the Riemann–Liouville and Grünwald definitions (Schmidt and Gaul, 2002). However, the Grünwald definition is considered in the present analysis for its simplicity in numerical implementation. The fractional derivative operator for a function of time $f (t)$ , approximated by the Grünwald definition, can be expressed as follows (Schmidt and Gaul, 2002)

\frac{d^{α} f (t)}{d t^{α}} = \frac{1}{{(Δ t)}^{α}} \sum_{j = 0}^{N_{t}} A_{j + 1} f (t - j Δ t)

(15)

where $Δ t$ is the time increment for the numerical scheme and the upper limit $N_{t}$ is the number of responses in memory and seems to be somewhat arbitrary. However, mathematically it is always less than the total number of time steps. $A_{j + 1}$ is the so-called Grünwald constant and can be expressed by Gamma function (Γ) or recurrence formula as follows (Schmidt and Gaul, 2002)

A_{j + 1} = \frac{Γ (j - α)}{Γ (- α) Γ (j + 1)} or A_{j + 1} = \frac{j - α - 1}{j} A_{j}

(16)

Substituting the Grünwald definition of fractional derivative (equation (15)) into equation (14) and noting that $A_{1} = 1$ (Schmidt and Gaul, 2002), the following discretized general relations in the time domain are obtained at any $(n + 1) th$ time step

\begin{matrix} {{\bar{ε}}_{b}^{N + 1}}_{n + 1} = (1 - c) \frac{E_{\infty} - E_{0}}{E_{\infty}} {ε_{b}^{N + 1}}_{n + 1} \\ - c \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{ε}}_{b}^{N + 1}}_{n + 1 - j}, \\ {{\bar{ε}}_{s}^{N + 1}}_{n + 1} = (1 - c) \frac{E_{\infty} - E_{0}}{E_{\infty}} {ε_{s}^{N + 1}}_{n + 1} \\ - c \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{ε}}_{s}^{N + 1}}_{n + 1 - j} \end{matrix}

(17)

where the dimensionless parameter c is given by $c = (τ^{α} / τ^{α} + (Δ t)^{α})$ .

Finally, substituting the relations given in equation (17) into equation (12), the discretized forms of the constitutive relations of the viscoelastic material appropriate for three-dimensional analysis are obtained as follows

\begin{matrix} {σ_{b}^{N + 1}}_{n + 1} = [C_{b}^{N + 1}] {ε_{b}^{N + 1}}_{n + 1} + [{\bar{C}}_{b}^{N + 1}] \\ \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{ε}}_{b}^{N + 1}}_{n + 1 - j}, \\ {σ_{s}^{N + 1}}_{n + 1} = [C_{s}^{N + 1}] {ε_{s}^{N + 1}}_{n + 1} + [{\bar{C}}_{s}^{N + 1}] \\ \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{ε}}_{s}^{N + 1}}_{n + 1 - j} \end{matrix}

(18)

where $[C_{b}^{N + 1}]$ , $[{\bar{C}}_{b}^{N + 1}]$ , $[C_{s}^{N + 1}]$ , and $[{\bar{C}}_{s}^{N + 1}]$ are explicitly given in Appendix 3.

2.3. Constitutive relation for the 1–3 PZC material

Since the top and bottom surfaces of the 1–3 PZC layer are electroded, the voltage difference is applied across the thickness and accordingly the electric field is generated only along the thickness direction of the actuating layer. Thus, in the present analysis, the constraining 1–3 PZC layer is subjected to the electric field $E_{z}$ along the z-direction only. Accordingly, the constitutive relations for the constraining layer of the SCLD treatment consistent with the present analysis are considered as follows (Ray and Pradhan, 2007)

\begin{matrix} {σ_{b}^{N + 2}} = [C_{b}^{N + 2}] {ε_{b}^{N + 2}} + [C_{bs}^{N + 2}] {ε_{s}^{N + 2}} - {e_{b}} E_{z}, \\ {σ_{s}^{N + 2}} = {[C_{bs}^{N + 2}]}^{T} {ε_{b}^{N + 2}} + [C_{s}^{N + 2}] {ε_{s}^{N + 2}} - {e_{s}} E_{z}, and \\ D_{z} = {e_{b}}^{T} {ε_{b}^{N + 2}} + {e_{s}}^{T} {ε_{s}^{N + 2}} + \in_{33} E_{z} \end{matrix}

(19)

where $D_{z}$ represents the electric displacement along the z-direction and $\in_{33}$ is the dielectric constant. The explicit forms of the transformed elastic coefficient matrices $[C_{b}^{N + 2}]$ and $[C_{s}^{N + 2}]$ , the elastic coupling constant matrix $[C_{bs}^{N + 2}]$ , and the piezoelectric constant matrices ${e_{b}}$ and ${e_{s}}$ appearing in equation (19) are available elsewhere (Ray and Pradhan, 2007).

2.4. Principle of virtual work

The principle of virtual work is employed to derive the governing equations of the overall smart structure and can be expressed as follows

\begin{matrix} \sum_{k = 1}^{N + 2} \int_{Ω_{k}} (δ {ε_{b}^{k}}^{T} {σ_{b}^{k}} + δ {ε_{s}^{k}}^{T} {σ_{s}^{k}} - δ {d_{t}}^{T} ρ^{k} {{\overset{\cdot\cdot}{d}}_{t}}) \\ d Ω_{k} - \int_{Ω_{N + 2}} δ E_{z} \in_{33} E_{z} d Ω_{k} - \int_{A} δ {d_{t}}^{T} {f} dA = 0 \end{matrix}

(20)

where $ρ^{k}$ is the mass density of the kth layer, ${f}$ is the externally applied surface traction acting over a surface area $A$ , and $Ω_{k}$ represents the volume of the kth layer. It may be noted that the layer numbers $N + 1$ and $N + 2$ represent the viscoelastic layer and the piezoelectric layer, respectively.

3. FE formulation

The overall plate is discretized by eight-noded isoparametric quadrilateral elements. Following equation (4), the generalized displacement vectors, associated with the ith $(i = 1, 2, 3, . . ., 8)$ node of the element, can be written as follows

\begin{matrix} {d_{ti}} = {[\begin{matrix} u_{0 i} & v_{0 i} & w_{0 i} \end{matrix}]}^{T} and {d_{ri}} \\ = {[\begin{matrix} θ_{xi} & θ_{yi} & θ_{zi} & ϕ_{xi} & ϕ_{yi} & ϕ_{zi} & γ_{xi} & γ_{yi} \end{matrix}]}^{T} . \end{matrix}

(21)

Thus, the generalized displacement vector at any point within the element can be expressed in terms of the nodal generalized displacement vectors ${d_{t}^{e}}$ and ${d_{r}^{e}}$ as follows

{d_{t}} = [N_{t}] {d_{t}^{e}} and {d_{r}} = [N_{r}] {d_{r}^{e}}

(22)

where

\begin{matrix} [N_{t}] = [\begin{matrix} N_{t 1} N_{t 2} \dots N_{t 8} \end{matrix}], [N_{r}] = [\begin{matrix} N_{r 1} N_{r 2} \dots N_{r 8} \end{matrix}], \\ N_{ti} = n_{i} I_{t}, N_{ri} = n_{i} I_{r}, \\ {d_{t}^{e}} = {[\begin{matrix} {d_{t 1}^{e}}^{T} {d_{t 2}^{e}}^{T} \dots {d_{t 8}^{e}}^{T} \end{matrix}]}^{T} and \\ {d_{r}^{e}} = {[\begin{matrix} {d_{r 1}^{e}}^{T} {d_{r 2}^{e}}^{T} \dots {d_{r 8}^{e}}^{T} \end{matrix}]}^{T} \end{matrix}

(23)

while $I_{t}$ and $I_{r}$ are the (3 × 3) and (8 × 8) identity matrices, respectively, and $n_{i}$ is the shape function of natural coordinates associated with the ith node. Making use of the relations given by equations (6) to (8), (10), (18), (19), and (22) in equation (20), and using the relation, $E_{z} = - V / h_{p}$ (Baz and Ro, 1995) with $V$ being the applied voltage across the thickness of the piezoelectric layer, the following open-loop equations of motion of an element integrated with the SCLD treatment are derived as follows

\begin{matrix} [M^{e}] {{\overset{\cdot\cdot}{d}}_{t}^{e}} + [K_{ttn}^{e}] {d_{t}^{e}} + [K_{trn}^{e}] {d_{r}^{e}} = {F^{e}} \\ - {{\bar{F}}_{tn}^{e}} - {F_{tpn}^{e}} V, \\ [K_{rtn}^{e}] {d_{t}^{e}} + [K_{rr}^{e}] {d_{r}^{e}} = - {{\bar{F}}_{rn}^{e}} - {F_{rp}^{e}} V \end{matrix}

(24)

The elemental mass matrix $([M^{e}])$ , the elemental nonlinear stiffness matrices ( $[K_{ttn}^{e}]$ , $[K_{trn}^{e}]$ , and $[K_{rtn}^{e}]$ ) and the elemental linear stiffness matrix $([K_{rr}^{e}])$ , the elemental memory load vectors due to the viscoelastic material ( ${{\bar{F}}_{tn}^{e}}$ and ${{\bar{F}}_{rn}^{e}}$ ), the elemental electroelastic coupling vectors ( ${F_{tpn}^{e}}$ and ${F_{rp}^{e}}$ ), and the elemental load vector ${F^{e}}$ appearing in equation (24) are given by

\begin{matrix} [M^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} \bar{m} {[N_{t}]}^{T} [N_{t}] dxdy, \bar{m} = \sum_{k = 1}^{N} ρ^{k} (h_{k + 1} - h_{k}) \\ + ρ^{N + 1} h_{v} + ρ^{N + 2} h_{p}, \\ [K_{ttn}^{e}] = [K_{tb}^{e}] + [K_{ts}^{e}] + [K_{tbn}^{e}], \\ [K_{trn}^{e}] = [K_{trb}^{e}] + [K_{trs}^{e}] + [K_{trbn}^{e}], \\ [K_{rtn}^{e}] = [K_{rtb}^{e}] + [K_{rts}^{e}] + [K_{rtbn}^{e}] and \\ [K_{rr}^{e}] = [K_{rrb}^{e}] + [K_{rrs}^{e}], \\ {{\bar{F}}_{tn}^{e}} = [{\bar{K}}_{ttn}^{e}] \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{d}}_{t}^{e}}_{n + 1 - j} \\ + [{\bar{K}}_{trn}^{e}] \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{d}}_{r}^{e}}_{n + 1 - j}, \\ {{\bar{F}}_{rn}^{e}} = [{\bar{K}}_{rtn}^{e}] \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{d}}_{t}^{e}}_{n + 1 - j} \\ + [{\bar{K}}_{rr}^{e}] \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{d}}_{r}^{e}}_{n + 1 - j}, \\ [{\bar{K}}_{ttn}^{e}] = [{\bar{K}}_{tb}^{e}] + [{\bar{K}}_{ts}^{e}] + [{\bar{K}}_{tbn}^{e}], \\ [{\bar{K}}_{trn}^{e}] = [{\bar{K}}_{trb}^{e}] + [{\bar{K}}_{trs}^{e}] + [{\bar{K}}_{trbn}^{e}], \\ [{\bar{K}}_{rtn}^{e}] = {[{\bar{K}}_{trb}^{e}]}^{T} + {[{\bar{K}}_{trs}^{e}]}^{T} + \frac{1}{2} {[{\bar{K}}_{trbn}^{e}]}^{T}, \\ [{\bar{K}}_{rr}^{e}] = [{\bar{K}}_{rrb}^{e}] + [{\bar{K}}_{rrs}^{e}], \\ {F_{tpn}^{e}} = {F_{tb}^{e}} + {F_{ts}^{e}} + {F_{tbn}^{e}}, {F_{rp}^{e}} \\ = {F_{rb}^{e}} + {F_{rs}^{e}}, {F^{e}} = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[N_{t}]}^{T} {f} dxdy \end{matrix}

(25)

The explicit form of the elemental stiffness matrices appearing in equation (25) and associated with the bending strain vector is given by

\begin{matrix} [K_{tb}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{tb}]}^{T} [D_{tb}^{p}] [B_{tb}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{tb}]}^{T} [D_{tbs}^{p}] [B_{ts}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{ts}]}^{T} {[D_{tbs}^{p}]}^{T} [B_{tb}] dxdy, \\ [K_{tbn}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} (\frac{1}{2} [B_{tb}]^{T} [D_{tb}^{p}] [B_{1}] [B_{2}] + [B_{2}]^{T} [B_{1}]^{T} [D_{tb}^{p}] \\ [B_{tb}] + \frac{1}{2} {[B_{2}]}^{T} {[B_{1}]}^{T} [D_{tb}^{p}] [B_{1}] [B_{2}]) dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{2}]}^{T} {[B_{1}]}^{T} [D_{tbs}^{p}] [B_{ts}] dx dy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} \frac{1}{2} {[B_{ts}]}^{T} {[D_{tbs}^{p}]}^{T} [B_{1}] [B_{2}] dxdy, \\ [K_{trb}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{tb}]}^{T} [D_{trb}^{p}] [B_{rb}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{tb}]}^{T} [D_{trbs}^{p}] [B_{rs}] dx dy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{ts}]}^{T} {[D_{rtbs}^{p}]}^{T} [B_{rb}] dx dy, \\ [K_{trbn}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{2}]}^{T} {[B_{1}]}^{T} [D_{trb}^{p}] [B_{rb}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{2}]}^{T} {[B_{1}]}^{T} [D_{trbs}^{p}] [B_{rs}] dxdy \\ [K_{rtb}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rb}]}^{T} {[D_{trb}^{p}]}^{T} [B_{tb}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rb}]}^{T} [D_{rtbs}^{p}] [B_{ts}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rs}]}^{T} {[D_{trbs}^{p}]}^{T} [B_{tb}] dxdy, \\ [K_{rtbn}^{e}] = \frac{1}{2} {[K_{trbn}^{e}]}^{T}, \end{matrix}

\begin{matrix} [{\bar{K}}_{tb}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rb}]}^{T} [D_{rrb}^{p}] [B_{rb}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rs}]}^{T} {[D_{rrbs}^{p}]}^{T} [B_{rs}] dxdy \\ + \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rb}]}^{T} [D_{rrbs}^{p}] [B_{rs}] dxdy, \\ [{\bar{K}}_{tb}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{tb}]}^{T} [{\bar{C}}_{b}^{v}] h_{v} [B_{tb}] dxdy, \\ [{\bar{K}}_{tbn}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} (\frac{1}{2} {[B_{tb}]}^{T} [{\bar{C}}_{b}^{v}] h_{v} [B_{1}] [B_{2}] + {[B_{2}]}^{T} {[B_{1}]}^{T} [{\bar{C}}_{b}^{v}] \\ h_{v} [B_{tb}] + \frac{1}{2} {[B_{2}]}^{T} {[B_{1}]}^{T} [{\bar{C}}_{b}^{v}] h_{v} [B_{1}] [B_{2}]) dxdy, \\ [{\bar{K}}_{trb}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{tb}]}^{T} [D_{1}] [B_{rb}] dxdy, \\ [{\bar{K}}_{trbn}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{2}]}^{T} {[B_{1}]}^{T} [D_{1}] [B_{rb}] dxdy, \\ [{\bar{K}}_{rrb}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rb}]}^{T} [D_{3}] [B_{rb}] dxdy, \\ {F_{tb}^{e}} = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{tb}]}^{T} {e_{b}} dxdy, \\ {F_{tbn}^{e}} = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{2}]}^{T} {[B_{1}]}^{T} {e_{b}} dxdy and {F_{rb}^{e}} = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rb}]}^{T} [H_{rb}^{p}] {e_{b}} dxdy \end{matrix}

(26)

and those associated with transverse shear strain vector are given by

\begin{matrix} [K_{ts}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{ts}]}^{T} [D_{ts}^{p}] [B_{ts}] dxdy, \\ [K_{trs}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{ts}]}^{T} [D_{trs}^{p}] [B_{rs}] dxdy, \\ [K_{rts}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rs}]}^{T} {[D_{trs}^{p}]}^{T} [B_{ts}] dxdy, \\ [K_{rrs}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rs}]}^{T} [D_{rrs}^{p}] [B_{rs}] dxdy, \end{matrix}

\begin{matrix} [{\bar{K}}_{rrs}^{e}] = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rs}]}^{T} [D_{4}] [B_{rs}] dxdy, \\ {F_{ts}^{e}} = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{ts}]}^{T} {e_{s}} dxdy and \\ {F_{rs}^{e}} = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[B_{rs}]}^{T} [H_{rs}^{p}] {e_{s}} dxdy \end{matrix}

(27)

In the above matrices, $a^{e}$ and $b^{e}$ indicate the length and the width of the element under consideration. The nodal strain–displacement matrices derived from the generalized strain–displacement relations are given in Appendix 4 and various rigidity matrices appearing in the above elemental stiffness matrices are presented in Appendix 5. Finally, the elemental equations of motion are assembled to derive the open-loop global equations of motion of the overall structure integrated with the SCLD patch at $(n + 1) th$ time step as follows

\begin{matrix} [M] {\overset{\cdot\cdot}{X}}_{n + 1} + [K_{ttn}] {X}_{n + 1} + [K_{trn}] {X_{r}}_{n + 1} \\ = {F}_{n + 1} - {{\bar{F}}_{tn}}_{n + 1} - {F_{tpn}} V, \\ [K_{rtn}] {X}_{n + 1} + [K_{rr}] {X_{r}}_{n + 1} = - {{\bar{F}}_{rn}}_{n + 1} \\ - {F_{rp}} V \end{matrix}

(28)

where $[M]$ is the global mass matrix; $[K_{ttn}]$ , $[K_{trn}]$ , and $[K_{rtn}]$ are the global nonlinear stiffness matrices; $[K_{rr}]$ is the global linear stiffness matrix; ${{\bar{F}}_{tn}}_{n + 1}$ and ${{\bar{F}}_{rn}}_{n + 1}$ are the global viscoelastic memory load; ${F_{tpn}}$ and ${F_{rp}}$ are the global electroelastic coupling vectors; ${X}_{n + 1}$ and ${X_{r}}_{n + 1}$ are the global nodal generalized displacement vectors; and ${F}_{n + 1}$ is the global nodal force vector. Using the relation between the strain vectors and the anelastic strain vector given by equation (17), the relation between the generalized global displacement vector and the global anelastic displacement vector can be obtained as follows

\begin{matrix} {\bar{X}}_{n + 1} = (1 - c) \frac{E_{\infty} - E_{0}}{E_{\infty}} {X}_{n + 1} \\ - c \sum_{j = 1}^{N_{t}} A_{j + 1} {\bar{X}}_{n + 1 - j}, \\ {{\bar{X}}_{r}}_{n + 1} = (1 - c) \frac{E_{\infty} - E_{0}}{E_{\infty}} {X_{r}}_{n + 1} \\ - c \sum_{j = 1}^{N_{t}} A_{j + 1} {{\bar{X}}_{r}}_{n + 1 - j} \end{matrix}

(29)

It is worth mentioning that the Grünwald coefficients $(A_{j})$ in equation (29) decreases with the increase in the value of j. Thus, the response of the viscoelastic material at a particular time depends very strongly on the recent past history of the response than on the remote past history of the response. Therefore, the Grünwald coefficients corresponding to the large value of j describe the fading memory of the viscoelastic material and truncation of the Grünwald series after some value of j does not affect the response.

4. Closed loop model

In order to activate the patch of the SCLD treatment with a control voltage, a simple negative velocity feedback control law has been implemented. Accordingly, the control voltage at any $(n + 1) th$ time step can be expressed in terms of the derivatives of the global nodal degrees of freedom as follows

V_{n + 1} = - K_{d} {\overset{\cdot}{w}}_{n + 1} = - K_{d} [U] {\overset{\cdot}{X}}_{n + 1}

(30)

in which $K_{d}$ is the control gain for the lth patch and $[U]$ is a unit vector defining the location of sensing the velocity signal for feedback. Substituting equation (30) into equation (28) and condensing the generalized degrees of freedom ${X_{r}}_{n + 1}$ , the equations of motion governing the closed loop dynamics of the substrate plate activated by the patch of the SCLD treatment can be obtained as follows

\begin{matrix} [M] {\overset{\cdot\cdot}{X}}_{n + 1} + [C_{n}] {\overset{\cdot}{X}}_{n + 1} + [K_{n}] {X}_{n + 1} \\ = {F}_{n + 1} + {{\bar{F}}_{n}}_{n + 1} \end{matrix}

(31)

where the nonlinear active damping matrix $[C_{n}]$ , the nonlinear stiffness matrix $[K_{n}]$ , and the nonlinear load vector due to the viscoelastic effect denoted by ${{\bar{F}}_{n}}_{n + 1}$ are given by

\begin{matrix} [C_{n}] = - K_{d} ({F_{tpn}} - [K_{trn}] {[K_{rr}]}^{- 1} {F_{rp}}) [U] \\ [K_{n}] = [K_{tn}] - [K_{trn}] {[K_{rr}]}^{- 1} [K_{rtn}], and {{\bar{F}}_{n}}_{n + 1} \\ = - ({{\bar{F}}_{tn}}_{n + 1} - [K_{trn}] {[K_{rr}]}^{- 1} {{\bar{F}}_{rn}}_{n + 1}) \end{matrix}

(32)

5. Validation of the model

The boundary conditions used for numerical simulation are presented in Table 1. A MATLAB^® code is developed using the FE model formulated in the previous section. In order to verify the validity of the present FE model, two examples are selected from the literature. The governing equation of the overall structure given by equation (31) contains nonlinear terms to be computed using an iterative method. In the first example, analysis is performed on a four-layer symmetric square cross-ply (0°/90°/90°/0°) plate integrated with the inactive SCLD patch of negligible thickness to compute the nonlinear transient response using the Newmark-beta method under uniform step load with an intensity of 1 N/mm² (Chen et al., 2000). High modulus glass-epoxy fiber-reinforced composite with the material properties as given in equation (33) is considered, while the geometric properties are used as a = b = 250 mm and h = 5 mm with each layer having equal thickness (Chen et al., 2000)

\begin{matrix} E_{1} = 25 E_{2}, E_{2} = 2.1 \times 10^{6}, G_{12} = G_{13} = 0.5 E_{2}, \\ G_{23} = 0.2 E_{2}, ν_{12} = 0.25, and ρ = 800 {kg / m}^{3} \end{matrix}

(33)

The results obtained with the present model are compared with the response given in Chen et al. (2000) in Figure 5 for the same plate without the SCLD patch. The computed result matches very closely with that of Chen et al. (2000) and confirms the accuracy of the implementation of the numerical integration method.

Table 1.

Various boundary conditions used in the FE model.

Type	At x = 0 and a	At y = 0 and b
Simply supported (SS1)	$v_{0} = w_{0} = θ_{y} = θ_{z} = ϕ_{y} = ϕ_{z} = γ_{y} = 0$	$u_{0} = w_{0} = θ_{x} = θ_{z} = ϕ_{x} = ϕ_{z} = γ_{x} = 0$
Simply supported (SS2)	$u_{0} = w_{0} = θ_{y} = θ_{z} = ϕ_{y} = ϕ_{z} = γ_{y} = 0$	$v_{0} = w_{0} = θ_{x} = θ_{z} = ϕ_{x} = ϕ_{z} = γ_{x} = 0$
Clamped-clamped (CC)	$\begin{matrix} u_{0} = v_{0} = w_{0} = θ_{x} = θ_{y} = θ_{z} \\ = ϕ_{x} = ϕ_{y} = ϕ_{z} = γ_{x} = γ_{y} = 0 \end{matrix}$	$\begin{matrix} u_{0} = v_{0} = w_{0} = θ_{x} = θ_{y} = θ_{z} \\ = ϕ_{x} = ϕ_{y} = ϕ_{z} = γ_{x} = γ_{y} = 0 \end{matrix}$
Clamped-free (CF)	$\begin{matrix} u_{0} = v_{0} = w_{0} = θ_{x} = θ_{y} = θ_{z} \\ = ϕ_{x} = ϕ_{y} = ϕ_{z} = γ_{x} = γ_{y} = 0 \end{matrix}$	–

FE model: finite element model.

Next, a viscoelastic simply supported beam (Galucio et al., 2004) is chosen to validate the proposed three-dimensional constitutive relation for the viscoelastic material using the FDM. To compute the linear transient transverse vibration of the viscoelastic beam, the structure in Figure 3 is first modified. The SCLD patch is extended over the whole top surface of the substrate plate and negligible thickness of the FFRC plate $(h)$ and the 1–3 PZC layer $(h_{p})$ is considered. Subsequently, the structure is reduced to the viscoelastic layer and considering the geometrical parameters and the material properties same as that of the beam in Galucio et al. (2004), the problem is solved using the present FEM. The following geometric data are adopted from Galucio et al. (2004), a = 10 m, b = 2 m, h_v = 0.5 m. The viscoelastic material properties (Galucio et al., 2004) are ρ = 500 kg/m³, ν = 0.3, $E_{0}$ = 19.6 MPa, $E_{\infty}$ = 98 MPa, $τ$ = 2.24 s, and $α$ = 0.75. The shear correction factor used for this analysis is assumed as (Galucio et al., 2004) $k_{c} = 10 (1 + ν) / (12 + 11 ν)$ . The dynamic response is computed at the center of the beam under uniform step load of 10 N/m applied all over the beam and is compared with that available in Galucio et al. (2004) as shown in Figure 6. It may be observed that the two response curves match excellently with each other. This validates the present three-dimensional constitutive relation for the viscoelastic material using the FDM and can be successfully implemented for three-dimensional analysis of passive and active constrained layer damping of geometrically nonlinear smart FFRC plates.

Finally, a comparison of the present three-dimensional FDM with the well-established GHM method is performed in case of a free layer damping (FLD) treatment of a $(0^{o} / 90^{o} / 0^{o})$ symmetric square cross-ply laminated composite plate with SS1-type boundary condition and under uniformly distributed loading of 500 N/m². The material of the each orthotropic layer is considered as graphite epoxy with material properties as follows

\begin{matrix} E_{1} = 172.5 GPa, E_{1} / E_{2} = 25, G_{12} = 0.5 E_{2}, G_{13} = G_{12}, \\ G_{23} = 0.2 E_{2}, ν_{12} = 0.25, ν_{13} = ν_{23} = ν_{12} and \\ ρ = 1600 kg / m^{3} \end{matrix}

(34)

The layerwise FSDT displacement field given in equations (1) and (2) is used up to $(N + 1) th$ layer for modeling the FLD treatment of the plate using GHM method. For the GHM model (Golla and Hughes, 1985; McTavish and Hughes, 1993) of the viscoelastic layer, the time-domain constitutive relation is given by

{σ_{s}^{N + 1}} = \int_{0}^{t} G (t - τ) \frac{\partial}{\partial τ} {ε_{s}^{N + 1}} \partial τ

(35)

In the GHM method, the time-dependent material relaxation function $G (t)$ is modeled as the material modulus function $s \tilde{G} (s)$ using a series of mini-oscillator terms in Laplace domain and is expressed as follows (Golla and Hughes, 1985)

s \tilde{G} (s) = G^{\infty} (1 + \sum_{r = 1}^{R} {\hat{α}}_{r} \frac{s^{2} + 2 {\hat{ς}}_{r} {\hat{ω}}_{r} s}{s^{2} + 2 {\hat{ς}}_{r} {\hat{ω}}_{r} s + {\hat{ω}}_{r}^{2}})

(36)

where $G^{\infty}$ corresponds to the equilibrium value of the modulus, that is, the final value of $G (t)$ . Each mini-oscillator term is a second-order rational function involving three positive constants ${\hat{α}}_{r}$ , ${\hat{ς}}_{r}$ , and ${\hat{ω}}_{r}$ . The parameters for a three-mini-oscillator GHM model and four-parameter FDM of 3M ISD112 are given by Galucio et al. (2005)

\begin{matrix} E_{0} = 1.5 MPa, E_{\infty} = 69.95 MPa, ν = 0.49, \\ α = 0.7915, τ = 1.4052 \times 10^{- 2} ms, \\ ρ^{v} = 1600 kg / m^{3} \end{matrix}

(37)

Using present FDM model, the response can be obtained by considering negligible thickness of the constraining 1–3 PZC layer, whereas the plate thickness is considered as $h_{0} = 3 mm$ with aspect ratio of 100 and the thickness of the viscoelastic layer is taken as $h_{v} = 200 μ m$ . Compared responses for the FLD treatment using both the GHM method and the FDM are displayed in Figure 7 and the responses are same for both the cases. The time required by the GHM method mostly depends on the number of parameters (such as parameter 1 or parameter 2 or 3). If the GHM parameter is 1, size of the system matrices becomes twofold, and if it is 2, size of the system matrices becomes threefold and so on. Thus, the computational time using GHM model is problem dependent and it is obvious that the GHM will require more computational time compared to the FDM as in case of FDM size of the system matrices remain same. For the present example problem, the ratio of required simulation time for the FDM and GHM model is approximately 1:50. Thus, the FEM involving FDM simulates much faster compared to the FEM involving the GHM method indicating the advantages of the present method.

6. Results and discussion

In this section, the numerical results obtained using the FE model have been presented. Numerical results are computed considering different CNT waviness, different stacking sequences, and boundary conditions for the FFRC substrate plates and shells integrated with a patch of SCLD treatment attached at the center of the top surface of the substrates. It may be noted from Figure 3 that the length and the width of the patch are taken as 50% of the length and the width of the substrate plates, respectively. PZT-5H/spur epoxy composite with 60% piezoelectric fiber volume fraction has been considered for the material of the constraining layer of the SCLD treatment. The elastic and the piezoelectric properties and the density of this constraining layer are (Ray and Pradhan, 2007)

\begin{matrix} C_{11}^{p} = 9.29 GPa, C_{12}^{p} = 6.18 GPa, C_{13}^{p} = 6.05 GPa, \\ C_{33}^{p} = 35.44 GPa, C_{23}^{p} = C_{13}^{p}, \\ C_{44}^{p} = 1.58 GPa, C_{66}^{p} = 1.54 GPa, C_{55}^{p} = C_{44}^{p}, \\ ρ^{p} = 5090 kg / m^{3}, \\ e_{31} = - 0.1902 C / m^{2}, e_{32} = e_{31}, e_{33} = 18.4107 C / m^{2}, \\ e_{24} = 0.004 C / m^{2}, e_{15} = e_{24} \end{matrix}

(38)

The thickness of the 1–3 PZC layer is considered as $h_{p} = 500 μ m$ for all the analyses. The viscoelastic material is chosen as 3M ISD112 and its material property is given in equation (37). The thickness of the constrained viscoelastic layer is chosen as $h_{v} = 250 μ m$ . The material properties of the base composite and the FFRC layers of the substrate plates are presented in Table 2. It may be noted that the material properties are computed using the analytical model presented by Kundalwal and Ray (2011, 2013). Here, the material properties are computed considering 60% carbon fiber volume fraction $(v_{f})$ . The radial distance between the starting and the endpoint of any CNT and wave frequency of the wavy CNTs are represented by $L_{n}$ and $ω$ , respectively. The waviness factor $(A / L_{n})$ is also computed for wavy CNTs. The CNT volume fraction in the FFRC is denoted by $V_{CNT}$ and the corresponding maximum possible values of $V_{CNT}$ can be computed as follows (Kundalwal and Ray, 2013)

{(V_{CNT})}_{max} = \frac{π d_{n}^{2} L_{nr}}{d (d_{n} + 1.7 \times 10^{- 9})} v_{f}

(39)

where $d_{n}$ and $d$ are the diameters of the CNTs and the carbon fibers, respectively, and $L_{nr}$ is the actual length or running length of the wavy CNTs. The diameter of the carbon fiber is taken as $d = 10 μ m$ , whereas the maximum amplitude of CNTs is assumed as $A = 100 d_{n}$ unless specified otherwise. As a consequence, $L_{n}$ and $A / L_{n}$ become 1.1472 µm and 0.068, respectively. It may be mentioned here that the maximum value of waviness of the CNTs is chosen as $ω = 24 π / L_{n}$ to avoid overlapping of nanotubes. The length (a) and the width (b) of the substrate plates are assumed to be equal (square plates) and in all the examples the overall thickness of the substrate plates is taken as $h = 3 mm$ . Also, each orthotropic layer of a substrate plate is considered to be of equal thickness. As the present model is aimed for thin plates, the aspect ratio $(a / h)$ is considered as 150 for all substrate plates. The shear correction factor is chosen to be $k_{c} = 5 / 6$ . Uniformly distributed load is suddenly applied all over the plate to set the plate into undergoing transient vibrations. The intensity of the applied load can be decided from the backbone curves, and a sample of such curves is presented in Figure 8 for $(0^{°} / 90^{°} / 0^{°})$ symmetric square cross-ply FFRC plates made of different materials with properties mentioned in Table 2. Sufficient nonlinearity is needed to be incorporated in the present analyses and as a consequence, the frequency ratio $(ω_{nl} / ω_{l})$ is kept more than 1.1 in all the following cases. Similar curves can be simulated for other stacking sequences and boundary conditions. However, they are not presented here. The control voltage supplied to the SCLD patch is negatively proportional to the velocity of the point at the center $\overset{\cdot}{w} (a / 2, b / 2, h / 2)$ of the plate. Depending on the applied load, the control gain is chosen arbitrarily such that the value of the maximum value of control voltage is nominal (250–400 V) and the vibrations are also controlled efficiently.

Table 2.

Material properties of base composite and FFRC with straight and wavy CNTs (60% fiber volume fraction is considered).

Material description	$C_{11}$	$C_{12}$	$C_{13}$	$C_{22}$	$C_{23}$	$C_{44}$	$C_{55}$	ρ(kg/m³)
Material description	(GPa)							ρ(kg/m³)
Base composite $(V_{CNT} = 0)$	145.00	7.78	7.78	14.32	8.63	2.84	3.44	1506
FFRC, straight CNTs $((V_{CNT})_{max} = 0.0215, ω = 0)$	145.84	9.00	9.00	23.96	12.12	5.90	4.02	1510
FFRC, wavy CNTs (2–3 plane) $((V_{CNT})_{max} = 0.0423, ω = 12 π / L_{n})$	146.85	9.31	9.31	25.76	13.15	6.28	4.80	1514.2
FFRC, wavy CNTs (2–3 plane) $((V_{CNT})_{max} = 0.0581, ω = 18 π / L_{n})$	148.05	9.48	9.48	26.40	13.54	6.40	5.64	1517.2
FFRC, wavy CNTs (2–3 plane) $((V_{CNT})_{max} = 0.0745, ω = 24 π / L_{n})$	150.26	9.67	9.67	26.83	13.81	6.48	6.88	1520.4
FFRC, wavy CNTs (1–3 plane) $((V_{CNT})_{max} = 0.0423, ω = 12 π / L_{n})$	172.62	12.80	12.80	21.82	11.19	5.29	9.67	1514.2
FFRC, wavy CNTs (1–3 plane) $((V_{CNT})_{max} = 0.0581, ω = 18 π / L_{n})$	195.64	13.04	13.04	22.20	11.48	5.34	9.77	1517.2
FFRC, wavy CNTs (1–3 plane) $((V_{CNT})_{max} = 0.0745, ω = 24 π / L_{n})$	219.59	13.12	13.12	23.09	12.00	5.53	9.90	1520.4

FFRC: fuzzy fiber-reinforced composite; CNTs: carbon nanotubes.

It is already mentioned earlier that the upper limit of the Grünwald series $(N_{t})$ is the number of responses in memory and is somewhat arbitrary but must be less than the number of time steps. From the definition of Grünwald constant $(A_{j})$ given in equation (16), it is seen that the value of $A_{j}$ depends on the fractional order $(α)$ of the derivative which is constant with time for a particular viscoelastic material, and thus increasing the number of Grünwald terms rapidly decreases the corresponding Grünwald constant substantiating the fading memory effect of the viscoelastic material. Thus, approximate truncated value of $N_{t}$ can be used so that sufficiently accurate computation of the memory load is possible without losing much memory information and the damping property of the viscoelastic material with elapsed time. The detailed discussion on this topic is elucidated elsewhere (Datta and Ray, 2015, 2018a). However, further results are computed considering all the previous response terms in memory.

The efficacy of the obliquely reinforced 1–3 PZC layer for controlling the vibrations of the plates is measured by the control authority of the SCLD patch for different piezoelectric fiber orientation angle $(ψ)$ in the vertical xz or yz planes. To measure the percentage reduction in the amplitude of vibration at the center of the plate, the performance index $(I_{d})$ is defined as follows

I_{d} = \frac{| a_{1} (a / 2, b / 2, h / 2) | - | a_{10} (a / 2, b / 2, h / 2) |}{| a_{1} (a / 2, b / 2, h / 2) |} \times 100

(40)

where $a_{1}$ and $a_{10}$ are the amplitudes of vibration at the 1st peak and the 10th peak. Keeping the control voltage constant at 300 V, the variation of the performance index $(I_{d})$ with the piezoelectric fiber orientation angle $(ψ)$ in the $xz$ plane is obtained when same amplitude of uniformly distributed load is applied. These performance curves are presented in Figures 9 and 10 for $(0^{°} / 90^{°} / 0^{°})$ symmetric square cross-ply FFRC plates with SS1-type boundary condition and $(45^{°} / - 45^{°})_{4}$ square angle-ply FFRC plates with SS2-type boundary condition, respectively. The figures reveal the effect of the presence of the straight and wavy CNTs, and it may also be noted that the curves are qualitatively similar as the SCLD patch provides maximum control authority in all the cases when vertically reinforced $(ψ = 0^{°})$ 1–3 PZC is used. At this juncture, it may be pointed out that the effect of CNT waviness is also prominent in these performance curves exhibiting maximum control effect when the waviness is in 1–3 plane. This may be attributed to the fact that the FFRC plate with CNT waviness in 1–3 plane offers maximum stiffness compared to the others due to its waviness aligned along the long base fibers. Performance curves for other lamina sequences and boundary conditions are also computed in similar way and in all the cases maximum control was obtained for vertically reinforced $(ψ = 0^{°})$ 1–3 PZC. Similar results are obtained with the variation of piezoelectric fiber orientation angle $(ψ)$ in the vertical yz plane. However, those curves are not presented here as they exhibit qualitative similarity. For further results in all cases, vertically reinforced $(ψ = 0^{°})$ 1–3 PZC is considered.

First, the effectiveness of the smart damping over the PCLD is investigated. In Figure 11, the responses at the center of the simply supported (SS1) $(0^{°} / 90^{°})$ antisymmetric square cross-ply plate made of base composite $(V_{CNT} = 0)$ are presented when the patch is passive $(K_{d} = 0)$ and active $(K_{d} \neq 0)$ under uniformly distributed load. Also, the actively controlled $(K_{d} \neq 0)$ response for the FFRC plate with straight CNTs $(ω = 0)$ is shown in the same figure under same loading. It may be observed that when the 1–3 PZC material is activated, improved damping characteristics and significant attenuation in vibration levels are obtained over the passive damping (PCLD). Also, Figure 11 reveals that the performance of the SCLD patch is better in case of controlling the FFRC substrate than that in case of composite substrate without CNTs. Furthermore, the effect of CNT waviness on the smart damping characteristics has been studied.

6.1. Effect of CNT waviness in different planes with same waviness

Here, several cases have been numerically studied considering plates with different stacking sequences and boundary conditions made of base composite, FFRC with straight CNTs $(ω = 0)$ , and FFRCs with wavy CNTs having the same CNT waviness $(ω = 24 π / L_{n})$ in 2–3 and 1–3 planes under same maximum value of applied load. First, the active damping responses of simply supported (SS1) $(0^{o} / 9 0^{o} / 0^{o})$ symmetric square cross-ply plates are computed under 2.2 kN/m² uniformly distributed loading. Figure 12 illustrates the control effect for a constant control gain $(K_{d} = 200)$ . It is evident from the figure that the FFRCs with wavy CNTs offer higher stiffness compared to those with straight CNTs and among all, the stiffness is highest when waviness is in 1–3 plane. Corresponding voltage requirement is shown in Figure 13, and although the maximum value of required voltage is quite close for these structures, comparatively lower maximum voltage is required for FFRCs with wavy CNTs containing waviness in the 1–3 plane. It is clear from Figure 12 that the performance of the SCLD patch is much more improved if the wavy CNTs are coplanar with the 1–3 plane.

The transient responses of simply supported (SS2) $(45^{°} / - 45^{°})_{4}$ antisymmetric angle-ply plates with eight layers under 2.5 kN/m² uniform load are presented in Figure 14 for active $(K_{d} = 200)$ control comparing the control authority of the SCLD patch for base composite without CNTs and FFRC substrates. It may be observed from the figure that the SCLD patch effectively controls the vibrations of the angle-ply plates enhancing the damping characteristics. The required control voltage is plotted in Figure 15 which indicates that for approximately same maximum value of control voltage the performance of the patch is significantly improved if the wavy CNTs are coplanar with the 1–3 plane.

Furthermore, similar analyses have been carried out on a clamped-clamped (CC) plates having four generally orthotropic layers $(30^{o} / 90^{o} / - 45^{o} / 60^{o})$ and a clamped-free (CF) angle-ply plates with six layers $(90^{o} / - 60^{o} / 45^{o} / - 30^{o} / 0^{o} / - 45^{o})$ and the responses are computed for 5 and 1.0 kN/m² uniform load, respectively. The responses are depicted in Figures 16 and 17, respectively, for active $(K_{d} \neq 0)$ control. For the CC plates, the control gain is used as $K_{d} = 200$ , whereas for the CF plates, $K_{d} = 400$ . From Figures 16 and 17, similar conclusions can be drawn that the SCLD treatment has significantly improved the smart damping characteristics of the CC and CF plates and the control authority of the treatment becomes maximum, if the wavy CNTs are coplanar with the 1–3 plane.

6.2. Effect of CNT waviness in same plane with different waviness

Here, two cases have been presented considering $(0^{o} / 9 0^{o} / 0^{o})$ symmetric square cross-ply plates with SS1 boundary condition made of FFRC with straight CNTs $(ω = 0)$ and FFRCs with wavy CNTs having the different CNT waviness in 2–3 and 1–3 planes under same maximum value of applied load. Three values of the CNT waviness have been considered $(ω = 12 π / L_{n}, 18 π / L_{n}, and 24 π / L_{n})$ . Figures 18 and 19 depict the effects of different CNT waviness in 2–3 and 1–3 planes, respectively, for smart damping of plates under 2.0 kN/m² uniformly distributed load. It is evident from the figures that the effect of smart damping increases in case of increasing waviness in both 2–3 and 1–3 planes. However, the effect is considerably more with increasing waviness when the CNTs are coplanar with 1–3 plane as the effective elastic properties are also increasing.

6.3. Effect of CNT waviness in different planes for same maximum deflection

Finally, the effect of CNT waviness is investigated for $(0^{o} / 9 0^{o} / 0^{o})$ symmetric square cross-ply plates with SS1 and CC boundary conditions and $(45^{o} / - 45^{o})_{4}$ antisymmetric angle-ply plates with SS2 and CC boundary conditions when the maximum amplitude of center deflection and the control gain is same. Controlled responses for $(0^{o} / 9 0^{o} / 0^{o})$ plates with SS1 boundary condition and for $(45^{o} / - 45^{o})_{4}$ plates with SS2 boundary conditions are displayed in Figures 20 and 21, respectively. In these cases, also the smart damping performance is better when CNT waviness is contained in 1–3 plane. Similar effect can be observed from Figures 22 and 23 representing controlled responses for $(0^{o} / 9 0^{o} / 0^{o})$ and $(45^{o} / - 45^{o})_{4}$ plates, respectively, with CC boundary condition. It may be noted that for CC boundary condition, with same maximum center deflection and control gain, the settling time required for $(45^{o} / - 45^{o})_{4}$ and $(0^{o} / 9 0^{o} / 0^{o})$ plates is very close to each other. However, to obtain same amplitude of maximum deflection, higher load is required for angle-ply plates than that required for cross-ply plates.

7. Conclusion

A smart FE model has been used to investigate the performance of the patch of the SCLD treatment for controlling geometrically nonlinear vibrations of laminated FFRC plates. The constraining layer of the SCLD patch is considered to be made of the vertically/obliquely reinforced 1–3 PZC. The constrained viscoelastic layer is modeled based on the three-dimensional FDM. The kinematics of deformations of the whole structure is assumed to be based on the layerwise FSDT, and von Kármán–type strain–displacement relation has been used to consider the geometric nonlinearity. A simple velocity feedback control law is used to model the active damping. The performance of the SCLD patch is presented, and from the results, it can be concluded that the maximum control is achieved if vertically reinforced 1–3 PZC is used irrespective of the material property, lamina sequence, and boundary conditions. Several numerical results have been presented to study the effect of CNT waviness on smart damping of plates considering different stacking sequences and boundary conditions. The numerical results for controlled responses reveal that the SCLD treatment significantly improves the damping characteristics of the structure over PCLD for suppressing their geometrically nonlinear transient vibrations. The performance of the SCLD treatment is significantly improved if the amplitudes of the wavy CNTs are aligned with the long base fibers of the FFRC. The present investigation suggests that the FDM of viscoelastic layer is a computationally efficient model for time-domain analysis of the SCLD of geometrically nonlinear vibrations of FFRC plates, and FFRC can be the candidate substrate material for superior lightweight smart structure.

Footnotes

Appendix 1

The matrices $[Z_{β}]$ (β = 1, 2, …, 6) appearing in equations (6) and (7) are given by

\begin{array}{l} [Z_{1}] = [\begin{matrix} z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 z & 0 & 0 & 0 \end{matrix}], \\ [Z_{2}] = [\begin{matrix} 1 & 0 & z & 0 & 0 & 0 & z^{2} & 0 & 0 & 0 \\ 0 & 1 & 0 & z & 0 & 0 & 0 & z^{2} & 0 & 0 \end{matrix}], \\ [Z_{3}] = [\begin{matrix} \frac{h}{2} & 0 & 0 & 0 & (z - h / 2) & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{h}{2} & 0 & 0 & 0 & (z - h / 2) & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{h}{2} & 0 & 0 & 0 & (z - h / 2) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 z & 0 & 0 & 0 \end{matrix}], \\ [Z_{4}] = [\begin{matrix} 0 & 0 & z & 0 & 1 & 0 & z^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & z & 0 & 1 & 0 & z^{2} & 0 & 0 \end{matrix}], \\ [Z_{5}] = [\begin{matrix} \frac{h}{2} & 0 & 0 & 0 & h_{v} & 0 & 0 & 0 & (z - h_{N + 2}) & 0 & 0 \\ 0 & \frac{h}{2} & 0 & 0 & 0 & h_{v} & 0 & 0 & 0 & (z - h_{N + 2}) & 0 \\ 0 & 0 & \frac{h}{2} & 0 & 0 & 0 & h_{v} & 0 & 0 & 0 & (z - h_{N + 2}) \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 z & 0 & 0 & 0 \end{matrix}], \\ [Z_{6}] = [\begin{matrix} 0 & 0 & z & 0 & 0 & 0 & z^{2} & 0 & 1 & 0 \\ 0 & 0 & 0 & z & 0 & 0 & 0 & z^{2} & 0 & 1 \end{matrix}] \end{array}

Appendix 2

The matrices $[\nabla_{bt}]$ , $[\nabla_{st}]$ , $[\nabla_{br 1}]$ , $[\nabla_{br 2}]$ , and $[\nabla_{sr 1}]$ appearing in equation (8) can be written as follows

\begin{matrix} [\nabla_{bt}] = {[\begin{matrix} \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial y} & 0 \\ 0 & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]}^{T}, [\nabla_{st}] = [\begin{matrix} 0 & 0 & \frac{\partial}{\partial x} \\ 0 & 0 & \frac{\partial}{\partial y} \end{matrix}], \\ [\nabla_{br 1}] = {[\begin{matrix} \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial y} & 0 \\ 0 & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]}^{T}, [\nabla_{br 2}] = {[\begin{matrix} \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial y} \\ 0 & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} \end{matrix}]}^{T}, \\ [\nabla_{sr 1}] = {[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{matrix}]}^{T} \end{matrix}

Appendix 3

The transformed elastic coefficient matrices in equation (10) have the following form

[{\bar{C}}_{b}^{k}] = [\begin{matrix} {\bar{C}}_{11}^{k} & {\bar{C}}_{12}^{k} & {\bar{C}}_{16}^{k} & {\bar{C}}_{13}^{k} \\ {\bar{C}}_{12}^{k} & {\bar{C}}_{22}^{k} & {\bar{C}}_{26}^{k} & {\bar{C}}_{23}^{k} \\ {\bar{C}}_{16}^{k} & {\bar{C}}_{26}^{k} & {\bar{C}}_{66}^{k} & {\bar{C}}_{36}^{k} \\ {\bar{C}}_{13}^{k} & {\bar{C}}_{23}^{k} & {\bar{C}}_{36}^{k} & {\bar{C}}_{33}^{k} \end{matrix}] and [{\bar{C}}_{s}^{k}] = [\begin{matrix} {\bar{C}}_{55}^{k} & {\bar{C}}_{45}^{k} \\ {\bar{C}}_{45}^{k} & {\bar{C}}_{55}^{k} \end{matrix}]

where ${\bar{C}}_{ij}^{k}$ (i, j = 1, 2, …, 6) are the transformed elastic coefficient.

The matrices $[C_{b}^{N + 1}]$ , $[{\bar{C}}_{b}^{N + 1}]$ , $[C_{s}^{N + 1}]$ , and $[{\bar{C}}_{s}^{N + 1}]$ in equation (18) are given by

\begin{matrix} [C_{b}^{N + 1}] = (1 + c \frac{E_{\infty} - E_{0}}{E_{0}}) [C^{N + 1}], [{\bar{C}}_{b}^{N + 1}] \\ = c \frac{E_{\infty}}{E_{0}} [C^{N + 1}], \\ [C_{s}^{N + 1}] = G_{0} (1 + c \frac{E_{\infty} - E_{0}}{E_{0}}) [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] and [{\bar{C}}_{s}^{N + 1}] \\ = c G_{\infty} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{matrix}

Appendix 4

The strain–displacement matrices appearing in equations (26) and (27) can be written as follows

\begin{matrix} [B_{tb}] = [\begin{matrix} B_{tb 1} & B_{tb 2} & \dots & B_{tb 8} \end{matrix}], \\ [B_{rb}] = [\begin{matrix} B_{rb 1} & B_{rb 2} & \dots & B_{rb 8} \end{matrix}], \\ [B_{ts}] = [\begin{matrix} B_{ts 1} & B_{ts 2} & \dots & B_{ts 8} \end{matrix}], \\ [B_{rs}] = [\begin{matrix} B_{rs 1} & B_{rs 2} & \dots & B_{rs 8} \end{matrix}], \\ [B_{1}] = {[\begin{matrix} \frac{\partial w_{0}}{\partial x} & 0 & \frac{\partial w_{0}}{\partial y} & 0 \\ 0 & \frac{\partial w_{0}}{\partial y} & \frac{\partial w_{0}}{\partial x} & 0 \end{matrix}]}^{T} and [B_{2}] = [\begin{matrix} B_{21} & B_{22} & \dots & B_{28} \end{matrix}] \end{matrix}

where the sub-matrices $[B_{tbi}]$ , $[B_{rbi}]$ , $[B_{tsi}]$ , $[B_{rsi}]$ , and $[B_{2}]$

\begin{matrix} [B_{tbi}] = {[\begin{matrix} n_{i, x} & 0 & n_{i, y} & 0 \\ 0 & n_{i, y} & n_{i, x} & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]}^{T}, [B_{tsi}] = [\begin{matrix} 0 & 0 & n_{i, x} \\ 0 & 0 & n_{i, y} \end{matrix}], \\ [B_{rbi}] = [\begin{matrix} [{\bar{B}}_{rbi}] & {\bar{O}}_{43} & {\bar{O}}_{42} \\ {\bar{O}}_{43} & [{\bar{B}}_{rbi}] & {\bar{O}}_{42} \\ {\bar{O}}_{33} & {\bar{O}}_{33} & [{\tilde{B}}_{rbi}] \end{matrix}], \\ [{\bar{B}}_{rbi}] = {[\begin{matrix} n_{i, x} & 0 & n_{i, y} & 0 \\ 0 & n_{i, y} & n_{i, x} & 0 \\ 0 & 0 & 0 & n_{i} \end{matrix}]}^{T}, \\ [{\tilde{B}}_{rbi}] = {[\begin{matrix} n_{i, x} & 0 & n_{i, y} \\ 0 & n_{i, y} & n_{i, x} \end{matrix}]}^{T}, \\ [B_{rsi}] = [\begin{matrix} [{\bar{B}}_{rsi}] & {\bar{O}}_{43} & {\bar{O}}_{42} \\ {\bar{O}}_{43} & [{\bar{B}}_{rsi}] & {\bar{O}}_{42} \\ {\bar{O}}_{23} & {\bar{O}}_{23} & n_{i} I_{2} \end{matrix}], \\ [{\bar{B}}_{rsi}] = {[\begin{matrix} n_{i} & 0 & 0 & 0 \\ 0 & n_{i} & 0 & 0 \\ 0 & 0 & n_{i, x} & n_{i, y} \end{matrix}]}^{T}, [B_{2 i}] = [B_{tsi}] \end{matrix}

where $n_{i, x} = (\partial n_{i} / \partial x)$ and $n_{i, y} = (\partial n_{i} / \partial y)$ , ${\bar{O}}_{43}$ , ${\bar{O}}_{42}$ , ${\bar{O}}_{33}$ , and ${\bar{O}}_{23}$ are 4 × 3, 4 × 2, 3 × 3, and 2 × 3 null matrices, respectively, and I₂ is 2 × 2 identity matrix.

Appendix 5

The various rigidity matrices appearing in equations (26) and (27) are given by

\begin{matrix} [D_{tb}^{p}] = \sum_{k = 1}^{N} \int_{h_{k}}^{h_{k + 1}} [{\bar{C}}_{b}^{k}] dz + [C_{b}^{N + 1}] h_{v} + [C_{b}^{N + 2}] h_{p}, \\ [D_{trb}^{p}] = \sum_{k = 1}^{N} \int_{h_{k}}^{h_{k + 1}} [{\bar{C}}_{b}^{k}] [Z_{1}] dz + \int_{h_{N + 1}}^{h_{N + 2}} [C_{b}^{N + 1}] [Z_{3}] dz \\ + \int_{h_{N + 2}}^{h_{N + 3}} [C_{b}^{N + 2}] [Z_{5}] dz, \\ [D_{rrb}^{p}] = \sum_{k = 1}^{N} \int_{h_{k}}^{h_{k + 1}} {[Z_{1}]}^{T} [{\bar{C}}_{b}^{k}] [Z_{1}] dz + \int_{h_{N + 1}}^{h_{N + 2}} {[Z_{3}]}^{T} [C_{b}^{N + 1}] \\ [Z_{3}] dz + \int_{h_{N + 2}}^{h_{N + 3}} {[Z_{5}]}^{T} [C_{b}^{N + 2}] [Z_{5}] dz, \\ [D_{ts}^{p}] = \sum_{k = 1}^{N} \int_{h_{k}}^{h_{k + 1}} [{\bar{C}}_{s}^{k}] dz + [C_{s}^{N + 1}] h_{v} + [C_{s}^{N + 2}] h_{p}, \\ [D_{trs}^{p}] = \sum_{k = 1}^{N} \int_{h_{k}}^{h_{k + 1}} [{\bar{C}}_{s}^{k}] [Z_{2}] dz + \int_{h_{N + 1}}^{h_{N + 2}} [C_{s}^{N + 1}] [Z_{4}] dz \\ + \int_{h_{N + 2}}^{h_{N + 3}} [C_{s}^{N + 2}] [Z_{6}] dz, \\ [D_{rrs}^{p}] = \sum_{k = 1}^{N} \int_{h_{k}}^{h_{k + 1}} {[Z_{2}]}^{T} [{\bar{C}}_{s}^{k}] [Z_{2}] dz + \int_{h_{N + 1}}^{h_{N + 2}} {[Z_{4}]}^{T} [C_{s}^{N + 1}] [Z_{4}] dz \\ + \int_{h_{N + 2}}^{h_{N + 3}} {[Z_{6}]}^{T} [C_{s}^{N + 2}] [Z_{6}] dz, \\ [D_{1}] = \int_{h_{N + 1}}^{h_{N + 2}} [{\bar{C}}_{b}^{N + 1}] [Z_{3}] dz, [D_{2}] = \int_{h_{N + 1}}^{h_{N + 2}} [{\bar{C}}_{s}^{N + 1}] [Z_{4}] dz, \\ [D_{3}] = \int_{h_{N + 1}}^{h_{N + 2}} {[Z_{3}]}^{T} [{\bar{C}}_{b}^{N + 1}] [Z_{3}] dz, [D_{4}] \\ = \int_{h_{N + 1}}^{h_{N + 2}} {[Z_{4}]}^{T} [{\bar{C}}_{s}^{N + 1}] [Z_{4}] dz, \\ [D_{tbs}^{p}] = [C_{bs}^{N + 2}] h_{p}, [D_{trbs}^{p}] = \int_{h_{N + 2}}^{h_{N + 3}} [C_{bs}^{N + 2}] [Z_{6}] dz, [D_{rtbs}^{p}] \\ = \int_{h_{N + 2}}^{h_{N + 3}} {[Z_{5}]}^{T} [C_{bs}^{N + 2}] dz, \\ [D_{rrbs}^{p}] = \int_{h_{N + 2}}^{h_{N + 3}} {[Z_{5}]}^{T} [C_{bs}^{N + 2}] [Z_{6}] dz, [H_{rb}^{p}] \\ = \int_{h_{N + 2}}^{h_{N + 3}} \frac{1}{h_{p}} {[Z_{5}]}^{T} dz, [H_{rs}^{p}] = \int_{h_{N + 2}}^{h_{N + 3}} \frac{1}{h_{p}} {[Z_{6}]}^{T} dz \end{matrix}

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Priyankar Datta

Manas Chandra Ray

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