Thermo-aeroelastic analysis of triangular composite plate with embedded shape memory alloy at supersonic flow

Abstract

This work investigates the effectiveness of a shape-memory alloy (SMA) in controlling the instabilities of triangular composite plates under supersonic flow. Lagoudas’ quadratic polynomial hardening theory models the SMA effect. First-order piston theory was used for the aerodynamic modeling, and the reference-temperature method was used for modeling the thermal heating. The buckling and post-buckling behaviors were studied for different boundary conditions with four different layups. In addition, buckling and post-buckling of the composite plate, with and without shape memory alloy wire, has been studied. The effect of SMA wire on aeroelastic instabilities is accurately studied. The embedded SMA wire significantly increased the stability region (postpone divergence and flutter velocities) and buckling temperature. Also, the time responses of the triangular composite plate are determined at different Mach numbers, showing that by increasing the Mach number, the SMA wire can control or decrease the vibration amplitudes.

Keywords

Triangular composite thermo-aeroelastic shape memory alloy supersonic flow

1. Introduction

Nowadays, using laminated composite structures to reduce the weight of aerospace vehicles is very common. Compared with traditional materials, composite materials have better fatigue resistance, corrosion resistance and noise reduction, higher stiffness and strength, and smaller specific gravity. In these vehicles, the aerodynamic forces combined with aerodynamic heating cause problems like buckling, divergence, flutter, and other instabilities. Innovative materials help engineers to reduce or suppress these issues. Shape memory alloy (SMA) materials are one of the subsets of intelligent materials. They control the vibration of structures and reduce other damages like thermal buckling, divergence, flutter, and limit cycle oscillations (Müller, 1979).

Many scientists tried to obtain and model the behavior of shape memory alloys. Müller (1979) presented a model of shape memory alloys based on the memory effect, thermomechanics of the system, and its static physics. Tanaka and Iwasaki (1985) presented a model similar to Müller’s model in phase transformations based on minimizing free energy. It was able to define the phenomenon of hysteresis by using the Clausius–Duhem unequal free energy equation. By expanding Tanaka’s theory and doing various experiments and research on SMA microstructures, Liang and Rogers (1990) found that four temperatures can be defined for shape memory alloys. Brinson (1993), by separating phase transformations into two temperature-dependent and stress-dependent parts, could explain and expand the equations more clearly than others. By presenting a model based on the total Gibbs free energy of each phase in proportion to the mass of that phase, Lagoudas and Shu (1999) presented an almost complete model for describing the behavior of SMA. In addition, Lagoudas has included the hardening rate of the alloy in his equations, which increased the accuracy of the presented model.

The experimental and numerical results of thermal post-buckling tests for various isotropic and composite plates with/without embedded SMA strips were investigated by Ho et al. (1997). They showed that SMA strips could reduce the buckling deformation of isotropic and composite panels. Lee and Lee (2000) used an ABAQUS code to show the effect of embedded SMA wires on buckling and post-buckling due to external and thermal loads. Tawfik et al. (2002) used embedded SMA wire to increase thermal buckling points and reduce the plates’ post-buckling deflection. Guo and Mei (2003) studied nonlinear aeroelastic modes, to obtain a reduced order model of nonlinear panel flutter at arbitrary supersonic yawed angle. Their results showed that when using nonlinear aeroelastic modes, the number of modal equations can be drastically reduced with little loss in accuracy. They also studied supersonic nonlinear panel flutter suppression using shape memory alloys. They show that the flutter response of the plates embedded with SMA can be reduced and even suppressed for an extensive operational range by combining dynamic pressure and temperature.

Kabir and Tatous Tehrani (2017) studied buckling and post-buckling of perfect and imperfect SMA-reinforced hybrid composite plates with a closed-form solution. Bayat and Ekhteraei Toussi (2017, 2020) used the layerwise solution to analytically analyze the thermal buckling and post-buckling of composite plates containing different layups of SMA wires under uniform and non-uniform temperature distribution. Mirzavand and Pourmohammad (2019) studied the thermal post-buckling of SMA-reinforced functionally graded (FG) cylinders. Eugeni et al. (2014) studied the post-buckling long-term dynamics of a forced nonlinear beam. Their analyses have emphasized the role of damping in the presence of a weakly chaotic response for a weak forcing load.

Razavilar et al. (2018) used Souza’s model for modeling the pseudoelastic behavior of the material. They showed that the SMA beam has a jumping phenomenon due to the hysteresis behavior of SMA material. The vibrations of thermally post-buckled functionally graded laminated beams based on a higher-order shear deformation theory and a two-step perturbation technique are investigated by Shen et al. (2019). The stability loss phenomenon in a rectangular plate with embedded SMA fibers is studied by Birman (1997). He showed that a non-uniform distribution of fibers through the width is more effective than the conventional uniform fiber distribution on instability control. An experimental study on the effect of SMA fibers on the dynamic response of a hybrid laminated beam is investigated by Rogers and Barker (1990), Ostachowicz et al. (2000), and Ostachowicz and Kaczmarczyk (2001) studied linear vibration, thermal stability, and flutter of SMA fiber-reinforced thin laminated composite plates by finite element method (FEM). Panda and Singh (2013) studied the effect of SMA fibers on the post-buckling vibration of laminated composite shells reinforced with SMA fibers using a linear FEM. Roh et al. (2004) examined a thermal snapping phenomenon in cylindrical panels by the FEM method.

Barzegari et al. (2012) studied the aeroelastic behavior of a cantilever wing with embedded SMA. They determined the effects of in-plane load due to recovery action of pre-strained SMA wires in supersonic flow by classical plate theory (CPT) and first-order shear deformation theory (FSDT). Librescu and Maalawi (2009) studied the aeroelastic design optimization of a slender, thin-walled wing-type structure against divergence. The main goal is to avoid torsional instability. The model formulation considers a large aspect ratio unswept wing of rectangular planform, while the flow conditions are restricted to those of subsonic incompressible ones. Results show that optimum patterns with decreasing wall thickness from the inboard portion toward the outboard one significantly improves the overall torsional stiffness level. Farsadi et al. (2021) studied nonlinear panel flutter and bifurcation behavior of functionally graded ceramic/metal wing-like tapered and skewed plates. The first-order shear deformation theory considers the transverse shear effect in the structural model. First-order linear piston theory is used to model the aerodynamic loading. Results demonstrate that the volume fraction and porosity coefficients significantly affect dynamic behavior and limit cycle oscillation amplitudes.

Donadon and de Faria (2016) studied the response of Laminated shells with embedded SMA wire under supersonic flow and showed that the stiffening effect induced by the changes in the fraction of martensite/austenite transformation phases of the shape memory alloy increases the rate of occurrence of flutter and destabilization. Lin et al. (2020) concluded that the SMA can change the vibration characteristics of the composite panel significantly, which will lead to different flutter behavior of the composite panel. Junior et al. (2017) examined the aeroelastic behavior of stiffened SMA hybrid composite and showed that using SMA wire only in the skin constitutive material improves the effectiveness parameter by up to 102.1% when the panel has a unitary aspect ratio.

Liu and Dowell (2005) studied the harmonic balance approach for an airfoil with a freeplay control surface. The control capability of SMA fibers on the free vibration and nonlinear thermal stability of a laminated beam is studied by Asadi et al. (2013a, 2013b). Khalili et al. (2013) analyzed the dynamic behavior of a hybrid laminated composite beam with embedded SMA fibers. They showed that the pseudoelastic characteristic of SMAs is a unique hysteresis energy dissipation behavior. In all of these studies, a simplified Brinson model was used to model the behavior of SMA fibers during uniform heating (Brinson 1993).

Zhou et al. (2022) investigated honeycomb core sandwich panels’ vibration and aeroelastic stability behavior with four-side supported boundaries in supersonic airflow. The quasi-steady first-order piston theory is used to model the aerodynamic pressure. Swain et al. (2022) presented the effect of delamination on the flutter characteristics of delaminated plates and the control of flutter velocity. The effect of delamination location and interface on the flutter characteristics of a laminated plate with various boundary conditions are investigated first, and an attempt is made to enhance the flutter velocity of the delaminated plate through an active control technique. Baitab et al. (2022) showed that embedding SMA is beneficial for mitigating the post-flutter vibrations. It also showed that shape memory alloy wires embedded in 3D composite structures significantly affect the aeroelastic performance of the structures by increasing the resultant bending moment of the cantilevered plate, leading to lower flutter speed. Mozafareiyan and Rezaeepazhand (2023) studied aero-thermoelastic nonlinear cracked plate. They concluded that the presence of crack or temperature alone increases the flutter speed/dynamic pressure, increasing the peak amplitude of the LCO and ultimately reducing the stability boundary.

Triangular plates are frequently used in different structures, such as large buildings, delta, and swept wings of aircraft (Tian et al., 2017; Visa et al., 2019; Zhang et al., 2020). Many studies about thermal buckling are done on rectangular plates, but there are few studies on triangular ones. In hypersonic flow, considerable deformation occurs due to significant heat and temperature differences. The present work aims to reduce these deformations by using shape memory alloy. Here, thermal buckling and post-buckling for triangular composite plates in supersonic flow are studied. Also, the starting point of instability and the effect of SMA force on the stability margin are studied for triangular composite plates. Among many researchers who studied the SMA effect and introduced its properties and behavior, Lagoudas’s theory is more accurate. It introduces some structural variables for strain-stress relations that can be solved without recursive operations (Hartl and Lagoudas, 2008). In most research, the thermal effect is studied at a steady temperature. However, this study considers the practical theory of the reference temperature method, which has fewer errors than previous theories in real cases and can be used for both laminar and turbulence flows (Anderson, 2006).

The triangular plates with simply supported (SSS) and clamped-simple-clamped (CSC) boundary conditions are considered in the present work. These two boundary conditions are selected to investigate the effect of boundary conditions on aeroelastic behavior and compare the results. The plate is modeled according to the classical plate theory while considering nonlinear strain according to Von Karman’s theory. The linear plate model is used to study linear behaviors such as natural frequencies, mode shape, buckling, and linear stabilities such as flutter and divergence. However, behaviors such as post-buckling, which shows the displacement of the plate after buckling or limit cycle amplitude, can only be predicted by the presence of nonlinear structural strains. The SMA wire is embedded in the composite plate to investigate the SMA effect on buckling, post-buckling, stability, and time response. The first-order piston theory is used for aerodynamics modeling. In the following, modeling and the equations of the triangular plate’s motions are presented; the numerical results and their analysis are provided. Finally, the conclusion and some suggestions for future works will be presented.

2. Thermo-aeroelastic modeling and the equations of motions

This section focuses on determining the governing equations for a triangular plate that contains an embedded SMA wire under the influence of aerodynamic and thermal forces. The equations of motion for the triangular plate are derived using classic plate theory (CLPT). To estimate temperature differences accurately, temperature reference theory is utilized. The Lagoudas theory is used to model the SMA, which is embedded in the triangular composite plate.

2.1. The classical laminated plate theory

Consider a four-layer triangular composite laminated plate as depicted in Figure 1, characterized by a ${[0, 90, 90, 0]}^{o}$ layering sequence. This plate has an altitude of $L$ , a base of $C_{r}$ , and a thickness denoted by $h$ . Each layer within the composite possesses an individual thickness, $h_{k}$ . The coordinates $X$ , $Y$ , and $Z$ refer to the longitudinal, transverse, and vertical dimensions, respectively. The angle $α$ represents the inclination between the $Y$ -axis and the triangle’s side. Additionally, a red line illustrates the shape memory alloy (SMA) wire, which is embedded along the $Y$ -axis across the plate. The embedded SMA wire, defined as a single element with a length of $L_{SMA}$ and a width of $C_{r_{SMA}}$ . The center of SMA wire is placed at location $C_{r} / 2$ and $L / 2$ .

For convenience in obtaining the equations of motion and solving the related equations, the equations of motion are mapped from the triangular plate to the unit square plate. The transformation equation from trapezoidal to rectangular coordinate is provided by Tian et al. (2017). Accordingly, the transformation equation from triangular to rectangular coordinate is given as:

ξ = \frac{x / C_{r} - (AR / 4) (1 + TR) \tan α (y / l)}{1 - (1 - TR) (y / l)}, η = \frac{y}{l},

J = [\begin{matrix} C_{r} (1 - (1 - TR) η) & \frac{AR}{4} (1 + TR) \tan α \\ 0 & l \end{matrix}]

(1)

$ξ$ and $η$ are dimensionless coordinates of a square plate. $AR$ is the aspect ratio and is defined as $AR = 4 (l / c_{r}) / (1 + TR)$ . $TR$ is the taper ratio and is defined as $TR = C_{t} / C_{r}$ . $C_{t}$ is the opposite side of the base of the trapezoidal plate. Matrix $J$ , is the Jacobian matrix that transforms the triangular domain into the unit square domain.

In classical laminated plate theory (CLPT) theory, the transverse displacement is independent of the transverse (or thickness) coordinate and the transverse normal strain $ϵ_{zz}$ is zero. Also, transverse shear strains $ϵ_{xz} = 0$ , and $ϵ_{yz}$ are set to zero. In terms of the Kirchhoff hypothesis, the displacement fields are expressed in the following form (Reddy, 2003):

\begin{matrix} u^{ESL} (x, y, z, t) = u_{0} (x, y, t) - z \frac{\partial w_{0}}{\partial x}, v^{ESL} (x, y, z, t) \\ = v_{0} (x, y, t) - z \frac{\partial w_{0}}{\partial y} \end{matrix}

w^{ESL} (x, y, z, t) = w_{0} (x, y, t)

(2)

where ( $u^{ESL}$ , $v^{ESL}$ , $w^{ESL}$ ) are the displacement components along the $(x, y, z)$ directions, respectively. The nonlinear strains in CLPT theory are expressed as (Reddy, 2003):

{ε}^{ESL} = {\begin{matrix} ε_{xx} & ε_{yy} & ε_{zz} & γ_{yz} & γ_{xz} & γ_{xy} \end{matrix}}^{T} = [ε^{0}] + z [ε^{1}]

ε_{xx}^{1} = - \frac{\partial^{2} w_{0}}{\partial x^{2}}, ε_{yy}^{1} = - \frac{\partial^{2} w_{0}}{\partial y^{2}}, ε_{zz}^{1} = 0, γ_{yz}^{1} = 0, γ_{xz}^{1} = 0,

\begin{matrix} γ_{xy}^{1} = - 2 \frac{\partial^{2} w_{0}}{\partial x \partial y}, ε_{xx}^{0} = \frac{\partial u_{0}}{\partial x} + \frac{1}{2} {(\frac{\partial w_{0}}{\partial x})}^{2}, \\ ε_{yy}^{0} = \frac{\partial v_{0}}{\partial y} + \frac{1}{2} {(\frac{\partial w_{0}}{\partial y})}^{2} \end{matrix}

ε_{zz}^{0} = 0, γ_{yz}^{0} = 0, γ_{xz}^{0} = 0, γ_{xy}^{0} = \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} + \frac{\partial w_{0}}{\partial x} \frac{\partial w_{0}}{\partial y}

(3)

where $ε^{0}$ is the membrane strain and $ε^{1}$ is the flexural (bending) strain, known as the curvature. Note that all strain components vary linearly through the laminate thickness, and they are independent of the material variations through the laminate thickness (Reddy, 2003).

The triangular composite plate is assumed to be monoclinic, which is a plate with just one symmetry. The relation between stress and strain for this composite is given as:

\begin{matrix} {\begin{matrix} \begin{matrix} σ_{xx} \\ σ_{yy} \\ σ_{zz} \end{matrix} \\ σ_{yz} \\ σ_{xz} \\ σ_{xy} \end{matrix}}^{(k)} = {[\begin{matrix} {\bar{c}}_{11} & {\bar{c}}_{12} & {\bar{c}}_{13} & 0 & 0 & {\bar{c}}_{16} \\ c_{21} & {\bar{c}}_{22} & {\bar{c}}_{23} & 0 & 0 & {\bar{c}}_{26} \\ {\bar{c}}_{31} & {\bar{c}}_{32} & {\bar{c}}_{33} & 0 & 0 & {\bar{c}}_{36} \\ 0 & 0 & 0 & {\bar{c}}_{44} & {\bar{c}}_{45} & 0 \\ 0 & 0 & 0 & {\bar{c}}_{54} & {\bar{c}}_{55} & 0 \\ {\bar{c}}_{61} & {\bar{c}}_{62} & {\bar{c}}_{63} & 0 & 0 & {\bar{c}}_{66} \end{matrix}]}^{(k)} {\begin{matrix} ε_{xx} - α_{xx} Δ T \\ ε_{yy} - α_{yy} Δ T \\ ε_{zz} - α_{zz} Δ T \\ γ_{yz} \\ γ_{xy} \\ γ_{xy} - 2 α_{xx} Δ T \end{matrix}} \end{matrix}

(4)

$σ_{ij}$ represent stress and $ε_{ij}$ express strain in $i - j$ direction. $α_{ij}$ is thermal coefficient and $Δ T$ is temperature differences. $C_{ij}$ shows material stiffnesses in their original form. Subscript of $xx$ , $yy$ , $zz$ express normal stress components in $X$ , $Y,$ and $Z$ direction and $yz$ , $xz$ , and $xy$ express shear stress in $Y - Z$ , $X - Z$ , and $X - Y$ directions, respectively.

2.2. Reference temperature model

The hypersonic fluid flow is assumed here, which moves along the $X$ axis according to Figure 1. For thermal heating modeling, a reference temperature model is considered, which can estimate panel friction and heat transfer in both laminar and turbulent flow (Anderson, 2006). This theory is based on a simple idea of the formulas derived from the theory of incompressible flow theory, in which the thermodynamic properties and heat transfer at the reference temperature represent the temperature somewhere inside the boundary layer (Anderson, 2006). This theory was first introduced by Rubesin and Johnson (1949) and then modified by Eckert (1956) to include a reference enthalpy. In this section, in some sense, the classical incompressible formulas were “corrected” for compressibility effects. Reference temperature (or reference enthalpy) methods have various applications in hypersonic flow analyses because of their simplicity.

Rubesin and Johnson (1949) have shown that it is possible to find a reference temperature $T^{*}$ between actual wall temperature ( $T_{w}$ ) and recovery temperature ( $T_{r}$ ), which $T_{r}$ is the temperature at a specific location on the assumed surface, when the convective heat transfer at this place is zero. Reference temperature will cause the variation of friction factors to vanish, and it can be expressed with reasonable accuracy by the following equation:

\frac{T^{*}}{T_{e}} = 1 + 0.032 M_{e}^{2} + 058 (\frac{T_{w}}{T_{e}} - 1)

(5)

$T^{*}$ shows reference temperature, $T_{e}$ , $M_{e}$ , and $T_{W}$ denote the temperature at the edge of the boundary layer, the local Mach number at the edge of the boundary layer, and the specified wall temperature, respectively. Wall temperature is determined as:

r = \frac{h_{aw} - h_{e}}{h_{0} - h_{e}} = \frac{T_{aw} - T_{e}}{T_{0} - T_{e}}

(6)

$h_{aw}$ , $h_{e}$ , and $h_{0}$ are enthalpies of the adiabatic, edge of the boundary layer, and primary, respectively. $T_{aw}$ and $T_{0}$ are adiabatic and primary temperatures. $r = \sqrt{P_{r}}$ , where $P_{r}$ is the Prandtl number and is assumed to be $0.715$ 0. By calculating the reference temperature, the induced force on the composite plate can be assessed.

2.3. Modeling shape memory alloy wire

This study assumes hypersonic flow, in which thermal stresses are induced. To reduce the effect of induced temperature difference, a shape memory alloy that reacts with temperature is embedded in the plate to control the thermal heating effect on the composite plate.

SMA has two properties: shape memory effect (SME) and Pseudoelasticity. In many previous studies, the static situation or steady-state conditions for loading the SMA are assumed, and the SME is used more because this phenomenon is primarily at relatively low temperatures. In the SME, the residual strain disappears after loading and unloading when the temperature goes up compared to the austenite finish temperature. The pseudoelasticity phenomenon happens when the temperature exceeds the austenite finish temperature. Loading and unloading on an SMA wire constitute a hysteresis cycle, which can create a damping cycle in its loop and gradually reduce the vibration of the plate. Pseudoelasticity is used when the plate is studied in a dynamic situation.

In this work, the heat and thermal conditions of the composite plate are assumed to be in a steady state, where the temperature starts at Martensite temperature and finishes at a steady state at Austenite temperature. Thus, the SME condition is presented. Although there are several methods for structural modeling of the SMA, Lagoudas’ (2008) theory is used because this theory can provide a reasonable estimate of SMA behavior in SME situations. In fact, in this theory, the evolution of the martensitic fraction is continuously determined over time, and the behavior of the SMA in a steady state is determined by calculating the martensite and austenite fractions. The force of the SMA wire should be calculated after phase transformation. SMA wire is embedded in the Martensite phase, and due to thermal heating, it transforms into the Austenite phase, and consequently, the strain will be reduced. This phenomenon depends on primary stress and temperature.

For SMA with the one-dimensional constitutive model, the thermoelastic response for pure austenite and martensite phase is described as (Lagoudas, 2008):

ε^{A} = S^{A} σ + α^{A} (T - T_{0})

ε^{M} = S^{M} σ + α^{M} (T - T_{0})

(7)

$ε^{A}$ , $S^{A}$ , $α^{A}$ express strain, compliance coefficients, and thermal coefficient in solid austenite, and $ε^{M}$ , $S^{M}$ , $α^{M}$ express them in the solid martensite phase.

For phase transformation modeling, two cases are presented: forward phase transformation (i.e. from austenite to martensite) and reverse phase transformation (from martensite to austenite). In both situations, the one-dimensional form of the thermodynamic energy $π$ is given by (Lagoudas, 2008):

\begin{matrix} π = | σ | H + \frac{1}{2} Δ S σ^{2} + Δ α (T - T_{0}) + ρ Δ s_{0} T - ρ Δ u_{0} \\ - [ρ b^{M} ζ + (μ_{1} + μ_{2})] = \pm Y \end{matrix}

(8)

where the + symbol is used for forwarding phase transformation and the − symbol is used for reverse phase transformation. $H$ is the maximum strain, $Δ S = S^{M} - S^{A}$ , $Δ α = α^{M} - α^{A}$ , $Δ s_{0}$ is specific entropy differences, $Δ u_{0}$ the express internal energy difference between austenite and martensite phases. $Y$ , $b^{M}$ , $b^{A}$ , $μ_{1}$ , and $μ_{2}$ are model parameters which experimentally determined. $Z$ express martensite rate which is between 0 and 1. SMA wire is modeled with the Lagoudas (2008) theory. In Lagoudas theory the quadratic polynomial hardening function is used, which expresses a type of model parameters that are estimated in terms of experimentally observable material parameters.

By substituting the quadratic polynomial hardening function in equation (8), and extracting strain $ϵ$ , we have:

\begin{matrix} ε = S σ + α (T - T_{0}) \\ + \frac{H sgn (σ)}{ρ b^{M}} [| σ | H + \frac{1}{2} Δ S σ^{2} + ρ Δ s_{0} (T - M_{s})] \end{matrix}

(9)

which $S$ is related to martensite fraction. So, by considering $S = S^{A} + ξ (S^{M} - S^{A})$ and again replacing SMA martensitic fraction:

ξ = \frac{1}{ρ b^{M}} [| σ | H + \frac{1}{2} Δ S σ^{2} + ρ Δ s_{0} (T - M_{s})]

(10)

By replacing the quadratic polynomial hardening functions in equation (8), the relation between stress and strain which is defined by Lagoudas (2008) will be the equation of the third degree (equation (14)). In there, the strain- stress relation is expressed as:

\begin{array}{l} ε = \frac{Δ S^{2}}{2 ρ b^{M}} (σ^{3} + [\frac{3 H}{Δ S} + \frac{Δ α Δ T}{Δ S}] σ^{2} \\ + [\frac{2 S^{A} ρ b^{M}}{Δ S^{2}} + \frac{2 ρ Δ s_{0}}{Δ S} (T - M_{s}) + \frac{2 H Δ α Δ T}{Δ S^{2}} + \frac{2 H^{2}}{Δ S^{2}}] σ \\ + \frac{2 α^{A} Δ T ρ b^{M}}{Δ S^{2}} + \frac{2 ρ Δ s_{0} Δ α Δ T}{Δ S^{2}} + \frac{2 H ρ Δ s_{0}}{Δ S^{2}} (T - M_{s})) \end{array}

(11)

So, the fourth characteristic of shape memory, that is, martensite fraction $ξ$ is needed and can be expressed in terms of other variables in the quadratic polynomial hardening function model. For solving stress in terms of strain, the third order equation (10) should be solved. This equation is estimated by Taylor expansion. For forwarding martensite the stress is given as:

σ = \frac{1}{δ_{2}} (- δ_{1} \pm \sqrt{δ_{1}^{2} - δ_{2} ε - δ_{2} ((δ_{3} - α) T - δ_{3} M_{s} + α T_{0})})

(12)

with

\begin{matrix} δ_{1} = \frac{2 H^{2}}{ρ Δ s_{0} (M_{s} - M_{f})}, δ_{2} = \frac{8 H Δ S}{ρ Δ s_{0} (M_{s} - M_{f})}, \\ δ_{3} = \frac{H}{M_{s} - M_{f}} \end{matrix}

Figure 2(a) shows the stress-strain curves for two different temperatures (Lagoudas, 2008) and Figure 2(b) is the approximate solution from equation (12). These figures can show that the approximate solution has high accuracy with a maximum error of 2%, hence the obtained equation gives an appropriate approximate solution.

The first step to obtaining SMA force is to determine the temperatures of transformation for constant stress levels. For this purpose, equation (13) is used to compute the values of the temperatures of non-zero stress transformation for the forward phase transformation ( $M_{s}^{σ}$ and $M_{f}^{σ}$ ), and for the reverse phase transformation ( $A_{f}^{σ}$ and $A_{s}^{σ}$ ). The martensite volume fraction $ζ$ , before austenite starts temperature in reverse martensitic is 1 and after martensite starts temperature in forwarding martensitic is $0$ .

\frac{1}{2} Δ S σ^{2} + σ H + ρ Δ s_{0} (M_{s}^{σ} - M_{s}) = 0

\frac{1}{2} Δ S σ^{2} + σ H + ρ Δ s_{0} (M_{f}^{σ} - M_{f}) - ρ b^{M} = 0

\frac{1}{2} Δ S σ^{2} + σ H + ρ Δ s_{0} (A_{f}^{σ} - A_{f}) - ρ b^{A} = 0

\frac{1}{2} Δ S σ^{2} + σ H + ρ Δ s_{0} (A_{s}^{σ} - A_{s}) = 0

(13)

Figure 3 shows the strain versus temperature plots for the isobaric loading-unloading conditions, for three different stress levels of 200, 250, and 300 MPa. In this figure, $M_{f}^{σ}$ , $A_{s}^{σ}$ , $A_{f}^{σ}$ , and $M_{s}^{σ}$ determine temperatures of martensite finished, austenite start, austenite finished, and martensite start, respectively. Triangular solid lines, circular dashed lines, and square dotted have been plotted to show the phase transformation under different constant stresses. In this article, the SMA wire embedded in Austenite start temperature phase and the effects of SMA hysteresis are neglected.

2.4. Aerodynamic forces modeling

Piston theory (Lighthill, 1953) provides a simple relation between pressure perturbation and structural motion. Each side of the composite plate is under a freestream flow. Since the upper and lower sides of the composite plate are under the aerodynamics load, the pressure difference between these two surfaces and the applied load is described as:

Δ P_{a} (x, y, t) = - 2 \frac{ρ_{\infty} U_{\infty}}{M_{\infty}} (\frac{\partial w}{\partial t} + U_{\infty} \frac{\partial w}{\partial x})

(14)

F_{Aero} = \int \int_{A} Δ P_{a} w_{i} dxdy

(15)

$P_{a}$ is the aerodynamic pressure on the plate, $ρ_{\infty}$ is the density of the air, $U_{\infty}$ is the relative velocity of the air around the panel, $M_{\infty}$ is Mach number, and $F_{Aero}$ gives the aerodynamic force on the plate due to the pressure difference.

By transforming the triangle domain to a rectangular domain, the aerodynamic force is written as:

F_{Aero} = \int \int - 2 \frac{ρ_{\infty} U_{\infty}}{M_{\infty}} | J | 〈 ω_{0} \frac{\partial w}{\partial t} + U_{\infty} (\frac{\partial w}{\partial x} + \frac{\partial w^{esl}}{\partial x}) 〉 wdxdy

(16)

$ω_{0}$ is the natural frequency.

2.5. The assumed mode method

According to the assumed mode method, the displacements $u, v$ , and $w$ are expressed as a product of the time-dependent generalized coordinates and the spatial functions (assumed mode shape). These functions must satisfy the geometric boundary conditions. The in-plane and transverse displacements for the assumed mode method are expressed in the following forms:

\begin{matrix} u_{0} (ξ, η, t) = \sum_{i} \sum_{j} u_{ij} (t) χ_{i} (ξ) ϑ_{j} (η), \\ v_{0} (ξ, η, t) = \sum_{k} \sum_{l} v_{kl} (t) γ_{k} (ξ) θ_{l} (η) \end{matrix}

w_{0} (ξ, η, t) = \sum_{m} \sum_{n} w_{mn} (t) ϕ_{m} (ξ) ψ_{n} (η)

(17)

in which $u_{ij} (t)$ , $v_{kl} (t)$ , $w_{mn} (t)$ are time-dependent generalized coordinates, $χ_{i} (ξ)$ , $γ_{k} (ξ)$ , and $ϕ_{m} (ξ)$ are assumed functions in the $ξ$ direction and $ϑ_{j} (η)$ , $θ_{l} (η)$ , and $ψ_{n} (η)$ are the assumed functions in the $η$ direction. For in-plane displacement, the necessary assumed function is selected from a simple rod model, and for transverse displacements from Euler-Bernoulli beam, with similar boundary conditions. The assumed functions for the in-plane displacements in $X$ and $Y$ directions from rod theory are given as:

χ_{i} (x), γ_{k} (x) = C_{1} \cos (ω x) + C_{2} \sin (ω x)

ϑ_{j} (y), θ_{l} (y) = C_{1} \cos (ω y) + C_{2} \sin (ω y)

(18)

The assumed mode functions $ϕ$ and $ψ$ for transverse displacement are selected from Euler-Bernoulli beam theory, and are defined as:

\begin{matrix} ϕ_{m} (x) = B_{1} \cosh (β x) + B_{2} \sinh (β x) + B_{3} \cos (β x) \\ + B_{4} \sin (β x) \end{matrix}

\begin{matrix} ψ_{n} (y) = B_{1} \cosh (β y) + B_{2} \sinh (β y) + B_{3} \cos (β y) \\ + B_{4} \sin (β y) \end{matrix}

(19)

For discretizing the equations of motion and determining the governing equations, the Lagrange-Euler method is used. For generalized coordinates of $q_{i} = (u_{ij}, v_{kl}, w_{mn})$ , the governing equations are determined as:

\begin{array}{l} ({[M]}_{P} + {[M]}_{S}) {\overset{\cdot\cdot}{\bar{q}}} + ([C] + {[C]}_{a e r o}) {\dot{\bar{q}}} \\ + ({[K]}_{P} + {[K]}_{S}) [\bar{q}] + {[N L]}_{P} + {[N L]}_{S} \\ = {F_{T}}_{P} + {F_{T}}_{S} + {F_{S}} \end{array}

(20)

where $[M]$ , $[C]$ , and $[K]$ are mass, damping, and stiffness matrices. $[NL]$ , $[F_{T}]$ , $[F_{S}]$ are structural nonlinear, thermal forces, and shape memory alloy forces matrices, respectively. Subscript of P, S, and aero denote plate, SMA, and aeroelastic quantities, respectively. More details are given in the Appendices.

3. Numerical results

This section delves into the thermal buckling analysis of various composite plate layups with different aspect ratios and boundary conditions. The effect of the SMA wire and its orientation is evaluated in each case. The study also investigates the effect of SMA on thermal post-buckling under different boundary conditions. Furthermore, the stability of triangular composite plates is analyzed for different layups and shape memory effects. Finally, the study determines the time result and phase diagram for different Mach numbers in SSS and CSC boundary conditions, before and after embedding the SMA wire, to showcase its effect on vibration control and stability region enhancement.

The results are presented in dimensionless form, but some values are assumed to obtain physically achievable results. Table 1 displays the material and geometrical properties of the triangular plate and the SMA wires, which will be used unless instructed otherwise.

Table 1.

Parameters of the triangular plate and SMA wire (Zhang et al., 2020).

Parameter	Definition	Value
$C_{r}$ (m)	Base of triangular	0.5
$h$ (m)	Height of triangular	0.004
$L$ (m)	Altitude of triangular	0.5
$α$	Angle of the triangle	$30 °$
$ν_{12}, ν_{23}, ν_{13}$	Poisson’s ratios	0.33
$α_{11}, α_{22}, α_{33} / ° K$	Thermal coefficients	7.5 × 10⁻⁶, 18 × 10⁻⁶, 18 × 10⁻⁶
$E_{Aus} (Gpa)$	Young’s modulus of SMA	32.5
$E_{Mar} (Gpa)$	Young’s modulus of SMA	23
$ρ_{sma} (kg / m^{3})$	Mass density of SMA	5100
$A_{F} (° K)$	Austenite finished temperature	$270$
$E_{1} (Gpa)$	Young’s modulus of the plate	29.4
$E_{2}, E_{3} (Gpa)$	Young’s modulus of the plate	15.2
$G_{1} (Gpa)$	Shear’s modulus of the plate	6.06
$G_{2}, G_{3} (Gpa)$	Shear’s modulus of the plate	3.03
$ρ (kg / m^{3})$	Mass density of plate	1330
$M_{s} (° K)$	Martensite starting temperature	$226$
$M_{f} (° K)$	Martensite finished temperature	$194$
$ρ Δ s_{0}$	Specific entropy differences	11.55 × 10⁻⁴
$A_{S} (° K)$	Austenite starting temperature	$241$
$α_{Aus} α_{Mar} / ° K$	Thermal coefficient	$22 \times 10^{- 6}$
Cr_SMA (m)	Width of SMA wire	0.001

3.1. Thermal buckling

When a plate undergoes thermal compressive stresses, the structure may undergo instability. Since it is caused by temperature loading, this type of instability is known as thermal buckling or thermal instability. At the temperature at which buckling occurs, the structure starts to deflect from its original configuration. This temperature is also known as the thermal bifurcation point.

This section presents the thermal buckling of the composite triangular plate based on the formulation mentioned in the previous section. Due to thermal heating at hypersonic speeds, the composite plate undergoes thermal buckling. The critical temperature at which buckling occurs is denoted as buckling temperature, and is determined from solving the eigenvalue solution. In equation (20), the equations of motion for all three directions are expressed. Since buckling is a linear phenomenon, by keeping the linear expressions for the transverse displacement $\bar{w}$ , $Δ T_{cr}$ is obtained by solving the following eigenvalue problem (equation (21)).

[K_{w}^{w} - Δ T_{cr} K_{w}^{T}] {\bar{w}} = 0

(21)

Figure 4 shows the thermal buckling temperature versus aspect ratio $C_{r} / L$ in two different boundary conditions CCS and SSS. In each figure, the effect of different layering has been compared.

To study the aspect ratio, the $L$ value is kept constant, and the $C_{r}$ value is changed. In Figure 4, the circular red line is for [0/90/0/90] layup, the rectangular blue line is for a symmetrical composite plate layup of [0/90/90/0], the circular green line is for [30/60] layup, and the triangular black line shows [−45/45] layup. This figure shows that in most cases, a composite plate with [30/60] layering has the highest critical temperature rather than the other layup. In this layup, composite layering is aligned with the side of the triangular plate. Also, this figure shows that critical buckling temperature differences are reduced when the aspect ratio increases. Figure 4 shows that layering has the highest stiffness and consequently higher buckling critical temperature in a lower aspect ratio with SSS boundary condition [0/90/90/0]. Generally, these results show that the thermal buckling temperature in the SSS boundary condition is lower than the CSC boundary condition.

In Figure 4(a), at higher values of $C_{r} / L$ , ${(Δ T)}_{cr}$ is unchanged, since increasing the aspect ratios will not significantly change the structure’s stiffness. In Figure 4(b), the structure’s stiffness is at a minimum value from aspect ratios between 2 and 4. Then, with increasing aspect ratios, the stiffness increases in all layering, and the thermal buckling temperature increases accordingly. Also, it should be noted that the boundary conditions in the CSC case are much stiffer than in the SSS case. In the boundary conditions of CSC, due to the higher stiffness in the boundaries, the increase in stiffness is much more significant, which causes a tremendous increase in the stability range.

The following presents the effect of shape memory alloy on the thermal buckling of triangular plates. The SMA wire is embedded in the middle of the plate, and in this study, it is assumed that it is parallel to the $X$ -direction. Figures 5 and 6 show the effect of SMA wire on the thermal buckling of the triangular composite plate for both CSC and SSS boundary conditions. In Figure 5, it is clear that the effect of SMA wire in low aspect ratio is high, and with increasing dimensionless aspect ratio, this effect is reduced. Although in this condition, the length of the SMA wire is increased, in a higher aspect ratio, by increasing the plate’s area, the temperature forces overcome the SMA wire and neutralize its effect. In low aspect ratios, the SMA wire affects thermal buckling temperature most. The temperature change in the shape-memory alloy wire causes an axial force, increasing the system’s stiffness in the in-plane and transverse directions.

Figure 6 shows the critical temperature at four different layering {(a) $[0 / 90 / 90 / 0]$ , (b) $[0 / 90 / 0 / 90]$ , (c) $[- 45 / 45]$ , (d) $[30 / 60]$ } for SSS boundary condition. The SMA wire is embedded along $X$ -axis in the middle of the composite plate. In this figure, the effect of SMA wire on critical buckling temperature in low $C_{r} / L$ ratio is high, and by increasing this ratio, its effect is reduced. Both Figures 5 and 6 show that with increasing the ratio of $C_{r} / L$ , the effect of SMA wire is not considerable and, the critical buckling temperature in both situations, with/without SMA, is the same.

shapes In the following the effect of the triangular plate angle ( $α$ ) with simple boundary conditions is studied. It is changed from $\frac{π}{8}$ to $\frac{5 π}{12}$ . The first mode shape for three different angles ( $\frac{π}{6}, \frac{π}{4}, \frac{π}{3}$ ) is shown in Figure 7.

The first six mode shapes of the simply supported triangular composite plate with the angle $α = \frac{π}{6}$ at $M_{a} = 0$ , are plotted in Figure 8. The lengths of the base and height of the triangular are assumed to be $0.5$ .

The effect of changing the angle of the triangular plate ( $α$ ) on the critical buckling temperature for two boundary conditions is shown in Figure 9. To study side angle of the triangle plate, aspect ratios are constant like Table 1 and just the angle of triangle will change. In each figure, four different layups are displayed. Figure 9(a) shows that by increasing the angle of the triangular plate, the critical buckling temperatures are increased for two layering [45/45] and [30/60] from $\frac{π}{8}$ to $\frac{π}{4}$ . This is due to alignment of two layers in the triangular in these cases. After that, the critical buckling temperature almost remained unchanged, because in higher plate angles the effect of the thermal bending load will be decreased and consequently the critical buckling temperature will remain constant.

In two other layups of [ $0 / 90 / 0 / 90$ ] and [ $0 / 90 / 90 / 0$ ] the critical buckling temperature reaches a peak at the angle $\tan^{- 1} 0.5$ , and then decreases, since, at this angle, the plate has the maximum amount of in-plane temperature load. At this angle, the triangular plate will be isosceles. It should be noted by increasing plate angle ( $α$ ) the effect of symmetry will be reduced and the differences in critical buckling temperature between $[0 / 90 / 0 / 90]$ and $[0 / 90 / 90 / 0]$ is reduced.

Figure 9(b) shows the critical buckling temperature versus the triangular plate angle for four different layups in CSC boundary conditions. It shows in the CSC boundary condition, that in all layering the critical buckling temperature will be independent of angle and it remains in a constant value in higher plate angles. At the lower plate’s angle, the $[30 / 60]$ layering has the maximum amount of critical temperature and at the higher angle $[0 / 90 / 90 / 0]$ layup is the most effective layering among the others. Also, Figure 9 shows that symmetry layering has a significant effect on increasing critical temperature.

Figure 10 shows the critical buckling temperature after embedding SMA wire with a composite plate in the SSS boundary condition. By placing this alloy along $X$ -axis (Figure 10(a)) and along $Y$ -axis (Figure 10(b)) the critical buckling temperature is increased at all plate angles. Figure (10(a)) shows that the critical buckling temperature decreases by increasing the angle of the plate because by rising the plate’s angle the length of borders will be increased and consequently the force of thermal bending moment is increased and the effect of SMA wire will be reduced.

In Figure 10(b) the SMA wire is embedded in the plate along $Y$ -axis and it is shown that the critical buckling temperature is exponentially decayed in all angles except for $\tan^{- 1} 0.5$ angle. Although the SMA wire could increase the critical buckling temperature, by increasing the angle of the plate and consequently reducing the length of the SMA wire, this effect will be reduced. Figure 11 shows in the SSS boundary condition, the effect of embedding SMA wire along the X-axis is more than embedding along Y-axis. This happened because the SMA wire was embedded along the aerodynamic forces and it could increase the stiffness of the system more than along Y-axis.

In the CSC boundary condition, the critical buckling temperature is plotted for two different embedding SMA wire angles (Figure 11). This figure shows that like the SSS boundary condition, SMA wire has a significant effect on increasing critical buckling temperature than the CSC case. In Figure 11(a) the SMA wire is embedded in the composite plate along the X-axis. This figure shows except [30/60] in other layups the critical buckling temperature at first is decreasing and then increased with an increase in sweep angles.

Since in Figure 9(b) for low aspect ratios, the buckling temperature has a sharp decrease, the SMA wire cannot compensate for this effect, but when this decrease is reduced, the effect of the SMA wire on increasing buckling temperature is clearer. Since in Figure 9(b), the most decrease in critical buckling temperature is in [ $30 / 60$ ] layering, the SMA cannot increase the critical buckling temperature after embedding with the composite plate.

Figure 11(b) shows the critical buckling temperature versus the sweep angle after embedding the SMA wire along $Y$ -axis. This figure shows SMA wire in [ $0 / 90 / 90 / 0$ ] layering has the highest effect on the critical buckling temperature than the other layups. The effect of increasing SMA wire length is not considered in this figure, because in the CSC boundary condition, the plate has harder stiffness and has higher critical buckling temperature. So, it overcomes the effect of increasing SMA wire. Also, this figure shows that, in the CSC boundary condition like the SSS boundary condition, the effect of embedding SMA wire along $X$ -axis is more than it is embedded along $Y$ -axis.

There are two main types of instabilities in aeroelastic structures, one is divergence and the other is flutter. In divergence, aerodynamic forces overcome structural forces and it is a static phenomenon similar to buckling. Flutter is a dynamic phenomenon and there is a simultaneous interaction of aerodynamic, structural, and dynamic forces. Therefore, understanding the phenomenon of buckling has a direct relationship with the phenomenon of divergence, which has been observed in many instability states investigated in following.

3.2. Post-buckling

Thermal post-buckling behavior is a nonlinear phenomenon that gives the structure’s deflection when the thermal load increases after the buckling load. If the applied thermal load is lower than the buckling point, the deflection of the structure is zero, but after that, the structure has deflection, which can be determined by post-buckling analysis. Here, the composite plate undergoes thermal load due to aerodynamic forces and friction between the plate and fluid. Figures 12 and 13 show the thermal post-buckling deflections for two boundary conditions, SSS and CSC, with embedded SMA wire at 0° and 90° angles. Thermal post-buckling for the composite plate with [0/90/0/90] layering is studied for the point in location of (0.3, 0.2).

Figure 12 shows the post-buckling deflection for CSC and SSS boundary conditions, respectively. In this figure, it is clear that the shape memory alloy wire has increased the buckling critical temperature value, and the post-buckling deflection of the plate with embedded SMA is lower than that of the plate without SMA.

Figure 13 shows thermal post-buckling for the triangular plate with embedded SMA wire along the Y-axis for two different boundary conditions of CSC and SSS. Figure 13(a) and (b) show the post-buckling deflection for CSC and SSS boundary conditions, respectively. The effect of SMA wire in increasing critical buckling temperature is obvious in these figures. Similar to Figure 12, in Figure 13(b), by increasing temperature the effect of the SMA wire is reduced.

3.3. Stability analysis

To assess the stability condition of the composite plate, eigenvalues of the linearized system from equation (20) are determined in terms of the dimensionless aerodynamic coefficient. At first, the equilibrium point is determined, and the nonlinear terms are linearized around this point. Then, the eigenvalues are calculated from linearized equations, and stability analysis is carried out. If the eigenvalues have a real positive value, the behavior of the plate is unstable at that aerodynamic pressure. There are two most common types of instability: divergence and flutter. At low speeds, aerodynamic forces reduce the amplitude of structural vibrations. However, the aerodynamic forces amplify the vibration at a velocity above a certain amount that depends on the structure’s stiffness. If the eigenvalue of the system is pure real with a positive value, the instability is divergence. However, if the eigenvalue is complex conjugate with a positive fundamental part, the instability is flutter. To linearize the presented nonlinear equations, all-time derivatives variables are set to zero. The equilibrium point is determined for obtained static equations, and then the Jacobian of the nonlinear equations is determined. The eigenvalues of the Jacobian matrix determine the type of instability of the system.

In Figure 14(a) for four different layerings [0/90/0/90], [0/90/90/0], [30/60], and [−45/45], the real parts of the eigenvalues are plotted versus dimensionless aerodynamic coefficient $(λ)$ . This figure shows that [0/90/0/90] layup has the lowest instability aerodynamics pressure, and [−45/45] has the highest aerodynamics instability pressure. In Figure 14(b), the real and imaginary parts of the eigenvalues are shown to express the type of unsteady point. Since the dominant eigenvalue is real in these cases, all unsteady points in these four mentioned layups are divergence instability.

Figure 15 shows the real part of the eigenvalue of the linearized system versus the dimensionless aerodynamic coefficient for the SSS boundary condition. As expected, the instabilities started much earlier than the CSC condition.

This figure shows that in the SSS boundary condition, the eigenvalues of the composite plate have positive real part with no imaginary part, which means the related instability is Divergence.

The effect of SMA wire on the stability analysis of the triangular plate with four different composites layering of $[0 / 90 / 0 / 90]$ , $[0 / 90 / 90 / 0]$ , $[- 45 / 45]$ , and $[30 / 60]$ at CSC boundary condition is shown in Figure 16. From this figure, it can be concluded that, in all of these cases, the shape memory alloy has a considerable effect on increasing the region of stability.

For $[0 / 90 / 0 / 90]$ layering, the plate is stable up to a dimensionless aerodynamic coefficient of 10.69, and after embedding with SMA wire, its stability margin increased more than 3.5 times to 38.29. For $[0 / 90 / 90 / 0]$ the stability margin is increased from 12.07 to about 40. In $[- 45 / 45]$ and $[30 / 60]$ layups, the SMA wire increased the margin of instability from 14.87 and 14.21 to about 50 and 43.62, respectively. These values show that the SMA wire has the most effect on $[0 / 90 / 90 / 0]$ layering in increasing the maximum amount of aerodynamic coefficient of the composite plate’s stability.

Figure 17 shows the type of instability of the triangular composite plate with CSC boundary conditions for different layups. It shows in all layering, after embedding SMA wire with composite plate, the positive real values of the eigenvalues of the system have imaginary values, and that means our instability point is changed to Flutter.

In Figure 18 the same layering is investigated for SSS boundary conditions, to show the effect of boundary conditions on instabilities. The rectangular blue points and triangular black points give the eigenvalue for the composite plate with and without SMA wires. The SMA wire is aligned along $X$ -axis. This figure shows the SMA wire has increased the domain of stability for $[0 / 90 / 0 / 90]$ and $[0 / 90 / 90 / 0]$ layups by 2.5 times and by 3 times for $[- 45 / 45]$ layups and by 3.7 times for $[30 / 60$ ] layering. In all figures, the SMA wire increased the bifurcation point even more than the instability point of the composite plate.

To determine the type of instability with/without embedded SMA wire for the triangular composite plate with SSS boundary conditions, the real and imaginary parts of the eigenvalue of the linearized system are shown in Figure 19. By examination of the dominant eigenvalues of the system, it is revealed that like the CSC condition, in the bifurcation point, the unstable eigenvalue of the linearized system has an imaginary part, so, the related instability is flutter.

3.4. Time response

This section presents the time response analysis of the triangular composite plates for two cases, with/without embedded SMA wire. The triangular composite plate layup is selected as [0/90/0/90], and the system’s response is determined at $ξ = 0.25$ and $η = 0.5 / 3$ . The SMA wire with a width of $0.001$ mm is embedded in the middle of the plate in two cases, in the first case along the Y-axis and the second case along the X-axis. The response of the composite plate is determined in four Mach numbers: 0.5, 1, 1.5, and 2. From equation (5), the temperature differences in these velocities are 9.6267°C, 38.5070°C, 86.6408°C, and $154.0282 ° C$ , respectively.

For a better understanding of the behavior of the nonlinear response, the Lyapunov exponent is determined. Lyapunov’s exponent characterizes the divergence rate of two infinitesimally near trajectories over time. Suppose the most prominent Lyapunov exponent is positive. In that case, it means the chaotic behavior of the nonlinear system, the zero Lyapunov exponent means a cyclic behavior, and the negative Lyapunov exponent means the equilibrium point.

At first, for a composite plate with SSS boundary conditions, the steady-state response of composite plates is shown in Figure 20 for four different Mach numbers. At $M_{a} = 0.5$ (blue line), the plate has a static or post-buckling deflection of 0.2058. In this Mach number, a divergence of the plate occurs. At this Mach number, the aeroelastic system has a real positive eigenvalue without an imaginary part, meaning the aerodynamic force will overcome the structural force and deflect it. Since the system has a stiffening nonlinearity, the aerodynamic force has reached a balance with the linear and nonlinear structural forces, and therefore, the system will reach a static deflection. At $M_{a} = 1$ (green line), the plate fluctuates irregularly. In this case, the Lyapunov exponent of the response is 119.95, which indicates that the composite plate has chaotic behavior at this Mach number. Also, due to divergence, the plate has an approximated static deflection of 0.738. At Mach = 1, the plate has dominated eigenvalue with an imaginary eigenvalue with a positive fundamental part, but due to nonlinear stiffness, the system will eventually fall into an aperiodic behavior.

In Figure 20, at $M_{a} = 1.5$ (brown line), the dimensionless amplitude of the plate is increased from 0.25 to 1.15, and the Lyapunov exponent is 79.43, which demonstrates this instability’s divergence and cyclic behavior. Also, in this Mach number, the dominant eigenvalue of the system is complex with a positive real part, which is in agreement with the oscillating behavior shown in Figure 20.

The red line in Figure 20 shows the plate with an SSS boundary condition having an oscillating response. The determination of its Lyapunov exponent gives the value of 80.108, which indicates chaos behavior is presented at this Mach number. Since the plate fluctuates on nonzero mean amplitude, the plate’s divergence is also presented. The plate has positive real eigenvalues with an imaginary part at this Mach number. The plate has chaotic behavior since the aerodynamic force did not reach equilibrium with linear and nonlinear structural forces.

Figure 21 shows the phase diagram for three Mach numbers 1, 1.5, and 2 in the SSS boundary condition. The black dots on this figure show the points of the Poincare section. From the Poincare section, it is clear that the composite plate probably has a chaotic behavior at the Mach = 1 and 2 because the number of points of contact with the Poincare section is too much, but in Figure 21(b), the plate has a 5-frequency behavior.

Shape memory alloy is embedded in the middle of the composite triangular plate along the $Y$ -axis. The steady-state response is shown in Figure 22(a). This figure shows that the SMA wire was completely successful in stabilizing the plate in $M_{a} = 0.5$ and its divergence is approximately zero. The green line in Figure 22(a) and (b) shows that at $M_{a} = 1$ , not only the amount of static deflection is reduced from $0.738$ to $0.53$ , but also the dimensionless amplitude is approximately is reduced from $0.1$ to $0.06$ . Lyapunov exponent at $M_{a} = 1$ is estimated as $20.54$ which shows that at this velocity, the plate has instability, and Figure 22(b) shows the plate has a 1 frequency cyclic behavior. The most significant reason that SMA couldn’t stabilize the system, could be the small width of SMA for the plate, but it could appropriately affect an increase in the stability margin of the plate.

At $M_{a} = 1.5$ the plate fluctuates between the dimensionless amplitudes of $0.15$ and $1$ (brown line Figure 22(a)–(c)). The static and dynamic deflections are approximately 15% and 40% lower than the deflections of the plate without SMA wire. At this Mach, the plate’s eigenvalue has a positive real value with an imaginary part that shows the instability of the linear system. Also, the Lyapunov exponent for time response is 82.87. Figure 22(c) indicates the plate has a 2-frequency behavior. By increasing the Mach number to 2, the effect of the SMA wire is almost equal to the previous Mach number. The red line in Figure 22(a) to (d) shows the plate with embedded SMA. At this condition, the plate oscillates with approximate static and dynamic amplitudes of $1.5$ and $0.375$ respectively. In these conditions, the static deflection was reduced by $16 %$ and the amplitude of the dynamic is reduced to a third. Also, the Poincare section in Figure 22(d) shows the plate has a 12-frequency behavior.

For a more detailed study on the effect of SMA wire on the time response of the plate, the SMA wire is embedded into the plate along the $X$ -axis and at the middle of the plate (Figure 23). Figure 23(a) shows that at $M_{a} = 0.5$ after embedding with SMA into the plate, almost all vibrations are completely damped. The green line in Figure 23(a), shows the behavior of the plate at $M_{a} = 1$ after embedding with the SMA wire along $X$ -axis. It shows the plate oscillates around $0.6529$ which is lower than the dynamic deflection of the plate without embedded SMA wire and higher than the case when the plate is embedded with SMA wire along $Y$ -axis. Also, the static deflection is zero in this case, while the plate with embedded SMA wire along Y-axis was not. Also, with comparison Figures 21(b), 22(c), and 23(b) it can be concluded that after embedding the SMA wire along the $Y$ -axis, the vibration frequency of the plate is reduced and after embedding the SMA wire along the X-axis this amount is reduced more significantly. Figure 23(a) to (c) show that plate at $M_{a} = 1.5$ fluctuate around $1.28$ with $0.03$ amplitude. At this Mach number, while the fluctuation point damped is not very considerable, the dimensionless amplitude is controlled obviously. The Lyapunov exponent in this situation is 67.4 which shows the instability. Figure 23(b) indicates the plate has a 1-frequency cyclic behavior.

The plate’s behavior after embedding with SMA wire at $M_{a} = 2$ is plotted with the red line in Figure 23(a) to (c). The plate is oscillating around $1.7$ with an approximate dynamic amplitude of $0.5$ . In this situation, not only the equilibrium point and its amplitude are higher than the plate embedded with SMA along $Y$ -axis, but also it had no significant change from the plate without the embedded SMA wire. There are two reasons for this result. The first reason is the angle of embedding the SMA wire with the plate, and the second one is that, since the thermal expansion of the SMA wire is higher than the plate, by increasing temperature thermal expansion of the SMA overcomes the thermal expansion of the plate. In other words, the force created by thermal expansion is more than the force induced to the plate by transformation. However, Figure 23(c) shows the plate’s behavior at Mach = 2 is 6-frequency cyclic behavior.

In the case of the CSC boundary conditions, it is expected that the displacement of the plate is much lower than in the SSS conditions because it has a higher stiffness. Figure 24 shows the time response of a triangular composite plate with CSC boundary conditions in four different Mach numbers. In contrast to SSS, at $M_{a} = 0.5$ the CSC plate hasn’t considerable deflection because at $M_{a} = 0.5$ the temperature difference is lower than the critical temperature (Figure 4(a)). At $Mach = 0.5$ , all eigenvalues of the system have a negative real part which means at this velocity, the plate with CSC boundary condition is stable.

At $M_{a} = 1$ the vibration of the plate is almost damped. The Lyapunov exponent is $- 1.717$ which means that the response of the system is settled down in an equilibrium point, or the system has a static deflection. Also, at $M_{a} = 1.5$ the static deflection of the plate is $0.399$ , which is about half of the deflection of the plate in the SSS boundary condition.

The red line in Figure 24 shows that at Mach number $2$ , the CSC plate has a constant static deflection, of $0.6739$ . In these Mach numbers (0.5, 1, 1.5), the dominate eigenvalue of the system is positive without an imaginary part, which means divergence instability. This happened because as it’s expected the stiffness of triangular composite plate in CSC boundary condition is greater than SSS.

Time response for CSC boundary conditions composite plate with embedded SMA wire, in three Mach numbers ( $0.5$ , $1$ , $1.5$ ) is plotted in Figure 25(a). It shows that the SMA wire could control the deflection in these Mach numbers. Figure 25(b) and (c) show that with the SMA wire, the plate fluctuates in a single loop in the steady state and the plate has one-period behavior. The dimensionless amplitude is about $0.7$ which shows by embedding the SMA wire along $X$ -axis, this alloy has a significant effect on reducing deflection. By increasing the Mach number, the SMA has an oppositive effect on the triangular composite plate. The reason is that at this Mach number the dominant eigenvalue of the system is complex with a real positive part (Figure 17(a)).

For the triangular composite plate with CSC boundary conditions and embedded SMA wire along the Y-axis, the time response is shown in Figure 26. The green line in Figure 26 shows the steady state time response at $M_{a} = 1$ , the plate has a static deflection of $0.022$ . At $M_{a} = 1.5$ with a brown line, the static displacement of the plate is about $0.194$ . The dominate eigenvalue of the plate for this Mach number is complex with a positive real part, which shows the plate has a period one behavior. For these Mach numbers $(0.5, 1, 1.5)$ , the amount of deflection is more than the case when the plate has embedded with SMA wire along $X$ -axis. So, the SMA along Y-axis has less effect on increasing the stiffness of the structure rather than on the $X$ -axis. At $M_{a} = 2$ the plate oscillates around an equilibrium deflection of $0.425$ with $0.15$ amplitude of oscillation. These results show that the SMA wire along Y-axis, wasn’t efficient in reducing the deflection of the plate, and the SMA wire along $X$ -axis was more successful to control the aeroelastic characteristics. Also, while the SMA wire along $Y$ -axis has reduced the plate’s deflection from $0.674$ to $0.425$ , it makes limit cycle oscillation in the triangular composite plate’s response. Figure 26(c) shows the phase diagram at $M_{a} = 2$ for the CSC boundary conditions and shows the plate has one-period behavior. Like Figure 25(b) and (c) at this Mach number, the plate has a complex eigenvalue with a positive real part.

4. Conclusion

This work presents the thermo-aerodynamic behavior of triangular composite plates in supersonic flow. The reference temperature method was used to study temperature conditions more realistically. Lagoudas quadratic polynomial hardening function is used for modeling the SMA effect, and a simple method is presented to approximate the stress-strain relation after phase transformation, with an error of less than 2%. The summary of the obtained results is as follows:

a. The buckling temperature was reduced by increasing the aspect ratio in the triangular composite plate without SMA wire.

b. Embedding SMA wire along the X-axis ( $0 °$ ) has a greater effect on increasing buckling temperature than embedding along the Y-axis ( $90 °$ ).

c. In both boundary conditions (CSC, SSS), $[0 / 90 / 0 / 90]$ had the weakest and $[30 / 60]$ had the highest stiffness. Also, the SMA wire considerably increased the instability margin, and its effect in the CSC boundary condition was greater than that in the SSS case.

d. By embedding the SMA wire, the plate’s deflection was reduced, and the chaotic behavior was changed to cyclic behavior. However, by embedding the SMA wire along the X-axis, the deflection is reduced more than when it is embedded along the Y-axis.

Footnotes

Appendices

In sections 2–5 after applying the assumed function method, and replacing equations (21)–(23) in equation (20), for extracting governing equation, the Lagrange method is used. In this method after differentiation, the structural matrix (mass, damping, stiffness), nonlinear matrix, and forces (SMA, temperature) matrix will be calculable.

Mass matrix calculated as $3 \times 3$ block matrix:

[M] = [\begin{matrix} M_{u}^{u} & 0 & M_{u}^{w} \\ 0 & M_{v}^{v} & M_{v}^{w} \\ M_{w}^{u} & M_{w}^{v} & M_{w}^{w} \end{matrix}]

In its arrays, subscript shows the direction and superscript shows the direction which affects.

M_{u}^{u} = c_{r} L I_{0} \int_{0}^{1} χ_{i} (ξ) χ_{d} (ξ) d ξ \int_{0}^{1} (1 - η) ϑ_{j} (η) ϑ_{e} (η) d η

\begin{matrix} M_{u}^{w} = - L I_{1} \int_{0}^{1} χ_{i} (ξ) \frac{\partial ϕ_{i} (ξ)}{\partial ξ} d ξ \int_{0}^{1} ϑ_{j} (η) ψ_{i} (η) d η \\ - \frac{4 I_{1} c_{r} L}{AR \tan α} 〈 \int_{0}^{1} χ_{i} (ξ) ϕ_{i} (ξ) d ξ \end{matrix}

\int_{0}^{1} (1 - η) ϑ_{j} (η) \frac{\partial ψ_{i} (η)}{\partial ξ} d η 〉

M_{v}^{v} = c_{r} L I_{0} \int_{0}^{1} γ_{k} (ξ) γ_{o} (ξ) d ξ \int_{0}^{1} (1 - η) θ_{l} (η) θ_{p} (η) d η

M_{v}^{w} = - c_{r} I_{1} \int_{0}^{1} γ_{k} (ξ) ϕ_{i} (ξ) d ξ \int_{0}^{1} (1 - η) θ_{l} (η) \frac{\partial ψ_{i} (η)}{\partial η} d η

\begin{matrix} M_{w}^{u} = - L I_{1} \int_{0}^{1} \frac{\partial ϕ_{i} (ξ)}{\partial ξ} χ_{i} (ξ) d ξ \int_{0}^{1} ψ_{i} (η) ϑ_{j} (η) d η \\ - 4 I_{1} \frac{c_{r} L}{AR \tan α} 〈 \int_{0}^{1} ϕ_{i} (ξ) χ_{i} (ξ) d ξ \end{matrix}

\int_{0}^{1} (1 - η) \frac{\partial ψ_{i} (η)}{\partial ξ} ϑ_{j} (η) d η 〉

M_{w}^{v} = - c_{r} I_{1} \int_{0}^{1} ϕ_{i} (ξ) γ_{k} (ξ) d ξ \int_{0}^{1} (1 - η) \frac{\partial ψ_{i} (η)}{\partial η} θ_{l} (η) d η

\begin{matrix} M_{w}^{w} = \frac{L}{c_{r}} I_{2} \int_{0}^{1} \frac{\partial ϕ_{i} (ξ)}{\partial ξ} \frac{\partial ϕ_{i} (ξ)}{\partial ξ} d ξ \int_{0}^{1} \frac{1}{[1 - η]} ψ_{i} (η) ψ_{i} (η) d η \\ + \frac{32 c_{r} l}{{(AR \tan α)}^{2}} I_{2} \\ 〈 \int_{0}^{1} ϕ_{i} (ξ) ϕ_{i} (ξ) d ξ \int_{0}^{1} [1 - η] \frac{\partial ψ_{i} (η)}{\partial η} \frac{\partial ψ_{i} (η)}{\partial η} d η 〉 \\ + c_{r} L I_{0} \int_{0}^{1} ϕ_{i} (ξ) ϕ_{i} (ξ) d ξ \int_{0}^{1} [1 - η] ψ_{i} (η) ψ_{i} (η) d η \end{matrix}

Also, for stiffness matrix, the calculated matrix is:

K_{ESL} = [\begin{matrix} K_{u}^{u} & K_{u}^{v} & K_{u}^{w} \\ K_{v}^{u} & K_{v}^{v} & K_{v}^{w} \\ K_{w}^{u} & K_{w}^{v} & K_{w}^{w} \end{matrix}]

In this case, because lack of space, just one direction (u) is written:

\begin{matrix} K_{u}^{u} = \frac{L}{cr} A_{11} 〈 \int_{0}^{1} \frac{d χ_{i}}{d ξ} \frac{d χ_{d}}{d ξ} d ξ \int_{0}^{1} \frac{1}{(- η + 1)} ϑ_{j} ϑ_{e} d η 〉 \\ + (8 \frac{cr}{AR \tan a} A_{16} + 16 \frac{crL}{{(AR \tan a)}^{2}} A_{11} + \frac{cr}{L} A_{66}) \\ 〈 \int_{0}^{1} χ_{i} χ_{d} d ξ \int_{0}^{1} (- η + 1) \frac{d ϑ_{j}}{d η} \frac{d ϑ_{e}}{d η} d η 〉 \\ + (4 \frac{L}{AR \tan a} A_{11} + A_{16}) \\ 〈 \int_{0}^{1} \frac{d χ_{i}}{d ξ} χ_{d} d ξ \int_{0}^{1} ϑ_{j} \frac{d ϑ_{e}}{d η} d η + \int_{0}^{1} χ_{i} \frac{d χ_{d}}{d ξ} d ξ \int_{0}^{1} \frac{d ϑ_{j}}{d η} ϑ_{e} d η 〉 \end{matrix}

\begin{matrix} K_{u}^{v} = (A_{12} + 4 \frac{L}{AR \tan a} A_{16}) \\ 〈 \int_{0}^{1} \frac{d χ_{i}}{d ξ} γ_{o} d ξ \int_{0}^{1} ϑ_{j} \frac{d θ_{p}}{d η} d η + \frac{L}{cr} A_{16} \int_{0}^{1} \frac{d χ_{i}}{d ξ} \frac{d γ_{o}}{d ξ} d ξ 〉 \end{matrix}

\begin{array}{l} \int_{0}^{1} \frac{1}{(- η + 1)} ϑ_{j} θ_{p} d η + (\frac{cr}{L} A_{26} + 16 \frac{c r L}{{(A R \tan a)}^{2}} A_{16} + 4 \frac{c r}{A R \tan a} (A_{12} + A_{66})) \\ 〈 \int_{0}^{1} χ_{i} γ_{o} d ξ \int_{0}^{1} (- η + 1) \frac{d ϑ_{j}}{d η} \frac{d θ_{p}}{d η} d η 〉 \\ + (4 \frac{L}{A R \tan a} A_{16} + A_{66}) 〈 \int_{0}^{1} χ_{i} \frac{d γ_{o}}{d ξ} d ξ \int_{0}^{1} \frac{d ϑ_{j}}{d η} θ_{p} d η 〉 \end{array}

\begin{array}{l} K_{u}^{w} = (- \frac{1}{c_{r}} B_{16} - 4 \frac{L}{c_{r} A R \tan a} B_{11}) \\ 〈 \int_{0}^{1} χ_{i} 1 \frac{d^{2} ϕ_{m}}{d ξ^{2}} d ξ \int_{0}^{1} \frac{1}{1 - η} \frac{d ϑ_{j}}{d η} ψ_{n} d η 〉 \\ + (- 8 \frac{1}{(A R \tan a)} B_{16} - \frac{1}{L} B_{12} - \frac{16 L}{{(A R \tan a)}^{2}} B_{11}) \\ 〈 \int_{0}^{1} \frac{d χ_{i}}{d ξ} ϕ_{m} d ξ \int_{0}^{1} ϑ_{j} \frac{d^{2} ψ_{n}}{d η^{2}} d η 〉 \\ + (- \frac{c_{r}}{L^{2}} B_{26} - 8 \frac{c_{r}}{(A R \tan a) L} B_{66} \\ - 4 \frac{c_{r}}{(A R \tan a) L} B_{12} - 64 \frac{c_{r} L}{{(A R \tan a)}^{3}} B_{11} - 48 \frac{c_{r}}{{(A R \tan a)}^{2}} B_{16}) \end{array}

\begin{matrix} 〈 \int_{0}^{1} χ_{i} ϕ_{m} d ξ \int_{0}^{1} (- η + 1) \frac{d ϑ_{j}}{d η} \frac{d^{2} ψ_{n}}{d η^{2}} d η 〉 \\ + (- \frac{2}{c_{r}} B_{16} - \frac{8 L}{c_{r} (AR \tan a)} B_{11}) \\ 〈 \int_{0}^{1} \frac{d χ_{i}}{d ξ} \frac{d ϕ_{m}}{d ξ} d ξ \int_{0}^{1} \frac{1}{(- η + 1)} ϑ_{j} \frac{d ψ_{n}}{d η} d η 〉 \\ + (- \frac{32 L}{{(AR \tan a)}^{2}} B_{11} - \frac{2}{L} B_{66} - \frac{16}{(AR \tan a)} B_{16}) \\ 〈 \int_{0}^{1} χ_{i} \frac{d ϕ_{m}}{d ξ} d ξ \int_{0}^{1} \frac{d ϑ_{j}}{d η} \frac{d ψ_{n}}{d η} d η 〉 \\ - \frac{L}{c_{r}^{2}} B_{11} 〈 \int_{0}^{1} \frac{d χ_{i}}{d ξ} \frac{d^{2} ϕ_{m}}{d ξ^{2}} d ξ \int_{0}^{1} \frac{ϑ_{j} ψ_{n}}{{(1 - η)}^{2}} d η 〉 \end{matrix}

where $A_{ij}$ are called extensional stiffnesses, $D_{ij}$ the bending stiffnesses, and $B_{ij}$ the bending-extensional coupling stiffnesses, which are defined in terms of the lamina stiffnesses ${\tilde{Q_{ij}}}^{k}$ as:

\begin{matrix} A_{ij} = \sum_{k = 1}^{N} {\tilde{Q_{ij}}}^{k} (z_{k + 1} - z_{k}), B_{ij} = \frac{1}{2} \sum_{k = 1}^{N} {\tilde{Q_{ij}}}^{k} (z_{k + 1}^{2} - z_{k}^{2}), \\ D_{ij} = 1 / 3 \sum_{k = 1}^{N} {\tilde{Q_{ij}}}^{k} (z_{k + 1}^{3} - z_{k}^{3}) \end{matrix}

We recall that the plane-stress-reduced stiffnesses $Q_{ij}$ in the material coordinate system are given in terms of the engineering constants as:

\begin{matrix} Q_{11} = \frac{E_{1}}{1 - ν_{12} ν_{21}}, Q_{12} = \frac{ν_{12} E_{1}}{1 - ν_{12} ν_{21}}, Q_{22} = \frac{E_{2}}{1 - ν_{12} ν_{21}}, \\ Q_{66} = G_{12}, Q_{44} = G_{23}, Q_{55} = G_{13} \end{matrix}

where $E_{1}$ denotes Young’s modulus of the material in the $x$ direction, $ν_{21}$ is the Poisson ratio:

ν_{21} = - \frac{ε_{11}}{ε_{22}}

and $E_{2}$ is Young’s modulus of the material in the y direction, $α$ is the ratio between triangular plate and y direction.

For dimensionless equations, dimensionless time and dimensionless parameters are defined below:

{\bar{u}}_{ij} = \frac{u_{ij}}{H}, {\bar{v}}_{kl} = \frac{v_{kl}}{H}, {\bar{w}}_{mn} = \frac{w_{mn}}{H}, τ = ω_{0} t = t \sqrt{\frac{D_{11}}{I_{0} c_{r}^{4}}}

By considering the dimensionless coefficients $Λ$ and $μ_{aero}$ for aerodynamic pressure as:

\begin{matrix} μ_{aero} = \frac{ρ_{\infty} h}{ρ_{p} cr M_{\infty}}, Λ = \frac{ρ_{\infty}}{ρ_{p}} \frac{M_{\infty} V s^{2}}{ω_{0}^{2}} \frac{cr}{h}, \\ ϒ = \frac{4}{crAR (1 + TR) \tan a} \end{matrix}

{(Λ μ_{aero})}^{\frac{1}{2}} = \frac{ρ_{\infty}}{ρ_{p}} \frac{Vs}{ω_{0}}

Considering $ς = \frac{(ρ_{a} cr)}{(ρ_{p} h)}$ equal to the mass ratio of air to plate and considering this coefficient constant relative to Mach (Xue, 1991).

\frac{ς}{M_{\infty}} = 0.01

Aerodynamic stiffness coefficient calculated as:

\begin{matrix} K_{Aer o_{w}} = - Λ c_{r} L \int_{0}^{1} ϕ_{r} \frac{\partial ϕ_{i}}{\partial x} d ξ \int_{0}^{1} ψ_{n} ψ_{s} d η - c_{r} L Λ ϒ \\ 〈 \int_{0}^{1} ϕ_{i} ϕ_{r} d ξ \int_{0}^{1} (1 - η) ψ_{s} \frac{\partial ψ_{n}}{\partial η} d η 〉 \end{matrix}

\begin{array}{l} {KF}_{w} = σ_{0} (\frac{L}{2 c_{r}} c o s^{2} θ \int_{0}^{1} \frac{d ϕ_{m}}{d ξ} \frac{d ϕ_{r}}{d ξ} d ξ \int_{0}^{1} \frac{1}{1 - η} ψ_{i} (y) ψ_{i} (y) d η \\ + (\frac{4 c_{r} s i n θ c o s θ}{(A R \tan a)} + \frac{c r}{2 L} s i n^{2} θ + \frac{8 c r L c o s^{2} θ}{{(A R \tan a)}^{2}}) \\ 〈 \int_{0}^{1} ϕ_{i} (x) ϕ_{i} (x) d ξ \int_{0}^{1} (- η + 1) \frac{d ψ_{n}}{d η} \frac{d ψ_{s}}{d η} d η 〉 \\ + \frac{1}{2} (s i n θ c o s θ + 4 \frac{L}{(A R \tan a)} c o s^{2} θ) \\ 〈 \int_{0}^{1} \frac{d ϕ_{m}}{d ξ} ϕ_{i} (x) d ξ \int_{0}^{1} ψ_{i} (y) \frac{d ψ_{s}}{d η} d η s i n 2 θ \\ + \int_{0}^{1} ϕ_{i} (x) \frac{d ϕ_{m}}{d ξ} d ξ \int_{0}^{1} \frac{d ψ_{s}}{d η} ψ_{i} (y) d η s i n 2 θ 〉 \end{array}

\begin{array}{l} K_{w}^{T} = - [\frac{L}{c r h} (2.5 α_{x x} A_{11} + 2.5 α_{y y} A_{12} + 1.5 α_{z z} A_{13} + 5 α_{x y} A_{16}) \\ 〈 \int_{0}^{1} \frac{d ϕ_{m}}{d ξ} \frac{d ϕ_{r}}{d ξ} d ξ \int_{0}^{1} \frac{ψ_{n} ψ_{s}}{1 - η} d η 〉 \\ + (\frac{c r L}{{(A R \tan a)}^{2}} (40 α_{x x} A_{11} + 40 α_{y y} A_{12} \\ 24 α_{z z} A_{13} + 80 α_{x y} A_{16}) + \frac{c r}{h (A R \tan a)} \\ (20 α_{x x} A_{16} + 20 α_{y y} A_{26} + 12 α_{z z} A_{36} + 40 α_{x y} A_{66}) \\ + \frac{c r}{L h} (2.5 α_{x x} A_{12} + 2.5 α_{y y} A_{22} + 1.5 α_{z z} A_{23} + 5 α_{x y} A_{26})) \\ 〈 \int_{0}^{1} ϕ_{m} ϕ_{r} d ξ \int_{0}^{1} (1 - η) \frac{d ψ_{n}}{d η} \frac{d ψ_{s}}{d η} d η 〉] \\ + \frac{L}{h (A R \tan a)} (10 α_{x x} A_{11} + 10 α_{y y} A_{12} + 6 α_{z z} A_{13} \\ + 20 α_{x y} A_{16}) \\ + \frac{1}{h} (2.5 α_{x x} A_{16} + 2.5 α_{y y} A_{26} + 1.5 α_{z z} A_{36} + 5 α_{x y} A_{66}) \\ 〈 \int_{0}^{1} ϕ_{m} \frac{d ϕ_{r}}{d ξ} d ξ \int_{0}^{1} \frac{d ψ_{n}}{d η} ψ_{s} d η + \int_{0}^{1} \frac{d ϕ_{m}}{d ξ} ϕ_{r} d ξ \int_{0}^{1} ψ_{n} \frac{d ψ_{s}}{d η} d η 〉 \end{array}

$α_{ij}$ shows the thermal coefficients in the ij-plane and thermal force in $u$ :

\begin{array}{l} F_{u}^{T} = Δ T 〈 [\frac{4 c r}{h (AR \tan a)} (2.5 α_{x x} A_{11} + 2.5 α_{y y} A_{12} + 1.5 α_{z z} A_{13} + 5 α_{x y} A_{16}) \\ + \frac{c r L}{h} (2.5 α_{x x} A_{16} + 2.5 α_{y y} A_{26} + 1.5 α_{z z} A_{36} + 5 α_{x y} A_{66})] \\ 〈 \int_{0}^{1} χ_{i} d ξ \int_{0}^{1} (1 - η) \frac{d ϑ_{j}}{d η} d η 〉 + \frac{L}{h} (2.5 α_{x x} A_{11} + 2.5 α_{y y} A_{12} + 1.5 α_{z z} A_{13} + 5 α_{x y} A_{16}) \\ \int_{0}^{1} \frac{d χ_{i}}{d ξ} d ξ \int_{0}^{1} ϑ_{j} d η 〉 \end{array}

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 iD

Morteza Dardel

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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