Flutter of variable stiffness composite laminates in supersonic flow with temperature effects

Abstract

The nonlinear thermal flutter behavior of variable stiffness composite laminates (VSCL) with curvilinear fibers in high supersonic flow is investigated. The first order shear deformation theory (FSDT) combining von Karman large-deflection strain-displacement relations, quasi-steady first-order piston theory aerodynamics and quasi-steady thermal stress theory are used to formulate the nonlinear panel flutter finite element equations of motion. The fiber orientation within a layer is assumed to vary linearly from $ϕ_{0}$ at the center to $ϕ_{1}$ at the vertical edges of the rectangular lamina. The flutter characteristics of variable stiffness composite laminates with different temperature distributions are then studied. The results show that the critical dynamic pressure decreases as $ϕ_{0}$ or $ϕ_{1}$ increases, whereas the limit cycle amplitude increases as $ϕ_{0}$ or $ϕ_{1}$ increases for the same dynamic pressure. The critical dynamic pressure and limit cycle amplitude both increase when the temperature gradient along panel thickness increases. Simple harmonic motions, unharmonic but periodic motions, and chaotic motions can be observed on VSCL under different temperatures. It also turns out that temperature distribution has similar influence on both the critical dynamic pressure and limit cycle amplitude of VSCL.

Keywords

Thermal flutter nonlinear panel flutter variable angle tow variable stiffness composites curvilinear fiber composite laminates temperature distribution

Introduction

Panel flutter is a self-excited oscillation phenomenon when subjected to inertia force, elastic force, and aerodynamic force. It can be encountered in the operation of aircraft and missiles at supersonic speed. The friction from the surrounding air causes increased heating of the structure. The presence of high thermal loads usually results in flutter motions at dynamic pressures lower than those without temperature effects. Thus, the effects of temperature as well as aerodynamic pressure must be considered in the proper analysis and design of aircraft or missile structures under high supersonic flow. Composite materials are being used increasingly in the aerospace, naval and mechanical industries, primarily for their high strength- and stiffness-to-weight ratios and their anisotropic material properties. Therefore, thermal flutter of composites in high supersonic flow has received widespread attention in recent years and a number of researches have been carried out. Dixon and Mei¹ used finite element method to study large amplitude panel flutter of thin rectangular composite laminates. The critical flutter values and limit cycle amplitudes of graphite/epoxy and boron/epoxy laminates were analysed and the effects of boundary conditions, lamination orientations and number of layers were investigated. Abbas et al.² studied the linear and nonlinear flutter of a simply supported composite laminate in supersonic flow. They used Hamilton's principle and Reissner functional to establish the equations of motion of the composite laminated panel and the thermal effect was also considered based on the adiabatic wall temperature. Zhou et al.³ proposed a finite element time domain method for determining the nonlinear flutter characteristics of composite panels at elevated temperatures, and verified the accuracy and convergence of the approach with examples. Cheng and Mei⁴ studied the nonlinear flutter of composite panels under hypersonic airflow using 24-DOF Bogner-Fox-Schmit rectangular plate element. The panel behaviors under uniform temperature and temperature varying in panel thickness were investigated. Guo and Mei⁵ investigated nonlinear flutter of composite panel at arbitrary supersonic yawed angle and elevated temperature environment by using finite element method, and proposed a new method using aeroelastic modes to reduce system equations. Song and Li⁶ analysed nonlinear aerothermoelastic characteristics of composite laminates considering the shock wave and the aerodynamic heating by using finite element method with a four-node rectangular element, and assessed the influences of the shock wave on the flutter behaviors. Zhou et al.⁷ proposed a unified solution for evaluating the aerothermoelastic flutter of composite plates with general boundary conditions under supersonic flow. Numerical results were compared with those of existing work and finite element method. Furthermore, the effects of thermal loads and boundary conditions on the flutter behaviors were examined. Xie et al.⁸ put forward a general higher-order shear deformation zig-zag theory to investigate the nonlinear aerothermoelastic behaviors of composite laminates. The effects of geometrical dimensions, temperature gradients and fiber orientations were observed, and the differences between the thermal flutter characteristics of composite laminates determined by different structural theories were compared in detail.

While the majority of existing studies on traditional composite (constant stiffness composite laminates-CSCL) assumed that the fiber orientation is constant within a single lamina, the advances in manufacturing techniques, such as the invention of the tow-placement machine, made it technically feasible and economically affordable to produce curved fiber within a lamina. The fibers in these plies follow a predefined curvilinear path such that the fiber angle and ply stiffness vary continuously through the plane of each ply. This capability is not present in either filament winding or tape layup machines which produce CSCL. Composites containing laminas with curvilinear fibers are called variable angle tow (VAT) composites or variable stiffness composite laminates (VSCL).

The curvilinear reinforcement was first studied by Cooper⁹ and Heller and Chiba¹⁰ in early 1970s. The curvilinear fiber format was put forward by Hyer and Charette¹¹ and Hyer and Lee¹² to substitute for straight fiber in order to improve the mechanical properties of a plate with holes. Gürdal and Olmedo,¹³ Lopes et al.,¹⁴ Gürdal et al.,¹⁵ and Lopes et al.¹⁶ proposed the concept of tow-placed variable stiffness composites and took the residual thermal stresses produced by solidification into account. The results showed that compared with the traditional composite laminates, the buckling behaviors of the variable stiffness ones were much better. Wu et al.^17,18 analysed the buckling and postbuckling of curvilinear fiber composites by using numerical simulation method. Khalafi and Fazilati¹⁹ investigated the parametric instability regions of variable stiffness composite laminated quadrilateral plates subjected to uniform in-plane loadings. The results of the developed formulation were compared with those available in the literature, and the effects of geometry layout, loading frequency and amplitude, changes in curvilinear fiber orientations, and material orthogonality on the parametric instability regions were researched. Fazilati and Khalafi²⁰ studied the free vibration of the variable stiffness composite laminated plates containing embedded cutout of desired shapes, and the effects of changes in the cutout geometry, location and orientation in conjunction with the curvilinear fiber placement were analysed.

Though variable stiffness composites have drawn much attention for their outstanding design flexibility, weight reduction and cost saving, the researches on their application in aeroelasticity have received relatively little attention in the literature. Stodieck et al.^21,22 studied the aeroelastic behaviors of a rectangular composite wing. Tow-steered composites were used to tailor the aeroelastic behaviors, and they showed a good performance over traditional unidirectional composite laminates. Stodieck et al.²³ assessed the potential wing weight savings of a full-size aeroelastically tailored wing. It turned out that optimized tow-steered laminates achieved much better mass reductions than optimized straight fiber composites. Haddadpour and Zamani²⁴ investigated the aeroelastic design of composite wings with curvilinear fiber which were modelled as thin-walled beams. The wing was optimized with a linear spanwise variation of the fiber orientation to maximize the aeroelatic instability speed. It was shown that much improved aeroelastic stability was achieved by the optimal variable stiffness wings compared with the conventional, constant-stiffness ones. Stanford et al.²⁵ studied aeroelastic tailoring of a cantilevered flat plate in low-speed flow, locating the Pareto front between static aeroelastic stresses and dynamic flutter boundaries using a genetic algorithm. Guimarães et al.²⁶ investigated the flutter behaviors of tow-steered composite panels using Ritz method combined with supersonic aerodynamic piston theory. The flutter stability boundaries for constant stiffness laminates and variable stiffness laminates were compared. Akhavan and Ribeiro²⁷ studied the aeroelastic instability of variable stiffness composite laminates in supersonic airflow. A third-order shear deformation theory and linear piston theory were used for structural and aerodynamic modelling, respectively. The p-version finite element method was adopted to discretize the aeroelastic model. The effects of boundary conditions, fibre angles and airflow direction on the flutter and divergence occurrence were investigated. Khalafi and Fazilati²⁸ developed an enhanced isogeometric finite element method to investigate the free vibration and the linear flutter characteristics of variable stiffness square and skew laminated plates. Their results were compared with those available in the literature to verify the accuracy and effectiveness. Fazilati and Khalafi²⁹ investigated the optimization of linear flutter behaviors of tow-steered laminated plates. The accuracy and reliability of the optimization process were verified, and the effects of changes in the flow direction, panel edge constraints, and layup characteristics were inspected. Ouyang and Liu³⁰ researched the nonlinear flutter behaviors of tow-steered composite laminates in high supersonic flow without temperature effects using finite element method, and investigated the effects of boundary conditions and fiber orientation on the nonlinear flutter behaviors. Rasool and Singha³¹ investigated the stability characteristics of variable stiffness composite plates under combined aerodynamic pressure and in-plane stress using finite element method. The effect of in-plane stresses on the stability behaviors was studied, and the limit cycle oscillation of variable stiffness plates subjected to aerodynamic pressure was researched. Zhang et al.³² proposed a computational method based on matrix perturbation theory to solve the aeroelastic sensitivity of the fiber angle of tow-steered composite wings. The optimal fiber paving path was obtained by adjusting the fiber angles in the highly sensitive region to increase the flutter velocity.

To the best of our knowledge, most of the existing flutter analysis with temperature effects are carried out on CSCL, while research work on flutter analysis of VSCL with temperature effects can hardly be found in the literature. This paper focuses on the thermal flutter behaviors of tow-steered composites under high supersonic flow. The flutter stability and nonlinear flutter responses are investigated with different temperature distributions such as uniform temperature, symmetric sinusoidal temperature along panel length and linear temperature along panel thickness.

Variable stiffness composite lamina

Figure 1 shows a rectangular lamina with length a and width b. The Cartesian coordinate system is employed to label the material points in the undeformed reference configuration. The notation of < $ϕ_{0}$ | $ϕ_{1}$ > signifies a fiber orientation, where $ϕ_{0}$ and $ϕ_{1}$ are the path orientations at the center and vertical edges (x = ±a/2), respectively.

If the fiber orientation within a layer is assumed to vary linearly from the center of the rectangular lamina, as adopted in Refs.,^12,14,27 then the fiber path orientation is

ϕ (x) = \frac{2 (ϕ_{1} - ϕ_{0})}{a} | x | + ϕ_{0}

(1)

and thus the reference path of the curvilinear fiber is

y (x) = {\begin{matrix} \frac{a}{2 (ϕ_{1} - ϕ_{0})} {\ln (\cos ϕ_{0}) - \ln [\cos (\frac{2 (ϕ_{1} - ϕ_{0})}{a} x + ϕ_{0})]} & (0 \leq x \leq \frac{a}{2}) \\ \frac{a}{2 (ϕ_{0} - ϕ_{1})} {\ln (\cos ϕ_{0}) - \ln [\cos (\frac{2 (ϕ_{0} - ϕ_{1})}{a} x + ϕ_{0})]} & (- \frac{a}{2} \leq x \leq 0) \end{matrix}

(2)

Finite element equations of motion

Based on von Karman assumption, the large deflection strain-displacement relations of composite laminated plate are³³

ε = [\begin{matrix} ε_{x} \\ ε_{y} \\ γ_{x y} \end{matrix}] = [\begin{matrix} \frac{\partial u_{0}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} \\ \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} \end{matrix}] + [\begin{matrix} \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2} \\ \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2} \\ \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \end{matrix}] + z [\begin{matrix} \frac{\partial θ_{y}}{\partial x} \\ \frac{\partial θ_{x}}{\partial y} \\ \frac{\partial θ_{y}}{\partial y} + \frac{\partial θ_{x}}{\partial x} \end{matrix}] = ε^{0} + z κ

(3)

\begin{matrix} ε^{0} = [\begin{matrix} \frac{\partial u_{0}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} \\ \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} \end{matrix}] + [\begin{matrix} \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2} \\ \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2} \\ \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \end{matrix}] & , & κ = [\begin{matrix} \frac{\partial θ_{y}}{\partial x} \\ \frac{\partial θ_{x}}{\partial y} \\ \frac{\partial θ_{y}}{\partial y} + \frac{\partial θ_{x}}{\partial x} \end{matrix}] \end{matrix}

(4)

γ = [\begin{matrix} γ_{y z} \\ γ_{x z} \end{matrix}] = [\begin{matrix} \frac{\partial w}{\partial y} \\ \frac{\partial w}{\partial x} \end{matrix}] + [\begin{matrix} θ_{x} \\ θ_{y} \end{matrix}]

(5)

where

u_{0}, v_{0}, w

are displacements of the mid-plane in the x, y and z directions, respectively;

θ_{x}

and

θ_{y}

are rotations in the yz and xz planes, respectively; ε is the total strain vector;

ε^{0}

is the membrane strain vector;

κ

is the flexural strains known as the curvatures of the mid-plane.

The shear correction factor, $k_{i j (k)}^{2} = k_{i j *}^{2} ϕ_{(k)}^{2}$ , is used herein. Where the classical (through-the-thickness) shear correction factor $k_{i j *}^{2} = \frac{π^{2}}{12}$ ; the finite element (in-plane) shear correction factor $ϕ_{(k)}^{2} = \frac{1}{1 + α (k_{i j *}^{2}) / 2}$ ; $α (k_{i j *}^{2}) \overset{df}{=} \frac{\sum_{t = 4}^{9} k_{stt}}{\sum_{t = 4}^{9} k_{btt}}$ ; $\sum_{t = 4}^{9} k_{stt}$ and $\sum_{t = 4}^{9} k_{btt}$ are the diagonal shear and bending stiffness coefficients associated with the rotational degrees of freedom, respectively.³⁴

The stress-strain relations of a lamina considering thermal effect are

\begin{array}{l} σ = \bar{Q} (ε - \bar{α} Δ T) \\ τ = {\bar{Q}}_{S} γ \end{array}

(6)

where

\bar{Q}

and

{\bar{Q}}_{S}

are the in-plane stiffness matrix and shear stiffness matrix, respectively;

Δ T

is temperature change;

\bar{α}

is the thermal expansion coefficient.

The forces and moments for the plate can be expressed as

\begin{array}{l} {\begin{matrix} N \\ M \end{matrix}} = [\begin{matrix} A & B \\ B & D \end{matrix}] {\begin{matrix} ε^{0} \\ κ \end{matrix}} - {\begin{matrix} N_{Δ T} \\ M_{Δ T} \end{matrix}} \\ R = A_{S} γ \end{array}

(7)

\begin{array}{l} (\begin{matrix} A, & B, & D \end{matrix}) = \sum_{k = 1}^{N_{L}} \int_{z_{k - 1}}^{z_{k}} \bar{Q} (1, z, z^{2}) d z \\ A_{S} = \sum_{k = 1}^{N_{L}} \int_{z_{k - 1}}^{z_{k}} {\bar{Q}}_{S} d z \\ (\begin{matrix} N_{Δ T}, & M_{Δ T} \end{matrix}) = \sum_{k = 1}^{N_{L}} \int_{z_{k - 1}}^{z_{k}} \bar{Q} \bar{α} Δ T (1, z) d z \end{array}

(8)

where A , B and D are stiffness matrices; A _S is the transverse shear stiffness matrix;

N_{Δ T}, M_{Δ T}

are loads caused by temperature; N , M , R are in-plane load, moment, and transverse load vectors, respectively; N_L is the number of layers.

For panel flutter under high supersonic flow (M_a > 1.6), the aerodynamic theory is assumed to be that of quasi-steady first-order piston aerodynamics. The aerodynamic pressure is presented as³⁵

p_{a} = - \frac{2 q}{β} (\frac{\partial w}{\partial x} + \frac{M_{a}^{2} - 2}{M_{a}^{2} - 1} \frac{1}{v_{\infty}} \frac{\partial w}{\partial t})

(9)

where

β = \sqrt{M_{a}^{2} - 1}

; dynamic pressure

q = \frac{ρ_{a} v_{\infty}^{2}}{2}

;

ρ_{a}

is the air density;

v_{\infty}

is the air-stream velocity and M_a is Mach number; t is time.

Based on principle of virtual work, the finite element formulation of equations of motion can be obtained³⁶

M \ddot{w} + C \ddot{w} + (K_{a} + K_{L} - K_{ΔT} + \frac{1}{2} N_{1} + \frac{1}{3} N_{2}) w = P_{Δ T}

(10)

where M is mass matrix; C is aerodynamic damping matrix;

K_{a}

is aerodynamic stiffness matrix;

K_{L}

is linear elastic stiffness matrix;

K_{Δ T}

is thermal stiffness matrix;

N_{1}

and

N_{2}

are the first and second order nonlinear stiffness matrices due to large deflection, respectively; w is the total nodal displacement vector;

P_{Δ T}

is thermal load vector.

Thermal panel flutter analysis in this paper is performed based on the following three assumptions:

the static deformation of the panel does not affect temperature distributions;

the response time of temperature field variation is much shorter than that of flutter response, hence, the temperature field is considered as constant during the thermal flutter analysis;

the effect of temperature on mechanical properties of materials is out of consideration.

A Matlab finite element program is developed using the three-node triangular Mindlin (MIN3) plate element.³⁴ The flutter behaviors are obtained by solving the equations of motion using Newmark method.

The non-dimensional parameters are computed using³⁷

λ = 2 q a^{3} / β E_{2} h^{3}, ω^{*} = ω a^{2} \sqrt{ρ h / E_{2} h^{3}}

(11)

where

λ

and

ω^{*}

are non-dimensional dynamic pressure and frequency; h is thickness of panel; ρ is density of panel; ω is vibrational frequency.

Numerical results

A variable stiffness composite laminated plate with the lamination scheme [0/< $ϕ_{0}$ | $ϕ_{1}$ >/<− $ϕ_{0}$ |− $ϕ_{1}$ >/90]_s is employed in this study. The dimensions of the plate are: a = b = 0.305 m, and panel thickness is 1.27 mm. Material properties are given in Table 1. The plate is simply supported along all edges.

Table 1.

Material properties of composite lamina.

E₁	155 GPa	ρ	1550 kg/m³
E₂	8.07 GPa	α₁	−0.07 × 10⁻⁶/°C
G₁₂	4.55 GPa	α₂	30.1 × 10⁻⁶/°C
ν₁₂	0.22

Mesh size study

The mesh size convergence study is carried out for a variable stiffness composite laminated plate with lamination scheme [0/<0|45>/<0|−45>/90]_s and [0/<90|−90>/<−90|90>/90]_s, and the non-dimensional critical dynamic pressures are given in Table 2. Using the mesh of 100 × 100 as reference, the relative errors of non-dimensional critical dynamic pressures of the two different lamination schemes with the mesh of 30 × 20 are only −0.24% and −0.67%, respectively. Taking into account precision and efficiency, the mesh of 30 × 20 is adopted in this study.

Table 2.

Effects of mesh size on the non-dimensional critical dynamic pressures.

Mesh	[0/<0\|45>/<0\|−45>/90]_s	Error (%)	[0/<90\|−90>/<−90\|90>/90]_s	Error (%)
10 × 6	336.85	−2.17	313.38	−3.68
10 × 10	337.11	−2.10	314.77	−3.25
20 × 10	339.16	−1.50	317.28	−2.48
30 × 20	343.50	−0.24	323.16	−0.67
40 × 40	343.95	−0.11	323.68	−0.51
60 × 60	344.17	−0.05	324.18	−0.36
100 × 100	344.33	–	325.34	–

Comparison study

To validate the model developed, a constant stiffness composite laminate [0/45/−45/90]_s with simply supported boundary conditions studied by Zhou in Ref.³ is selected as a reference for comparison. The critical temperature $Δ T_{c r}$ is 21.387 °F (11.882°C). The analysis was performed by using a 30 × 20 mesh. The limit cycle amplitudes with different temperatures are shown in Figure 2. It can be seen that the present results are in good agreement with those of Ref.³

To evaluate the capability of the flutter model for variable stiffness composite laminated plate, a square four-layer symmetric VSCL plate studied by Khalafi in Ref.28 is considered. The material properties are: E₁/E₂ = 10, G₁₂/E₂ = 0.33, ν₁₂ = 0.3. The critical flutter aerodynamic pressure is investigated and the results are depicted in Figure 3 with the comparison with those of Khalafi in Ref.28 A good consistency can be observed from Figure 3.

Thermal flutter stability of variable stiffness composite laminates

To investigate the effect of temperature on critical dynamic pressure, three temperature distributions were considered:

Uniform temperature $Δ T_{1} = T_{0}$ ;

Symmetric sinusoidal temperature along panel length $Δ T_{2} (x) = T_{0} \sin \frac{π (x + a / 2)}{a}$ ;

Linear temperature along panel thickness $Δ T_{3} (z) = T_{0} (1 + \frac{k z}{h})$ , where 0≤k≤2, k=0 means there is no temperature gradient through thickness. These distributions were employed in Refs.^36,38 for thermal flutter analysis.

The non-dimensional critical dynamic pressures of [0/±45/90]_s and [0/<0|45>/<0|−45>/90]_s are presented in Figure 4(a) and 4(b), respectively. It can be seen that the effects of temperature on non-dimensional dynamic pressures are significant for both composite laminates with straight fibers and curvilinear fibers. The thermal loads result in the decrease of non-dimensional dynamic pressure. Temperature distribution in composite laminates has some influence in the dynamic pressure. For the same $T_{0}$ , uniform temperature has the most significant effect on the critical dynamic pressure, while symmetric sinusoidal temperature along panel length has the least effect due to a lower average temperature in the laminate. Temperature gradient through thickness which changes average temperature also has some effects on non-dimensional critical dynamic pressure. The higher the temperature gradient k, the higher the non-dimensional critical dynamic pressure. It is also revealed that temperature changes the critical dynamic pressure in the same way for both laminate [0/±45/90]_s and laminate [0/<0|45>/<0|−45>/90]_s.

The effects of fiber orientation on critical dynamic pressure are investigated as well. Seven lamination schemes were employed: VSCL [0/<0|45>/<0|−45>/90]_s, [0/<15|45>/<−15|−45>/90]_s, [0/<30|45>/<−30|−45>/90]_s, [0/<45|60>/<−45|−60>/90]_s, [0/<45|75>/<−45|−75>/90]_s, [0/<45|90>/< −45|−90>/90]_s and CSCL [0/±45/90]_s. The non-dimensional critical dynamic pressures of these laminates under different temperatures are presented in Figures 5 to 7, respectively. It is clear that the non-dimensional critical dynamic pressure can be changed by using curvilinear fiber in the composite laminated plate. A varying fiber orientation changes the stiffness of the plate and consequently affects the dynamic characteristic of the laminated plate. For the same temperature distribution, if $ϕ_{1}$ is kept at 45 and $ϕ_{0}$ is increased, the non-dimensional critical dynamic pressure decreases; while if $ϕ_{0}$ remains at 45, increasing $ϕ_{1}$ results in a decreasing non-dimensional critical dynamic pressure. The non-dimensional critical dynamic pressure decreases significantly and changes in an almost linear fashion with thermal loads increase. All these variation tendencies are identical to composite laminates under different temperature distributions.

Nonlinear thermal flutter response of variable stiffness composite laminates

The critical temperature $Δ T_{c r}$ of laminate [0/±45/90]_s and laminate [0/<0|45>/<0|−45>/90]_s with uniform temperature is 11.786°C and 9.8828°C, respectively. Figures 8 to 10 show time history and phase plane plot of the constant stiffness laminate [0/±45/90]_s at the position ξ = 0.75, η = 0.5 (ξ = x/a, η = y/b). Figure 8 shows the limit cycle oscillation of the plate at λ = 280 and $Δ T / Δ T_{c r} = 1.0$ . A harmonic limit-cycle motion is predicted as evident by the time history shown in Figure 8(a).The phase plane plot in Figure 8(b) depicts that the plate motion is dominated by a simple harmonic motion. The periodic motion of the same plate at λ = 230 and $Δ T / Δ T_{c r} = 2.0$ is shown in Figure 9. The motion is unharmonic but periodic as seen in Figure 9(a). The periodic motion with several dominant orbits is shown in the phase plane plot (Figure 9(b)). Figure 10 illustrates the plate motion at λ = 240 and $Δ T / Δ T_{c r} = 3.0$ . It can be seen that the motion is now chaotic. Figures 11 to 13 show the time history and phase plane plot of the variable stiffness laminate [0/<0|45>/<0|−45>/90]_s at ξ = 0.75, η = 0.5 with uniform temperature. Figure 11 shows the limit cycle oscillation of the plate at λ = 320 and $Δ T / Δ T_{c r} = 1.0$ . The time history (Figure 11(a)) shows a harmonic limit-cycle motion, and the phase plane plot (Figure 11(b)) shows that the plate motion is dominated by a simple harmonic motion. The periodic motion of the same plate at λ = 280 and $Δ T / Δ T_{c r} = 2.0$ is shown in Figure 12. The motion is unharmonic but periodic as seen in Figure 12(a). The periodic motion with three dominant orbits is shown in the phase plane plot (Figure 12(b)). Figure 13 illustrates the plate motion at λ = 250 and $Δ T / Δ T_{c r} = 3.0$ . It can be seen that the motion is chaotic, and clearly snap-through motions can be observed in Figure 13(a). Figure 14 depicts the deflection of [0/<0|45>/<0|−45>/90]_s at t = 0.315s with $Δ T / Δ T_{c r} = 1.0$ and $Δ T / Δ T_{c r} = 2.0$ , and the maximum appears near to the point ξ = 0.75, η = 0.5.

The effect of temperature on limit cycle amplitude of both straight fiber laminate [0/±45/90]_s and curvilinear fiber laminate [0/<0|45>/<0|−45>/90]_s is presented in Figure 15. $T_{0} = 8$ °C is assumed and the results shown in Figure 15 are the plate motion at η = 0.5 under different temperature distributions. The location of the maximum amplitude is near to ξ = 0.75. The effect of thermal loads on limit cycle amplitude is significant both for composite laminates with curvilinear fibers and straight fibers. The increase of thermal loads results in the increase of amplitude. Temperature distribution has some effects on the amplitude as well. Symmetric sinusoidal temperature along panel length produces lower amplitude than the other ones, because its average temperature in the laminate is lower than that of the other two.

To study the effect of fiber orientation on nonlinear flutter response, the VSCL [0/< $ϕ_{0}$ | $ϕ_{1}$ >/<− $ϕ_{0}$ |− $ϕ_{1}$ >/90]_s is employed and two fiber orientations are considered: (1) $ϕ_{1}$ = 45 while $ϕ_{0}$ varies from 0° to 45° (Figure 16(a)); (2) $ϕ_{0}$ = 45 while $ϕ_{1}$ varies from 45° to 90° (Figure 16(b)). The limit cycle amplitudes under different temperature distributions are presented in Figures 17 to 19, respectively. It is obvious that the limit cycle amplitude of the composite laminated plate can be changed by changing the $ϕ_{0}$ or $ϕ_{1}$ in the ply with curvilinear fiber. The limit cycle amplitude increases when the the dynamic pressure increases, and the change rule is consistent for a given fiber orientation. Under a given temperature distribution and dynamic pressure, when $ϕ_{1}$ is fixed at 45 and $ϕ_{0}$ is in the ascending order, the limit cycle amplitude is ascending; if $ϕ_{0}$ is kept at 45 and $ϕ_{1}$ is in the ascending order, the limit cycle amplitude is ascending. Figure 17(a) and 17(b) show that the limit cycle amplitude at $T_{0} = 8$ °C is much higher than that at $T_{0} = 0$ °C. Figures 18 and 19 show the limit cycle amplitudes of composite laminates with symmetric sinusoidal temperature along panel length and linear temperature along panel thickness, respectively. By comparing Figures 18 and 19 with Figure 17(a), it can be seen that different temperature distributions have similar influence on nonlinear flutter response of the composite laminates. Figure 19 also shows that under the same dynamic pressure, limit cycle amplitude can be affected by the temperature gradient k. The higher the temperature gradient, the higher the limit cycle amplitude.

Conclusions

A supersonic flutter analysis of tow-steered composites with temperature effects was performed. The von Karman large deflection strain-displacement relations and the first order piston theory were employed to account for the structural and aerodynamic nonlinearities, respectively. The effects of different temperature distributions on flutter of tow-steered composites were discussed. The following points can be concluded:

The plate motion of tow-steered composite laminates is dominated by a simple harmonic motion when the temperature is low, while by unharmonic periodic motion or chaotic motion as the temperature rises.

The non-dimensional critical dynamic pressure relates with the thermal loads, temperature distributions along panel length or thickness, or fiber orientation change. The different temperature distributions along panel length result in the changes of non-dimensional critical dynamic pressure, while the increasing of temperature gradient through thickness k results in the increase at a small rate. The non-dimensional critical dynamic pressure decreases significantly and changes in a linear fashion when the thermal load increases, and it decreases as $ϕ_{0}$ or $ϕ_{1}$ increases under a given temperature distribution.

The limit cycle amplitude relates with the thermal loads, temperature distributions along panel length or thickness, or fiber orientation change. The limit cycle amplitude changes with the different temperature distributions along panel length, while it increases at a small rate with the increasing of temperature gradient through thickness k. The limit cycle amplitude increases with the increase of thermal loads, and also with the increase of $ϕ_{0}$ or $ϕ_{1}$ .

The results show that all these variation tendencies of non-dimensional critical dynamic pressure and limit cycle amplitude apply to variable stiffness composite laminates with different temperature distributions.

Footnotes

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

Yi Liu

References

Dixon

Mei

Finite element analysis of large-amplitude panel flutter of thin laminates. Aiaa J 1993; 31: 701–707.

Abbas

Ibrahim

Gibson

RF.

Nonlinear flutter of orthotropic composite panel under aerodynamic heating. Aiaa J 1993; 31: 1478–1488.

Zhou

Xue

Mei

Finite element time domain-modal formulation for nonlinear flutter of composite panels. Aiaa J 1994; 32: 2044–2052.

Cheng

Mei

Finite element modal formulation for hypersonic panel flutter analysis with thermal effects. Aiaa J 2004; 42: 687–695.

Guo

Mei

Application of aeroelastic modes on nonlinear supersonic panel flutter at elevated temperatures. Comput Struct 2006; 84: 1619–1628.

Song

FM.

Aerothermoelastic analysis of nonlinear composite laminated panel with aerodynamic heating in hypersonic flow. Compos Part B-Eng 2014; 56: 830–839.

Zhou

Hua

Aero-thermo-elastic flutter analysis of supersonic moderately thick orthotropic plates with general boundary conditions. Int J Mech Sci 2018; 141: 46–57.

Xie

Zhang

, et al. Nonlinear aerothermoelastic analysis of composite laminated panels using a general higher-order shear deformation zig-zag theory. Int J Mech Sci 2019; 150: 226–237.

Cooper

AAG.

Trajectory fiber reinforcement of composite structures. PhD Thesis, St. Louis: Washington University, 1972.

10.

Heller

Chiba

Alleviation of the stress concentration with analogue reinforcement. Exp Mech 1973; 13: 519–525.

11.

Hyer

Charette

RF.

Use of curvilinear fiber format in composite structure design. Aiaa J 1991; 29: 1011–1015.

12.

Hyer

Lee

HH.

The use of curvilinear fiber format to improve buckling resistance of composite plates with central holes. Compos Struct 1991; 18: 239–261.

13.

Gürdal

Olmedo

In-plane response of laminates with spatially varying fiber orientations: variable stiffness concept. Aiaa J 1993; 31: 751–758.

14.

Lopes

Camanho

Gürdal

, et al. Progressive failure analysis of tow-placed, variable-stiffness composite panels. Int J Solids Struct 2007; 44: 8493–8516.

15.

Gürdal

Tatting

CK.

Variable-stiffness composite panels: effects of stiffness variation on the in-plane and buckling response. Compos Part A – Appl S 2008; 39: 911–922.

16.

Lopes

Gürdal

Camanho

PP.

Variable-stiffness composite panels: buckling and first-ply failure improvements over straight-fiber laminates. Comput Struct 2008; 86: 897–907.

17.

Weaver

Raju

, et al. Buckling analysis and optimisation of variable angle tow composite plates. Thin Wall Struct 2012; 60: 163–172.

18.

Raju

Weaver

PM.

Postbuckling analysis of variable angle tow composite plates. Int J Solids Struct 2013; 50: 1770–1780.

19.

Khalafi

Fazilati

Parametric instability behavior of tow steered laminated quadrilateral plates using isogeometric analysis. Thin Wall Struct 2018; 133: 96–105.

20.

Fazilati

Khalafi

Effects of embedded perforation geometry on the free vibration of tow-steered variable stiffness composite laminated panels. Thin Wall Struct 2019; 144: 106287.

21.

Stodieck

Cooper

Weaver

, et al. Improved aeroelastic tailoring using tow-steered composites. Compos Struct 2013; 106: 703–715.

22.

Stodieck

Cooper

Weaver

, et al. Optimization of tow-steered composite wing laminates for aeroelastic tailoring. Aiaa J 2015; 53: 2203–2215.

23.

Stodieck

Cooper

Weaver

, et al. Aeroelastic tailoring of a representative wing box using tow-steered composites. Aiaa J 2017; 55: 1425–1439.

24.

Haddadpour

Zamani

Curvilinear fiber optimization tools for aeroelastic design of composite wings. J Fluid Struct 2012; 33: 180–190.

25.

Stanford

Jutte

KC.

Aeroelastic benefits of tow steering for composite plates. Compos Struct 2014; 118: 416–422.

26.

Guimarães

TAM

Castro

SGP

Cesnik

CES

, et al. Supersonic flutter and buckling optimization of tow-steered composite plates. Aiaa J 2019; 57: 397–407.

27.

Akhavan

Ribeiro

Aeroelasticity of composite plates with curvilinear fibres in supersonic flow. Compos Struct 2018; 194: 335–344.

28.

Khalafi

Fazilati

Supersonic panel flutter of variable stiffness composite laminated skew panels subjected to yawed flow by using NURBS-based isogeometric approach. J Fluid Struct 2018; 82: 198–214.

29.

Fazilati

Khalafi

Aeroelastic panel flutter optimization of tow-steered variable stiffness composite laminated plates using isogeometric analysis. J Reinf Plast Comp 2019; 38: 885–895.

30.

Ouyang

Liu

Panel flutter of variable stiffness composite laminates in supersonic flow. Acta Aeronautica et Astronautica Sinica 2018; 39: 221539.

31.

Rasool

Singha

MK.

Aeroelastic analysis of pre-stressed variable stiffness composite panels. J Vib Control 2020; 26: 724–734.

32.

Zhang

Chen

Aeroelastic tailoring method of tow-steered composite wing using matrix perturbation theory. Compos Struct 2020; 234: 111696.

33.

Reddy

JN.

Mechanics of laminated composite plates and shells: theory and analysis. Boca Raton: CRC Press, 2004.

34.

Tessler

Hughes

TJR.

A three-node Mindlin plate element with improved transverse shear. Comput Method Appl M 1985; 50: 71–101.

35.

Abdel-Motaglay

Chen

Mei

Nonlinear flutter of composite panels under yawed supersonic flow using finite elements. Aiaa J 1999; 37: 1025–1032.

36.

Xue

Mei

Finite element nonlinear panel flutter with arbitrary temperatures in supersonic flow. Aiaa J 1993; 31: 154–162.

37.

Pidaparti

RMV

Yang

HTY.

Supersonic flutter analysis of composite plates and shells. Aiaa J 1993; 31: 1109–1117.

38.

Xue

Mei

Shore

CP.

Finite element two-dimensional panel flutter at high supersonic speeds and elevated temperature. In: 31st Structures, structural dynamics and materials conference, Long Beach, CA, USA, 02 April–04 April 1990. Paper no. AIAA-90-0982, pp. 1464–1475.