Flutter Analysis of A Laminated Composite Plate with Temperature Dependent Material Properties

Keywords

Flutter; Composite Plate; Piston Theory; Temperature-Dependent Materials.

Introduction

Panel flutter is a self-excited oscillating phenomenon and involves interactions among elastic, inertia, and aerodynamic forces in supersonic flow. The panel flutter differs from wing flutter only in that the aerodynamic force resulting from the airflow acts only on one side of the panel. In the framework of small deflection linear structural theory, there is a critical speed of the airflow (or dynamic pressure λcr ) beyond which the panel motion becomes unstable and grows exponentially with time.

Plate/shell panels are a popular and a useful form of structural components with significant applications in aerospace vehicles, such as high-speed aircrafts, rockets and spacecrafts. Future design concepts civil and military aircrafts are likely to imply the achievement of extremely light weight structural configurations. Therefore these panels are being constructed using advanced fiber-reinforced composite materials to achieve minimum weight design. When a flight vehicle travels at high supersonic speeds, it only will experience flutter due to dynamic pressure but also will be affected by increased temperature owing to the aerodynamic heating. Structural components and/or mechanical elements, such as high-speed aircraft and spacecraft components are subjected to thermal loads due to high temperature, high gradient temperature, and cyclical changes of temperature, etc. The usage of new types of materials such as fiber-reinforced composite materials and functionally gradient materials is on the increase. It has become important to perform more accurate analyses of the thermo mechanical behavior of the above structures, elements and materials. The influence of temperature dependent material properties on thermal stresses at elevated temperature and/or high gradient temperature is quite significant.

The research progress and some of the references can be found in the literature as can be seen in Fung [1] and Dowell [2,3]. Bismarck-Nasr [4] reviewed the finite element analysis of aeroelasticity of plates and shells. The most used aerodynamic theory is piston theory that was first suggested by Ashley and Zartarian [5]. Mei [6] developed a finite element approach to panel flutter. Dowell and Voss [7] studied on theoretical and experimental panel flutter. Rosettes and Tong [8] applied a hybrid stress finite element method and used linearized piston theory. Their results denoted that flutter characteristics are strongly dependent on the composite fiber angle and anisotropy. Srinivasan and Babu [9] studied the panel flutter of laminated composites by using the integral equations method. Wang [10] studied high supersonic/ hypersonic flutter of prismatic composite plate/shell panels.

The effect of temperature on panel flutter behavior is presented by some researchers. Liaw [11] studied the geometrically nonlinear supersonic flutter characteristics of laminated composite thin-plate structures subjected to thermal loads. Lee et al. [12] investigated vibration and flutter analysis of stiffened composite plate considering thermal effect. Guo and Mei [13] presented a finite element time domain formulation using aeroelastic modes for the analysis of nonlinear flutter of isotropic and composite panels at arbitrary supersonic yawed angle and elevated temperature environment. Sohn and Kim [14] and Ibrahim et al. [15, 16] investigated thermal buckling and nonlinear flutter behaviors of functionally graded panels subjected to combined thermal and aerodynamic loads. Sohn and Kim [14] also analyzed the effects of volume fraction distributions, boundary conditions, temperature changes and aerodynamic pressures on panel flutter characteristics. Xue and Mei [17] presented a finite element formulation for analysis of large amplitude limit-cycle oscillations of panels with the influence of non-uniform temperature distribution. Librescu et al. [18] analyzed flutter and post-flutter behavior of long flat panels in a supersonic/hypersonic flow field exposed to a high-temperature field. Abbas et al. [19] investigated nonlinear flutter of isotropic and specially orthotropic panels in supersonic airflow under aerodynamic heating.

Only a few investigations have dealt with the temperature dependent material properties on panel/shell behavior. Mecitoglu [20] denoted the results of the free vibrations of a conical shell with temperature dependent material properties. Prakash and Ganapathi [21] investigated numerically the influence of thermal environment on the supersonic flutter behavior of flat panels made of functionally graded materials using the finite element procedure. Olson [22] presented an analysis of the supersonic flutter of a finite circular cylindrical shell with temperature dependent material properties.

Therefore, in this study, the aeroelastic behavior of a laminated composite rectangular plate with temperature dependent material properties was considered. The equations are solved using the finite element method. The laminated composite plates are discretized with the Semiloof thin shell elements with eight-nodes and a total of thirty-two degrees of freedom. The effects of aspect ratio, temperature distribution and lamination on the flutter boundary have been examined.

Governing Equations

An exhaustive formulation of the analytical method within the linear elastic and aerodynamic theories can be found in Dowell’s monograph [3]. The following considerations show the main steps in modeling the aerothermoelastic problem and its formulation within the finite element method.

Figure 1 defines the coordinate system to be used in developing the laminated aeroelastic plate analysis. The Cartesian coordinate system is assumed to have its origin on the middle surface of the plate, so that the middle surface lies in the xy plane. The displacements at a point in the x, y, z directions are u, v, and w, respectively. a and b are the dimensions of the rectangular plate, t is the plate thickness and M is the Mach number. The basic assumptions relevant to the laminated composite plate are [23]:

1. The plates consist of orthotropic laminae bonded together, with the principal material axes of the orthotropic laminae oriented along arbitrary directions with respect to the xy axes.
2. The thickness of the plate, t, is much smaller than the lengths along the plate edges, a and b.
3. The displacements, u, v, and w are small compared with the plate thickness.
4. The in-plane strains εx, εy And γxy are small compared with unity.
5. Tranverse shear strains γxz And γyz are negligible.
6. Tangential displacements u and v are linear functions of the z coordinate.
7. The transverse normal strain εz is negligible.
8. Each ply obeys Hooke’s law.
9. The plate thickness t is constant.
10. Transverse shear stresses τxz And τyz vanish on the plate surfaces defined by z = ± t / 2.

Assumption 5 is a result of the assumed state of plane stress in each ply, whereas assumptions 5 and 6 together define the Kirchhoff deformation hypothesis that normals to the middle surface remain straight and normal during deformation. According to the Weierstrass theory (also shown in reference [24]), and assumptions 6 and 7, the displacements can be expressed as:

where u0 and v0 are the tangential displacements of the middle surface along the x and y directions, respectively. Due to assumption 7, the tranverse displacement at the middle surface, w0 (x, y), is the same as the tranverse displacement of any point having the same x and y coordinates, so w0 (x, y) = w(x,y). Substituting Equation (1) in the strain-displacement equations for the transverse shear strain and using assumption 5, we find that:

Substituting Equations (1) and (2) in the strain-displacement relations for the in-plane strains, we find that:

where the strains on the middle surface are:

and the curvatures of the middle surface are:

кx is a bending curvature associated with bending of the middle surface in the xz plane and кy is a bending curvature associated with bending of the middle surface in the yz plane. кxy is a twisting curvature associated with out-of-plane twisting of the middle surface, which lies in the xy plane before deformation.

Since Equation (3) gives the strains at any distance z from the middle surface, the stresses along arbitrary xy axes in the kth lamina of a laminate may be found by substituting Equation (3) into the lamina stress-strain relationships considering thermal effects as follows:

where the subscript k refers to the kth lamina. Q¯ij are the components of the transformed lamina stiffness matrix which are defined as follows:

where θ is the lamina orientation angle and the Q¯ij are the components of the lamina stiffness matrix, which are related to the engineering constants by:

where E1 and E2 longitudinal modulus of elasticity associated with the x and y direction, respectively. G12 is the shear modulus associated with the xy plane and ν12 is the Poisson’s ratio.

We can write the constitutive relations as the matrix expressions:

and in the case of thermal strains

The force and moment resultants, per unit length, are defined as:

which lead to the constitutive relations for a laminated panel: Here, refers to the kth lamina’s stress.

{N} and {M} are the force and moment resultants with temperature dependent material properties, respectively. The laminate stiffness matrices are given by:

The effects of elevated temperatures on the aeroelastic behavior of high-speed aircraft structures and launch vehicles as an especially important design consideration that needs further investigation and development. In the present study, considering elasticity modulus respect to temperature with linear approach, can be written as:

where T1 is the room temperature and T2 is 180°C. E1T1 and E1T2 are longitudinal modulus of elasticity associated with the direction 1 at temperature T1 and T2, respectively. E2, which is the longitudinal modulus of elasticity associated with the direction 2 and shear modulus G12 can be written as the same. Temperature T is respect to x and can be written as:

where TL and TT temperatures at leading (x=0) and trailing (x=a) edges.

The derivation of the aeroleastic equations is obtained using Hamilton’s principle. A system of equations of motion for a nonconservative elastic system can be obtained using a variation of the form:

where T is the kinetic energy, U is the potential energy of the system, and δW is the virtual work done by the aerodynamic forces acting on the structure from time τ0 to τ1. Potential and kinetic energy can be expressed as:

In the analysis of Reference [25], only rotary inertias of the faces about the elastic axis were considered, force reads,

where q¯ = ρV2 / 2 is the free stream dynamic pressure, β =√ M2 −1 , V is the free stream velocity, M is the free stream Mach number, and is the air density. For sufficiently high Mach number, the expression (17) can be approximated as:

The expression of the aerodynamic load given in (17) is known in the literature as the quasi-steady case. Further, if the aerodynamic damping is neglected in Equation (18), we get the quasi-static Ackeret’s expression,

The boundary conditions for a rectangular plate as shown in Figure 1 can be written as;
For clamped edges:

For simply supported edges:

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Figure 1

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Figure 1. Geometry of the aeroelastic plate.

Method of Solution

In the present study, Semilo of shell element is chosen as a finite element for the solution. As shown in Figure 2 , Semilo of element has 9 nodes and a total of 45 degrees of freedom. By applying shear constraints [26], it becomes a finite element model which has 8 nodes and a total of 32 degrees of freedom.

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Figure 2

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Figure 2. Semiloof element.

Displacement vectors at global and local coordinates can be defined as:

In general case, displacements, strains and stresses can be written in matrix form and substituting into Hamilton’s principle (14), and minimizing the functional, the following matrix equation can be obtained for each element:

where λ¯ is dynamic pressure parameter and defined as:

where q¯ is the free stream dynamic pressure. [ke], [me] and [ae] are the element stiffness, mass and aerodynamic stiffness matrices, respectively. For whole system, applying boundary conditions and using standard assembly technique for the finite element method,

can be obtained. [K], [M] and [A] are the system stiffness, mass and aerodynamic stiffness matrices, respectively. {Q} is the vector of the system nodal degrees of freedom. The system of Equation (27) assumes solutions in the form,

considering ω complex number, the Equation (28) reads

substituting Equation (29) into the Equation (27)

the problem becomes an eigenvalue problem. Therefore, the determinant of the expression (30) has to be zero also shown in Equation (31).

ω2 correspond to the squares of natural frequencies of the plate. The flutter occurs when any two natural frequencies coalesce where is purely an imaginary number.

Numerical Results

A general computer program was developed for the present composite shell finite element formulation as applied to supersonic panel flutter analysis. As part of the evaluation process, the natural frequencies were first obtained and compared with Srinivasan and Babu’s [9] study. Unsymmetrically laminated composite twolayered square plate with all edges clamped (also shown in Figure 3) has been studied by Srinivasan and Babu [9]. The material properties are E 1 = 213.8GPa , E 2 =18.6GPa G12 = 5.165GPa , ν = 0.28 and ρ =1920 kg / m3 .

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Figure 3

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Figure 3. Aeroelastic two-layered composite plate.

The dimensionless dynamic pressure parameter and flutter frequency, respectively, defined as:

where D = E1 t 3 / [12( 1−ν12 ν21 )] also known as flexure rigidity. equals to one when the airflow acts only on one side of the panel. If the airflow acts on two sides, n equals to two.

For convergence study, the natural frequencies obtained in the present investigation have been compared with those given by Srinivasan and Babu [9]. The authors’ results agree well with this study, as can be seen from Table 1.

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Table 1

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Table 1. Natural frequencies of all edges clamped rectangular plate (Hz).

The material properties for the main model which the authors will investigate in this paper are E1 =62 GPa, E2 =24.8 GPa, G1,2 =17.38 GPa, V1,2 =0.23 and ρ =1920 kg / m Firstly, this square plate considered, t/a=0.01 two layered and meshed 5x5, 7x7 and 10x10. The natural frequencies obtained and compared with ANSYS results. Present results agree well with ANSYS, as can be seen from Table 2. Using a 10x10 mesh, a well agreement is found. Thus, a 10x10 mesh was used to model the square plates. The flutter boundaries determined for an [0/90/0/90/0/90/0/90] eight-layered, composite square plate with all edges simply supported. Figure 4 shows the flutter boundary of an eight-layered composite square plate.

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Figure 4

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Figure 4. Flutter analysis of eight-layered composite square plate with all edges simply supported.

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Table 2

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Table 2. Natural frequencies of all edges clamped rectangular plate compared with ANSYS (Hz).

In the present study, the effects of lamination on the flutter boundary have been examined and critical resultants have been determined for 4, 6, 8 and 12-layered composite square plates with all edges simply supported. The numbers of layer for a constant plate thickness affect the flutter boundary until a certain laminate number as shown in Table 3.

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Table 3

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Table 3. The effect of layer numbers on the flutter boundary.

Table 4 shows the effect of aspect ratio on the flutter boundary for the angle-ply laminated composite square plates where t/a=0.01 . The flutter boundary increases with the aspect ratio as shown clearly in Figure 5. The first two mode shapes of the square plate that flutter occurs can be seen in Figure 6.

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Table 4

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Table 4. The effect of aspect ratio on the flutter boundary.

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Figure 5

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Figure 5. The effect of aspect ratio on the flutter boundary.

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Figure 6

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Figure 6. Mode shapes for the first two modes.

The influence of temperature dependent material properties on thermal stresses at elevated temperature and/or high gradient temperature is quite significant. Therefore, in this study, the aeroelastic behaviour of a laminated composite rectangular plate with temperature dependent material properties was considered. Elasticity modulus of fibers and resin decrease when the temperature increases. In the present study, natural frequencies and flutter analysis were obtained also considering this heating effect at high speeds. Table 5 and Table 6 show, respectively, natural vibration frequencies at the uniform and linearly varying elevated temperature. The analysis results denoted that high temperatures on the plate results in a decrease of the flutter boundary, as can be seen in Table 7. The flutter analyses have been done applying to leading and trailing edges different temperatures and considering the linearly varying temperature in the flow direction. The uniform and varying temperatures have similar effects on the flutter boundary as shown in Table 7 and 8.

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Table 5

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Table 5. Natural free vibration frequencies at the uniform temperatures.

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Table 6

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Table 6. Natural free vibration frequencies at linearly varying elevated temperatures.

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Table 7

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Table 7. Flutter boundary at the uniform temperatures.

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Table 8

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Table 8. Flutter boundary at linearly varying elevated temperatures.

Conclusion

The aeroelastic behaviour of a laminated composite rectangular plate with temperature dependent material properties was investigated. Plate equations for homogenous linear elastic material and small deformations are derived in the frame of the Kirchhoff theory. Uniform and linear temperature distributions are considered on the layered composite plate and it is assumed that the material properties of fiber and matrix vary with the temperature. The aerodynamic forces are obtained by the piston theory. Equations of motion are derived in the variational form by the use of the Hamilton principle. The Equations are solved using the finite element method. The laminated composite plates are discretized with the Semilo of thin shell elements with eight nodes and a total of thirty-two degrees of freedom. A general computer program was developed for the present composite shell finite element formulation as applied to supersonic panel flutter analysis. The free vibration results compared with the results of literature and ANSYS software program. Present results agree well with ANSYS and literature also shown. The effects of aspect ratio, temperature distribution and lamination on the flutter boundary have been examined. The flutter occurs at a high dynamic pressure for the plate with high aspect ratio. The high temperatures on the plate results in a decrease of the flutter boundary. The number of laminate for a constant plate thickness effects the flutter boundary until a certain laminate number.

The present results illustrate the effects of aspect ratio, temperature and number of layers on the flutter boundary. Extensions can also be made to include aerodynamic and structural nonlinearities. The effects of aerodynamic damping, nonlinear temperature gradients and moisture on the flutter boundary can also be investigated in further studies.