Thermally induced stability and vibration of initially stressed laminated composite plates

Chun-Sheng Chen*, Wei-Ren Chen**, Hung-Wei Lin*** *Department of Mechanical Engineering, Lunghwa University of Science and Technology, Guishan Shiang 33306, Taiwan, E-mail: cschen@mail.lhu.edu.tw. **Department of Mechanical Engineering, Chinese Culture University, Taipei 11114, Taiwan, E-mail: wrchen@faculty.pccu.edu.tw. ***Department of Electrical Engineering, Lee Ming Institute of Technology, Taishan 24305, Taiwan, E-mail: hwlin@mail.lit.edu.tw.


Introduction
For the past decades, the composite materials are widely used in spacecraft and engineering industries because of their higher tensile strength and lower weight.Composite plate structures are often applied at elevated temperature environments.In such thermal circumstances, the thermal induced compressive stresses will be developed in the composite plates and consequently lead to the change in mechanical behaviors.The thermally induced behavior of composite plate plays an important role in the design of structural components in thermal environments.Thus, the studies on thermal vibration and buckling of composite plates are increasing considerably in recent years.
Many investigations on thermally induced behaviors of composite plates are concerned with the thermal stability and vibration.The critical buckling temperatures of laminated plates based on a finite strip method were studied by Dawe and Ge [1].In the pre-buckling stage, an in-plane thermal stress analysis was conducted first, and a buckling analysis was followed using the determined in-plane stress distribution.Wang et al. [2] presented the local thermal buckling of laminated plate using the delaminated buckling model.The analytical predictions for the critical temperature yielding the local delamination buckling are shown to correlate well with experimental results.Shian and Kuo [3] developed a thermal buckling analysis method for composite sandwich plates.The results show that the buckling mode of sandwich plate depends on the fiber orientation in the faces and the aspect ratio of the plate.Thermal buckling analysis of cross-ply laminated hybrid composite plates with a hole subjected to a uniform temperature rise was investigated by Avci et al. [4].The effects of hole size, lay-up sequences and boundary conditions on the thermal buckling temperatures were investigated.The equivalent mechanical loading concept was used to study various thermal buckling problems of simple laminated plate configurations by Jones [5].The results were given in the form of buckling temperature change from the stress-free temperature against plate aspect ratio curves.Matsunaga [6] presented the thermal buckling of laminated plates using the principle of virtual work.Several sets of truncated mth order approximate theories were applied to solve the eigenvalue problems.Modal transverse shear and normal stresses could be calculated by integrat-ing the equilibrium equations.
The governing equations for determining thermal buckling of imperfect sandwich plates were developed by Zakeri and Alinia [7].The buckling thermal stress remains unchanged for aspect ratios greater than five.The structural optimization of a laminated plate subjected to thermal and shear loading was considered by Teters [8].The optimization criteria depend on two variable design parameters of composite properties and temperature.Thermal buckling analysis of composite laminated plates under uniform temperature rise was investigated by Shariyat [9].A numerical scheme and a modified instability criterion are used to determine the buckling temperature in a computerized solution.A thermal buckling response of symmetric laminated plates subjected to a uniformly distributed temperature load was presented by Kabir et al. [10].The numerical results were presented for various significant effects such as length-to-thickness ratio, plate aspect ratio and modulus ratio.Thermal buckling behavior of imperfect laminated plates based on first order plate theory was studied by Pradeep and Ganesan [11].A decoupled thermo-mechanical analysis is used to deal with the thermal buckling and vibration behavior of sandwich plates.The variation of natural frequency and loss factor with temperature was studied by Owhadi and Shariat [12].The plate was assumed to be under the longitudinal temperature rise.The effects of initial imperfections on buckling loads were discussed.A perturbation technique was used by Verma and Singh [13] to find the buckling temperature of laminated composite plates subjected to a uniform temperature rise.It was found that small variations in material and geometric properties of the composite plate significantly affect the buckling temperature of the laminated composite plate.Wu [14] investigated the stresses and deflections of a laminated plate under thermal vibration using the moving least squares differential quadrature method.The method provides rapidly convergent and accurate solutions for calculating the stresses and deflections.
The thermal buckling behavior of the laminated plates subjected to uniform and/or non-uniform temperature fields was studied by Ghomshei [15].The influence parameters of plate aspect ratio, cross-ply ratio and stiffness ratio on the critical temperature were presented.Rath [16] the free behavior of laminated plates subjected to varying temperature and moisture.A simple laminated plate model is developed for the vibration of composite plates subjected to hygrothermal loading.The results showed the effects of geometry, material and lamination parameters of woven fiber laminate on the vibration of composite plates for different temperature.Ghugal [17] presented the flexural response of cross-ply laminated plates subjected to thermo-mechanical loads.The shear deformation theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate.Thermal stresses for three-layer symmetric cross-ply laminated plates subjected to uniform linear and nonlinear and thermo-mechanical loads are obtained.The governing equations for laminated beams subjected to uniform temperature rise are derived by Fu [18].The effects of the transverse shear effects and boundary conditions on the thermal buckling and postbuckling of the beams are discussed.A differential quadrature method is applied to obtain the maximum buckling temperature of laminated composite by Malekzadeh [19].The direct iterative method in conjunction with genetic algorithms is used to determine the optimum fiber orientation for the maximum buckling temperature.
From the literature reviewed, researches on the vibration and buckling of initially stressed laminate plates under thermal environmental condition seem to be lacking.The vibration and stability behaviors of initially-stressed laminate plates have been investigated by Chen et al. [20][21] in recent years.The studies revealed that the initial stress in structures may significantly influence the behaviors of laminated plates.Therefore, while studying the thermal buckling and vibration behavior of laminate plates, the effect of initial stress should be taken into account.In this paper, the equilibrium equations for a laminated plate subjected to the arbitrary initial stress and thermal condition are established by using variation method.The temperature field is assumed to be uniform plus linearly distributed through the plate thickness.The effects of various parameters on the critical temperature, natural frequencies and buckling loads in thermal environments are presented.

Equilibrium equations
Following a similar technique described by Brunelle and Robertson [22] and Chen et al. [20][21], Hamilton's principle is applied to derive nonlinear equations of the composite plate including the effects of rotary inertia and transverse shear.For an initially stressed body which is in equilibrium and subjected to a time-varying incremental deformation, the Hamilton's principle can be expressed as where 00 0 1 ;; 2 where U, K, We and Wi are the strain energy, kinetic energy, work of external forces and internal forces, respectively; ij and ij are the stresses and strains; vi are the displacements referred to the spatial frame;  is the density; Xi is the body force per unit initial volume and pi is the external force per unit initial surface area.The application of the minimum total energy principle leads to the general equations and boundary conditions.Assume that the stresses and applied forces are constant, and substitute Eq. ( 2) into Eq.( 1).Then taking the variation and integrating the kinetic energy term by parts with respected to time, Eq. ( 1) becomes: If a rectangular plate is considered, the equations can be rephrased in xy coordinates.The incremental displacements are assumed to be of the following forms: where ux, uy and w are the displacements of the middle surface in the x, y and z direction, respectively;x and y denotes the rotation angle about y and x axis, respectively.The two edges of a rectangular plate are set along x and y axes, respectively.The stress-strain relations are taken to be those of uncoupled linear thermal elasticity.Hence, the constitutive relations for the kth lamina including the thermal effect can be written as: where Cij are the elastic constants of lamina; ij are thermal expansion coefficients and T is the temperature rise.The stress-displacement relations are found to be: Substitute Eqs. ( 4)-( 6) into Eq.( 3), perform all necessary partial integrations and group the terms by the five independent displacement variations, ux, uy, w, x and y, to yield the following five governing equations:   ; ( , ) where all the integrals are integrated through the thickness h of the plate from -h/2 to h/2.

Solution of the governing equations
Because the stability and vibration behaviors of the investigated initially stressed laminate composite plate are affected by various parameters, it would be difficult to present results for all cases.Thus, only the initially-stressed simply supported cross-ply laminate plate under the combined uniform and linear thermal loading is investigated.The lateral loads and body forces fx, fy, fz, mx and my are taken to be zero.The only nonzero initial stress is assumed to be (Fig. 1) which comprises of the constant uniaxial stress n and bending stress m.Hence, the nonzero axial stress result- ants are Nxx = hn, Mxx = Sh 2 n / 6 and * xx M = h 3 n / 12.The factor S = m / n denotes the ratio of a bending stress to a normal stress.For the cross-ply plate, the stiffness coefficients C16, C26 and C45 will be equal to zero in Eqs. ( 6) and (7).Fig. 1 A simply supported plate with initial stress The combined uniform and linear temperature distribution is of the form as where To is the uniform temperature rise and Tg is the temperature gradient.The nonzero thermal stress resultants are -, For a simply supported laminated plate, the boundary conditions along the x-constant edges are: and along the y-constant edges are: For the simply supported plate, the displacement fields satisfying the geometric boundary conditions are given as follows: All summations are summed up from m, n = 1 to  .For a buckling problem, t i mn e  is neglected in Eq. (17).Substituting the initial stress (13), temperature distribution ( 14) and displacement fields ( 17) into the gov-erning Eqs. ( 8)- (12), and collecting the coefficients for any fixed values of m and n leads to the following eigenvalue equation: in which parameter  refers to the corresponding frequency or buckling coefficient.For the vibration problems, the coefficients of the symmetric matrix [C] and [G] are expressed as follows: =-- For the thermal buckling problem, the coefficients of matrix [C] are given by neglecting thermal induce stresses resultant terms in the stiffness matrix in Eq. ( 18) and the coefficients of matrix [G] are: As to the buckling load problems, the coefficients of the symmetric matrix [C] are given by neglecting the initial stress resultant terms of the matrix [C].The coefficients of the matrix [G] are:

Results and discussion
For verifying the present computer program, the close agreements between the present results and those in Matsunaga [23], Liu and Huang [24] for cross-ply plates as shown in Tables 1-2 demonstrate the accuracy and effectiveness of the present method.Parametric studies are carried out to examine the effects of various variables on the vibration and stability response of laminate plates under thermal environments.The following non-dimensional natural frequency (= ), buckling coefficient ( ) and thermal buckling coefficient   Tg / To = 0) Fig. 2 presents the effect of modulus ratio on the thermal buckling temperature of plates with different stack layers.The critical temperature increases monotonically as the modulus ratio or/and layer number increase.The critical temperatures of eight-layer plates under different temperature gradient Tg are given in Table 3.The increasing temperature gradient reduces the thermal buckling temperature, and its influence on the critical temperature is less than the layer number.The effects of modulus ratio and span ratio on critical temperature parameters are shown in Fig. 3.The buckling temperature of plate with a smaller span ratio is always higher than that with a larger span ratio, especially for the plate with a higher modulus ratio.Thus, with a higher modulus, higher stacking number of layer, lower span ratio and lower gradient temperature, the laminated plate has a higher thermal buckling temperature.The effect of buckling coefficient on the natural frequency of plates under various uniform temperature rises can be observed in Fig. 4. The natural frequency decreases with the increasing initial compressive stress and temperature rise.The buckling load can be obtained when the natural frequency approaches zero.Meanwhile, the plate under a lower temperature rise has a greater buckling coefficient.Fig. 5 shows the effect of modulus ratio on the natural frequency of laminated plates.The laminate plate with higher modulus ratio has a larger vibration frequency and higher buckling load.
The buckling load and natural frequency of laminate plate with different layer numbers and modulus ratios under uniform temperature rise are shown in Tables 4-5.The plate with larger stack layer number or/and higher modulus ratio has a higher critical buckling load and natu-ral frequency.It can also be observed that the buckling load and natural frequency decreases steadily with the increasing uniform temperature rise.Thus, the two-layered plate with smallest modulus ratio and under higher temperature rise will possess the smallest buckling load and natural frequency.The effect of different temperature gradient on buckling load and natural frequency of plates is presented in Tables 6-7.When the linear gradient temperature increases the buckling load and natural frequency coefficient slightly decrease.The laminated plate with lower modulus ratio and under higher uniform temperature and temperature gradient has a smaller critical buckling and vibration frequency.The influence of temperature gradient on the buckling load and natural frequency for laminate plates is less apparent than that of uniform temperature.8-9.It is evi-dent that the compressive initial stress (Kf < 0) produces a decreasing effect on the critical temperature and natural frequency, and the tensile initial stress has a reverse effect.Likewise, the initially stressed laminate plate with higher modulus ratio has a larger critical temperature than the one with lower modulus ratio.The effect of bending stress ratio on the critical buckling coefficient for initially stressed plates under uniform temperature is given in Table 10.As can be seen, the increasing bending stress ratio decreases the critical buckling load.The influence of bending stress on the natural frequency of initially stressed plates under a fixed uniform temperature is presented in Table 11.The vibration frequency decreases with the increase in bending stress.However, the natural frequency is not affected by the increasing bending stress when the plate is subject to the pure bending stress only.The lowest natural frequency can be observed for the plate with a lower modulus ratio and under a higher bending stress.

Conclusions
The vibration and buckling behaviors of initially stressed and thermally stressed laminate plates have been described and discussed in this paper.The results demonstrate the influence of the modulus ratio, number of layer, initial stress and thermal stress on the vibration and buckling behaviors of laminate plates.Following the above discussions, the preliminary results are summarized as follows: 1.The modulus ratio, number of layer and uniform temperature has an apparent influence on natural frequency and buckling load.They are slightly affected by the temperature gradient rise and bending stress.2. With the increasing modulus ratio and number of layer, the critical temperature, buckling load and natural frequency increase.The uniform temperature has a reverse effect.3. The compressive stress significantly reduces the natural frequency and critical temperature but the tensile stress produces an opposite effect.
11) where fx, fy, fz, mx and my are the lateral loadings.The arbitrary initial stresses are included in the stress resultants Nij, Mij and resultants.The coefficients associated with material parameters, initial stress, thermal stress resultants and rotary inertia are defined as: m x / a sin n y / b e u h V sin m x / a cos n y / b e w h W sin m x / a sin n y / b e cos m x / a sin n y / b e sin m x / a cos n y / b e and used throughout the vibration and stability study.If the stress is tensile, then the buckling coefficient Kf is positive.There is no initial stress when Kf = 0 and S = 0.The critical buckling temperature is denoted by Tcr.

Table 4
Effect of layer number and modulus ratio on critical buckling coefficient of plates under various uniform temperature rise (a / b = 1, a / h = 10, S = 0, Tg / To = 0)

Table 9
Effect of initial stresses on the natural frequency of plates under linear temperature rise (a / b = 1, a / h = 10, n = 8, S = 0, To / Tcr = 0.5)