Effects of hygrothermal conditions on free vibration behaviour of laminated composite structures

Free vibration behaviour of laminated composite structure under combined hygrothermal load is investigated in this article. The responses are obtained numerically through a finite element model using hygrothermal dependent composite material properties. The convergence behaviour of the present numerical model has been established and validated by comparing the responses with those available open literatures. The effects of various hygrothermal conditions and geometrical parameters (curvature ratio, side-to-thickness ratio and aspect ratio) are examined in details.


Introduction
Laminated structures made of composites are exposed to combined hygrothermal loading both during manufacturing as well as in service. It is well known that, the mechanical properties (Young's modulus and rigidity modulus) of laminated composites are significantly influenced due to hygrothermal and high strain rate loading. The corrugated fiber properties due to uneven environment and fiber volume fractions may alter the mechanical responses substantially. The structural components made up of laminated composites found in aerospace, marine, automotive, agriculture, sports, and biomedical industries are exposed to vibration which experiences fatigue/cyclic loading during their service life. In recent years researchers quest to analyse (analytical, numerical and/or experimental) the complex loading in conjunction with degraded material properties of laminated structures accurately for finished products. It is noted that, free vibration behavior of laminated shell panels under hygrothermal loading have been analysed by using various displacement models say, classical laminated shell theory (CST), the first order shear deformation theory (FSDT), and the higher order shear deformation theory (HSDT). In this regard some of the important contributions are discussed briefly in the following line to address the objective and necessity of the present study. Sai Ram and Sinha [1] investigated the effects of moisture and temperature on the free vibration responses of laminated composite plates using finite element method (FEM). A quadratic isoparametric finite element (FE) formulation is presented by Parhi et al. [2] for the hygrothermal free vibration and transient analysis of multiple delaminated plates and shells. Their formulation is based on the FSDT. Patel et al. [3] analyzed the static and dynamic behavior laminated composite plates using FEM. Their model is based on the modified HSDT and includes hygrothermal dependent material properties. Huang and Zheng [4] analysed nonlinear free and forced vibration of moderately thick laminated plates using HSDT and von-Karman type nonlinear kinematics under transient loading. Shen et al. [5] reported dynamic behaviour of laminated composite plate resting on elastic foundation under  [9] using FSDT and von-Karman nonlinearity. The severity due to combined variation in temperature and moisture concentrations on the nonlinear vibration behaviour of laminated composite structure is reported by Ashraf [10]. The effects of temperature and moisture content on natural frequencies of woven fiber glass/epoxy delaminated composite plate is analysed through combined numerical (FEM) and experimental steps by Panda et al. [11]. They modelled the composite plate based on the FSDT kinematics by considering the degraded hygrothermal properties. Panda and Mahapatra [12] studied the nonlinear thermal free vibration behaviour of laminated composite spherical shell panel using the HSDT and Green-Lagrange nonlinearity.
We note that limited studies related to linear/nonlinear vibration behaviour of laminated composite flat/curved panels under hygrothermal environment have been reported in the open literature. Based on the authors' knowledge, studies related to free vibration behaviour of laminated composite spherical shell panel under hygrothermal loading by using HSDT mid-plane kinematics are also scarce. The present work aims to analyse the free vibration behaviour of laminated composite shear deformable spherical shell panels under hygrothermal loading using a mathematical model based on the HSDT. The temperature and/or moisture dependent material properties are considered in order to obtain a realistic response. The sets of governing equations are obtained using Hamilton's principle and discretised using FEM steps. The effect of various geometrical parameters on the free vibration behaviour of laminated composite spherical panels under combined hygrothermal loading is investigated through subsequent numerical illustrations. Figure 1 shows the geometry and lay-up of a laminated spherical shell panel of length a, width b and thickness h with composed of 'N' number of equally thick anisotropic layers. R1 and R2 are the radii of curvature of the shell panel at the mid surface (Z=0). In order to derive the hygro-thermo elastic mathematical model of laminated composite spherical shell panel, a HSDT based displacement field is adopted which is assumed to be in following form:

Hygro-thermo elastic equations
The hygro-thermo elastic constitutive matrix equation of generalized stress tensor considering plane stress condition for any general k th isotropic composite lamina and any fibre orientation angle θ, can be expressed as

Work done, kinetic energy and strain energy
The in-plane hygro thermal forces are obtained using the steps in [14] and given by where,   N ,   M and   P , are the resultant vectors of compressive in-plane hygrothermal forces, moments and the higher order terms due to combined temperature and moisture variation.
The work done (W) due to in-plane hygrothermal load is given by The kinetic energy of a vibrating laminated plate is expressed in following step where,    ,    and    are the density, global displacement vector and first order derivative of the global displacement vector with respect to time.
where,   f is the function of thickness coordinate.
The total strain energy of the laminated shell panel is expressed as Substituting the values of stress and strain components from Equations (3) and (4)

Finite element implementation
The FEM has proved to be a powerful and widely appreciable method for the vibration analysis of laminated composite structures whose analytical solutions are difficult to obtain. A nine-noded isoparametric element having nine degrees of freedom per node is employed in the present model.  (10) where, Ni is the nodal interpolation function for i th node and NN is the numbers of node per element.
Employing FEM the strain vector in Eq.

System governing Equation
The system governing equation of free vibrated laminated orthotropic spherical shell panel under hygrothermal loading can be obtained using Hamilton's principle as follows. where, L = [T-(U+W)] Now employing the Equation (10) into Equation (6) and (9), the elemental form is obtained as in [12] and subsequently Equation (12) can be expressed as in [15] as where,     is the displacement vector,   M and   K are the global mass and stiffness matrices, respectively. Now, the numerical solutions are obtained using a direct iterative method.    The fundamental frequency in nondimensional form is obtained using the equation

Convergence and Comparison Study
In order to establish the convergence behaviour the free vibration under unlike environmental conditions, responses computed using the present model is plotted with respect to mesh divisions in figure 1. The geometrical, material properties, lamination scheme and support conditions are taken same as [6] and [9]. The results show very good convergence rate with respect to mesh refinement. Based on the convergence study, it is concluded that a (5×5) mesh is adequate to obtain desired responses throughout the analysis. For validation study, simply supported square (a=b=0.5 m), laminated composite spherical shells (R1=R2, h=5mm, a/h=100) are considered. The material properties are taken same as [2]. The responses obtained using the present model along with the reference values are presented in Table 2. It is clearly observed that the present results are in good agreement with the reference values. The difference in results is because the fact that the reference uses FSDT based mathematical model where as the present model is developed based on HSDT mid plane kinematics, which is more realistic in nature.

Parametric Study
In order to address the effect of various geometrical parameter like curvature ratio (R/a = 5, 10, 50, 100), thickness ratio (a/h = 5, 10, 50, 100) and aspect ratio (a/b = 1, 1.5, 2, 2.5) on the nondimensional fundamental frequency of laminated composite spherical shell panel under hygrothermal environment is computed and discussed in detail in the following examples.

Example 1.
In this example, a simply supported square (a/b=1) eight layer anti-symmetric ([0°/90°]4) cross-ply laminated composite spherical panel (a/h=20) is used for the computational purpose under different hygrothermal load. The responses are presented in figure 2. It is observed that the nondimensional fundamental frequency parameter increases as temperature and moisture concentrations increases for each case. It can also be seen that frequency parameter decreases as the curvature ratio increases. Figure 3 shows the effect of thickness ratio (a/h) and hygrothermal conditions on the nondimensional linear frequency responses of angle-ply ([±45]2), square (a/b=1) laminated composite spherical (R/a=50) panel under CFCF support condition. It is observed that the nondimensional fundamental frequency responses are higher for thin shell panels in comparison to the thick panels.

Example 3.
Square hybrid ([±30 /0/75]) laminated spherical panels (R/a=50, a/h=100), under CSCS support condition and different environmental conditions are considered. The computed results are presented in figure 4. It can be seen that the nondimensional fundamental frequency parameter increases with hygrothermal load and aspect ratios.

Conclusion
Effect of hygrothermal load and geometrical parameter on the free vibration behaviour of laminated composite spherical shell panels have been investigated in the present study. A mathematical model based on the HSDT mid-plane kinematics is developed and discretised using FEM. The convergence and validation of the present model has been established. The significance of the present HSDT based model for the analysis of laminated structures under combined hygrothermal loading condition is observed. The subsequent numerical illustrations indicate that considering the effect of hygrothermal conditions is inevitable for laminated composite structures, particularly when they are exposed to a severe environmental condition. Irrespective of the hygrothermal load and support conditions, the nondimensional fundamental frequency responses decrease with curvature ratios whereas they increase with increase in thickness ratios and aspect ratios under all sorts of hygrothermal conditions.