Stress analysis of infinite laminated composite plate with elliptical cutout under different in plane loadings in hygrothermal environment

This paper presents the solution of stress distribution around elliptical cutout in an infinite laminated composite plate. Analysis is done for in plane loading under hygrothermal environment. The formulation to obtain stresses around elliptical hole is based on Muskhelishvili’s complex variable method. The effect of fibre angle, type of in plane loading, volume fraction of fibre, change in temperature, fibre materials, stacking sequence and environmental conditions on stress distribution around elliptical hole is presented. The study revealed, these factors have significant effect on stress concentration in hygrothermal environment and stress concentration changes are significant with change in temperature.

φ (z1) , ψ (z2) stress functions in z plane λ biaxial loading condition φ 1 (z1) , ψ 1 (z2) first stage stress function φ 0 (z1) , ψ 0 (z2) second stage stress function φ 0 (ζ ) , ψ 0 (ζ ) second stage stress function in ζ plane f 1 f 2 hole reactions H 1 to H 4 complex constants H 1 to H 4 complex conjugate E f 1 and E f2 the elastic module of fiber material along the fiber and transverse direction ν f 1 and ν f2 Poisson ratios of fiber material along the fiber and transverse direction V f and Vm fiber and matrix volume fraction β f 1 and β f2 moisture coefficients of fiber along the fiber and transverse direction α f 1 , α f 2 thermal expansion coefficients of fiber along x and y-direction αm and Em thermal expansion coefficient and modulus of elasticity of matrix material βm and Vm moisture coefficient and volume fraction of matrix material E 1 and E 2 modulus of elasticity along longitudinal and transverse direction of fiber α 11 and α 22 thermal expansion coefficients of laminated reinforced composite along x and y-direction α 1 coefficient of thermal expansion of composite material x, y, z cartesian coordinates ∆T increment in temperatures ∆C increment in moisture

Introduction
The use of the laminated composite plates has extensively increased in advanced and modern engineering applications. It is due to the fact that composite laminates possesses very high strength to weight and stiffness to strength ratio. Due to these excellent properties, composite laminate structures are light in weight and give superior performance in structural engineering applications and are very popular in weight sensitive applications like spacecrafts. As a part of service requirement in certain applications, holes or cutouts are made in the structures. When these structures with holes or openings are subjected to different types of loadings, they give rise to stress concentrations. In engineering applications like pressure vessels, openings of circular or elliptical shape, they are provided for inspection purpose. Window structure in aeroplanes is one of the examples of service requirement. Stress investigation of infinite plates with holes or opening or cutouts is an interesting area of analysis. Investigation is needed to study the effect of loading, environmental conditions, like temperature and moisture, etc. on stress concentration. Better estimation of stress concentration factor helps to improve optimum performance of structures made from composite laminates. Stress concentration around elliptical holes plays an important role in the failure and fracture of the structures. Savin [1] obtained the solution for stress distribution around circular, elliptical, triangular and rectangular holes in isotropic as well as anisotropic plates using Mushkhelisvili's complex variable. Daoust et al. [2] analyzed stress distribution around a triangular hole in anisotropic material and revealed the effect of length/height ratios of the triangular hole, degrees of bluntness at the triangle vertex and load orientation on stress concentration. Xin Lin Gao [3] solved problem of infinite isotropic plate with elliptical hole and revealed effect of arbitrary biaxial loadings on stress concentration. Simha et al. [4] analyzed stress concentration around irregular hole having fixed hole geometry with perimeter being varied and subjected to different in plane loading. Theocaris et al. [5] investigated triangular hole under tension at infinity. Effect of round of corners on stress concentration was revealed. Kakhandki et al. [6] revealed stress distribution around irregular shaped hole in an orthotropic laminate using savin's approach. The results are validated with the FEM solution. Rao et al. [7] used Savin formulation to get stresses around holes in anisotropic plates under inplane loading. It considers the effect of arbitrary shape of hole and arbitrarily oriented loading. Failure strength of the laminate is also obtained using different laminate failure criterion. Jafari et al. [8] revealed the effect of varying parameters, like cutout shape and bluntness, load direction or cutout orientations on the stress distributions and SCF in the perforated plates. Sharma and Dave [9] obtained general solution to SIF for hypocycloidal hole in infinite symmetric anisotropic plate. Study revealed effect of stacking sequence, hole geometry, material properties and loading angle on stress intensity factors for hypocycloidal hole. Darwish et al. [10] investigated stress concentration factor (SCF) in countersunk rivet holes in orthotropic laminated plates subjected to uniaxial tension. The effect of geometric and material parameters such as plate thickness, straight-shank radius, countersunk angle, countersunk depth, plate width, and the laminate ply angles on the SCF is investigated. A relation fitting in these parameters is obtained to get SCF around rivet hole. Patel et al. [11] revealed the effect of stacking sequence, ply groups, loading angles, materials and corner radii on the failure strength and moment distribution in symmetric laminated composite plate containing a square hole. The layer wise stresses and failure strengths are evaluated and the failure strengths of laminates are calculated. Khamseh and Waas [12] done the experimental analysis to investigate failure mechanisms in laminated composite plates containing circular cutouts, under biaxial in plane compressive loading. Biaxial tests were carried out with 48 ply graphite/epoxy composites by varying fibre orientation. Sharma [13] investigated an optimum design of a 4, 8 and 16 layered graphite/epoxy and glass/epoxy symmetric laminated plate containing an elliptical hole subjected to in-plane loading conditions using complex variable method and genetic algorithm (GA). The optimum stacking sequence is obtained using genetic algorithm. Li et al. [14] investigated a plate weakened by an oblique penetration of a circular cylindrical hole using finite element method. Parametric studies of effects of angle of inclination, plate thickness, and width are revealed & empirical relation is developed. Hasebe et al. [15] investigated the problem of a crack originated at point of a circular hole with rigidly stiffened displacement. For this case stress distributions and SIF are calculated using complex variable method. Puhui et al. [16] investigated stresses around elliptic hole using, Lekhnitskii's complex potential method in unsymmetric composite laminate plate. W. Becker et al. [17], solved a problem of finding moments around elliptical hole in unsymmetric laminate plate. The new complex potential is developed.
Many researchers have investigated the effect of environmental conditions on the behaviour of laminated composite plate. Analytical results using Chebyshev series for the non-linear flexural response of hygrothermomechanical loaded elastically supported thick laminated rectangular plates were investigated by Upadhyay et al. [18]. Buckling behaviour of the composite laminate plate under hygrothermo mechanical loading was analyzed using C 0 FEM by Kumar et al. [19]. Non-linear static behaviour of a piezoelectric sandwich plate having elastic foundation and hygrothermo mechanical loading was investigated by Lal et al. [20] by using SFSDT and investigated the ef-fect of random system characteristics with fibre volume fraction, stacking sequence, type of geometry, etc. on the transverse largest deflection. Static analysis of a sandwich plate having a soft compressible core was revealed by a higher-order Zig-zag theory by Pandit et al. [21]. Hadjiloizi et al. [22,23] build a micro-mechanical model to analyze smart composite magnato-thermoelastic thin plate with varying thickness using 3D formulation in Part-I [22]. In second part [23], the theory is applied using examples of thin laminated magnetoelectric plates having constant thickness. Hsieh [24] investigated the effect of heat flow and moisture transfer on stresses and moments around elliptical hole in composite plate. Rahnama [25] investigated SCF in unidirectional composite plate having broken fibres and hygrothermal condition. Effect of fibre volume fraction on SCF in wet and dry conditions is studied. Sai Ram [26] analyzed moments around cutouts in hygrothermal environment with transverse loading in unsymmetric laminate plate.
Dimitri et al. [27] used SFEM and IGA to obtain SCF in isotropic plates with inclusions like circular holes, u and v notches. Fantuzzi et al. [28] obtained numerically mode I SIF in composite structures using SFEM based on GDQ method. SFEM with domain decomposition technique is used by Viola et al. [29] to study stresses in infinite plate with circular inclusions and uniaxial loading.
From the literature survey, it is observed that the researches have done the work on the analysis of laminated composite plates containing cutouts of different shapes and subjected to inplane moments or forces to get stresses and bending moments around cutouts. Hygrothermal effects on SCF are investigated by number of researchers by considering transverse loading. To the best of the author's knowledge for in plane loading, effect of various parameters like stacking sequence, fiber orientation, loading conditions, fiber volume fraction, and cutout dimensions on the normalized stresses around elliptical hole under hygrothermal conditions were not investigated.
In the current analysis stress distribution around elliptical hole in composite laminate subjected to in plane loading is investigated. The effect of laminate stacking sequence, fiber orientation, loading conditions, fiber volume fraction, type of fibre material under hygrothermal conditions on stresses around elliptical cutout is studied.

Mathematical formulation
An anisotropic infinite plate with an elliptical cutout having major axis as a & minor axis as b is subjected to in plane loading at infinity as shown in Figure 1. To account for all cases of loading, general loading condition of Gao [3] is considered. λ is a loading factor & β is angle made by X'Y' coordinate system with XY coordinate system. Figure 2 shows different in plane loading cases considered in present analysis by choosing appropriate λ and β.

Complex variable representation
Plate thickness is very small as compared with other two dimensions and loading is in plane hence approximating problem as plane stress problem with stresses in out of plane direction as zero. Plate faces are stress free. The generalized Hooke's law for a thin anisotropic plate for arbitrary loading along X'Y' axis relates stresses σx, σy, τxy and strains ϵx, ϵy, ϵxy, For the multilayer laminate with given fiber orientation angle is calculated using method given in [30].
The stresses in terms of the Airy's stress function U are given by, Keeping Eq. (2) in Eq. (1) & putting the result in compatibility equation, we get the 4 th order biharmonic characteristic equation of the system.
The Eq. (3) has four roots (µ j j = 1, 2, 3, 4) whose values depends on [︀ a ij ]︀ are known as complex parameters. Using Muskhelishvili's complex variable approach [32], the stress components in XY plane are given by, To get stress functions φ (z1) & ψ (z2), conformal mapping is used. The area outside the elliptical hole with major diameter a & minor diameter b in z plane is mapped outside the circle having unit radius in ζ plane using mapping function ω(ζ ), R is size factor For unit circle, ζ = cos θ + i sin θ, using in Eq. (5) along with complex parameter µ 1 and µ 2 , we get final mapping function, Stress function φ (z1) & ψ (z2) are obtained by using superposition principle as shown in Figure 3. In first stage, the plate with the fictitious hole is considered and first stage stress functions φ 1 (z1) and ψ 1 (z2) are obtained using boundary conditions shown in Figure 3a. First stage stress functions are used to find the boundary conditions f 1 and f 2 at hole boundary. In second stage, negative of f 1 and f 2 are applied at hole boundary Figure 3b and second stage stress function φ 0 (z1) and ψ 0 (z2) are obtained. As seen in Figure 3c, addition of first & second stage gives original configuration of infinite plate with hole subjected to loading at infinity. As per superposition principle, Stress function for first stage is given by, Using Gao's [3] biaxial loading factor λ, stresses applied at infinity along X'& Y' axis are given by, (refer to Figure 1) Putting Eq. (9) in Eq. (4) and solving Eq. (4) simultaneously to get constants B, B* & C*. The boundary condition in stage 1 are given by Keeping Z 1 and Z 2 from Eq. (6) into Eq. (9) and using Eq. (9) in Eq. (11) we get Where, Γ is boundary of unit circle in ζ plane and λ 1 and λ 2 are complex constants and does not affect stress solution hence removed from the equation. Keeping −f 1 = f 0 1 and −f 2 = f 0 2 in Eq. (14) and evaluating integral we get Keeping values from Eq. (9) & Eq. (15) in Eq. (8) we get stress functions φ (z1) and ψ (z1). Putting these stress function in Eq. (4), we get the following equations of stresses around the hole.

Properties of laminate
The micromechanical approach is used in the present paper to get the material properties of the composite laminates to account the effect of temperature and environment conditions. The properties retention ratio for the matrix is given as [20] Rm = Where, t is the temperature at which properties of a composite laminate is to be estimated, t d , tevn and tw are glass transition temperature for dry condition, reference environmental temperature and glass transition temperature at wet conditions respectively.
Using the micromechanical approach, the elastic constants of reinforced composite laminate are given by following equations [20] Where 'f ' and 'm' are subscript used for fiber and matrix material of composite lamina. The matrix thermal property can be estimated with a thermal characteristics retention ratio given as The thermal expansion coefficients using a micromechanical approach can be written as Kumar et al. [19] (27)

Results and discussion
In the present solution, stresses around the elliptical hole in an infinite symmetric laminate plate subjected to in plane Define temperature, Moisture content, Fibre Volume fraction Obtain , ij A     for laminate under consideration using Jones [30]   loading at infinity is obtained. The cross and angle ply laminates along with different stacking sequences are considered for the analysis. The material properties of composite laminate used in the present paper are obtained by using micromechanical approach to account for temperature & environment effects.
Flow chart in Figure 4 gives procedure of getting stress distribution around elliptical cutout under hygrothermal environment.
Properties of carbon fiber (T-300) & matrix material listed below are used for calculating stress distribution results around the elliptical hole. The effects of loading, materials, temperature, humidity, fiber volume fraction, laminate geometry and stacking sequence of lamina on stress distribution around the elliptical hole can be analyzed effectively by using the solution presented in this paper. Figure 5(a) shows the results of stress distribution around elliptical hole in 16 ply laminate plate of Graphite epoxy, subjected to equibiaxial tension σ ∞ x = σ ∞ x = 1. The maximum tangential stress, σ θmax = 3.8588 occur at 4.5 ∘ & 184.5 ∘ which is in good agreement with result presented in Sharma [11]. Figure 5(b) shows the stress distribution results from Sharma [11]. Figure 6 shows the variation in σ θ around elliptical hole due to temperature change in [0/90]s laminate. The maximum values of tangential stresses are, σ θmax = 9.8190, 10.3304, 11.1480, 12.8781 for ∆t = 0 ∘ , 50 ∘ , 100 ∘ , 150 ∘ . These values occur at 0 ∘ & 180 ∘ angle as expected for cross ply laminate. For ∆t = 150 ∘ , percentage change in σ θmax is 31.15% & for ∆t = 50 ∘ it is 4.85%. So it is clear that for higher temperature rise effect on stress concentration is significant. Table 1 depicts the variations in σx max σy max τxy max due to the change in the temperature ∆t. Cross ply laminate with elliptical hole having a/b=2 is subjected to equibiaxial tenstion. Shear stress τxy max is least affected due to change in the temperature. The Location of these maximum stresses for given loading condition are different.
Laminate plate with elliptical hole is subjected to equiaxial loading and results for σ θ max listed in Table 2. From Table 2, it is clear that stacking sequence affect the σ θ max considerably. To limit the stress concentration for a given loading, selection of proper lamina angles and stack-     ing sequence is very important. The variation in stress concentration due to change in temperature is 31% & 29% respectively for first two ply groups with ∆t = 150 ∘ . From Figure 7, it is clear that stacking sequence has significant effect on stress concentration with change in temperature. So selection of proper stacking sequence and lamina fibre orientation is required to minimize the effect of change in temperature on stress concentration. The stacking sequence affects the laminate properties and hence the stresses around the hole. Figure 8a, 8b shows the effect of change in temperature on stress concentration in a cross ply and angle ply laminate plate subjected to uniaxial tension in x and y direction , biaxial tension and shear loading at infinity. From Figure8a it is clear that, for tensile loading in x direction & shear loading at infinity in cross ply laminate has least effect on stress concentration due to change in temperature. In cross ply laminate subjected to tensile loading along y axis or equibiaxial tensile loading, there is increase in stress concentration by 28% and 31% respectively. Figure8b depicts the effect of temperature change on stress concentration for angle ply laminate. For tensile loading in x direction variation in stress concentration is less but for other three types of loading it is considerable. From Figure8a and8b it is also clear that type of loading and fibre orientation has significant effect on stress concentration. Stress concentration in angle ply is less as compared with cross ply for different in plane loading. Table 3 lists the results of σ θ max for a cross ply laminate subjected to tensile loading with load angle varied  from 0 ∘ to 90 ∘ . From table it is clear that, the location of σ θ max varies with the change in loading angle in x direction. From Figure 9 it is clear that, in cross ply laminate σ θ max decreases up to 30 ∘ loading angle & then increases. For a temperature change of above 100 ∘ effect on stress concentration is significant. Table 4 shows the results of σ θ max for a angle ply laminate subjected to tensile loading with load angle varied from 0 ∘ to 90 ∘ . From Figure 10 it is clear that, in angle ply laminate σ θ max increases up to 60 0 loading angle & then decreases. Comparing Table 3 & Table 4 along with Figure 9 and 10, it is observed that, effect of loading angle and temperature change is more significant in cross ply laminate as compared with angle ply laminate. Table 5 shows the effect of fibre fraction on σ θ max for different values of ∆t. Results are obtained for cross and angle plate laminate subjected to tensile loading along x axis (σ ∞ x = 1). From the Table 5, it is clear that volume fraction of a fibre has a least effect on stress concentration as compared with the change in temperature. If fibre fraction is increased from 0.4 to 0.8 at ∆t = 0 ∘ , there is 4.5 % change in the σ θ max but for Vf =0.1 & ∆t changed from 0 ∘ to 150 ∘ there is 24% change in the σ θ max . Table 6 contains the results of moisture content effect on the σ θ max . Cross ply and angle ply laminate plates subjected to tensile loading along x axis, y axis and biaxial loading is considered. From the Table 6, it is clear that change in moisture content has less effect on stress concentration for the cases of different types of stacking sequences, loading. For tensile loading along x direction, there is 0.26% change in stress concentration if ∆c is varied from 0 to 0.2.      Figure 11 shows results σ θ max for cross ply laminate subjected to tensile loading along x axis for different cases of a/b ratio. From figure it is clear that, change in fibre fraction volume has least effect on the stress concentration. It is important to study the effect of temperature on stress concentration around elliptical hole in laminate plate by choosing different fibre materials. The fibre materials listed in Table 7 are used for the analysis.
Material properties of fiber materials listed in Table 7 are used to study the effect of temperature change on stress concentration for these materials. Figure 12 shows the distribution of σ θ around elliptical hole in cross ply laminate plate with four different fibre materials. From Figure 12, it is clear that stress concentration is greatly affected due to material properties. Loading considered for the analysis is tensile along the x axis. We get σ θ max = 3.   Figure 13 shows the distribution of around elliptical hole in angle ply laminate plate with four different fibre materials. From Figure 13, it is clear that location of σ θ max is different for different materials in ([45/−45]s) laminate plate. Loading considered for the analysis is tensile along the x axis. We get σ θ max = 2.6028 at 32 ∘ , 148 ∘ , 212 ∘ & 328 ∘ for carbon fibre, σ θ max = 1.6653 at 43 ∘ , 137 ∘ , 223 ∘ & 317 ∘ for Kevlar fibre, σ θ max = 1.8814 at 36.5 ∘ , 143.5 ∘ , 216.5 ∘ & 323.5 ∘ S-glass fibre, σ θ max = 2.0836 at 34.5 ∘ , 145.5 ∘ , 214.5 ∘ & 325.5 ∘ boron fibre.
From Figure 14 it is clear that change in the temperature has a significant effect on σ θ max . Fibre materials differ in thermal coefficient of expansion in longitudinal and transverse direction which reflects in temperature effect on σ θ max . If ∆t is varied from 0 ∘ to 150 ∘ , the percentage change in stress concentration is 31%, 28.4%, 23.3% & 32.3% for carbon, Kevlar, S -glass & boron fibres. So elastic modules & thermal coefficient of expansion of fibre material has a significant effect on stress concentration change due to change in temperature of the material.

Conclusions
This paper presents the solution of stress distribution around elliptical hole in infinite laminate subjected to in plane loading with hygrothermal considerations. Outcomes of the present work can be concluded as: • The effect of temperature increase on stress concentration is significant for ∆t= 100 or above. • Proper stacking sequence of laminates minimizes the effect of change in temperature on stress concentration. • The type of the fiber material has significant effect on stress concentration along with the combination of loading condition. • In cross ply laminate ([0/90]s) with elliptical hole, the stress concentration observed for different loading, materials etc. is higher than the angle ply laminate ([45/−45]s). • The effect of temperature change on stress concentration is more significant in cross ply laminates as compared with angle ply laminates. • The Effect of change in moisture content on stress concentration is least significant for all type of loading & stacking sequence. • The change in stress concentration due to variation in fiber volume fraction is very small. • a/b ratio affects the stress concentration around elliptical hole.

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