1D analysis of laminated composite and sandwich plates using a new fifth‐order plate theory

Abstract In the present study, a new fifth‐order shear and normal deformation theory FOSNDT is developed for the analysis of laminated composite and sandwich plates under cylindrical bending. The theory considered the effects of transverse shear and normal deformations. To account for the effect of transverse shear deformation, in‐plane displacement uses polynomial shape function expanded up to fifth‐order in‐terms of the thickness coordinate. Transverse displacement uses derivative of shape function to account for the effect of transverse normal deformations. Therefore, the present theory involves six independent unknown variables. The theory satisfies traction free boundary conditions at top and bottom surfaces of the plate and does not require the shear correction factor. The principle of virtual work is used to obtain the variationally consistent governing differential equations and associated boundary conditions. Analytical solutions for simply supported boundary conditions are obtained using Navier’s solution technique. Non‐dimensional displacements and stresses obtained using the present theory are compared with existing exact elasticity solutions and lower and higher‐order theories to prove the efficacy of the present theory. The comparison shows that the displacements and stresses predicted by the present theory are in good agreement with those obtained by using the exact solution.


INTRODUCTION
The demand for high-strength, high-modulus and low density composite materials have generated an increased number of applications in many industries such as in aircraft, spacecraft, civil engineering, mechanical engineering, marine and many more.
The development of plate theory has a long history.Many well-known engineers, scientists, and mathematicians have made their contribution in the development of beam, plate and shell theories such as Jacob II Bernoulli, Leonard Euler, Joseph-Louis Lagrange Simeon Denis Poisson, Claude-Louis Navier and Gustav Robert of various higher-order plate theories and the solution techniques are recently reviewed by Sayyad and Ghugal 2015a .
Reddy 1984 has developed a simple higher-order shear deformation theory HSDT for laminated composite beams and plates.This HSDT is further used by many researchers for the solution of various solid mechanics problems.Kant and Kommineni 1994 have established a refined higher-order shear deformation theory for linear and geometrically non-linear behavior of fiber reinforced angle ply laminated composite and sandwich plates based on finite element formulation using a Lagrangian approach.Soldatos andWatson 1997 andShu andSoldatos 2000 developed the hyperbolic shear deformation theory for the cylindrical bending of cross-ply and angle-ply laminates.
Chakrabarti and Sheikh 2005 have developed a finite element model for the bending analysis of soft core sandwich plates.A study of global-local higher-order theories for laminated composite plates is performed by Zhen and Wanji 2007 by presenting the general formulas of n th order global local higher-order theory.Fares and Elmarghany 2008 have presented a refined zig-zag nonlinear FSDT of laminated composite plates using the Galerkin method.Ferreira et al. 2011 applied the Carrera's unified formulation CUF for predicting the free vibration, static deformation and buckling behavior of thin and thick cross-ply laminated plates.Carrera and Zappino 2016 proposed several models based on 1D, 2D and 3D kinematics for free vibrations of shell structures using Lagrange polynomials.Pagani et al. 2016 have developed refined computational model based on layer-wise approach using CUF for the analysis of laminated structures.Sarangan and Singh 2016 have presented higherorder closed form solutions for the static, buckling and free vibration analysis of laminated composite and sandwich plates based on new shear deformation theories using Navier's closed form solution technique.Kant and Shiyekar 2008 obtained Navier type closed form solutions for the cylindrical bending of piezoelectric laminates subjected to electro-mechanical loading using higher-order shear and normal deformation theory.Sayyad and Ghugal 2015b applied a n th order shear deformation theory for the cylindrical bending of composite laminates.Ghugal and Sayyad 2011 presented trigonometric shear and normal deformation theory for the free vibration of thick isotropic square and rectangular plate which was further extended by Sayyad and Ghugal 2016 for the cylindrical bending of multilayered composite laminates and sandwiches.A critical review of literature on bending, buckling and free vibration analysis of shear deformable isotropic, laminated composite and sandwich beams based on equivalent single layer theories, layerwise theories, zig-zag theories and exact elasticity solution has recently been presented by Sayyad andGhugal 2017a . Sayyad andGhugal 2017b have also developed a displacement based unified shear deformation theory for the analysis of shear deformable advanced composite beams and plates.

The plate under consideration for the present study
A cross-ply laminated composite plate made of orthotropic fibrous composite material having length 'a' and width 'b' in the in the x and y directions respectively is considered as shown in Figure 1.The y direction of the plate is assumed to be infinitely long compared to other two dimensions, therefore, strains in the y direction are assumed to be zero The thickness of the plate is measured in z-direction and at z 0, the mid plane of the plate is located.The plate under consideration consists of N number of layers bonded together.The plate is carrying an out of plane load q x , acting on its top surface.i.e.   3 The transverse displacement w in zdirection is assumed to be a function of x and z coordinates to include the effect of transverse normal deformations   4 The theory enforces the parabolic variation of the transverse shear stress across the thickness of the plate.Thus, the theory obviates the need of the shear correction factor.5 The body forces are not considered in the analysis.

Kinematics of the present theory
Based on the aforementioned assumptions and features, the displacement field of the present theory FOSNPT can be expressed as where u and w are the x and z-directional displacements of any point on the plate, u 0 and w 0 are the in-plane displacements of mid-plane in x and z-directions respectively; and x x   are rotations of the normal to the middle plane about y axis which account the effect of transverse shear deformation.
and z z   represent higher-order transverse cross-sectional deformation modes i.e. effect of transverse normal deformations.The non-zero strain components associated with the present displacement field are obtained by using the linear theory of elasticity.

Constitutive Equations
The constitutive equations for the k th lamina are given by 11 13 13 33 55 0 0 0 0 where ij Q are the reduced elastic constants in x-z plane, x  is the normal stress along x-direction, z  is the stress acting along z-direction and xz  is shear stress along z-direction.The following relationship between the reduced elastic constants and the engineering elastic constants are used.

Governing Equations and Boundary Conditions
Variationally consistent governing differential equations and associated boundary conditions are derived by using the principle of virtual work.For the plate under consideration, the principle of virtual work takes the following form.
where is the virtual displacement i.e. infinitesimal change in the position coordinates of the points under consideration.q x represents transverse load acting on the top surface of the plate.By substituting virtual strain from Eq. 5 into the Eq. 8 one can obtain The governing equations can be derived from Eq. 9 by integrating the displacement variables by parts and setting the coefficients of 0 0 , , , , and       to zero separately, and the following equations can be obtained: where the extension, bending, bending-extension, bending-twisting stiffnesses used in the equations 12 -17 can be obtained as The boundary conditions along edges x 0, x a are of the following form: 1 2 0 0 0 1 2 0 or 0; 0 or / 0; / 0 or 0; 0 or 0 0 or 0; 0 or 0; 0 or 0 For a simply supported laminated composite plate, the kinematic boundary conditions are given below: To determine the unknown displacement variables, the Navier's solution technique is implemented.To satisfy the aforementioned boundary conditions the displacements and rotations are assumed in Fourier trigonometric form where , , , , and u w     are the unknowns to be determined.According to Navier's solution scheme, transverse load is also expanded in Fourier trigonometric form where m q is the coefficient of Fourier series expansion and m is the positive integer.For sinusoidal load, 0 m q q  and m 1.
After knowing the values of unknown displacement variables 0 0 , , , , and Eq. 23 , one can obtain all the displacements and stress components within the laminated composite plate using equations 4 through 6 .

Estimation of transverse shear stress and normal stress
Through-thickness distributions of transverse shear and normal stresses for composite laminates are important for delamination type failure.The evaluation of transverse shear stresses from the constitutive relations leads to discontinuity at the inter face of two adjacent layers of a laminate and thus violates the equilibrium conditions.Thus, elasticity equilibrium equation neglecting the body force is used to derive expression for the transverse stress in the k th lamina of composite laminate.
From equation 25 the transverse stress xz  can be evaluated through integration with respect to the laminate thickness coordinate z .The in-plane stress x  obtained by using equation 4 is substituted in equation 25 .The constants of integrations C can be determined by substituting the boundary conditions.It is expected that this procedure will produce an accurate transverse shear stresses.

NUMERICAL RESULTS AND DISCUSSION
Aluminum alloy and fibrous composite materials are being used increasingly for numerous space applications.3.1 Aluminum alloy: Aluminum is one of the most widely used metals in modern aircraft construction.It is vital to the aviation industry because of its high strength to weight ratio and its comparative ease of fabrication.The outstanding characteristic of aluminum is its light weight.Aluminum melts at the comparatively low temperature of 1250 0 F. It is nonmagnetic and is an excellent conductor.Following material properties Aluminum 3003-H14 are used for numerical study.
Material 1 Krishna Murty, 1984 : 2 Fibrous composite materials: Engineers are interested in these materials because of their favorable mechanical characteristic of high strength/high stiffness to weight ratio and potential for zero or near-zero coefficient of thermal expansion.The use of high modulus Graphite-Epoxy composite parts for space applications is already well established.Using Graphite-Epoxy parts for space vehicles and structures has many advantages including: 1 Critical weight savings 2 Improved control of thermal distortions 3 Increased structural stiffness.Following properties of Graphite-Epoxy composite material are used for the numerical study.

 
The through-the-thickness profiles for in-plane displacement u , in-plane normal stress x  and transverse shear stress xz  for laminated and sandwich plates subjected to a sinusoidal load are plotted in Figures 3 through

14.
High-strength aluminum alloy is an important airframe material since 1920s.Therefore, the present theory is tested for the plate made of aluminum alloy material 1 .Comparison of non-dimensional displacements and stresses of aluminum alloy plate subjected to sinusoidal load are tabulated in Table 1.For the comparison purpose, numerical results by using HSDT of Reddy 1984, FSDT of Mindlin 1951 and CPT are obtained.The numerical results are presented for thick a/h 4 , moderately thick a/h 10 and thin plates a/h 100 .From Table 1, it is pointed out that numerical results obtained by using the present theory and HSDT of Reddy 1984 are in excellent agreement with each other whereas FSDT and CPT underestimate the displacements and stresses due to neglect of shear and normal deformations.
The comparison of non-dimensional displacements and stresses for the two-layer 0 0 /90 0 laminated composite plates is shown in Table 2.The plate is subjected to a sinusoidal load Figure 2a and made up of orthotropic material 2. Both the layers are of equal thickness i.e. h/2.Through-the-thickness distributions of inplane displacement and stresses are plotted in Figures 3-5 and variation of transverse displacement with respect to a/h ratio is plotted in Figure 6.Exact elasticity solutions presented by Pagano 1969 are taken as basis for the comparison of numerical results obtained by using the present theory FOSNDT , HSDT of Reddy 1984, SSNPT of Sayyad and Ghugal 2016, FSDT of Mindlin 1951 and CPT.HSDT, FSDT and CPT do not consider the effect of transverse normal deformation ε z 0 whereas the present theory and SSNPT considers the effect of transverse normal deformation ε z 0 .It can be observed from Table 2 that the present theory shows considerable improvement in the in-plane displacement and stresses compared to those obtained by using HSDT and SSNPT.The percentage error predicted using the present theory is less in many cases as compared to HSDT, SSNPT, FSDT and CPT.This is in fact due to inclusion of fifth order term in-terms of the thickness coordinate in the displacement field.Figures 4 and 5 shows stresses are always maximum in 0 0 layer and minimum in 90 0 layers.The transverse shear stress which is an important indicator to the onset of delamination are obtained using equations of equilibrium to ascertain the continuity at the layer interface.Through-the-thickness distribution of transverse displacement is not uniform when itis obtained using the present theory and SSNPT whereas it is uniform when obtained by using HSDT, FSDT and CPT.
Table 3 compares numerical values of non-dimensional displacements and stresses obtained by using the present theory and other higher-order theories for three-layer 0 0 /90 0 /0 0 symmetric laminated composite plate subjected to a sinusoidal load see Figure 2b .The plate is made of material 2 and overall thickness is equally distributed among all the layers i.e. h/3.The examination of Table 3 reveals that present results are in excellent agreement with those obtained by using the exact elasticity solution of Pagano 1969 .In this problem also considerable improvement in the results is observed due to refinement of the polynomial shape function.Large percentage error is observed when these quantities are obtained by using FSDT and CPT due to neglect of shear and normal deformations.Through-the-thickness distributions of in-plane displacement and stresses are plotted in Figures 7-9.Variation of transverse displacement with respect to a/h ratios is plotted in Figure 10.Table 4 compares the numerical values of non-dimensional displacement and stresses of three-layer 0 0 /core/90 0 symmetric sandwich plate subjected to a sinusoidal load see Figure 2c .Thickness of top and bottom face sheets is 0.1h each whereas thickness of middle soft core is 0.8h.Face sheets of the plate are made of a fibrous composite material 3 whereas the core is made of material 4. For the sandwich plates in cylindrical bending, the exact elasticity solution is not available in the literature; hence present results are compared with published results.Present results are in good agreement with the HSDT of Reddy 1984 andSSNPT of Sayyad andGhugal 2016 .Figure 10 shows variation of transverse displacement with respect to aspect ratio for the three-layer 0 0 /90 0 /0 0 symmetric laminated composite plate subjected to sinusoidal load.Figures 11-13 plots the throughthe-thickness distributions of in-plane displacement, in-plane normal stress and transverse shear stress.The examination of Figure 12 reveals that the in-plane normal stress developed in the middle core is very small compared to that in top and bottom face sheets.This is in fact due to core material is soft compared to material of face sheets.The transverse shear stress is obtained using equations of equilibrium of the theory of elasticity to ascertain the stress continuity at the layer interface.Variation of transverse displacement with respect to a/h ratios is plotted in Figure 14.  Figure 6: Variation of transverse displacement w with respect to aspect ratio for two-layer 0 0 /90 0 antisymmetric laminated composite plate subjected to sinusoidal load.
Figure 7: Through thickness variation of in-plane displacement u at x 0, z for three-layer 0 0 /90 0 /0 0 symmetric laminated composite plate subjected to sinusoidal load.a/h 4  A new fifth-order shear and normal deformation theory for the cylindrical bending of laminated composite and sandwich plates have been developed in this paper.To account for the effect of transverse shear deformation, in-plane displacement uses polynomial shape function expanded up to fifth-order in-terms of the thickness coordinate.The present theory involves six-degrees-of-freedom.The theory satisfies traction free boundary conditions at top and bottom surfaces of the plate and does not required the shear correction factor.For simplicity, this theory is applied for the analysis of laminated composite and sandwich plates deformed in cylindrical bending.Non-dimensional displacements and stresses obtained using the present theory are compared with existing exact elasticity solutions and lower and higher-order theories.From the comparison of numerical results, it is concluded that the present theory is in good agreement with exact elasticity solution of Pagano and shows considerable improvement in the numerical results obtained by using higher-order shear deformation theory of Reddy.This validate that the effect of transverse shear and normal deformations both plays important role in the analysis of laminated composite structures.

Figure 1 :2
Figure 1: Geometry and co-ordinate system of the layered plate deform in cylindrical bending

Figure 2 :
Figure 2: Simply supported laminated plates subjected to sinusoidal load

Figure 8 :Figure 9 :Figure 10 :Figure 12 :Figure 13 :Figure 14 :
Figure8: Through thickness variation of in-plane normal stress x  at x a/2, z for three-layer 0 0 /90 0 /0 0 symmetric laminated composite plate subjected to sinusoidal load a/h 4  Substitution of Eqs.21 and 22 into governing equations 12 through 17 leads to the following form

Table 1 .
Comparison of In-Plane Displacement, Transverse Displacement, In-Plane Normal Stress and Transverse Shear Stress for Aluminum Alloy Plate Subjected to Sinusoidal Load under Cylindrical Bending