Ocean Engineering 159 (2018) 237–252
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Numerical and experimental validation of nonlinear deflection and stress responses of pre-damaged glass-fibre reinforced composite structure Chetan Kumar Hirwani, Subrata Kumar Panda * Department of Mechanical Engineering, National Institute of Technology, Rourkela, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Nonlinear bending Delaminated composite panel Experimental validation Mechanical load HOST
In the present article, nonlinear static deflections of internally damaged shear deformable laminated composite curved (single and doubly) shell panel are investigated numerically under the quasi-static loading and validated with experimental results. For the numerical analysis, a general nonlinear mathematical model of the laminated composite curved shell panel including the effect of the internal damage is derived with the help of two higherorder kinematic theories and Green-Lagrange nonlinear strain. The desired governing equation is obtained by minimizing the energy expression and solved via nonlinear finite element steps. The numerical responses are computed via a generic MATLAB code developed based on the present mathematical formulation. The degree of accuracy of the current numerical model has been checked and the subsequent validation is established by comparing the present results with available published results. In addition, an experimental investigation has also been carried out for the comparison purpose via three-point bend test on the laminated Glass/Epoxy composite with artificial delamination. Lastly, numbers of numerical examples are solved to demonstrate the implicit behavior of the currently developed higher-order nonlinear model for the analysis of the pre-damaged layered structure.
1. Introduction Since last few decades, the application of layered composites material has been increased rapidly in the manufacturing the weight sensitive and high-performance structures in civil construction, marine, aircraft, aerospace and automobile industries due to their excellent property and ability to be tailored for particular applications (Gay and Hoa, 2007). Most commonly the laminated composite structure does not have any reinforcement in the thickness direction which leads to the high inter-laminar shear stress and subsequent separation of the consecutive layers. The separation of the layer generally termed to be delamination or debonding may arise due to the several reasons and in several ways. Some of the common reasons are entrapment of air pockets or foreign particles, incomplete curing, material and geometric discontinuity during the manufacturing and eccentric and low-velocity impact loading during service life (Sridharan, 2008). The presence of internal debonding could affect the structural integrity and the structural performance significantly. Hence, the thorough understanding and quantitative
measurement of the influence of the debonding on the stiffness and structural responses of the laminated structure is very much essential. In order to identify the necessary gaps of the earlier studies on the delaminated structure, a comprehensive review of the previously published literature is discussed in the following lines and subsequent objective of the present article also framed accordingly. The present review is mainly focused on the two major issues i.e., the solution techniques adopted for the desired structural responses and the mid-plane kinematics used for the mathematical modeling purpose of the laminated/delaminated structure. In the early nineties, the transverse deflection behavior of laminated composite curved shell panels (single and doubly) are investigated numerically by Reddy and Chandrashekhara (1985) using finite element (FE) model based on the extended Sander's shell theory including the von-Karman type of geometrical nonlinear strain-displacement relations. Similarly, FE steps are further utilized to examine the nonlinear bending and the stress behavior of the laminated composites plate by Barbero and Reddy (1990) using the generalized laminated plate theory (GLPT) in
* Corresponding author. Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela 769008, Sundergarh, Odisha, India. E-mail addresses:
[email protected],
[email protected] (S.K. Panda). https://doi.org/10.1016/j.oceaneng.2018.04.035 Received 12 September 2017; Received in revised form 8 March 2018; Accepted 10 April 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.
C.K. Hirwani, S.K. Panda
Ocean Engineering 159 (2018) 237–252
delamination are modeled using the sub-laminate approach by Manoach et al. (2016). The progressive failure analysis of the damaged layered composite structure with and without stiffener is presented by Riccio et al. (2016, 2017a, 2017b). The system of exact kinematic condition (SEKC) steps have been employed to evaluate the dynamic characteristics by Szekrenyes (2015) to study the effect of delamination through the width of the plate to main the kinematic continuity between the laminated and debonded composite. Further, the updated version of SEKC approach is introduced by Szekrenyes (2016) to examine the mechanical responses of delaminated composite plates using different mid-plane kinematic (first, second and third-order) plate theories. Similarly, the effect of the location and the length of the delamination are investigated by using new artificial immune system method (Bazardehi et al., 2012; Mohebbi et al., 2013). The extensive review of the literature reveals that the various work has been attempted and reported on the linear/nonlinear flexural strength of the laminated composite curve and flat panel is mainly using von-Karman type of geometrical nonlinear strains in the framework of FOST and HOST (higher-order shear deformation theory) mid-plane kinematics. In addition, the damaged laminated composite panel model also focused on the edge debonding rather than internal debonding. However, both types of debonding phenomena in the laminate structures are equally important and cannot be avoided. To the best of the authors' knowledge, no FE modeling has been discussed or reported yet in the open literature which accounts the large deformation in Green-Lagrange type of nonlinear strain kinematics in association with HOST mid-plane theory considering the effect of internal damage using the sub-laminate concept. Hence, in the present work authors' aim to develop a generic type of FE model for the laminated curved shell panel with internal damage in the framework of two HOST kinematic model and GreenLagrange nonlinear strain to evaluate the nonlinear transverse deflection under the quasi-static type of mechanical load. The necessary consistency behavior and the validity of the said higher-order models have been established by solving an adequate number of examples. Further, to show the inevitability of the proposed HOST kinematics the results are compared with experimental results of laminated Glass/Epoxy composite plate with and without damage. Lastly, the importance and effect of the geometrical and material parameters along with various size, position and location of delamination on the nonlinear transverse deflection under two different mechanical load are computed and conferred in details.
conjunction with von-Karman nonlinear strain. The FE solutions of the nonlinear static and the transient responses of the laminated composite plate are reported by Chang and Huang (1991) based on the higher-order mid-plane theory and von-Karman geometrical nonlinearity. Also, the nonlinear FE steps are employed to compute the bending strength of the doubly curved laminated composite as well as the sandwich shell panels by Kant and Kommineni (1992, 1994) using the higher-order kinematic theory and von-Karman strains. The nonlinear (static and dynamic) FE solutions of the laminated composites plate are reported by Ganapathi et al. (1996) for the laminated structure using an eight noded quadrilateral element and von-Karman strain-displacement relations. Further, a new plate/shell triangular element is developed by Zhang and Kim (2005) to compute the geometrical nonlinear deflection parameters of the laminated plate based on the first-order mid-plane kinematics. Later, the first-order shear deformation theory (FOST) type of mid-plane kinematics along with von-Karman geometrical nonlinear strains are utilized by Baltacıoglu et al. (2010) for the computation of the flexural strength of the orthotropic laminated structure via numerical methods. The effect of combined pressure and thermal loading on the nonlinear flexural behavior of the laminated composite plate resting on an elastic foundation is analysed by Shen (2000). The closed-form solution of the fundamental frequency and flexural behavior of the sandwich shell panel is presented first time by Biglari and Jafari (2010) with the help of newly developed three-layered higher-order mixed theory. Gupta et al. (2013) reported the FE solutions of the nonlinear flexural responses of the laminated composite plate with growing damage via the FOST kinematics and von-Karman nonlinear strains. Likewise, the static, buckling, flutter and dynamic responses of the functionally graded material (FGM) plate structure using the FE associated non-uniform rational B-spline method by Valizadeh et al. (2013) in the framework of the FOST kinematics. In contrast to the finite element method (FEM), Galerkin Kriging in association with the meshfree approach also employed to compute the buckling and the flexural characteristic of the laminated composite plate by Bui and Nguyen (2013). The nonlinear FE solutions of the large deflection transverse bending characteristics of the FGM sandwich plate is computed using the refined plate theory and von-Karman nonlinearity by Kaci et al. (2013). Further, the four variable refined plate theory is developed first time to investigate the static responses of the FGM plate by Benatta et al. (2014) including the geometrical nonlinearity under the pressure loading. Similarly, the FE technique has been implemented to examine the modal responses of thick and thin laminated structure by Tornabene et al. (2017) in the framework of the equivalent layerwise theory. Unlike the other available techniques, the explicit Dynamic Relaxation method has been employed by Alamatian and Rezaeepazhand. (2016) for the analysis of the nonlinear static characteristic of the laminated composite plate with variable cross-sections. Malta and Martin (Malta et al., 2017) investigated numerically the effect of compressive load on the five-layer flexible pipe using 3D nonlinear FE model. Now, the articles reviewed in this section for the structure or the structural components with internal damage and solved using various numerical/analytical technique including the large (finite) deformation effect. The influence of the generation and the propagation of the transverse crack on the stiffness behavior of the angle-ply laminated composites plate is reported by Amara et al. (2006). Further, the delaminated composite beam is analysed numerically by Hicks et al. (2003) using the energy method in association with the FE approach and compared with subsequent experimental results. Similarly, the effect of delamination on the laminated tube and bar structures are computed using the simulation model by Carneiro and Savi (Vieira and Savi, 2000). Wang and Shenoi (2003) proposed a new modeling approach based on the linear curved beam theory including the fracture mechanics for the analysis of the delaminated composite curved beam. Likewise, the dynamic characteristics of the laminated composite structure with
2. Theory and formulation 2.1. Geometry of the proposed shell panel The physical model of the doubly curved laminated composite shell panel (Fig. 1) assumed to be consist of ‘N’ number of orthotropic layers of
Fig. 1. Geometry of the curved panel.
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Ocean Engineering 159 (2018) 237–252
9 ζ > uðξ1 ; ξ2 ; ζÞ ¼ u0 1 þ þ ζϕ1 þ ζ2 ψ 1 þ ζ3 θ1 > > > > R1 > = HOST 2: ζ 2 3 vðξ1 ; ξ2 ; ζÞ ¼ v0 1 þ þ ζϕ2 þ ζ ψ 2 þ ζ θ2 > > > R2 > > > ; wðξ1 ; ξ2 ; ζÞ ¼ w0 þ ζϕ3
(2)
where, u; v and w are the displacements of any point along the curvilinear coordinate directions i.e., ξ1 ; ξ2 and ζ respectively. The displacement functions say, u0 ; v0 and w0 are the displacements defined at the mid-plane of the panel structure including the rotations of normals, ϕ1 and ϕ2 about the corresponding ξ2 and ξ1 axes, respectively. The remaining mathematical functions in the displacement fields say, ϕ3 ; ψ 1 ; ψ 2 ; θ1 and θ2 are the necessary higher-order terms of Taylor's series expansion defined at the mid-plane to maintain the parabolic variation of the shear stresses. However, the present formulation deals with the shallow shell panel which holds the relation R1ξ ¼ R1ξ ≪1 . So, the terms 1 þ Rξξ and 1 2 1 1 þ Rξξ can be equated to unity in equations (1) and (2) and modified 2
form can be given as:
Fig. 2. Laminate with delamination.
uniform thickness and converted to the necessary mathematical form for the analysis purpose. The geometrical parameters of the shell panel geometries are length a, width b, thickness h in ξ1 ; ξ2 and ζ directions, respectively utilized for defining the shell geometry including the principal radii of curvature, Rξ1 and Rξ2 along the ξ1 and ξ2 directions, respectively at the mid-plane (ζ ¼ 0) of the shell and twist radius of curvature Rξ1 ξ2 is assumed to be infinite. The global coordinate system ξ1 ; ξ2 ; ζ is located at the mid-plane of the laminated shell panel with ζ in the vertical direction. The different configuration of the panel geometries can be achieved by choosing the curvature suitably say, cylindrical shell ðRξ1 ¼ R; Rξ2 ¼ ∞Þ, spherical ðRξ1 ¼ R; Rξ2 ¼ RÞ, elliptical ðRξ1 ¼ R; Rξ2 ¼ 2RÞ, hyperboloid ðRξ1 ¼ R; Rξ2 ¼ RÞ and plate ðRξ1 ¼ ∞; Rξ2 ¼ ∞Þ.
(3)
9 uðξ1 ; ξ2 ; ζÞ ¼ u0 þ ζϕ1 þ ζ2 ψ 1 þ ζ3 θ1 = 2 3 vðξ1 ; ξ2 ; ζÞ ¼ v0 þ ζϕ2 þ ζ ψ 2 þ ζ θ2 ; wðξ1 ; ξ2 ; ζÞ ¼ w0 þ ζϕ3
(4)
Further, the delaminated shell panel has been modeled mathematically using the kinematic model as in Ju et al. (1995) with few modifications to maintain the same variables as well as the necessary continuity in equations (3) and (4). Similar to the laminated segment, the mid-plane displacement functions of the delaminated segment (already having “p” number of delamination as in Fig. 2) is defined another coordinates O; ξ0 1 ; ξ0 2 and ζ 0 can be expressed as:
2.2. Deformation kinematics
9 2 3 u0 ðξ0 1 ; ξ0 2 ; ζ0 Þ ¼ u0 0 þ ζ 0 ϕ0 1 þ ζ0 ψ 0 1 þ ζ0 θ0 1 = 2 3 v0 ðξ0 1 ; ξ0 2 ; ζ 0 Þ ¼ v0 0 þ ζ0 ϕ0 2 þ ζ 0 ψ 0 2 þ ζ0 θ0 2 ; w0 ðξ0 1 ; ξ0 2 ; ζ0 Þ ¼ w0 0
(5)
9 2 3 u0 ðξ0 1 ; ξ0 2 ; ζ0 Þ ¼ u0 0 þ ζ 0 ϕ0 1 þ ζ0 ψ 0 1 þ ζ0 θ0 1 = 2 3 v0 ðξ0 1 ; ξ0 2 ; ζ 0 Þ ¼ v0 0 þ ζ0 ϕ0 2 þ ζ 0 ψ 0 2 þ ζ0 θ0 2 ; w0 ðξ0 1 ; ξ0 2 ; ζ0 Þ ¼ w0 0 þ ζ0 ϕ0 3
(6)
Here, the displacement functions are also replicate the same values as in the laminate case i.e., u0 ; v0 and w0 are the displacements of a general point in the direction of ξ0 1 ; ξ0 2 andζ0 respectively including u0 0 ; v0 0 and w0 0 are the displacements of a point on the mid-plane and ϕ0 1 and ϕ0 2 are the rotations of the normal to the mid-plane about ξ0 2 and ξ0 1 axis, respectively. The parameters ϕ0 3 ; ψ 0 1 ; ψ 0 2 ; θ0 1 and θ0 2 are higher order terms of Taylor series expansion defined at the mid-plane.
For the modeling purpose, two well established higher-order kinematics are utilized in this investigation to display the state space variables of any point within the material continuum of the laminated composite curved shell panel and termed as HOST-1 and HOST-2. The displacement kinematics of the HOST-1 and HOST-2 is considered to be nine and ten degrees of freedom (DOF), respectively. Now, the HOST-1 and HOST-2 are expressed as same as in (Reddy and Liu, 1985): 9 ζ > uðξ1 ; ξ2 ; ζÞ ¼ u0 1 þ þ ζϕ1 þ ζ2 ψ 1 þ ζ3 θ1 > > > > R1 > = HOST 1: ζ 2 3 vðξ1 ; ξ2 ; ζÞ ¼ v0 1 þ þ ζϕ2 þ ζ ψ 2 þ ζ θ2 > > > R2 > > > ; wðξ1 ; ξ2 ; ζÞ ¼ w0
9 uðξ1 ; ξ2 ; ζÞ ¼ u0 þ ζϕ1 þ ζ2 ψ 1 þ ζ3 θ1 = 2 3 vðξ1 ; ξ2 ; ζÞ ¼ v0 þ ζϕ2 þ ζ ψ 2 þ ζ θ2 ; wðξ1 ; ξ2 ; ζÞ ¼ w0
2.3. Strain-displacement relation The generalized nonlinear deformation of the laminated structure with and without internal damage is modeled using the following Green's strain in Lagrangian space as in Reddy (2004):
(1)
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εij ¼
8 εξ1 ξ1 > > > > ε > < ξ2 ξ2
εζζ
εξ2 ζ > > > > ε > : ξ1 ζ
εξ1 ξ2
9 > > > > > =
Ocean Engineering 159 (2018) 237–252
8 > > > > > > > > > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > > > > > > > > > =
8 > > > > > > > > > > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > > > > > > > > > > =
" 2 2 # ∂u 2 ∂v ∂w þ þ þ ¼ ∂ζ ∂ζ ∂ζ > > > > > > > > > > ∂v ∂w v > > > > > > > > > > > > > > > þ ∂u w ∂u ∂v w ∂v ∂w v ∂w > > > ; > > > > > R ∂ ζ ∂ ξ 2 > > > > 2 þ þ þ þ > > > > > > > > R R R ∂ ξ ∂ ζ ∂ ξ ∂ ζ ∂ ξ ∂ ζ > > > > 12 2 2 2 2 2 > > > > > > > > ∂ u ∂ w u > > > > > > > > þ > > > > ∂ u w ∂ u ∂ v w ∂ v ∂ w u ∂ w > > > > > > > > ∂ζ ∂ξ1 R1 þ þ þ þ > > > > > > > > > > > > R R R ∂ ξ ∂ ζ ∂ ξ ∂ ζ ∂ ξ ∂ ζ > 1 12 1 1 1 1 > > > > > > > ∂ u ∂ v w > > > > > > > > : > ; > þ þ2 : ; ∂ u w ∂ u w ∂ v w ∂ v w ∂ w u ∂ w v R12 ∂ξ2 ∂ξ1 þ þ þ þ þ þ ∂ξ1 R1 ∂ξ2 R12 ∂ξ1 R12 ∂ξ2 R2 ∂ξ1 R1 ∂ξ2 R2
∂w ∂ζ
εij ¼ fεl g þ fεnl g
1 2
(8)
(7)
where, fσ ij g, ½Qij and fεij g are the stress tensor, the elastic coefficient matrix and the strain tensor, respectively (Jones, 1975).
where, fεl g and fεnl g are the linear and nonlinear strain tensors, respectively. Now, the individual mid-plane strains are obtained by substituting the corresponding displacement fields i.e., equations (3) and (4) into equation (7) and conceded as:
2.5. Energy calculation The total strain energy of the laminated shell panel due to the me-
8 1 9 8 2 9 8 3 9 kξ1 ξ1 > kξ1 ξ1 > kξ1 ξ1 > > > > > > > > > > > > > > > > 1 2 > > > > > > > > > > > k k kξ32 ξ2 > > > > > ξ2 ξ2 > ξ2 ξ2 > > > > > > < 0 = < 0 = < 0 > = ζζ 2 3 þ z k1 þz þ z εij ¼ 2 3 0 k k ε > > > > > > > > ξ ξ ξ ζ ζ ζ 2 2 2 ξ2 ζ > > > > > > > > > > > > > > > > > > > > 0 > > > > > > > > > kξ11 ζ > kξ21 ζ > kξ31 ζ > ε > > > > > > > > ζ ξ > > > > : 1 > ; ; : 2 > ; : 3 > ; : 01 > kξ1 ξ2 kξ1 ξ2 kξ1 ξ2 εξ1 ξ2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 8 0 9 εξ ξ > > > > 01 1 > > > εξ2 ξ2 > > > > > > = < ε0 >
εl
8 8 8 8 9 9 9 9 8 9 9 8 > > > > nl1 > nl2 > nl3 > nl4 > nl5 > > > > > > > > > > k k k k k > > > > > > > > > > > ξ1 ξ1 ξ1 ξ1 ξ1 ξ1 ξ1 ξ1 ξ ξ kξ1 ξ1 nl6 > > > > > > > > > > > > > 1 1 nl1 > > > > > > > > nl2 > nl3 > nl4 > nl5 > nl > > > > > > > > > > > > 6 kξ2 ξ2 kξ2 ξ2 kξ2 ξ2 > kξ2 ξ2 kξ2 ξ2 > > > > > > > > > > > kξ2 ξ2 > > > > > > > > > > > > < < < < < = = = = < nl2 nl3 nl4 = nl5 nl1 = 1 kζζ kζζ kζζ kζζ kζζ εζζ 21 31 41 51 61 0 þ z þ z þ z þ z þ z þ þz nl2 nl3 nl4 nl5 nl1 nl0 0 2k 2k k 2k 2k 2 ε > > > > > > > > > > > > > > 2 > ξ2 ζ > 2 > ξ2 ζ > 2 > ξ2 ζ > 2 > ξ2 ζ > 2 > ξ2 ζ > 2 > ξ2 ζ > 2> > > > > > > > > > > > > > > > > > > > > > > > 0 > > > > > > > > > > > > > > 2εxz nl0 > 2kξ1 ζ nl2 > 2kξ1 ζ nl3 > kξ1 ζ nl4 > 2kξ1 ζ nl5 > 2kxz nl1 > > > > > > > > > > > > > > > > > > > > > > > > > > > ; : nl 6 nl nl nl nl nl nl : 2ε 0 ; : 2k 1 ; > > > > 2 > 3 > 4 > 5 > 2k : : : : ; ; ; ; ξ ξ 2k 2k k 2k 1 2 xy ξ1 ξ2 ξ1 ξ2 ξ1 ξ2 ξ1 ξ2 xy |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 8 > εξ ξ nl0 > > > 1 1 nl0 > > > εξ2 ξ2nl 0 1
> > > > > > =
εnl
(9)
εnl
or fεg ¼ fεl g þ fεnl g ¼ ½Hl fεl g þ ½Hnl fεnl g. Here, ½Hl and ½Hnl are the thickness coefficient matrices associated with the linear and nonlinear mid-plane strains. The details of the thickness coefficient matrices can be seen in the references (Singh and Panda, 2014; Mahapatra and Panda, 2015). In this study, the strain-displacement relations are provided only for the HOST-2 cases for the sake of brevity. It is noteworthy to mention that the expressions could be formed for the HOST-1 model by dropping the necessary terms from the displacement and strain-displacement relations.
chanical loading is computed using the necessary stress and strain tensors and reported as: 1 U¼ ∬ 2
(
N X
) T ζ ∫ ζkk1 εij σ ij dζ dξ1 dξ2
(11)
k¼1
2.6. Work done The total work done due to the applied transverse mechanical load (F) is expressed in the following form:
2.4. Laminate constitutive relations W ¼ ∫ fδgT fFgdA
k
σ ij
k ¼ Qij εij
(12)
A
The generalized stress-strain relations for any material continuum which kth layer is oriented at an arbitrary angle 'θ' about any arbitrary axes are given by:
2.7. Finite element formulation The most of the numerical analysis of laminated structure is investigated using the FEM due to the easy applicability to the problems asso-
(10)
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Ocean Engineering 159 (2018) 237–252
ciated with the material and geometrical complexities. Hence, a suitable isoparametric FEM step is implemented for the modeling of the current shell structure including delamination for the discretisation purpose. In this study, a nine noded quadrilateral Lagrangian element with nine and ten degrees of freedom per node is considered according to the type of model i.e., the HOST-1 and the HOST-2, respectively. The details regarding the nodal polynomial functions of the present isoparametric element can be seen from the source (Cook et al., 2000). Now, the displacement filed vector over each of the element can be expressed as follows: fδg ¼
9 X
½Ni fδi g
Fig. 3. Element at connecting boundary.
visualize that there are three nodes in the sharing boundaries say, 1, 4 and 8. For the common nodes 1, 4 and 8, the displacement field variable are represented as
(13)
i¼1
where, ½Ni and fδi g are nodal interpolation functions and the nodal displacement vectors, respectively. The nodal displacement vector for the proposed models, i.e., the HOST-1 and the HOST-2 are represented as
fδi g ¼ fu0i v0i w0i ϕ1i ϕ2i ϕ3i ψ 1i ψ 2i θ1i θ2i gT
fδi g ¼ fu0i v0i w0i ϕ1i ϕ2i ψ 1i ψ 2i θ1i θ2i gT and fδi g ¼ fu0i v0i w0i
fδ' i g ¼ fu0 0i v0 0i w0 0i ϕ0 1i ϕ0 2i ϕ0 3i ψ 0 1i ψ 0 2i θ0 1i θ0 2i gT
ϕ1i ϕ2i ϕ3i ψ 1i ψ 2i θ1i θ2i g , respectively. After introducing the FE steps, the linear and the nonlinear mid-plane strain vectors can be rewritten in terms of nodal displacement vectors and conceded to the following form: T
fεl g ¼ ½Bfδi g ; fεnl g ¼ ½A½Gfδi g
u'0i ¼ u0i þ eϕ1i þ e2 ψ 1i þ e3 θ1i ; v'0i ¼ v0i þ eϕ2i þ e2 ψ 2i þ e3 θ2i w'0i ¼ w0i ; ϕ'1i ¼ ϕ1i ; ϕ'2i ¼ ϕ2i ; ϕ'3i ¼ ϕ3i ; ψ '1i ¼ ψ 1i ; ψ '2i ¼ ψ 2i ; θ'1i ¼ θ1i ; θ'2i ¼ θ2i
(14)
n X
" #" # !
ζ
' δi ¼ ½λfδi g
∫ ζkk1 ½BT Dl
(19)
for i ¼ 1; 4 and 8
(20)
where, ½λ is the coefficient matrix, the detail components of this matrix for both the models can be seen in Hirwani et al. (2016). Now, using equations (3) and (4), the following transformations of the element E-2 is provided in the following lines:
k¼1
n 'o ' δ ¼ ½f δ
(15) ½k ¼ ½kl þ ½knl1 þ ½knl2 þ ½knl3 |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} kl k nlk k ζ ζ and ½Dl ¼ ∫ ζkk1 ½Hl T Q ½Hl ; ½Dnl1 ¼ ∫ ζkk1 ½Hl T Q ½Hnl ; k k ζ ζ ½Dnl2 ¼ ∫ ζkk1 ½Hnl T Q ½Hl ; ½Dnl3 ¼ ∫ ζkk1 ½Hnl T Q ½Hnl ;
(18)
where, ‘e’ denotes the distance between the mid-plane of the laminate (E1) and the delaminated (E-2) elements. For the presently proposed conditions, i.e., equation (19) helps to maintain the continuity throughout and allow the models to the detention of the global mechanical responses. However, the present formulation does not consider any local effect like the perturbation of the stress field around the point of initiation of the deboning. But the consideration of the system of exact kinematic conditions (SEKC) describes as in Szekrenyes (2013) account for the local effect for the similar type of analysis. Now, equation (19) is rewritten in the matrix form and expressed as:
" #" # n X 1 ζ ∫ ζkk1 ½BT Dnl1 A B dz dA þ ∫ 2 A k¼1 A k¼1 " # ! " #" # ! n X 1 ζ ∫ ζkk1 ½GT ½AT Dnl2 B dz dA þ ∫ G dz dA þ ∫ 2A A k¼1 " #" # ! n X ζ ∫ ζkk1 ½GT ½AT Dnl3 A ½Gdz dA
k ¼∫
for E 2
(17)
Now, the continuity conditions for the displacement at the sharing boundaries are expressed as:
where, [B] and [G] are the strain-displacement matrices of the linear and the nonlinear strain vectors, respectively. However, [A] is the displacements dependent matrix i.e., the linear displacement will be taken as the input to form the necessary matrix. The individual terms of all individual matrices i.e., [A], [B] and [G] can be seen in Singh and Panda (2014), Mahapatra and Panda (2015) for the corresponding kinematic models as said previously. The elemental stiffness ([k]) can be further expressed as in the following line: " #
for E 1
(21)
where, (16)
In order to maintain the brevity, the detailed derivation regarding the stiffness calculation of delaminated segment is not provided. However, the elemental stiffness ([k’]) matrix of the delaminated segment can easily be obtained by reiterating the steps from equations (7)–(16).
n 'o T δ ¼ δ1 δ'2 δ'3 δ4 δ'5 δ'6 δ'7 δ8 δ'9
(22)
½f ¼ Diag ½λ I I λ I I I λ I
(23)
where, the matrix [I] is (9 9) and (10 10) identity matrices for HOST1 and HOST-2, respectively. It is important to discuss that, [λ] will have the considerable value for the sharing node otherwise it is an identity matrix as mentioned in equation (23). Now, the transformations for the element stiffness and mass matrices of the debonded element (E-2) can be expressed as:
2.8. Conditions for the displacement continuity In the earlier sections, the element in the laminated and the delaminated segments are derived separately with the assumption that all the variables in the displacement kinematics are independent. However, at the sharing boundaries of both the segment, the displacement continuity condition for the displacement field must be satisfied. In order to understand the same in a more clear way, a panel with the elements in both the laminated and delaminated segment is presented in Fig. 3. For the easy readability, two elements are named as element E-1 and element E-2 in the laminate and delaminate segments, respectively. It is easy to
h 'i k ¼ ½f T k ' ½f
(24)
After the successful implementation of continuity conditions, the elemental stiffness matrix ½k' and nodal displacement vector fδ'i g will be '
replaced by ½k and fδi g for the node at sharing boundaries. Further, the '
modified elemental stiffness ½k will be utilized to frame the global stiffness matrices via assembly steps as in the FEM. 241
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2.9. System governing equations
3. Results and discussion
The necessary equilibrium equation for the static analysis of the laminated composites shell panel is obtained by minimizing the total potential energy expression and conceded as:
The nonlinear static deflections of debonded composite curved shell panel under the influence two types of quasi-static mechanical load are computed numerically with the help of presently developed HOST based nonlinear FE models. The rate of convergence of the proposed higherorder FE solutions have been checked for different mesh densities and compared with those available published results to verify the accuracy. Also, the transverse deflection responses are computed experimentally for the comparison purpose with the help of three-point-bend test using the in-house fabricated woven glass fiber reinforced epoxy composite plate including the artificial debonding at National Institute of Technology Rourkela (NIT Rourkela). For the numerical calculation, the elastic properties of the laminated composite are evaluated experimentally using uniaxial tensile test. In addition, different kind of numerical examples has been solved to examine the influences of the geometrical parameter, debonding size, location and position on nonlinear static deflection strength of the laminated curved panel. As mentioned previously, two types of mechanical loading have been considered for the present computational analysis namely, uniformly distributed load (UDL) and sinusoidal load (SDL) where UDL and SDL follow the relation q ¼ q0 and q ¼ q0 sin πaξ1 sin πbξ2 , respectively. The material properties used
∂
Y
¼ ∂ðU WÞ ¼ 0
(25)
Q where, is the potential energy functional expression due to the external transverse mechanical load. Now, the final form of the bending equilibrium equation of the laminated structure under the transverse mechanical load is expressed by substitution equations (11) and (12) into equation (25): ½Kfδg ¼ fFg
(26)
where, ½K and fFg are the global stiffness matrix and global force vectors, respectively. 2.10. End constraint conditions Now, equation (26) is solved by imposing the following end constraints at each side of the panel: Simply-supported case: v 0 ¼ w 0 ¼ ϕ2 ¼ ϕ3 ¼ ψ 2 ¼ θ 2 ¼ 0
at ξ1 ¼ 0 and a
u0 ¼ w0 ¼ ϕ1 ¼ ϕ3 ¼ ψ 1 ¼ θ1 ¼ 0
at ξ2 ¼ 0 and b
in the current numerical analysis including the experimentally evaluated properties are provided in Table 1. The present study utilized three different sizes of debonding and represented as a ratio of side length i.e., c/a, where ‘c’ and ‘a’ are the lengths of the delamination and parent panel along the longitudinal directions, respectively. In this analysis, three debonding sizes are utilized say, c/a ¼ 0.25, 0.5 and 0.75 including the
Clamped case: u0 ¼ v0 ¼ w0 ¼ ϕ1 ¼ ϕ2 ¼ ϕ3 ¼ ψ 1 ¼ ψ 2 ¼ θ1 ¼ θ2 at ξ1 ¼ 0 and a; at ξ2
Table 2 Comparison of linear and nonlinear central deflection of a simply-supported 8layer unidirectional (0 )8 square laminate subjected to UDL loading.
¼ 0 and b
Load (N)
Table 1 Material property.
Zhang and Kim, 2006 RDKQ-NL24
Zhang and Kim, 2006 RDKQ-NL20
Argyris and Tenek, 1994
HOST1
HOST2
0.0906 0.2718 0.4529
0.0826 0.2477 0.4128
0.091 0.272 0.454
0.0966 0.2897 0.4828
0.0829 0.2487 0.4144
0.0779 0.1708 0.2287
0.0712 0.1574 0.2118
0.084 0.177 0.236
0.0773 0.1611 0.195
0.0727 0.1695 0.2334
Linear
Material properties
Material set-1 [MAT (1)]
Material set-2 [MAT (2)] Experimental
Material set-3 [MAT (3)]
Eξ1 Eξ2 ¼ Eζ Gξ1 ξ2 Gξ2 ζ Gξ1 ζ υξ1 ξ2 ¼ υξ2 ζ ¼ υξ1 ζ
525,000 MPa 21,000 MPa 10,500 MPa 10,500 MPa 10,500 MPa 0.25
8.739 GPa 7.926 GPa 3.75 GPa 1.875 GPa 3.75 GPa 0.17
25 E2 1 GPa 0.5 E2 0.2 E2 0.5 E2 0.25
0.4 1.2 2 Nonlinear 0.4 1.2 2
Fig. 4. (a)–(b). Convergence study with different mesh densities (HOST-1 and HOST-2). 242
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laminate case (c/a ¼ 0) where the value is set to be zero. Further, the deflection responses are presented in nondimensional form say, w ¼ w=h where, w is the maximum deflection at the center of the panel else stated otherwise.
Table 3 Nondimensional central deflection of the simply-supported cross-ply [(0 /90 ), (0 /90 /0 ) and (0 /90 /90 /0 )] spherical shell panel under SDL. Spherical shell panel Lamination scheme
a/h
R/a ¼ 20 Reddy and Liu (Reddy and Liu, 1985)
0 /90 0 /90 /0 0 /90 /90 /0
10 100 10 100 10 100
3.1. Convergence and comparison of the present results
Present
FOST
HOST
HOST-1
HOST-2
12.309 7.127 6.6756 3.615 6.6099 3.6104
12.094 7.1236 7.1016 3.617 7.1237 3.6133
12.2964 7.1178 7.2331 3.6391 7.2524 3.6344
12.2665 7.133 7.0671 3.6696 7.1205 3.7396
12.161 10.621 7.125 4.342 7.1474 4.343
12.3621 10.6329 7.2539 4.3716 7.2735 4.3712
12.4959 10.7664 7.0821 4.2543 7.1336 4.2696
The convergence rate of the present nonlinear numerical solution has been established by solving an arbitrary example with the help of proposed higher-order FE models (HOST-1 and HOST-2). The nonlinear deflection parameters are computed for the square (a ¼ b ¼ 0.25 m) fourlayer antisymmetric cross-ply (0 /90 )2 cylindrical shell panel (a/h ¼ 50) with simply supported boundaries under two mechanical loading (UDL and SDL). It is also necessary to mention that the responses are computed for four different curvature ratios (R/a ¼ 5, 10, 20 and 50) to address the effect of curvature on the deflection responses and the mesh densities. The material property MAT (1) is used and the computed responses are presented in Fig. 4(a)–(b). From the figures, it could be easily concluded that the results are converging well with different mesh densities and an (8 8) mesh sufficient to compute the desired responses and is utilized further throughout the analysis. Further, the transverse deflections of the simply-supported square eight-layer (0 )8 laminated composite plate are computed numerically under UDL type of mechanical load. The computed deflections are obtained via the presently proposed higher-order kinematic models, compared with the available open literature results (Zhang and Kim, 2006; Argyris and Tenek, 1994) and depicted in Table 2. For the solution of this example, the material properties and the geometrical parameters are taken as same as the references. In addition, another validation study is reported with the reference (Reddy and Liu, 1985) for the simply supported spherical shell panel with three different stacking sequences [(0 /90 ), (0 /90 /0 ) and (0 /90 /90 /0 )] under the SDL type of loading and presented in Table 3. The results are also calculated using the same input parameters as in the reference (Reddy and Liu, 1985) and compared with that of the FOST and the HOST solutions from the same reference. In continuation to the earlier studies, further the deflection responses of the laminated composite cylindrical shell panel with and without internal debonding (c/a ¼ 0.25) under the mechanical UDL is examined using the newly developed numerical models. In this study, the nondimensional transverse central line deflection along the panel length is examined and subsequent comparisons are made with FOST result of Nanda (2014) and presented in Fig. 5. The geometrical configurations and the composite properties for this numerical example are taken same
R/a ¼ 100
0 /90
0 /90 /0
0 /90 /90 /0
10 100 10 100 10 100
12.373 10.653 6.6939 4.337 6.628 4.3368
Fig. 5. Comparison of non-dimensional deflection along the length of simplysupported square cylindrical shell panel with and without delamination.
Fig. 6. (a). Incorporation of artificial delamination using Teflon tape. (b). Pre-curing of the fabricated composite plate using hot press. 243
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as in reference (Nanda, 2014). From the comparison tables and the figure, it is understood that the present linear and or nonlinear results of the flat/curved structure with and without internal damage are within expected line and few deviations between the results exist due to the difference in modeling and solution techniques, respectively. 3.2. Debonded laminated plate fabrication steps In order to conduct the experimental investigation Glass/Epoxy composite plate with and without artificial delamination have been fabricated at National Institute of Technology Rourkela (NIT Rourkela) using the hand lay-up method. Now, the detail of the fabrication process is provided in the following lines. First of all, the necessary materials such as woven glass fiber in a required size and number, epoxy (Lapox L-12), hardener (K-6), wooden mould, roller, mould release spray (polyvinyl alcohol), plastic sheet have been arranged. Further, the epoxy and the corresponding hardener (10:1 wt ratio) are mixed together with proper care i.e. avoiding the bubbles formation in the mixture. The fabrication steps start with the application of mould release agent on the plastic sheet and further, it is laid down on the flat wooden platform. Now, the thin layer of the gel is coated with the help of brush on the plastic sheet and the woven glass fiber is placed and pressed with the help of roller to avoid any void formations. The process is repeated till the plane of delamination where artificial delamination needs to place. The artificial delamination of required size at the particular location is incorporated using a Teflon tape (Fig. 6(a)). After the incorporation of the delamination, same or different numbers of woven glass fiber are placed following the same steps as discussed previously. After completing the above steps, another plastic sheet is placed at the top of the laminate. Now the laminate is processed for the pre-curing in the hot press (Fig. 6(b)) at 60 C for 30 min and then left for post-curing in the ambient condition for two days. Finally, the plastic sheet at the top and bottom of the laminate are removed and the composite is stored in an air tied atmosphere.
Fig. 7. Visual representation of different size of delamination.
Fig. 8. Presentation of interface delamination.
The above-described process is utilized to fabricate the three sets of the flat laminated composite using eight layers of woven glass fabric. The first set of glass fiber reinforced polymer (GFRP) composite are obtained by fabricating the laminate with three different sizes of artificial debonding (c/a ¼ 0, 0.25, and 0.5) at the centre mid-plane of the laminate as shown in Fig. 7. Similarly, a debonding (c/a ¼ 0.25) is seeded at the centre of the different interfaces I-1, I-2, I-3 and I-4 (Fig. 8) to get the second set of the delaminated flat composite. Further, the third set is prepared by incorporating the artificial damage (c/a ¼ 0.25) at the different locations (left-1, left-2, middle, right-1 and right-2) as depicted in Fig. 9.
Fig. 9. Different position of the delamination for experimental analysis.
3.3. Material property evaluation
the help of mathematical relation Gξ1 ξ2 ¼
4 E45
E1ξ E1ξ 1
2
2υξ1 ξ2 Eξ1
1 . It is
necessary to mention that Poisson's ratio is taken to be 0.17 from the earlier published open literature (Mohanty et al., 2012). The final set of material property is found out by averaging the three result of each sample to avoid the discrepancy as well as ensure the accuracy of the desired properties and presented in Table 1 (MAT (2)).
Now, the elastic properties of the fabricated GFRP composite plate are obtained with the help of uniaxial tensile test. The standard specimens for the testing are prepared as per the ASTM standard ASTM D 3039/ 3039M and their details are given in Fig. 10(a). The specimen is cut along three different directions say longitudinal, transverse and 45 inclined to the longitudinal direction to get the desired Young's modulus say, Eξ1 ; Eξ2 and E45 respectively. Further, the test is performed using the Universal Testing Machine (UTM) INSTRON-1195 (Fig. 10(b)) at NIT Rourkela by setting the loading rate of 1 mm/min to avoid the distortion of the specimens. As discussed previously, the shear modulus is evaluated with
3.4. Experimental evaluation of static deflection and comparison with numerical result First of all, the sample specimens for the three-point bend test is 244
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Fig. 10. (a). Sample for the tensile test. (b). UTM INSTRON 1195.
Fig. 11. (a). Sample for the three-point bend test. (b). UTM INSTRON 5967.
delaminated composite are computed numerically with the help of presently developed higher-order FE models. It is important to mention that the numerical responses are obtained for plate dimension (48mm 16mm 3 mm) using the experimental material property MAT (2) in Table 1. Further, the co-ordinate for the different location of the delamination is provided in Table 4. The numerically computed result for all the three sets of delaminated composite i.e. size, position and location are now compared with those of the corresponding
prepared according to the ASTM standard ASTM 790 and presented in Fig. 11(a). The test is performed for all the three sets of the laminated/ delaminated composite using a 30 kN capacity Universal Testing Machine (UTM) INSTRON-5967 (Fig. 11(b)) at NIT Rourkela. For the test purpose, the loading rate of 2 mm/min is kept fixed for all the experimentation and the deflections are imported from the machine and saved corresponding to the data sheet of each sample. Now, for the comparison purpose, the nonlinear static deflections of the all the set of laminated/ 245
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Table 4 Co-ordinate for different location of delamination. Location
Table 7 Comparison study on nonlinear static deflection of eight-layer Glass/Epoxy laminated composite plate for different location of the delamination.
Co-ordinate (mm) p
Middle Left shift-1 Left shift-2 Right shift-1 Right shift-2
q
(18, 6) (6, 6) (12, 6) (30, 6) (24, 6)
r
(30, 6) (18, 6) (24, 6) (42,6) (36, 6)
s
(18, 10) (6, 10) (12, 10) (30, 10) (24, 10)
(30, (18, (24, (42, (36,
10) 10) 10) 10) 10)
Table 5 Comparison study of nonlinear static deflections of eight-layer Glass/Epoxy laminated composite plate for different size of delamination. Delamination size (c/a)
Load (N)
Experiment (mm)
HOST-1
HOST-2
0
100 110 120 130 140
0.636 0.701 0.764 0.833 0.897
0.7581 0.8336 0.9090 0.9843 1.0596
0.6684 0.7257 0.7789 0.8279 0.8677
100 110 120 130 140
0.702 0.778 0.862 0.941 1.02
0.8604 0.9457 1.0309 1.1158 1.2006
0.7498 0.8106 0.8699 0.9022 0.9701
100 110 120 130 140
0.836 0.965 1.033 1.082 1.171
1.1163 1.2254 1.3340 1.4121 1.5096
0.9192 0.9837 1.0452 1.0979 1.1863
0.25
0.5
Load (N)
Experiment (mm)
HOST-1
HOST-2
I-1
100 110 120 130 140
0.702 0.778 0.862 0.941 1.02
0.8604 0.9457 1.0309 1.1158 1.2006
0.7498 0.8106 0.8699 0.9022 0.9701
I-2
100 110 120 130 140
0.683 0.751 0.801 0.870 0.925
0.8085 0.8888 0.9690 1.0491 1.1290
0.7190 0.7718 0.7993 0.8698 0.9380
I-3
100 110 120 130 140
0.675 0.738 0.781 0.825 0.89
0.7586 0.8342 0.9096 0.9850 1.0603
0.6880 0.7419 0.7934 0.8214 0.8889
100 110 120 130 140
0.648 0.692 0.768 0.801 0.879
0.7176 0.7894 0.8611 0.9329 1.0046
0.6531 0.7055 0.7694 0.7995 0.8685
I-4
Load (N)
Experiment (mm)
HOST-1
HOST-2
Middle
100 110 120 130 140
0.702 0.778 0.862 0.941 1.02
0.8604 0.9457 1.0309 1.1158 1.2006
0.7498 0.8106 0.8699 0.9022 0.9701
Left shift-1
100 110 120 130 140
0.668 0.714 0.786 0.853 0.861
0.7895 0.8680 0.9465 1.0249 1.1031
0.6959 0.7385 0.8024 0.8579 0.8676
Left shift-2
100 110 120 130 140
0.698 0.769 0.837 0.895 0.954
0.8341 0.9171 0.9999 1.0826 1.1652
0.7288 0.7873 0.8444 0.8938 0.9472
Right shift-1
100 110 120 130 140
0.668 0.714 0.786 0.853 0.861
0.7895 0.8680 0.9465 1.0249 1.1031
0.6959 0.7385 0.8024 0.8579 0.8676
Right shift-2
100 110 120 130 140
0.698 0.769 0.837 0.895 0.954
0.8341 0.9171 0.9999 1.0826 1.1652
0.7288 0.7873 0.8444 0.8938 0.9472
3.5. Additional numerical analysis The convergence and subsequent comparison studies including the experimental validation indicates the capability of the presently developed higher-order nonlinear kinematic models to solve the variety of geometrical nonlinear problem of damaged laminated flat/curved composite structure. Based on the above conclusion, the models are now employed to solve a series of numerical illustrations to demonstrate the applicability of the said models and have the better insight of the certain design parameter effect on the nonlinear bending characteristics. In this section, for all the illustration a square laminate (a ¼ b ¼ 0.25 m) under the transverse load of magnitude 0.1 MPa is considered using the MAT (1) material property else stated otherwise. The nonlinear deflection parameters are presented in the nondimensional form using the same formulae said previously ðw ¼ w=hÞ.
Table 6 Comparison study on nonlinear static deflections of eight-layer Glass/Epoxy laminated composite plate for delamination at different interfaces. Interface
Delamination location
3.6. Influence of side to thickness ratio on nonlinear static bending responses of delaminated elliptical shell panel In this illustration, a four-layer cross-ply (0 /90 /0 /90 ) elliptical shell panel (R/a ¼ 20) with the simply-supported constraint is analysed for different size of delamination under the mechanical UDL and SDL type of loading. The nondimensional transverse central displacements for the five different side to thickness ratios (a/h ¼ 10, 20, 30, 40 and 50) is computed employing the proposed higher-order models and shown in Fig. 12(a)–(b). It is observed from the results that the deflection parameters increased with increasing of the side to thickness ratio irrespective of the size of delamination. With the increase in side to thickness ratio, the thickness of the panel and overall stiffness decreases result in the deflection responses increases. Additionally, a significant increase in nonlinear deflection parameter has also been observed with increase in the size of the deboning and the results are within the expected line.
experimental results and provided in the corresponding tables i.e. Tables 5–7, respectively. From the tables, it can be easily pointed out that the responses obtained via the HOST-2 are showing close values for all three cases i.e. the size, location and position of delamination when compared to the HOST-1. All most each case the differences are below 10% when the delamination size increases, however, the results show negligible error (1%) for the location as well as the position. It is mainly due to the fact that the HOST-2 model is derived by taking the thickness stretching effect and large deformation behavior through GreenLagrange strain-displacement relation.
3.7. Influence of curvature ratio on nonlinear bending responses of delaminated composite hyperboloid shell panel The nondimensional transverse central displacement of clamped four246
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Fig. 12. (a)–(b). Influence of side to thickness ratio on nonlinear bending response of delaminated composites elliptical shell panel under UDL and SDL loading.
layer cross-ply (0 /90 /0 /90 ) hyperboloid shell panel with different size of internal damage subjected to SDL and UDL loading is examined for five different curvature ratio (R/a ¼ 5, 10, 20, 50 and 100) and presented in Fig. 13(a)–(b). It is evident from the figures that the nonlinear deflection parameters increased irrespective of the size of internal damage when the curvature ratio increases. It is because of the wellknown fact that the deep shell panel has the higher bending strength as compared to shallow shell panel and in this example the same situation occur when curvature ratio increases for the constant side length. Further, the deflection parameters are following an increasing trend when the size of internal damage at the centre mid-plane increases for both the mechanical loading i.e., SDL and UDL loading. Also, it is worthy to mention that the debonding of the consecutive layers of the laminate reduces the overall stiffness of the structural panel and it results the increase of the deflection values.
using both the higher-order models for different aspect ratios (a/b ¼ 1, 1.5, 2, 2.5 and 3) and plotted in Fig. 14(a)–(b). The results indicate the notable effect of the aspect ratios on the transverse central displacement i.e., the central displacement values decreased with the increase in aspect ratios for both types of mechanical loading (UDL and SDL). The deflection values are higher for the UDL loading in comparison to the SDL type of loading. It is due to the fact that the load magnitude is uniform in case of UDL throughout the panel surface whereas the load varies from zero to maximum from the boundary to the mid-plane for sinusoidal loading. However, the effect of the size of internal damage is same as discussed in the previous example for all five aspect ratios. 3.9. Influence of modular ratio on nonlinear bending responses of delaminated composite cylindrical shell panel The nonlinear bending strength of the clamped symmetric cross-ply (0 /90 /90 /0 ) cylindrical shell panel (R/a ¼ 20 and a/h ¼ 50) with different size of delamination at the centre mid-plane of the laminated under UDL and SDL loading is investigated for the five different modular ratios and shown in Fig. 15(a)–(b). The figures show the reduction in nonlinear transverse displacement with an increase in the modular ratios as expected for the present case. It is because when the modular ratio increased the global stiffness increase which in turn decreases the
3.8. Influence of aspect ratio on nonlinear bending responses of delaminated composite spherical shell panel The effect of aspect ratio on the nonlinear static responses of the simply-supported symmetric cross-ply (0 /90 /90 /0 ) delaminated spherical shell panel (R/a ¼ 20 and a/h ¼ 50) under UDL and SDL loading is studied numerically in this example. The results are calculated
Fig. 13. (a)–(b). Influence of curvature ratio on nonlinear bending response of delaminated composites hyperboloid shell panel under UDL and SDL loading. 247
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Fig. 14. (a)–(b). Influence of aspect ratio on nonlinear bending response of delaminated composites spherical shell panel under UDL and SDL loading.
Fig. 15. (a)–(b). Influence of modular ratio on nonlinear bending response of delaminated composites cylindrical shell panel under UDL and SDL loading.
transverse deflection values. Subsequently, the figures also show the significant increase in the displacement parameter as the area of debonding increases between two consecutive layers irrespective of the loading conditions.
3.11. Influence of delamination position on nonlinear bending responses of delaminated composites elliptical shell panel The debonding not always initiated at the centre of mid-plane of the laminated structure, but it may arise at any interfaces as well in a real situation. So it is a matter of concern that the debonding at every interface is equally important, or the emphasis has to be given for the debonding at any particular interface. In order to understand such case, an example of elliptical shell panel with the delamination (c/a ¼ 0.5) at the different interfaces is investigated under UDL and SDL types of loading. The detailed position of the delamination can be seen in Fig. 8. For the computation, a clamped eight-layer (0 /90 /0 /90 ) s elliptical shell panel is investigated numerically by setting the thickness and curvature ratios as 50 and 20, respectively. The nonlinear transverse centerline displacement along the length and the width of the elliptical shell panel are calculated and plotted in Fig. 17(a)–(d). The deformation characteristic is found to be maximum for the middle interface (I-1) and it starts falling as the debonding moves in other interfaces i.e., I-2, I-3 and I-4 and it is lowest for the interface I-4. The above trend clearly indicates that the debonding at the mid-plane is more significant than any other interfaces away from the mid-plane.
3.10. Influence of size of debonding on nonlinear bending responses of delaminated composite spherical shell panel In order to demonstrate the effect of the delamination size on the nonlinear static responses, a simply-supported spherical shell panel (0 / 90 /90 /0 )s example is solved using the design parameters a/h ¼ 50 and R/a ¼ 50 under two loadings (UDL and SDL). The responses (transverse centerline deflection along length and width) are evaluated for the different sizes of debonding areas (Fig. 7) with the help of presently developed nonlinear FE models and depicted in Fig. 16(a)–(d). The interesting outcome reveals that the delamination size has the significant effect on the deflection parameter i.e. as the increase in the debonding sizes the overall structural integrity (stiffness) decrease which in turn increases the deflection parameter. Further, it is important to discuss that the effect of debonding size is independent of loading.
248
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Fig. 16. Influence of size of debonding on nonlinear bending response of delaminated composites spherical shell panel under UDL and SDL loading.
demonstrated. For the analysis, a simply-supported cylindrical shell panel with cross-ply laminations (0 /90 /90 /0 ) is examined using the side to thickness and the curvature ratios as 50 and 60, respectively under two types of mechanical loading UDL and SDL. The stress values are computed through the thickness using both the HOST models and shown in Fig. 20(a)–(b). The responses are calculated using the MAT (3) type of properties and setting the load 100 kN. From the figures, the increasing trend of stress values is found with the increase in delamination sizes. It is worthy to discuss here that the present results calculated using the constitutive relations. In this example, the stresses are expressed in the 2 nondimensional form using the formulae as ðσ Þnd ¼ σ ah =Eξ2 ,
3.12. Influence of location of debonded area on nonlinear bending responses of delaminated hyperboloid composite shell panel The significance of the debonding size and the position is already demonstrated in the earlier example. Now, the effect of location of the debonding area on the nonlinear static responses of doubly curved shell panel has been illustrated in this example by shifting the debonded area into five different locations (left, right, middle, top and bottom) on the mid-plane of the laminated plate as shown in Fig. 18. The responses are computed numerically for the simply-supported hyperboloid shell panel (a/h ¼ 50, R/a ¼ 20) with rectangular (a ¼ 2b) delamination (c/ a ¼ 0.25) and antisymmetric cross-ply (0 /90 /0 /90 ) s lay-up sequences under the UDL and SDL type of loading. The nonlinear transverse centerline displacement along the length (X/a) and width (Y/b) are calculated using presently developed FE models and presented in Fig. 19(a)–(d). The figures indicate the significant effect of the change in location of the debonding area on the nonlinear responses. In addition, it is interesting to note that the responses follow the same trend and equal magnitude for the left or right shift of delamination irrespective of the models and loading conditions.
respectively. 4. Conclusive remarks The nonlinear static deflections of the laminated composite shell structures are computed numerically with the help of new nonlinear mathematical model based on the higher-order kinematic theories including Green-Lagrange nonlinear strains and all the nonlinear higherorder strain terms. The responses are evaluated via a generic computer code prepared in MATLAB environment. The convergence and the validation study of the proposed nonlinear numerical models have been performed by computing an adequate number of numerical examples. In addition, the responses are computed experimentally with the help of
3.13. Influence of size of debonding on in-plane stress of cylindrical shell panel In this example, the influence of delamination sizes on the stress is 249
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Fig. 17. Influence of delamination position on nonlinear bending response of delaminated composites elliptical shell panel under UDL and SDL loading.
numerical study the present results are compared with those of the experimental data. Lastly, the applicability of the developed models has been explored by solving few new examples which show the influences of the material and geometrical parameter on the nonlinear static behavior of the curved shell panel. On the basis of the numerical and the experimental investigation, the following conclusive remarks have been put forward for the better understanding. The validation studies indicate that the results of the presently developed nonlinear higher-order models are in line with those of the earlier published result. Further, the confidence on the present models has also been added by comparing the present static deflections with that of the experimental nonlinear bending responses of internally damaged Glass/Epoxy laminate. Various numerical interpretations indicate the significant effect of the damage on the nonlinear static behavior i.e., the presence of debonding reduces the total structural deformation bearing capacity and increases the deflection responses. In addition, the presence of the debonding at the mid-plane (middle interface) of the laminate is more crucial than any other interface away from the mid-plane.
Fig. 18. Different location of delamination in panel.
three-point bend test on in-house fabricated woven Glass/Epoxy laminate composite plate including different sizes of the delamination. In order to add more confidence as well as the superiority of the proposed
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Fig. 19. Influence of location of debonding area on nonlinear bending response of delaminated composites hyperboloid shell panel under UDL and SDL loading.
Fig. 20. Influence of size of debonding on in-plane stress of delaminated composites cylindrical shell panel.
The debonding nearer to the constrained boundary does not affect the overall stiffness and static responses significantly in comparison to centrally located debonding.
The result indicates that the stress and the deflection parameters affected considerably due to increase in the size of internal debonding i.e., following an increasing trend with the increase in the size of the debonding irrespective of the type of loading. 251
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