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Buckling and post buckling characteristics of laminated composite plates with damage under thermo-mechanical loading
Buckling and post buckling characteristics of laminated composite plates with damage under thermo-mechanical loading V.M. Sreehari, D.K. Maiti PII: DOI: Reference:
S2352-0124(16)00011-4 doi: 10.1016/j.istruc.2016.01.002 ISTRUC 84
To appear in: Received date: Revised date: Accepted date:
24 October 2015 18 January 2016 18 January 2016
Please cite this article as: Sreehari VM, Maiti DK, Buckling and post buckling characteristics of laminated composite plates with damage under thermo-mechanical loading, (2016), doi: 10.1016/j.istruc.2016.01.002
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ACCEPTED MANUSCRIPT Buckling and post buckling characteristics of laminated composite plates with damage under thermo-mechanical loading
Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, WB- 721302, India
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a,b
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Sreehari. V. Ma * and Maiti. D. Kb
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ABSTRACT
The present work addresses the development of a finite element formulation for handling
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bending, buckling, and post buckling analysis of composite laminated structures with damage. The inverse hyperbolic shear deformation theory (IHSDT) was applied in the finite element formulation. The effect of damage is analysed for thin composite plates. An anisotropic damage
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formulation was used to simulate the damage, which is based on the concept of stiffness
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reduction. Computer programming is developed in the MATLAB environment. The excellent agreement of the results obtained in the present method with those from references shows that
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the technique is effective and precise. Parametric studies in the buckling behaviour of a damaged composite plate are presented. Critical buckling temperatures are computed for a damaged plate using the present model. Thermal post buckling equilibrium paths are traced for various parametric variations for composite plates with mild damage and compared the results with that of undamaged cases. The validation of IHSDT has been demonstrated for buckling analysis in thermal environment for composite plates with an internal flaw. The present work is worthwhile compared with previous works due to the choice of finite element method and inverse hyperbolic shear deformation theory for analyzing the influence of damage on buckling and post buckling behavior of laminated plates.
*
Corresponding author: Tel.: +918001581060 Address:Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, WB- 721302, India. E-mail address: [email protected]
ACCEPTED MANUSCRIPT Keywords: Composite plate; Finite element method; Shear deformation theory; Buckling load; Thermal post buckling; Anisotropic damage.
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1. Introduction
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Composite plates play an essential role in current and future aerospace, civil and mechanical structural system due to their improved performance. The diverse problems in
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structural analysis necessitate the proper knowledge of structural behaviour. Hence, the necessity of a thorough study of buckling behaviour of composite structures using the finite element
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method considering different features increases the interest. Ozben Tamer [1] had reported the critical buckling load of fiber reinforced composite plate. Choudhary and Tungikar [2] presented Finite Element Analysis for geometrically nonlinear behaviour of laminated composite plates.
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Chai and Hoon [3] have conducted studies on the effect of the various in-plane loads on laminated composite plates. Onkar et al. [4] studied the buckling of laminated composite plates by stochastic finite element analysis. Bao and Jiang [5] presented a critical review of bending and
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buckling of flat, rectangular, orthotropic thin plates. Kapania and Raciti [6] presented a review of laminated structures, highlighting shear effects and buckling. Many researchers [7-9] had discussed the effects of bending-stretching coupling matrix in the case of a regular, antisymmetric (i.e., not generally unsymmetric), angle-ply plate, and obtained buckling loads for such problems. In addition to buckling analysis, a lot of significant investigations were carried out on post buckling behavior of laminated plates by numerous researchers [10, 11] and they comprehensively presented the state of knowledge in this field. Development of accurate theoretical models is necessary to predict the structural responses in composite laminated plates. Increased use of composites in primary structures in the past few years has led to the development of a variety of plate theories by researchers. Even though three dimensional elasticity solutions predict structural behavior accurately, its higher computational complexities
ACCEPTED MANUSCRIPT led to the developments of two dimensional plate theories. Among the two dimensional theories, equivalent single layer theories are computationally efficient and they include Classical
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Laminated Plate Theory (CLPT), First order Shear Deformation Theory (FSDT) and higher order shear deformation theories. In the past decades, researchers [12, 13] have established various
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higher order polynomial shear deformation theories built on equivalent single layer theory. Over the last few years, several shear deformation theories having non-polynomial nature and
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expressed in terms of shear-strain function have been proposed. Recently, Grover et al. [14]
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proposed new nonpolynomial shear deformation theories (NPSDT). Regarding to the thermal buckling, it is important to explore few significant works done in this area. Tauchert [15], Tauchert and Huang [16] developed the Navier type solution for composites with antisymmetric
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and symmetric ply orientations under thermal loadings. Thangaratnam et al. [17] employed
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FSDT and found critical buckling temperature. Many researchers [18, 19] have conducted structural analysis under thermal environment using finite element method.
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Even though such structures are highly reliable, internal flaws are almost inevitable at some phase of its operative life span. The occurrence of damage significantly affects the consistent operation of the structure. The early detection of structural failure is one of the major tasks in the aerospace industry. Damage may be due to some unfavourable conditions and leading to the shortening of operating life of the structures. Effective methods are to be developed to find the effects of damage in any structural parts. Peck and Springer [20] assumed delamination to occur between two nearby laminates and perfect bonding between all other laminates. Wadee and Vollmecke [21] obtained the same decrement in stiffness from the defect irrespective of its depth and observed stable post buckling behaviour when delamination growth was not considered. It is important to note that no growth was witnessed at the first critical load for the global buckling example. They found out the transitional depth ratio to be located
ACCEPTED MANUSCRIPT between 10 and 20 % of the depth of the plate, where the mode of buckling changes and buckling-driven delamination growth only occurs where the pre-existing delaminations are closer
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to the surface only. In the present analysis, damage propagation is not included. The damage may be due to cracks, fatigue, corrosion, or caused during manufacturing processes. Damage process
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is the one which causes weakening in mechanical properties. An anisotropic damage formulation is used in this study to simulate the damage, which is based on the concept of stiffness reduction.
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The concept of stiffness reduction was utilized by Kucukler et al. [22] for evaluating the lateral-
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torsional buckling of steel structures. A continuous parameter that can be correlated to the density of defects is presented in general continuum damage mechanics. This technique is useful in describing the weakening of the material before the initiation of micro-cracks. Valliappan et
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al. [23] developed the elastic constitutive relationship for anisotropic damage mechanics and
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explained the employment of these constitutive relations in finite element analysis. Martha et al. [24], Belytschko and Black [25] explained certain damage incorporated problems by “hard-
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wiring” the damage as a portion of the geometry of the structure. Yet, a more useful parametric modeling of damage is found in a variety of literatures [26-28]. Pidaparti [29] found out the free vibration characteristics of a composite plate and established that the methodology by Valliappan et al. has more effect than some other formulations. Zhang et al. [30] also appreciated this method and concluded that symmetrisation schemes do not produce as good agreement with experimental results as demonstrated by the non-symmetrical model and hence, whenever possible the symmetrisation treatment should be avoided in anisotropic damage modeling. Zhang and Cai [31] investigated continuum damage mechanics and used appropriate damage variable to represent the macroscopic effects of microscopic damages and developed elastic constitutive models. However, most studies [32, 33] on free vibration and static stability behaviour with zones of damage were restricted to simple beams and plates than composites.
ACCEPTED MANUSCRIPT It is noted that very little attention has been given to the mechanism of failure and its influence on the response behaviour of the structure under load. Despite the fact that literature is
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available on structural responses, no such literature is available on the effects of region of damage in buckling and thermal post buckling of laminated composite plates using FEM
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considering anisotropic damage formulation. The aim of the present paper is to establish accurate theoretical model necessary to predict buckling responses in damaged composite structures. In
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the present investigation, authors’ purpose is to study the bending, buckling, and post buckling
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behaviour of laminated composite plates with confined regions of damage using FEM where the governing equations are developed based on Inverse Hyperbolic Shear Deformation Theory (IHSDT). Grover et al. [14] demonstrated by comparing to other prevailing shear deformation
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theories that new NPSDTs have only a small total percentage error. They demonstrated the
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efficiency and enhanced performance of IHSDT compared to similar existing higher order shear deformation theories involving shear strain function. This bending theory describes the stresses
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and deflections more accurately and authors have explained its potential impact in buckling analysis previously [19]. The effort of solving higher order differential terms in the displacement field equations has been removed because IHSDT provides us with a non-polynomial displacement field equation and thereby helps to analyze complex problems with reduced computational complexity. This theory fulfils zero transverse shear stress at the top surface and at the bottom surface of the plate and the transverse shear stress distribution is non-linear across the thickness. In the current work, the finite element damage formulation considers effects of transverse shear, geometric non linearity of composite plates, thermal effect, etc. while predicting buckling and we have assumed that perfect bonding exists between all laminates. The presented methodology provides an alternative way to understand the effect of damage on the stability of the composite. The present work is valuable compared with previous works due to the selection of finite element method and inverse hyperbolic shear deformation theory for analyzing the
ACCEPTED MANUSCRIPT impacts of damage on buckling and post buckling behavior of laminated composite plate. The outcomes from this investigation will benefit to address some remarkable problems relating to
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composite structures. Once we have clearly determined the effect of damage, we can reduce the deterioration of properties in composite structures. For example, we can use methods such as
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usage of optimized ply orientations in design stage or employment of smart patches later to
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reduce the effects of damage. 2. Mathematical modeling
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2.1. Element selection
A finite element formulation is developed for handling bending and buckling analysis of laminated composite plates with damage. The fundamental idea of finite elements is that the
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structure is considered as an assembly of elements connected at nodes. We are considering an 8-
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noded iso-parametric quadrilateral serendipity element (as shown in Fig. 1) with C0-continuity and with 56 degrees of freedom in the present work. An isoparametric element has an advantage
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that element geometry and displacements are represented by same set of shape functions. The benefit of 8 noded element is that all the nodes are located on element sides and hence there are no internal nodes and shape functions have quadratic variation in x and y direction. The shape functions used are,
N1 0.25(1 )(1 )(1 );
N 5 0.5(1 )(1 2 )
N 2 0.5(1 2 )(1 );
N 6 0.25(1 )(1 )(1 )
N 3 0.25(1 )(1 )(1 );
N 7 0.5(1 2 )(1 )
N 4 0.5(1 )(1 2 );
N8 0.25(1 )(1 )(1 )
2.2. Displacement field The chosen displacement field for structural analysis of laminated composite plate is on the basis of IHSDT. A laminated plate containing N orthotropic layers is considered as in Fig. 2. The
ACCEPTED MANUSCRIPT thickness, length, and width of plate are h, a, and b respectively as shown in Fig. 3. The principle
Displacement model given in Eq. (1) was proposed in Ref. [14]. w0 sinh 1 ( rz / h) - z h x
( x, y ) x 2 r 4
v( x, y, z ) v0 ( x, y ) - z
w0 sinh 1 ( rz / h) - z h y
( x, y ) y 2 r 4
(1)
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2r
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w( x, y, z ) w0 ( x, y )
2r
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u ( x, y , z ) u0 ( x, y ) - z
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material co-ordinates ( x1k , x2k , x3k ) of the kth lamina is at an angle k to the general x co-ordinate.
In the above displacement field, u0, v0, and w0 are the midplane displacements while θx, θy are the shear displacements. The transverse shear stress factor, represented by r, is fixed as 3 per
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the reverse technique in post processing phase shown in Ref. [14]. The continuity requirement
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can be changed from C1 to C0 for simplification. It can be done by taking
w w and as distinct x y
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independent degrees of freedom as: w w x and y x y
(2)
For the finite element analysis total potential energy is given as, U
NE
U
(e)
(3)
e 1
For obtaining the total potential energy, the elemental elastic stiffness matrix is given as, N
N
K ( e ) wp wq BT D B det J p 1 q 1
(4)
where wp , wq are the weights used in the Gaussian quadrature numerical integration method, det
ACCEPTED MANUSCRIPT D J denotes the determinant of Jacobian matrix, and is the stress-strain stiffness matrix. [B] is
0
0
y
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
N i 0
N i
0
0
0
0
0
0
0
0
0
N i
0
N i
0
N i
N i
x
N i
x
0
y
x
N i
0
N i
0
0
0
N i
y
x
y
Ni
0
Ni
0
0
0
0
0
0
0
0
0
Ni
h
r 2 4
0
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0
y
D
0
y
0
x
where i = 1, 2, 3,..8 and Ω from Eq. (1) is
0 0 0 N i y N i x 0 N i y N i x 0 0 Ni 0 0
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N i
0
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N i
0
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0
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N i x 0 N i y 0 0 0 [ B] 0 0 0 0 0 0 0
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the matrix containing derivatives of shape functions as shown below.
- 2r
2.3. Strain-displacement relations The displacement w produces some additional extension in the x and y directions of the middle surface resulting in von Karman strain-displacement relationship as given below,
ACCEPTED MANUSCRIPT u 1 w x 2 x
2
x
v 1 w y 2 y
2
y
u v w w y x x y u w z x v w z y
(5)
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yz
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xz
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xy
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The above relation can be written in matrix form as, 0 L in which the first term is the linear expression and second gives the non-linear terms. Similar to the equation of linear stress, we can write for non-linear components as
d L BL q
, where {q}
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represents the non-linear components of displacement vector for an element.
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The non-linear strain components can be also written as, 0 w w x 1 A I y w 2 w y x
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w x 1 L 0 2 w y
The derivatives of w can be related to the nodal parameters as
(6)
I Gq. Now, the
geometric stiffness matrix is obtained as in Eq. (7) by following the procedure described by Zienkiewicz [12].
K
G V
T
Tx T xy
Txy G dV Ty
Where Tx, Ty, and Txy represents the in-plane loads.
(7)
ACCEPTED MANUSCRIPT 2.4. Stress-strain relations
Q16
0
Q22
Q26
0
Q62
Q66
0
0
0
Q44
0
0
Q54
0 x - x T 0 y - y T - T 0 xy xy xz Q45 yz Q55
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Q12
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x Q11 Q 21 y xy Q61 xz 0 yz 0
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relations for a lamina about any arbitrary axes is given by
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While investigating the buckling behaviour in thermal environment, the stress-strain
(8)
Where α represents the coefficient of thermal expansion and is given by,
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where m cos , n sin
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αx= α1m2 + α2n2 ; αy= α2m2 + α1n2 ; αxy= 2 (α1-α2) mn
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2.5. Buckling and post buckling behaviour As a result of buckling, an eigenvalue problem can be formulated as shown below, ([K] +λ [Kσ]) {Δ} = {0}
(9)
where [K] is the global stiffness matrix, [Kσ] is the global geometric stiffness matrix, λ is the in-plane magnification factor and {Δ} is the global displacement field. The eigenvalue problem is solved to extract minimum eigenvalues or critical buckling loads. Once the buckling condition is attained, the nonlinear stiffness matrix [Knl] is included into the formulation (Zienkiewicz [12]) as,
([K] + [Knl] +λ [Kσ]) {Δ} = {0}
(10)
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Equation (10) demonstrates the nonlinear equilibrium for laminated composite plate. Small
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displacement rise compared to previous equilibrium state is considered and Eq. (10) is solved as an eigenvalue problem, for finding the thermal post buckling response path. As demonstrated by
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Singha et al. [18], a sequence of eigenvalue problems is solved to find the thermal post buckling
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path.
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2.6. Introducing the effects of damage
In this paper, we are considering damage located at the centre of a composite plate. The damaged area is a patch having a size 4% of the whole plate area as in Fig. 4. The damage in
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composites is anisotropic in nature. Anisotropic damage is the damage which is scattered in
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numerous orthogonal directions and the amount of damage in any orthogonal direction is independent of the other. In the present analysis, the consequence of an internal flaw is computed by means of an anisotropic damage formulation, which is established on the concept of stiffness
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change. Using an idealized model having a reduced elastic property in the damage region, the effects of a region of damage are incorporated into the formulation. Thus the effect of internal flaw is parametrically included into the previous formulation (for undamaged composite) using a parameter Γf. In essence, this is an effective parameter which represents reduction in effective area. The parameter is shown in Eq. (11).
f
Af - A*f
(11)
Af
Where f = 1, 2 and it denotes the in-plane orthogonal directions and Af* is the effective area of plate with damage. Γ1 denotes the damage in the fibre direction while Γ2 denotes the damage in the orthogonal direction in the same plane. For example, in a (0/90) layup, the fibre is
ACCEPTED MANUSCRIPT aligned in co-ordinate axis 1 in the top layer and orthogonal here implies a co-ordinate axis 2, (i.e., in a direction 90 degree to the previous). In the second/bottom layer, the fibre is aligned in
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co-ordinate axis 2, and there the orthogonal implies co-ordinate axis 1 direction. This technique of modeling damage using parameters in any anisotropic material was suggested by Valliappan
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et al. [23]. A stress-strain stiffness matrix for a two dimensional damaged laminate using this
*
* * D
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formulation is in Eq. (13).
Where,
D
1- 1 1- 2 E 1- 1221 2 12 2 1- 2 E 1- 1221 2
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2 1- 1 E 1 1- 1221 1- 1 1- 2 E 1 21 1- 12 21 D* 0 0 0
(12)
0
0
0
0
0
1- 1 1- 2 1- 1221 1- 1221 2 G12 2 2 1- 1 1- 2 1- 1221 1- 1221
0
0
1- 2
0
0
0
2
0 0 0 2 1- 1 G13 0
2
0
2
G23
(13)
A stress-strain matrix for a laminate having internal flaw is evaluated and further on proceeding as in undamaged case formulation, an eigenvalue problem for a composite plate having an internal flaw can be obtained.
ACCEPTED MANUSCRIPT 3. Numerical results and discussion
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3.1. Problem definition The structures studied here are mostly square plates, length (a) = breadth (b), having the
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thickness ratio = 100. The damaged area is a patch having a size 4% of the whole plate area as in Fig. 4, located at the center of the plate with each element having a particular damage ratio.
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Numerous parametric studies have been conducted for the composite plate having damage. The
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boundary conditions employed in the analysis are ‘all edges clamped’ and ‘all edges simply supported’. Computer programming is developed in MATLAB environment. The numerical results were obtained and compared with the available literature. Transverse deflection and
a2 Pnd Pcr 3 E2 h
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100h3 E2 w w and 4 q a 0
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critical buckling load are non dimensionalised using the relations given in Eq. (14).
(14)
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The various Material Models used are (taken from the literature from which comparison studies are done),
Material Model 1 (MM1): E₁/E₂ = 25, G₁₂ = G₁₃ = 0.5 E₂, G₂₃ = 0.2 E₂, v₁₂ = 0.25. Material Model 2 (MM2): E₁/E₂ = 40, G₁₂ = G₁₃ = 0.6 E₂, G₂₃ = 0.5 E₂, v₁₂ = 0.25.
3.2. Convergence and validation study Convergence of the solution with a mesh refinement for a laminated composite plate having laminate sequence (0/90/90/0) with a/h ratio 10 and 100 is shown in Fig. 5. As the fineness of mesh increases the convergence of the results is found to be fairly accurate. Based on convergence study, a (10 × 10) mesh has been used in most cases of later study. For validation purposes, some of the results of buckling analysis from present work are compared with the
ACCEPTED MANUSCRIPT results available in literature [13] for the undamaged composite plate. The excellent agreement of the results obtained in the present method with those from references shows that the technique is
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effective and precise. The comparison of uniaxial critical buckling loads with various side-tothickness ratios is computed for a four layer cross-ply plate (0/90/90/0) as in Table 1. Material 2
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properties were taken for analysis. Results in Table 1 validate the present finite element formulation as well as Matlab programming. Table 2 shows the comparison of the non
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dimensional buckling loads obtained by the present analysis with the reference [13] values for a
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square, laminated composite plate for in-plane uniaxial loading condition. The material 2 property is considered in the analysis. Analysis is performed for four-layer (n = 4) antisymmetric angle ply plates. Analysis is carried out for a laminated composite plate having laminate
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sequence (5/-5/5/-5). It is clearly evident from the results that, IHSDT can be successfully used
3.3. Effects of damage
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in the analysis of laminated composite plates.
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After incorporating the effects of damage, some results for a simple plate were compared with the results obtained by Prabhakara and Datta [32]. The present model studies with conventional small defects or damage. A plate with centre damage (damage at centre location in terms of length/ width and which occurs throughout the thickness) is considered. We have introduced the damage parameters in the centre 4 elements (intensity of damage per element will be given wherever necessary) for a plate with 10×10 mesh size. The buckling coefficients for a plate with 3 different aspect ratios are found and compared with those of reference as shown in Table 3, and validates the finite element formulation incorporating damage and the methodology.
ACCEPTED MANUSCRIPT 3.3.1. Effect of internal flaw in bending A laminated composite plate having four layers (0/90)s with central damage is analysed
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and central transverse deflections are found. We have obtained the central transverse deflection
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of the whole plate at specific percentages of elemental damage areas. The results obtained are shown in Fig. 6. For plates having higher E1/E2, the deflection is found to be less. As the
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3.3.2. Orthogonality of damage parameter
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percentage of damaged area increases in an element, the central deflection increases also.
As the model of damage being considered is anisotropic in nature, it is vital to understand the nature of orthotropy shown by this formulation. It may be inferred that the contribution of
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damage in the direction of fiber, 1 may not be equal to that in the orthogonal direction to the
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fiber, 2 . In order to know this, one of these orthogonal parameters must be fixed while the other must be varied to obtain the characteristics of the structure. Change in the buckling load of a
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composite plate with central damage, (0/90/0) ply orientation, with variation of one of the damage parameters while the other is fixed as shown in Fig. 7. It can be noticed that damage in the direction of fiber, i.e., 1 , shows a steeper descent of buckling features when compared to the influence of 2 . Owing to this, for all further computational purposes, the value of 2 is fixed at 0.1, while the intensity of damage is denoted by the damage ratio, Г₁/Г₂. The damage ratio can take values from 0 to 9. Mild damage can be indicated by a damage ratio between 0 and 3 whereas heavy damage can be indicated by a damage ratio between 7 and 9. 3.3.3. Effect of internal flaw in buckling under mechanical load Composite plates with internal flaw are analysed for uniaxial as well as biaxial loading conditions (for material 1 property, simply supported at all ends) and the results observed are
ACCEPTED MANUSCRIPT plotted in Fig. 8. Buckling loads are found at particular percentages of elemental damage areas using the present anisotropic damage modeling. It is seen that the critical buckling load decreases
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as the damage area increases.
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The variation of non dimensional buckling load for thin composite plates under uniaxial loadings with damage patches in the central region having damage intensities of Γ1/Γ2 = 2 and
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Γ1/Γ2 = 3 is presented in Table 4. It is found from the present analysis that the internal flaws will
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significantly affect the buckling characteristics of laminated composite plates. Critical buckling loads of symmetric (0/90/90/0) laminates under uniaxial and biaxial compression for damaged and undamaged cases are shown in Fig. 9. Analysis is conducted with side to thickness ratio
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varying from 10 to 100. For this we have taken a laminated composite plate with simply
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supported at all ends and for material 1 property. For damaged cases the nondimensional buckling loads are less than those from undamaged cases in both uniaxial and biaxial loadings. It is found that critical loads are higher for uniaxial loading. Furthermore as thickness decreases,
cases.
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the nondimensional buckling load increases. The variation is much less above a/h = 40 for all
3.3.4. Effect of internal flaw in critical buckling temperature Buckling as well as post buckling study is performed using IHSDT for laminated composite plates with central damage exposed to constant in-plane thermally produced loadings. The critical buckling temperature of damaged plate has been obtained and compared with those of the undamaged case. The buckling temperature is found for an isotropic square plate with α = 2x10-6 oC-1 and a/h = 100, for both ‘all edges simply supported’ and ‘all edges clamped’ boundary conditions. Computations performed for two damage intensities, Γ1/Γ2 = 3 and Γ1/Γ2 = 7, are given. The results are presented in Table 5.
ACCEPTED MANUSCRIPT The critical buckling temperature is found and plotted for a square symmetric cross-ply (0/90/90/0) laminated composite plate with and without damage in Fig. 10. The plate considered
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is under simply supported boundary conditions and uniform temperature. As E1/E2 increases from 10 to 40, the critical buckling temperature decreases. The decrement is higher for damaged
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plates, as observed from the steeper curve of the damaged case.
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3.3.5. Effect of internal flaw in thermal post buckling
The thermal post buckling equilibrium path is traced for composite plates with an internal flaw and compared the results with that of the undamaged case for various parametric variations.
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A square laminated plate of 4-layers (45/-45/45/-45) with simply supported ends is considered
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for the analysis. Studies are presented for a/h = 50, a/h = 100, E1/E2 = 20, E1/E2 = 40, α2/ α1 = 5, and α2/α1 = 10 (Tcr = (Tcr)a/h=
100
for Fig. 11 and similarly for further figures) and the results
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obtained are shown in Figures 11 to 16. For a mild damage with Г₁/Г₂ < 3, there will not be much change in the flaw size under loading as it will be in the elastic region. The post-buckling predictions presented here are with damage ratio 1. In Figs 11 and 12, thermal post buckling equilibrium paths are traced for two different a/h conditions, a/h = 50 and a/h = 100, respectively. It can be observed from the plot that the critical buckling temperature increases as a/h decreases. Similar analysis can be performed (Figs 13 and 14) using Young’s modulus ratio instead of span-to-thickness ratio for obtaining the thermal post buckling equilibrium path. However, here as E1/E2 increases, the buckling temperature also increases. From any of the above plots it can be understood that for the same loading, higher displacements are observed in damaged composite plates compared to undamaged plates. As the load increases, the difference in displacements with and without internal flaw cases increases.
ACCEPTED MANUSCRIPT For thin plates, the difference is seen mostly at a later post buckling state. The difference in displacements with and without internal flaw is more when E1/E2 increases. The thermal post
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buckling equilibrium path is traced for composite plates with an internal flaw and compared the results with that of undamaged case as shown in Fig. 15 and Fig. 16 for α2/ α1= 5 and α2/α1 = 10
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respectively. As the ratio of thermal expansion coefficients α2/α1 increases, the buckling temperature decreases. Either from Fig. 11 or Fig. 12, it can be understood that for same loading,
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higher displacements are observed in damaged composite plates compared to undamaged plates.
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As load increases, the difference in displacements with and without internal flaw cases increases. For α2/α1 = 5 case, the difference is seen mostly at a later post buckling state. The difference in displacements with and without internal flaw is more when α2/α1 increases.
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4. Conclusion
Buckling and post buckling of a laminated composite plate with damage has been studied. An FEM formulation using inverse hyperbolic shear deformation theory for a composite plate
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having an internal flaw is implemented. Computer programming is developed in MATLAB environment. From the comparison studies, it is seen that the results obtained by the present technique are quite agreeable with reference solutions and this demonstrates the efficiency and the robustness of the method. It is also demonstrated through this investigation that the inverse hyperbolic shear deformation theory can be precisely used in the study of laminated composite structures with internal flaws. It is found from the present analysis that the internal flaws will significantly affect the critical buckling temperatures and post buckling paths of laminated composite plates. The other major contributions of present work are summarized in the following points.
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The displacement field taken in present analysis removed the difficulty of solving higher order differential terms in the displacement field equations by providing us with a
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nonpolynomial displacement field equation and helps to analyze under reduced
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computational complexity which is a great advantage. The damage model taken in this work has a wider scope of application due to the simplicity of stress-strain relations, i.e., constitutive relation for a region of damage retains its basic form. Transverse central
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deflections for a composite plate are determined for particular percentages of damage
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area and it is noted that as the percentage of damage area increases, the transverse central deflection increases.
The finite element damage formulation considered here (anisotropic damage formulation)
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has a strong orthogonality. Damage in the direction of fibre affects the buckling behavior
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of composite plates more than damage in any other direction. It is observed that when the damage intensity rises, the buckling load reduces. Also as the
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percentage of damaged area increases, the buckling load decreases. Effects of material anisotropy, boundary condition, and span-thickness ratio on the buckling behaviour of a composite plate with an internal flaw are studied.
Critical buckling temperature of a damaged plate has been obtained and compared with those of undamaged case.
Thermal post buckling equilibrium path is traced for composite plates with a mild damage and compared the results with that of undamaged case. It is noticed that the variations in displacements increases for higher values of a/h, E1/E2, and α2/α1 values.
ACCEPTED MANUSCRIPT References [1] Ozben Tamer. Analysis of critical buckling load of laminated composites plate with different
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boundary conditions using FEM and analytical methods. Comput Mater Sci 2009;45:1006-15. [2] Choudhary SS, Tungikar VB. A simple Finite Element for non-linear analysis of composite
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plates. Int J Eng Sci Technol 2011;3:4897-907.
[3] Chai GB, Hoon KH. Buckling of generally laminated composite plates. Compos Sci Technol
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1992;45:125-33.
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[4] Onkar AK, Upadhyay CS, Yadav D. Generalized buckling of composite laminated plates using the stochastic finite element method. Int J Mech Sci 2006;48(7):780-9. [5] Bao G, Jiang W. Analytic and finite element solutions for bending and buckling of orthotropic
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rectangular plates. Int J Solids Struct 1997;34(14):1797-822.
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[6] Kapania RK, Raciti S. Recent advances in analysis of laminated beams and plates. Part ISheareffects and buckling. AIAA J 1989;27(7):923-35.
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[7] Harris GZ. The buckling and post-buckling behavior of composite plates under biaxial loading. Int J Mech Sci 1975;17:187-202. [8] Leissa AW. Conditions for laminated plates to remain flat under inplane loading. Compos Struct 1986;6(4):261–70.
[9] Leissa AW. An Overview of Composite Plate Buckling. In Compos Struct 4. edited by IH Marshall. Elsevier Appl Sci London, 1987;1-29. [10]
Turvey GJ, Marshall IH, editors. Buckling and postbuckling of composite plates.
Springer Sci and Business Media; 1995. [11]
Shen HS. Thermal buckling and postbuckling of laminated plates. In: Shanmugam NE,
Wang CM, editors. Analysis and design of plated structures- Vol 1: Stability, Woodhead publishing, Cambridge England; 2006, p. 170-211.
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Zienkiewicz OC. The Finite Element Method, 3rd ed. Tata McGraw-Hill publishing;
1971. Reddy JN. Mechanics of laminated composite plates and shells- theory and analysis. 2nd
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[13]
ed. CRC Press; 2004.
Grover Neeraj, Singh BN, Maiti DK. New nonpolynomial shear-deformation theories for
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[14]
structural behavior of laminated-composite and sandwich plates. AIAA J 2013;51(8):1861-71. Tauchert TR. Thermal buckling of thick antisymmetric angle-ply laminates. J Therm
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[16]
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Stresses 1987;10:113-24.
Tauchert TR, Huang NN. Thermal buckling of symmetric angle-ply laminated plates. In:
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[17]
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laminated plates. J Comput Struct 1989;32(5):1117-24. Singha MK, Ramchandra LS, Bandyopadhyay JN. Thermal postbuckling analysis of
[19]
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laminated composite plate. J Compos Struct 2001;54:453-8. Sreehari VM, Maiti DK. Buckling and post buckling analysis of laminated composite
plates in hygrothermal environment using an Inverse hyperbolic shear deformation theory. J Compos Struct 2015;129:250-5. [20]
Peck SO, Springer, GS. The behavior of delaminations in composite plates—analytical
and experimental results. J Compos Mater 1991;25:907-29. [21]
Wadee MA, Vollmecke C. Semi-analytical modelling of buckling driven delamination in
uniaxially compressed damaged plates. IMA J Appl Math 2011;76(1):120-45. [22]
Kucukler M, Gardner L, Macorini L. Lateral-torsional buckling assessment of steel
beams through a stiffness reduction method. J Constr Steel Res 2015;109:87-100. [23]
Valliappan S, Murti V, Wohua Z. Finite element analysis of anisotropic damage
mechanics problems. Eng Fract Mech 1990;35(6):1061-71.
ACCEPTED MANUSCRIPT [24]
Martha LF, Wawrzynek PA, Ingraffea AR. Arbitrary crack representation using solid
modelling. Eng with Comput 1993;9(2):63-82. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing.
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[25]
Int J Numer Meth in Eng 1999;45(5):601-20.
Ladeveze P, LeDantec E. Damage modelling of the elementary ply for laminated
composites. Compos Sci and Tech 1992;43(3):257-67.
Talreja R. A continuum mechanics characterization of damage in composite materials. In
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[27]
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[26]
[28]
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Proc R Soc of Lond Ser A, Math and Phys Sci 1985;399(1817):195-216. Murami V, Upadhyay C. Towards a generalized macro-level damage model for
unidirectional composites. Adv Mat Res 2008;47:869-72. Pidaparti R. Free vibration and flutter of damaged composite panels. J Compos Struct
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[29]
[30]
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1997;38(1-4):477-81.
Wohua Zhang, Chen Yunmin, Jin Yi. Effects of symmetrisation of net-stress tensor in
[31]
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anisotropic damage models. Int J Fract 2001;109:345–63. Wohua Zhang, Yuanqiang Cai. Continuum damage mechanics and numerical
applications. Zhejiang University Press. Springer; 2010. [32]
Prabhakara DL, Datta PK. Vibration and static stability characteristics of rectangular
plates with a localized flaw. J Compos Struct 1993;49(5):825-36. [33]
Rahul R, Datta PK. Static and dynamic characteristics of thin plate like beam with
internal flaw subjected to in-plane harmonic load. Int J Aeronautical and Space Sci 2013;14(1):19-23.
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Table captions
Table 1. Non dimensionalized buckling load for varied (a/h) ratios.
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Table 2. Comparison of non dimensional buckling load for (5/-5/5/-5) plates.
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Table 3. Buckling coefficients of a damaged composite plate for various aspect ratios. Table 4. Non dimensional buckling loads for different damage intensities.
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Table 5. Critical buckling temperatures for undamaged and damaged composite plates.
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Figure captions
Fig. 2. Lamina thickness.
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Fig. 1. Eight noded iso-parametric element.
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Fig. 3. Dimensions of the plate.
Fig. 4. Mesh Discretization with damage in center 4 elements. Fig. 5. Convergence of non dimensional critical buckling load. Fig. 6. Influence of damage area on the transverse central deflection. Fig. 7. Buckling load variation of a centrally damaged plate, with individual damage parameters Γ1 and Γ2, while other is kept fixed at 0.1. Fig. 8. Buckling load variation as damage area increases. Fig. 9. Critical buckling loads for symmetric laminates under uniaxial and biaxial compression for undamaged and damaged cases. Fig. 10. Critical buckling temperature for 4 layered symmetric composite plate for undamaged and damaged cases.
ACCEPTED MANUSCRIPT Fig. 11. Effect of internal flaw in thermal post buckling response for a/h = 50. Fig. 12. Effect of internal flaw in thermal post buckling response for a/h = 100.
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Fig. 13. Effect of internal flaw in thermal post buckling response for E1/E2 = 20.
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Fig. 14. Effect of internal flaw in thermal post buckling response for E1/E2 = 40. Fig. 15. Effect of internal flaw in thermal post buckling response for α2/α1 = 5.
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Fig. 16. Effect of internal flaw in thermal post buckling response for α2/ α1 = 10.
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Present
Thick plate (a/h = 5) Moderate thick plate (a/h = 50) Thin plate (a/h = 100)
11.92
FSDT Reddy [13] 11.58
35.35
35.36
35.96
35.96
HSDT Reddy [13] 11.99
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Side-to-thickness ratios
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Table 1 Non dimensionalized buckling load for varied (a/h) ratios.
35.35
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35.95
Table 2 Comparison of non dimensional buckling load for (5/-5/5/-5) plates.
10.67 20.99 32.04
n=6 Present Reddy [13] 11.38 11.08 22.73 22.59 36.26 36.33
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Reddy
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5 10 100
n=2 Present [13] 10.97 21.08 32.01
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a/h
Table 3 Buckling coefficients of a damaged composite plate for various aspect ratios. AR 0.8 1 1.6
Present 3.48 4.32 1.71
Prabhakara and Datta [32] 3.59 4.45 1.77
Variation 3.06% 2.92% 3.38%
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Γ1/Γ2 = 2
Γ1/Γ2 = 3
22.67
21.02
50
20.52
18.67
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Elemental damage area (in percentage) 40
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Table 4 Non dimensional buckling loads for different damage intensities.
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Table 5 Critical buckling temperatures (in 0C) for undamaged and damaged composite plates. Simply Supported Source
63.14
Singha et al. [18]
63.26
--
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Present
Damaged Γ1/Γ2 = 3 60.06
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Undamaged
Γ1/Γ2 = 7 55.86 --
Clamped Undamaged 167.23 167.85
Damaged Γ1/Γ2 = 3 165.56 --
Γ1/Γ2 = 7 161.53 --
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Fig. 1. Eight noded iso-parametric element.
Fig. 2. Lamina thickness.
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Fig. 3. Dimensions of the plate.
Fig. 4. Mesh Discretization with damage in center 4 elements.
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a/h=10
30
a/h=100
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35
25
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20 15 10
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Non dimensional buckling load
40
5 0 0
2
4
6
8
10
12
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Mesh size
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Fig. 5. Convergence of non dimensional critical buckling load.
Maximum transverse deflection (mm)
6 5 4
E1/E2 = 10
3
E1/E2 = 40
2 1 0 0
10
20
30
40
50
60
Percentage of defect area
Fig. 6. Influence of damage area on the transverse central deflection.
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35
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30 25 20 15
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Variation of Γ2; Γ1=0.1
10
Variation of Γ1; Γ2=0.1
5
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Non dimensional buckling load
40
0 0
0.2
0.4 0.6 Damage Parameter Γ1, Γ2
0.8
1
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Fig. 7. Buckling load variation of a centrally damaged plate, with individual damage parameters Γ1 and Γ2, while other is kept fixed at 0.1.
Non dimensional buckling load
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25
Uniaxial
20
Biaxial
15
10
5
0 0
10
20
30
40
50
Percentage of defect area
Fig. 8. Buckling load variation as damage area increases.
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20
15
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10
5
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Non dimensional buckling loads
25
0 20
40
60 a/h
80
100
120
D
0
uniaxial (undamaged case) uniaxial (damaged case) biaxial (undamaged case) biaxial (damaged case)
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Fig. 9. Critical buckling loads for symmetric laminates under uniaxial and biaxial compression for undamaged and damaged cases.
88
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86
Undamaged
84
Damaged
82 80
Tcr
78 76 74 72 70
0
10
20
30
40
50
E1/E2
Fig. 10. Critical buckling temperature for 4 layered symmetric composite plate for undamaged and damaged cases.
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6 a/h = 50
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5
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4 T/Tcr 3 2
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Without Damage With Damage
1
0
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0 0.2
0.4
0.6 w/h
0.8
1
1.2
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D
Fig. 11. Effect of internal flaw in thermal post buckling response for a/h = 50.
2.5
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2
a/h = 100
1.5
T/Tcr
1
Without Damage
0.5
With Damage
0 0
0.2
0.4
0.6
0.8
1
w/h
Fig. 12. Effect of internal flaw in thermal post buckling response for a/h = 100.
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2.5 E1/E2 = 20
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2
T/Tcr 1
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1.5
Without Damage 0.5 0 0.2
0.4
0.6 w/h
0.8
1
1.2
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0
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With Damage
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Fig. 13. Effect of internal flaw in thermal post buckling response for E1/E2 = 20.
2.5
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2
E1/E2 = 40
1.5
T/Tcr
1
Without Damage
0.5
With Damage
0 0
0.2
0.4
0.6 w/h
0.8
1
1.2
Fig. 14. Effect of internal flaw in thermal post buckling response for E1/E2 = 40.
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2.5
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α2/ α1 = 5 2
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1.5 T/Tcr 1
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Without Damage
0.5
With Damage 0 0.2
0.4
0.6 w/h
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0
0.8
1
1.2
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Fig. 15. Effect of internal flaw in thermal post buckling response for α2/α1 = 5.
2.5
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2
α2/ α1 = 10
1.5
T/Tcr
1
Without Damage
0.5
With Damage 0 0
0.2
0.4
0.6
0.8
1
1.2
w/h
Fig. 16. Effect of internal flaw in thermal post buckling response for α2/ α1 = 10.
ACCEPTED MANUSCRIPT Highlights
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Analysis of buckling and post buckling of composite plates with damage using FEM. Governing equations are based on inverse hyperbolic shear deformation theory. Effect of damage is simulated by an anisotropic damage formulation. Critical buckling temperature for a damaged composite plate has been obtained. Effect of mild damage on thermal post buckling paths is presented.
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● ● ● ● ●
S2352-0124(16)00011-4 doi: 10.1016/j.istruc.2016.01.002 ISTRUC 84
To appear in: Received date: Revised date: Accepted date:
24 October 2015 18 January 2016 18 January 2016
Please cite this article as: Sreehari VM, Maiti DK, Buckling and post buckling characteristics of laminated composite plates with damage under thermo-mechanical loading, (2016), doi: 10.1016/j.istruc.2016.01.002
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ACCEPTED MANUSCRIPT Buckling and post buckling characteristics of laminated composite plates with damage under thermo-mechanical loading
Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, WB- 721302, India
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a,b
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Sreehari. V. Ma * and Maiti. D. Kb
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ABSTRACT
The present work addresses the development of a finite element formulation for handling
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bending, buckling, and post buckling analysis of composite laminated structures with damage. The inverse hyperbolic shear deformation theory (IHSDT) was applied in the finite element formulation. The effect of damage is analysed for thin composite plates. An anisotropic damage
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formulation was used to simulate the damage, which is based on the concept of stiffness
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reduction. Computer programming is developed in the MATLAB environment. The excellent agreement of the results obtained in the present method with those from references shows that
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the technique is effective and precise. Parametric studies in the buckling behaviour of a damaged composite plate are presented. Critical buckling temperatures are computed for a damaged plate using the present model. Thermal post buckling equilibrium paths are traced for various parametric variations for composite plates with mild damage and compared the results with that of undamaged cases. The validation of IHSDT has been demonstrated for buckling analysis in thermal environment for composite plates with an internal flaw. The present work is worthwhile compared with previous works due to the choice of finite element method and inverse hyperbolic shear deformation theory for analyzing the influence of damage on buckling and post buckling behavior of laminated plates.
*
Corresponding author: Tel.: +918001581060 Address:Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, WB- 721302, India. E-mail address: [email protected]
ACCEPTED MANUSCRIPT Keywords: Composite plate; Finite element method; Shear deformation theory; Buckling load; Thermal post buckling; Anisotropic damage.
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1. Introduction
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Composite plates play an essential role in current and future aerospace, civil and mechanical structural system due to their improved performance. The diverse problems in
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structural analysis necessitate the proper knowledge of structural behaviour. Hence, the necessity of a thorough study of buckling behaviour of composite structures using the finite element
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method considering different features increases the interest. Ozben Tamer [1] had reported the critical buckling load of fiber reinforced composite plate. Choudhary and Tungikar [2] presented Finite Element Analysis for geometrically nonlinear behaviour of laminated composite plates.
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Chai and Hoon [3] have conducted studies on the effect of the various in-plane loads on laminated composite plates. Onkar et al. [4] studied the buckling of laminated composite plates by stochastic finite element analysis. Bao and Jiang [5] presented a critical review of bending and
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buckling of flat, rectangular, orthotropic thin plates. Kapania and Raciti [6] presented a review of laminated structures, highlighting shear effects and buckling. Many researchers [7-9] had discussed the effects of bending-stretching coupling matrix in the case of a regular, antisymmetric (i.e., not generally unsymmetric), angle-ply plate, and obtained buckling loads for such problems. In addition to buckling analysis, a lot of significant investigations were carried out on post buckling behavior of laminated plates by numerous researchers [10, 11] and they comprehensively presented the state of knowledge in this field. Development of accurate theoretical models is necessary to predict the structural responses in composite laminated plates. Increased use of composites in primary structures in the past few years has led to the development of a variety of plate theories by researchers. Even though three dimensional elasticity solutions predict structural behavior accurately, its higher computational complexities
ACCEPTED MANUSCRIPT led to the developments of two dimensional plate theories. Among the two dimensional theories, equivalent single layer theories are computationally efficient and they include Classical
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Laminated Plate Theory (CLPT), First order Shear Deformation Theory (FSDT) and higher order shear deformation theories. In the past decades, researchers [12, 13] have established various
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higher order polynomial shear deformation theories built on equivalent single layer theory. Over the last few years, several shear deformation theories having non-polynomial nature and
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expressed in terms of shear-strain function have been proposed. Recently, Grover et al. [14]
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proposed new nonpolynomial shear deformation theories (NPSDT). Regarding to the thermal buckling, it is important to explore few significant works done in this area. Tauchert [15], Tauchert and Huang [16] developed the Navier type solution for composites with antisymmetric
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and symmetric ply orientations under thermal loadings. Thangaratnam et al. [17] employed
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FSDT and found critical buckling temperature. Many researchers [18, 19] have conducted structural analysis under thermal environment using finite element method.
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Even though such structures are highly reliable, internal flaws are almost inevitable at some phase of its operative life span. The occurrence of damage significantly affects the consistent operation of the structure. The early detection of structural failure is one of the major tasks in the aerospace industry. Damage may be due to some unfavourable conditions and leading to the shortening of operating life of the structures. Effective methods are to be developed to find the effects of damage in any structural parts. Peck and Springer [20] assumed delamination to occur between two nearby laminates and perfect bonding between all other laminates. Wadee and Vollmecke [21] obtained the same decrement in stiffness from the defect irrespective of its depth and observed stable post buckling behaviour when delamination growth was not considered. It is important to note that no growth was witnessed at the first critical load for the global buckling example. They found out the transitional depth ratio to be located
ACCEPTED MANUSCRIPT between 10 and 20 % of the depth of the plate, where the mode of buckling changes and buckling-driven delamination growth only occurs where the pre-existing delaminations are closer
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to the surface only. In the present analysis, damage propagation is not included. The damage may be due to cracks, fatigue, corrosion, or caused during manufacturing processes. Damage process
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is the one which causes weakening in mechanical properties. An anisotropic damage formulation is used in this study to simulate the damage, which is based on the concept of stiffness reduction.
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The concept of stiffness reduction was utilized by Kucukler et al. [22] for evaluating the lateral-
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torsional buckling of steel structures. A continuous parameter that can be correlated to the density of defects is presented in general continuum damage mechanics. This technique is useful in describing the weakening of the material before the initiation of micro-cracks. Valliappan et
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al. [23] developed the elastic constitutive relationship for anisotropic damage mechanics and
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explained the employment of these constitutive relations in finite element analysis. Martha et al. [24], Belytschko and Black [25] explained certain damage incorporated problems by “hard-
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wiring” the damage as a portion of the geometry of the structure. Yet, a more useful parametric modeling of damage is found in a variety of literatures [26-28]. Pidaparti [29] found out the free vibration characteristics of a composite plate and established that the methodology by Valliappan et al. has more effect than some other formulations. Zhang et al. [30] also appreciated this method and concluded that symmetrisation schemes do not produce as good agreement with experimental results as demonstrated by the non-symmetrical model and hence, whenever possible the symmetrisation treatment should be avoided in anisotropic damage modeling. Zhang and Cai [31] investigated continuum damage mechanics and used appropriate damage variable to represent the macroscopic effects of microscopic damages and developed elastic constitutive models. However, most studies [32, 33] on free vibration and static stability behaviour with zones of damage were restricted to simple beams and plates than composites.
ACCEPTED MANUSCRIPT It is noted that very little attention has been given to the mechanism of failure and its influence on the response behaviour of the structure under load. Despite the fact that literature is
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available on structural responses, no such literature is available on the effects of region of damage in buckling and thermal post buckling of laminated composite plates using FEM
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considering anisotropic damage formulation. The aim of the present paper is to establish accurate theoretical model necessary to predict buckling responses in damaged composite structures. In
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the present investigation, authors’ purpose is to study the bending, buckling, and post buckling
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behaviour of laminated composite plates with confined regions of damage using FEM where the governing equations are developed based on Inverse Hyperbolic Shear Deformation Theory (IHSDT). Grover et al. [14] demonstrated by comparing to other prevailing shear deformation
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theories that new NPSDTs have only a small total percentage error. They demonstrated the
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efficiency and enhanced performance of IHSDT compared to similar existing higher order shear deformation theories involving shear strain function. This bending theory describes the stresses
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and deflections more accurately and authors have explained its potential impact in buckling analysis previously [19]. The effort of solving higher order differential terms in the displacement field equations has been removed because IHSDT provides us with a non-polynomial displacement field equation and thereby helps to analyze complex problems with reduced computational complexity. This theory fulfils zero transverse shear stress at the top surface and at the bottom surface of the plate and the transverse shear stress distribution is non-linear across the thickness. In the current work, the finite element damage formulation considers effects of transverse shear, geometric non linearity of composite plates, thermal effect, etc. while predicting buckling and we have assumed that perfect bonding exists between all laminates. The presented methodology provides an alternative way to understand the effect of damage on the stability of the composite. The present work is valuable compared with previous works due to the selection of finite element method and inverse hyperbolic shear deformation theory for analyzing the
ACCEPTED MANUSCRIPT impacts of damage on buckling and post buckling behavior of laminated composite plate. The outcomes from this investigation will benefit to address some remarkable problems relating to
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composite structures. Once we have clearly determined the effect of damage, we can reduce the deterioration of properties in composite structures. For example, we can use methods such as
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usage of optimized ply orientations in design stage or employment of smart patches later to
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reduce the effects of damage. 2. Mathematical modeling
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2.1. Element selection
A finite element formulation is developed for handling bending and buckling analysis of laminated composite plates with damage. The fundamental idea of finite elements is that the
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structure is considered as an assembly of elements connected at nodes. We are considering an 8-
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noded iso-parametric quadrilateral serendipity element (as shown in Fig. 1) with C0-continuity and with 56 degrees of freedom in the present work. An isoparametric element has an advantage
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that element geometry and displacements are represented by same set of shape functions. The benefit of 8 noded element is that all the nodes are located on element sides and hence there are no internal nodes and shape functions have quadratic variation in x and y direction. The shape functions used are,
N1 0.25(1 )(1 )(1 );
N 5 0.5(1 )(1 2 )
N 2 0.5(1 2 )(1 );
N 6 0.25(1 )(1 )(1 )
N 3 0.25(1 )(1 )(1 );
N 7 0.5(1 2 )(1 )
N 4 0.5(1 )(1 2 );
N8 0.25(1 )(1 )(1 )
2.2. Displacement field The chosen displacement field for structural analysis of laminated composite plate is on the basis of IHSDT. A laminated plate containing N orthotropic layers is considered as in Fig. 2. The
ACCEPTED MANUSCRIPT thickness, length, and width of plate are h, a, and b respectively as shown in Fig. 3. The principle
Displacement model given in Eq. (1) was proposed in Ref. [14]. w0 sinh 1 ( rz / h) - z h x
( x, y ) x 2 r 4
v( x, y, z ) v0 ( x, y ) - z
w0 sinh 1 ( rz / h) - z h y
( x, y ) y 2 r 4
(1)
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2r
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w( x, y, z ) w0 ( x, y )
2r
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u ( x, y , z ) u0 ( x, y ) - z
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material co-ordinates ( x1k , x2k , x3k ) of the kth lamina is at an angle k to the general x co-ordinate.
In the above displacement field, u0, v0, and w0 are the midplane displacements while θx, θy are the shear displacements. The transverse shear stress factor, represented by r, is fixed as 3 per
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the reverse technique in post processing phase shown in Ref. [14]. The continuity requirement
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can be changed from C1 to C0 for simplification. It can be done by taking
w w and as distinct x y
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independent degrees of freedom as: w w x and y x y
(2)
For the finite element analysis total potential energy is given as, U
NE
U
(e)
(3)
e 1
For obtaining the total potential energy, the elemental elastic stiffness matrix is given as, N
N
K ( e ) wp wq BT D B det J p 1 q 1
(4)
where wp , wq are the weights used in the Gaussian quadrature numerical integration method, det
ACCEPTED MANUSCRIPT D J denotes the determinant of Jacobian matrix, and is the stress-strain stiffness matrix. [B] is
0
0
y
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
N i 0
N i
0
0
0
0
0
0
0
0
0
N i
0
N i
0
N i
N i
x
N i
x
0
y
x
N i
0
N i
0
0
0
N i
y
x
y
Ni
0
Ni
0
0
0
0
0
0
0
0
0
Ni
h
r 2 4
0
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0
y
D
0
y
0
x
where i = 1, 2, 3,..8 and Ω from Eq. (1) is
0 0 0 N i y N i x 0 N i y N i x 0 0 Ni 0 0
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N i
0
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N i
0
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0
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N i x 0 N i y 0 0 0 [ B] 0 0 0 0 0 0 0
PT
the matrix containing derivatives of shape functions as shown below.
- 2r
2.3. Strain-displacement relations The displacement w produces some additional extension in the x and y directions of the middle surface resulting in von Karman strain-displacement relationship as given below,
ACCEPTED MANUSCRIPT u 1 w x 2 x
2
x
v 1 w y 2 y
2
y
u v w w y x x y u w z x v w z y
(5)
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yz
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xz
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xy
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The above relation can be written in matrix form as, 0 L in which the first term is the linear expression and second gives the non-linear terms. Similar to the equation of linear stress, we can write for non-linear components as
d L BL q
, where {q}
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represents the non-linear components of displacement vector for an element.
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The non-linear strain components can be also written as, 0 w w x 1 A I y w 2 w y x
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w x 1 L 0 2 w y
The derivatives of w can be related to the nodal parameters as
(6)
I Gq. Now, the
geometric stiffness matrix is obtained as in Eq. (7) by following the procedure described by Zienkiewicz [12].
K
G V
T
Tx T xy
Txy G dV Ty
Where Tx, Ty, and Txy represents the in-plane loads.
(7)
ACCEPTED MANUSCRIPT 2.4. Stress-strain relations
Q16
0
Q22
Q26
0
Q62
Q66
0
0
0
Q44
0
0
Q54
0 x - x T 0 y - y T - T 0 xy xy xz Q45 yz Q55
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Q12
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x Q11 Q 21 y xy Q61 xz 0 yz 0
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relations for a lamina about any arbitrary axes is given by
PT
While investigating the buckling behaviour in thermal environment, the stress-strain
(8)
Where α represents the coefficient of thermal expansion and is given by,
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where m cos , n sin
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αx= α1m2 + α2n2 ; αy= α2m2 + α1n2 ; αxy= 2 (α1-α2) mn
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2.5. Buckling and post buckling behaviour As a result of buckling, an eigenvalue problem can be formulated as shown below, ([K] +λ [Kσ]) {Δ} = {0}
(9)
where [K] is the global stiffness matrix, [Kσ] is the global geometric stiffness matrix, λ is the in-plane magnification factor and {Δ} is the global displacement field. The eigenvalue problem is solved to extract minimum eigenvalues or critical buckling loads. Once the buckling condition is attained, the nonlinear stiffness matrix [Knl] is included into the formulation (Zienkiewicz [12]) as,
([K] + [Knl] +λ [Kσ]) {Δ} = {0}
(10)
ACCEPTED MANUSCRIPT
Equation (10) demonstrates the nonlinear equilibrium for laminated composite plate. Small
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displacement rise compared to previous equilibrium state is considered and Eq. (10) is solved as an eigenvalue problem, for finding the thermal post buckling response path. As demonstrated by
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Singha et al. [18], a sequence of eigenvalue problems is solved to find the thermal post buckling
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path.
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2.6. Introducing the effects of damage
In this paper, we are considering damage located at the centre of a composite plate. The damaged area is a patch having a size 4% of the whole plate area as in Fig. 4. The damage in
D
composites is anisotropic in nature. Anisotropic damage is the damage which is scattered in
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numerous orthogonal directions and the amount of damage in any orthogonal direction is independent of the other. In the present analysis, the consequence of an internal flaw is computed by means of an anisotropic damage formulation, which is established on the concept of stiffness
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change. Using an idealized model having a reduced elastic property in the damage region, the effects of a region of damage are incorporated into the formulation. Thus the effect of internal flaw is parametrically included into the previous formulation (for undamaged composite) using a parameter Γf. In essence, this is an effective parameter which represents reduction in effective area. The parameter is shown in Eq. (11).
f
Af - A*f
(11)
Af
Where f = 1, 2 and it denotes the in-plane orthogonal directions and Af* is the effective area of plate with damage. Γ1 denotes the damage in the fibre direction while Γ2 denotes the damage in the orthogonal direction in the same plane. For example, in a (0/90) layup, the fibre is
ACCEPTED MANUSCRIPT aligned in co-ordinate axis 1 in the top layer and orthogonal here implies a co-ordinate axis 2, (i.e., in a direction 90 degree to the previous). In the second/bottom layer, the fibre is aligned in
PT
co-ordinate axis 2, and there the orthogonal implies co-ordinate axis 1 direction. This technique of modeling damage using parameters in any anisotropic material was suggested by Valliappan
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et al. [23]. A stress-strain stiffness matrix for a two dimensional damaged laminate using this
*
* * D
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US
formulation is in Eq. (13).
Where,
D
1- 1 1- 2 E 1- 1221 2 12 2 1- 2 E 1- 1221 2
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2 1- 1 E 1 1- 1221 1- 1 1- 2 E 1 21 1- 12 21 D* 0 0 0
(12)
0
0
0
0
0
1- 1 1- 2 1- 1221 1- 1221 2 G12 2 2 1- 1 1- 2 1- 1221 1- 1221
0
0
1- 2
0
0
0
2
0 0 0 2 1- 1 G13 0
2
0
2
G23
(13)
A stress-strain matrix for a laminate having internal flaw is evaluated and further on proceeding as in undamaged case formulation, an eigenvalue problem for a composite plate having an internal flaw can be obtained.
ACCEPTED MANUSCRIPT 3. Numerical results and discussion
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3.1. Problem definition The structures studied here are mostly square plates, length (a) = breadth (b), having the
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thickness ratio = 100. The damaged area is a patch having a size 4% of the whole plate area as in Fig. 4, located at the center of the plate with each element having a particular damage ratio.
US
Numerous parametric studies have been conducted for the composite plate having damage. The
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boundary conditions employed in the analysis are ‘all edges clamped’ and ‘all edges simply supported’. Computer programming is developed in MATLAB environment. The numerical results were obtained and compared with the available literature. Transverse deflection and
a2 Pnd Pcr 3 E2 h
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100h3 E2 w w and 4 q a 0
D
critical buckling load are non dimensionalised using the relations given in Eq. (14).
(14)
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The various Material Models used are (taken from the literature from which comparison studies are done),
Material Model 1 (MM1): E₁/E₂ = 25, G₁₂ = G₁₃ = 0.5 E₂, G₂₃ = 0.2 E₂, v₁₂ = 0.25. Material Model 2 (MM2): E₁/E₂ = 40, G₁₂ = G₁₃ = 0.6 E₂, G₂₃ = 0.5 E₂, v₁₂ = 0.25.
3.2. Convergence and validation study Convergence of the solution with a mesh refinement for a laminated composite plate having laminate sequence (0/90/90/0) with a/h ratio 10 and 100 is shown in Fig. 5. As the fineness of mesh increases the convergence of the results is found to be fairly accurate. Based on convergence study, a (10 × 10) mesh has been used in most cases of later study. For validation purposes, some of the results of buckling analysis from present work are compared with the
ACCEPTED MANUSCRIPT results available in literature [13] for the undamaged composite plate. The excellent agreement of the results obtained in the present method with those from references shows that the technique is
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effective and precise. The comparison of uniaxial critical buckling loads with various side-tothickness ratios is computed for a four layer cross-ply plate (0/90/90/0) as in Table 1. Material 2
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properties were taken for analysis. Results in Table 1 validate the present finite element formulation as well as Matlab programming. Table 2 shows the comparison of the non
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dimensional buckling loads obtained by the present analysis with the reference [13] values for a
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square, laminated composite plate for in-plane uniaxial loading condition. The material 2 property is considered in the analysis. Analysis is performed for four-layer (n = 4) antisymmetric angle ply plates. Analysis is carried out for a laminated composite plate having laminate
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sequence (5/-5/5/-5). It is clearly evident from the results that, IHSDT can be successfully used
3.3. Effects of damage
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in the analysis of laminated composite plates.
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After incorporating the effects of damage, some results for a simple plate were compared with the results obtained by Prabhakara and Datta [32]. The present model studies with conventional small defects or damage. A plate with centre damage (damage at centre location in terms of length/ width and which occurs throughout the thickness) is considered. We have introduced the damage parameters in the centre 4 elements (intensity of damage per element will be given wherever necessary) for a plate with 10×10 mesh size. The buckling coefficients for a plate with 3 different aspect ratios are found and compared with those of reference as shown in Table 3, and validates the finite element formulation incorporating damage and the methodology.
ACCEPTED MANUSCRIPT 3.3.1. Effect of internal flaw in bending A laminated composite plate having four layers (0/90)s with central damage is analysed
PT
and central transverse deflections are found. We have obtained the central transverse deflection
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of the whole plate at specific percentages of elemental damage areas. The results obtained are shown in Fig. 6. For plates having higher E1/E2, the deflection is found to be less. As the
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3.3.2. Orthogonality of damage parameter
US
percentage of damaged area increases in an element, the central deflection increases also.
As the model of damage being considered is anisotropic in nature, it is vital to understand the nature of orthotropy shown by this formulation. It may be inferred that the contribution of
D
damage in the direction of fiber, 1 may not be equal to that in the orthogonal direction to the
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fiber, 2 . In order to know this, one of these orthogonal parameters must be fixed while the other must be varied to obtain the characteristics of the structure. Change in the buckling load of a
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composite plate with central damage, (0/90/0) ply orientation, with variation of one of the damage parameters while the other is fixed as shown in Fig. 7. It can be noticed that damage in the direction of fiber, i.e., 1 , shows a steeper descent of buckling features when compared to the influence of 2 . Owing to this, for all further computational purposes, the value of 2 is fixed at 0.1, while the intensity of damage is denoted by the damage ratio, Г₁/Г₂. The damage ratio can take values from 0 to 9. Mild damage can be indicated by a damage ratio between 0 and 3 whereas heavy damage can be indicated by a damage ratio between 7 and 9. 3.3.3. Effect of internal flaw in buckling under mechanical load Composite plates with internal flaw are analysed for uniaxial as well as biaxial loading conditions (for material 1 property, simply supported at all ends) and the results observed are
ACCEPTED MANUSCRIPT plotted in Fig. 8. Buckling loads are found at particular percentages of elemental damage areas using the present anisotropic damage modeling. It is seen that the critical buckling load decreases
PT
as the damage area increases.
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The variation of non dimensional buckling load for thin composite plates under uniaxial loadings with damage patches in the central region having damage intensities of Γ1/Γ2 = 2 and
US
Γ1/Γ2 = 3 is presented in Table 4. It is found from the present analysis that the internal flaws will
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significantly affect the buckling characteristics of laminated composite plates. Critical buckling loads of symmetric (0/90/90/0) laminates under uniaxial and biaxial compression for damaged and undamaged cases are shown in Fig. 9. Analysis is conducted with side to thickness ratio
D
varying from 10 to 100. For this we have taken a laminated composite plate with simply
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supported at all ends and for material 1 property. For damaged cases the nondimensional buckling loads are less than those from undamaged cases in both uniaxial and biaxial loadings. It is found that critical loads are higher for uniaxial loading. Furthermore as thickness decreases,
cases.
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the nondimensional buckling load increases. The variation is much less above a/h = 40 for all
3.3.4. Effect of internal flaw in critical buckling temperature Buckling as well as post buckling study is performed using IHSDT for laminated composite plates with central damage exposed to constant in-plane thermally produced loadings. The critical buckling temperature of damaged plate has been obtained and compared with those of the undamaged case. The buckling temperature is found for an isotropic square plate with α = 2x10-6 oC-1 and a/h = 100, for both ‘all edges simply supported’ and ‘all edges clamped’ boundary conditions. Computations performed for two damage intensities, Γ1/Γ2 = 3 and Γ1/Γ2 = 7, are given. The results are presented in Table 5.
ACCEPTED MANUSCRIPT The critical buckling temperature is found and plotted for a square symmetric cross-ply (0/90/90/0) laminated composite plate with and without damage in Fig. 10. The plate considered
PT
is under simply supported boundary conditions and uniform temperature. As E1/E2 increases from 10 to 40, the critical buckling temperature decreases. The decrement is higher for damaged
US
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plates, as observed from the steeper curve of the damaged case.
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3.3.5. Effect of internal flaw in thermal post buckling
The thermal post buckling equilibrium path is traced for composite plates with an internal flaw and compared the results with that of the undamaged case for various parametric variations.
D
A square laminated plate of 4-layers (45/-45/45/-45) with simply supported ends is considered
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for the analysis. Studies are presented for a/h = 50, a/h = 100, E1/E2 = 20, E1/E2 = 40, α2/ α1 = 5, and α2/α1 = 10 (Tcr = (Tcr)a/h=
100
for Fig. 11 and similarly for further figures) and the results
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obtained are shown in Figures 11 to 16. For a mild damage with Г₁/Г₂ < 3, there will not be much change in the flaw size under loading as it will be in the elastic region. The post-buckling predictions presented here are with damage ratio 1. In Figs 11 and 12, thermal post buckling equilibrium paths are traced for two different a/h conditions, a/h = 50 and a/h = 100, respectively. It can be observed from the plot that the critical buckling temperature increases as a/h decreases. Similar analysis can be performed (Figs 13 and 14) using Young’s modulus ratio instead of span-to-thickness ratio for obtaining the thermal post buckling equilibrium path. However, here as E1/E2 increases, the buckling temperature also increases. From any of the above plots it can be understood that for the same loading, higher displacements are observed in damaged composite plates compared to undamaged plates. As the load increases, the difference in displacements with and without internal flaw cases increases.
ACCEPTED MANUSCRIPT For thin plates, the difference is seen mostly at a later post buckling state. The difference in displacements with and without internal flaw is more when E1/E2 increases. The thermal post
PT
buckling equilibrium path is traced for composite plates with an internal flaw and compared the results with that of undamaged case as shown in Fig. 15 and Fig. 16 for α2/ α1= 5 and α2/α1 = 10
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respectively. As the ratio of thermal expansion coefficients α2/α1 increases, the buckling temperature decreases. Either from Fig. 11 or Fig. 12, it can be understood that for same loading,
US
higher displacements are observed in damaged composite plates compared to undamaged plates.
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As load increases, the difference in displacements with and without internal flaw cases increases. For α2/α1 = 5 case, the difference is seen mostly at a later post buckling state. The difference in displacements with and without internal flaw is more when α2/α1 increases.
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D
4. Conclusion
Buckling and post buckling of a laminated composite plate with damage has been studied. An FEM formulation using inverse hyperbolic shear deformation theory for a composite plate
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having an internal flaw is implemented. Computer programming is developed in MATLAB environment. From the comparison studies, it is seen that the results obtained by the present technique are quite agreeable with reference solutions and this demonstrates the efficiency and the robustness of the method. It is also demonstrated through this investigation that the inverse hyperbolic shear deformation theory can be precisely used in the study of laminated composite structures with internal flaws. It is found from the present analysis that the internal flaws will significantly affect the critical buckling temperatures and post buckling paths of laminated composite plates. The other major contributions of present work are summarized in the following points.
ACCEPTED MANUSCRIPT
The displacement field taken in present analysis removed the difficulty of solving higher order differential terms in the displacement field equations by providing us with a
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nonpolynomial displacement field equation and helps to analyze under reduced
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computational complexity which is a great advantage. The damage model taken in this work has a wider scope of application due to the simplicity of stress-strain relations, i.e., constitutive relation for a region of damage retains its basic form. Transverse central
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deflections for a composite plate are determined for particular percentages of damage
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area and it is noted that as the percentage of damage area increases, the transverse central deflection increases.
The finite element damage formulation considered here (anisotropic damage formulation)
D
has a strong orthogonality. Damage in the direction of fibre affects the buckling behavior
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of composite plates more than damage in any other direction. It is observed that when the damage intensity rises, the buckling load reduces. Also as the
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percentage of damaged area increases, the buckling load decreases. Effects of material anisotropy, boundary condition, and span-thickness ratio on the buckling behaviour of a composite plate with an internal flaw are studied.
Critical buckling temperature of a damaged plate has been obtained and compared with those of undamaged case.
Thermal post buckling equilibrium path is traced for composite plates with a mild damage and compared the results with that of undamaged case. It is noticed that the variations in displacements increases for higher values of a/h, E1/E2, and α2/α1 values.
ACCEPTED MANUSCRIPT References [1] Ozben Tamer. Analysis of critical buckling load of laminated composites plate with different
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boundary conditions using FEM and analytical methods. Comput Mater Sci 2009;45:1006-15. [2] Choudhary SS, Tungikar VB. A simple Finite Element for non-linear analysis of composite
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plates. Int J Eng Sci Technol 2011;3:4897-907.
[3] Chai GB, Hoon KH. Buckling of generally laminated composite plates. Compos Sci Technol
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1992;45:125-33.
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[4] Onkar AK, Upadhyay CS, Yadav D. Generalized buckling of composite laminated plates using the stochastic finite element method. Int J Mech Sci 2006;48(7):780-9. [5] Bao G, Jiang W. Analytic and finite element solutions for bending and buckling of orthotropic
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rectangular plates. Int J Solids Struct 1997;34(14):1797-822.
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[6] Kapania RK, Raciti S. Recent advances in analysis of laminated beams and plates. Part ISheareffects and buckling. AIAA J 1989;27(7):923-35.
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[7] Harris GZ. The buckling and post-buckling behavior of composite plates under biaxial loading. Int J Mech Sci 1975;17:187-202. [8] Leissa AW. Conditions for laminated plates to remain flat under inplane loading. Compos Struct 1986;6(4):261–70.
[9] Leissa AW. An Overview of Composite Plate Buckling. In Compos Struct 4. edited by IH Marshall. Elsevier Appl Sci London, 1987;1-29. [10]
Turvey GJ, Marshall IH, editors. Buckling and postbuckling of composite plates.
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Shen HS. Thermal buckling and postbuckling of laminated plates. In: Shanmugam NE,
Wang CM, editors. Analysis and design of plated structures- Vol 1: Stability, Woodhead publishing, Cambridge England; 2006, p. 170-211.
ACCEPTED MANUSCRIPT [12]
Zienkiewicz OC. The Finite Element Method, 3rd ed. Tata McGraw-Hill publishing;
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[13]
ed. CRC Press; 2004.
Grover Neeraj, Singh BN, Maiti DK. New nonpolynomial shear-deformation theories for
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structural behavior of laminated-composite and sandwich plates. AIAA J 2013;51(8):1861-71. Tauchert TR. Thermal buckling of thick antisymmetric angle-ply laminates. J Therm
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[17]
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laminated plates. J Comput Struct 1989;32(5):1117-24. Singha MK, Ramchandra LS, Bandyopadhyay JN. Thermal postbuckling analysis of
[19]
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laminated composite plate. J Compos Struct 2001;54:453-8. Sreehari VM, Maiti DK. Buckling and post buckling analysis of laminated composite
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Valliappan S, Murti V, Wohua Z. Finite element analysis of anisotropic damage
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ACCEPTED MANUSCRIPT [24]
Martha LF, Wawrzynek PA, Ingraffea AR. Arbitrary crack representation using solid
modelling. Eng with Comput 1993;9(2):63-82. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing.
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[25]
Int J Numer Meth in Eng 1999;45(5):601-20.
Ladeveze P, LeDantec E. Damage modelling of the elementary ply for laminated
composites. Compos Sci and Tech 1992;43(3):257-67.
Talreja R. A continuum mechanics characterization of damage in composite materials. In
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[27]
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[26]
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Proc R Soc of Lond Ser A, Math and Phys Sci 1985;399(1817):195-216. Murami V, Upadhyay C. Towards a generalized macro-level damage model for
unidirectional composites. Adv Mat Res 2008;47:869-72. Pidaparti R. Free vibration and flutter of damaged composite panels. J Compos Struct
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[29]
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1997;38(1-4):477-81.
Wohua Zhang, Chen Yunmin, Jin Yi. Effects of symmetrisation of net-stress tensor in
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anisotropic damage models. Int J Fract 2001;109:345–63. Wohua Zhang, Yuanqiang Cai. Continuum damage mechanics and numerical
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Rahul R, Datta PK. Static and dynamic characteristics of thin plate like beam with
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Table captions
Table 1. Non dimensionalized buckling load for varied (a/h) ratios.
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Table 2. Comparison of non dimensional buckling load for (5/-5/5/-5) plates.
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Table 3. Buckling coefficients of a damaged composite plate for various aspect ratios. Table 4. Non dimensional buckling loads for different damage intensities.
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Table 5. Critical buckling temperatures for undamaged and damaged composite plates.
D
Figure captions
Fig. 2. Lamina thickness.
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Fig. 1. Eight noded iso-parametric element.
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Fig. 3. Dimensions of the plate.
Fig. 4. Mesh Discretization with damage in center 4 elements. Fig. 5. Convergence of non dimensional critical buckling load. Fig. 6. Influence of damage area on the transverse central deflection. Fig. 7. Buckling load variation of a centrally damaged plate, with individual damage parameters Γ1 and Γ2, while other is kept fixed at 0.1. Fig. 8. Buckling load variation as damage area increases. Fig. 9. Critical buckling loads for symmetric laminates under uniaxial and biaxial compression for undamaged and damaged cases. Fig. 10. Critical buckling temperature for 4 layered symmetric composite plate for undamaged and damaged cases.
ACCEPTED MANUSCRIPT Fig. 11. Effect of internal flaw in thermal post buckling response for a/h = 50. Fig. 12. Effect of internal flaw in thermal post buckling response for a/h = 100.
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Fig. 13. Effect of internal flaw in thermal post buckling response for E1/E2 = 20.
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Fig. 14. Effect of internal flaw in thermal post buckling response for E1/E2 = 40. Fig. 15. Effect of internal flaw in thermal post buckling response for α2/α1 = 5.
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Fig. 16. Effect of internal flaw in thermal post buckling response for α2/ α1 = 10.
ACCEPTED MANUSCRIPT
Present
Thick plate (a/h = 5) Moderate thick plate (a/h = 50) Thin plate (a/h = 100)
11.92
FSDT Reddy [13] 11.58
35.35
35.36
35.96
35.96
HSDT Reddy [13] 11.99
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Side-to-thickness ratios
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Table 1 Non dimensionalized buckling load for varied (a/h) ratios.
35.35
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35.95
Table 2 Comparison of non dimensional buckling load for (5/-5/5/-5) plates.
10.67 20.99 32.04
n=6 Present Reddy [13] 11.38 11.08 22.73 22.59 36.26 36.33
D
Reddy
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5 10 100
n=2 Present [13] 10.97 21.08 32.01
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a/h
Table 3 Buckling coefficients of a damaged composite plate for various aspect ratios. AR 0.8 1 1.6
Present 3.48 4.32 1.71
Prabhakara and Datta [32] 3.59 4.45 1.77
Variation 3.06% 2.92% 3.38%
ACCEPTED MANUSCRIPT
Γ1/Γ2 = 2
Γ1/Γ2 = 3
22.67
21.02
50
20.52
18.67
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Elemental damage area (in percentage) 40
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Table 4 Non dimensional buckling loads for different damage intensities.
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Table 5 Critical buckling temperatures (in 0C) for undamaged and damaged composite plates. Simply Supported Source
63.14
Singha et al. [18]
63.26
--
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Present
Damaged Γ1/Γ2 = 3 60.06
D
Undamaged
Γ1/Γ2 = 7 55.86 --
Clamped Undamaged 167.23 167.85
Damaged Γ1/Γ2 = 3 165.56 --
Γ1/Γ2 = 7 161.53 --
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PT
ACCEPTED MANUSCRIPT
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D
Fig. 1. Eight noded iso-parametric element.
Fig. 2. Lamina thickness.
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D
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US
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PT
ACCEPTED MANUSCRIPT
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Fig. 3. Dimensions of the plate.
Fig. 4. Mesh Discretization with damage in center 4 elements.
ACCEPTED MANUSCRIPT
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a/h=10
30
a/h=100
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35
25
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20 15 10
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Non dimensional buckling load
40
5 0 0
2
4
6
8
10
12
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D
Mesh size
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Fig. 5. Convergence of non dimensional critical buckling load.
Maximum transverse deflection (mm)
6 5 4
E1/E2 = 10
3
E1/E2 = 40
2 1 0 0
10
20
30
40
50
60
Percentage of defect area
Fig. 6. Influence of damage area on the transverse central deflection.
ACCEPTED MANUSCRIPT
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35
CR I
30 25 20 15
US
Variation of Γ2; Γ1=0.1
10
Variation of Γ1; Γ2=0.1
5
MA N
Non dimensional buckling load
40
0 0
0.2
0.4 0.6 Damage Parameter Γ1, Γ2
0.8
1
PT E
D
Fig. 7. Buckling load variation of a centrally damaged plate, with individual damage parameters Γ1 and Γ2, while other is kept fixed at 0.1.
Non dimensional buckling load
AC CE
25
Uniaxial
20
Biaxial
15
10
5
0 0
10
20
30
40
50
Percentage of defect area
Fig. 8. Buckling load variation as damage area increases.
ACCEPTED MANUSCRIPT
PT CR I
20
15
US
10
5
MA N
Non dimensional buckling loads
25
0 20
40
60 a/h
80
100
120
D
0
uniaxial (undamaged case) uniaxial (damaged case) biaxial (undamaged case) biaxial (damaged case)
PT E
Fig. 9. Critical buckling loads for symmetric laminates under uniaxial and biaxial compression for undamaged and damaged cases.
88
AC CE
86
Undamaged
84
Damaged
82 80
Tcr
78 76 74 72 70
0
10
20
30
40
50
E1/E2
Fig. 10. Critical buckling temperature for 4 layered symmetric composite plate for undamaged and damaged cases.
ACCEPTED MANUSCRIPT
6 a/h = 50
PT
5
CR I
4 T/Tcr 3 2
US
Without Damage With Damage
1
0
MA N
0 0.2
0.4
0.6 w/h
0.8
1
1.2
PT E
D
Fig. 11. Effect of internal flaw in thermal post buckling response for a/h = 50.
2.5
AC CE
2
a/h = 100
1.5
T/Tcr
1
Without Damage
0.5
With Damage
0 0
0.2
0.4
0.6
0.8
1
w/h
Fig. 12. Effect of internal flaw in thermal post buckling response for a/h = 100.
ACCEPTED MANUSCRIPT
2.5 E1/E2 = 20
PT
2
T/Tcr 1
CR I
1.5
Without Damage 0.5 0 0.2
0.4
0.6 w/h
0.8
1
1.2
MA N
0
US
With Damage
PT E
D
Fig. 13. Effect of internal flaw in thermal post buckling response for E1/E2 = 20.
2.5
AC CE
2
E1/E2 = 40
1.5
T/Tcr
1
Without Damage
0.5
With Damage
0 0
0.2
0.4
0.6 w/h
0.8
1
1.2
Fig. 14. Effect of internal flaw in thermal post buckling response for E1/E2 = 40.
ACCEPTED MANUSCRIPT
2.5
PT
α2/ α1 = 5 2
CR I
1.5 T/Tcr 1
US
Without Damage
0.5
With Damage 0 0.2
0.4
0.6 w/h
MA N
0
0.8
1
1.2
PT E
D
Fig. 15. Effect of internal flaw in thermal post buckling response for α2/α1 = 5.
2.5
AC CE
2
α2/ α1 = 10
1.5
T/Tcr
1
Without Damage
0.5
With Damage 0 0
0.2
0.4
0.6
0.8
1
1.2
w/h
Fig. 16. Effect of internal flaw in thermal post buckling response for α2/ α1 = 10.
ACCEPTED MANUSCRIPT Highlights
PT E
D
MA N
US
CR I
PT
Analysis of buckling and post buckling of composite plates with damage using FEM. Governing equations are based on inverse hyperbolic shear deformation theory. Effect of damage is simulated by an anisotropic damage formulation. Critical buckling temperature for a damaged composite plate has been obtained. Effect of mild damage on thermal post buckling paths is presented.
AC CE
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