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Buckling, postbuckling and progressive failure analyses of composite laminated plates under compressive loading
Accepted Manuscript Buckling, postbuckling and progressive failure analyses of composite laminated plates under compressive loading Ömer Namdar, Haluk Darendeliler PII:
S1359-8368(17)30253-6
DOI:
10.1016/j.compositesb.2017.03.066
Reference:
JCOMB 5000
To appear in:
Composites Part B
Received Date: 20 January 2017 Revised Date:
25 March 2017
Accepted Date: 26 March 2017
Please cite this article as: Namdar E, Darendeliler H, Buckling, postbuckling and progressive failure analyses of composite laminated plates under compressive loading, Composites Part B (2017), doi: 10.1016/j.compositesb.2017.03.066. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Buckling, Postbuckling and Progressive Failure Analyses of Composite Laminated Plates under Compressive Loading Ömer Namdara,*, Haluk Darendelilerb
Structural Analysis Engineer, ELAN-AUSY GmbH, Channel 2, Harburger Schlossstrasse 24, 21079 Hamburg, Germany, E-mail: [email protected]
b
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a
Department of Mechanical Engineering, Middle East Technical University, 06800, Ankara Turkey, Email: [email protected]
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* Corresponding Author Tel: +49 (0) 40 28466660, E-mail address: [email protected]
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Abstract
In this study, buckling, post-buckling and progressive failure of composite laminated plates have been investigated numerically and experimentally. Buckling load, load-displacement relations for post buckling and maximum out-of-plane displacements of the plates are determined. Furthermore, the numerical results are compared with experimental findings for two different laminates made of
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woven fabric and uni-directional tapes. The comparisons show that there is a good agreement between numerical and experimental results obtained for buckling load and post-buckling behavior especially for the laminates with uni-directional tapes.
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Keywords: Buckling, Post-buckling, Progressive Failure, Composite Structures
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1. Introduction
Investigation of the buckling and post-buckling behavior of the composite laminated plates is
an important issue to observe their strength and stiffness characteristics since the structural performance of a composite material depends on its composition, orientation, fiber shape, matrix and fiber material properties and quality of bondings between fiber and matrix [1]. The critical buckling loads of the composite structures have been investigated extremely in the literature and there are many studies about post-buckling behavior of composite laminates [2-15]. The studied structures were generally subjected to mono-axial compression although other types of loading
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ACCEPTED MANUSCRIPT conditions, such as shear, were considered [19]. The shear contribution to the buckling response of composite structures was also investigated [20]. Analytical and numerical methods have been employed for predicting the buckling load and post-buckling behavior of the composite structures [3,10,18,21,22] and the outcomes such as critical buckling loads, post-buckling behavior and failure
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loads have been compared with the experimental results [6,7,9,12-18,22]. The effect of geometric imperfections on the buckling behavior has also been taken into consideration by carrying out numerical analyses and experiments [3,11,13-15,23]. Additionally, the investigation of secondary instability and mode-jumping of composite structures have been performed in several studies
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[3,4,17]. In some investigations, finite shell elements have been developed and used [6,12,15], whereas in most of the analyses commercial finite element programs have been employed by
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implementing user-developed subroutines [2,5,9,13,14].
In literature, the post-buckling investigations have been performed mostly for the composite structures which have been manufactured from uni-directional and woven fabric pre-pregs. In recent studies, the post-buckling behavior and progressive failure analyses of the variable-stiffness composite structures which were produced by tow placement technology have also been studied
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[9,13]. The progressive failure analyses have been performed using the continuum damage model and propagation of damage has been correlated with experiments [5,8,9,24-28]. Moreover, analytical and closed-form solutions of functionally graded plates and beams have been studied for
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various types of loading conditions [29-33].
In this paper, buckling, post-buckling and failure characteristics of composite laminated plates
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which made by uni-directional and woven fabric laminae have been investigated numerically and experimentally. The composite specimens have been manufactured by using prepreg forms which consist of thermoset epoxy and carbon reinforcements and the specimens have been configured as angle-ply symmetric and balanced laminates. Eigenvalue analysis, non-linear Riks and Newton Raphson methods have been employed to analyze buckling, post-buckling behavior and failure [34]. The progressive failure calculations have been carried out by using the Hashin’s Failure Criterion [35]. A test fixture was designed and manufactured to determine the large deformation of composite laminated plates during post-buckling under compressive in-plane loading. The results
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ACCEPTED MANUSCRIPT which had been obtained from numerical analyses and experiments were compared with each other.
2. Materials
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The laminated plates that used in this study had been manufactured by using two different prepreg laminae. These are AS4/8552 carbon fiber reinforced epoxy prepreg UD tape and AS4/8552 carbon fiber reinforced epoxy prepreg 5HS Fabric. Both prepreg laminae consist of same fiber and matrix materials. AS4/8552 UD Tape is a uni-directional fiber reinforced composite which
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all fibers are aligned in a single direction. AS4/8552 5HS is a fabric lamina, which the textile structure is formed by interlaced fibers that are 90° angle with each other [36]. The thickness
3.
Experimental Procedure
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values and mechanical properties of each lamina are given in Table 1.
A number of experiments were performed to evaluate the results of the numerical analyses by using four composite plates; UD-1 and UD-2 made from AS4/8552 UD Tape and FABRIC-1 and
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FABRIC-2 made from AS4/8552 5HS Fabric. The stacking sequences and ply thickness values of the manufactured laminated plates are given in Table 2. The stacking of the plies has been selected such that either of the plates are balanced or symmetric to prevent the secondary instabilities
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under compressive load.
The test fixture has been designed and manufactured to investigate the large displacement post-
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buckling behavior of composite laminated plates under compressive in-plane loading (Figure 1). The clamped boundary conditions were applied for the top and bottom edges of plates whereas the side edges of plates are simple supported. To provide the expected boundary condition in the tests, 30 mm length of the specimens from upper and lower edges were stuck in the metallic blocks of top and bottom fixtures. Furthermore, knife edge supports were located 10 mm inward from the side edges of the specimens. The supports were arranged such that the dimensions of the part of the plate under the effect of buckling was 400
× 330
. This configuration was also applied
exactly into the FE model in order to simulate the realistic behavior of the specimen.
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ACCEPTED MANUSCRIPT The composite laminates, which were tested in this study, had been cured at 175 ± 10
temperature and at 680 ± 50
pressure in an autoclave. Total duration of the autoclave
process was about 130-180 minutes. Furthermore, the vessel of autoclave had been purged of oxygen using an inert gas to prevent thermal combustion or charring of the materials which were
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cured [36]. After the curing process, specimens were cut with the help of a CNC milling machine.
4. Numerical Method
The buckling process of composite laminated plates has been analyzed by the commercial finite
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element software Abaqus [34]. The critical buckling loads and mode shapes for the first buckling mode of laminates have been obtained by employing the Linear Eigenvalue Extraction method.
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The first buckling mode shapes of the laminates with small amplitudes such as 0.1-5 % of the panel thicknesses have been used as the initial geometric imperfections and implemented into FE model for the non-linear analyses. Riks and Newton-Raphson methods have been employed for the non-linear analyses of composite laminated plates to investigate their buckling and post-buckling performances. The progressive failure analyses have been carried out to examine the damages on
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the laminates by using Newton-Raphson method since Riks method has long computation time and convergence difficulties with progressive failure option. To prevent divergence of the computation, to mitigate instabilities and to eliminate rigid body
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modes of the FE model, the adaptive automatic stabilization scheme has been employed for the buckling analyses which have been performed by using Newton-Raphson Method [34]. When the
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FE solution goes unstable due to the buckling behavior, large displacements occur and corresponding strain energy releases. Part of the strain energy is dissipated by the relevant scheme using the damping factor that is determined by means of the dissipated energy fraction. The convergence is controlled by comparing the energy dissipated by viscous damping and the total strain energy. The optimal dissipated energy fraction is determined through a trial and error procedure by evaluating the FE results obtained from Riks method. The analyses of progressive failure also create some convergence difficulties due to severe softening behavior and stiffness degradation. Abaqus software provides a scheme called as viscous regularization, which regulate the tangent stiffness matrix to be positive for small time increments
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ACCEPTED MANUSCRIPT and stabilizes damage evolution problems by introducing viscosity coefficients into the material model, to prevent divergence during the progressive failure analysis [34]. The built-in boundary condition was applied by restraining all degrees of freedom at the lower edge of finite element model. At the top edge, only the vertical displacement was allowed and an
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incremental vertical compressive load was applied. At both of side edges only the displacement in the out of plane direction is constrained and the rotation along the vertical edges was permitted to simulate the knife edge supports.
The composite laminates have been modeled by using quadrilateral shell element; S4R, reduced
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integration. The S4R is a finite-membrane-strain element which is defined as general-purpose shell with 4 nodes and six degrees of freedom [34]. The 460 mm by 350 mm finite element model
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consists of 6510 shell elements, and 6674 nodes. The elements are approximately 5 mm x 5 mm in size however the regions of the plates between knife edges and metallic blocks are modeled with a finer mesh by using an element size of 3 mm x 5 mm.
5.
Progressive Failure Analysis
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The damage analysis used in this study has been based on material stiffness degradation of model. The stress levels, which damage initiations were expected depending on the allowable strength values of materials in fiber, matrix and shear directions, have been determined by using
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Hashin’s Failure Criterion [35] defined as follows: Tensile Fiber Mode,
≥1
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+
>0
Compressive Fiber Mode, !
≥1
Tensile Matrix Mode, "
+
≥1
(1) <0 (2)
>0
Compressive Matrix Mode,
(3) <0
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ACCEPTED MANUSCRIPT "!
#
"! 2
where
− 1& +
+
2
≥1
(4)
is the normal stress in the fiber direction,
is the normal stress in the transverse to the
fiber direction. To determine the damage initiation, the stress level which indicates the initiation of material
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stiffness degradation, the effective stress tensor ' ( is calculated for all of the material points at each time step. The components of effective stress tensor are assumed to be stress values acting over the area of a section that still remains undamaged. Hence the effective stress components )*+ are used ' ( = -'
*+
and computed from the following relation [34, 37]
) ' ( = . ) 0 , ' = . ̂
1 4 351 − 6 7 3 0 0 , - = 3 3 3 0 2
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where ' ( , ' and the damage operator M are given as
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in the Hashin’s criterion for
0
1 51 − 6 7 0
: 9 9 0 9 9 1 9 51 − 6 78
(5)
0
(6)
; = <' ( = <-'
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The relationship between the strain tensor, ;, and effective stress tensor, ' ( , is given as [34, 37] (7)
' = => ;
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where < is the compliance matrix. Then taking the inverse of the strain stress equation (8)
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where => is the constitutive tensor for the damaged laminae and given as:
51 − 6 7A 1 51 − 6 751 − 6 7B A => = @ ? 0
51 − 6 751 − 6 7B A 51 − 6 7A 0
0 0 D 51 − 6 7C ?
(9)
In the above equation, ? = 1 − 51 − 6 751 − 6 7B B , A is the Young’s modulus in the fiber direction, A is the Young’s modulus in the matrix direction, C is the shear modulus and B
and
B are Poisson’s ratios. The variable of shear damage, 6 , is calculated using the damage variables
of fiber and matrix as follows [34].
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= 1 − 51 − 6E 751 − 6F 751 − 6E 751 − 6F 7
The damage variables, 6 , 6
and 6
(10)
are determined at each integration point for all plies based
on effective stress levels and take different values for fiber and matrix for both tension and compression [34,37]. The damage variables, 6E , 6F , 6E and 6F for a definite failure mode are
obtained by using following expression [26, 34]. GI̅ JG̅ − GK̅ L , G̅ ≥ GK̅ G̅JGI̅ − GK̅ L
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6=
(11)
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where GK̅ and GI̅ are the equivalent displacement levels which correspond to initiation of damage and completely damaged material, respectively, for the related failure mode.
The slope of the equivalent stress, M, versus equivalent displacement, G̅, curve is positive for the
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linear elastic material up to GK̅ , then a negative slope is achieved and the evaluation of the damage
variables is started for the particular failure mode (Figure 2). Equivalent stress and equivalent displacement values are calculated by using formulas given below [34]. ≥ 07 6 = 6E , G̅ = G̅IE
G̅IE = NF O〈Q 〉 + Q M IE =
〈
〉〈Q 〉 + G̅IE ⁄NF
Fiber compression 5 ) 〈−
Matrix tension 5 ) 〈
(15)
≥ 07, 6 = 6E , G̅ = G̅ TE
G̅ TE = NF O〈Q 〉 + Q M TE =
(13)
(14)
〉〈−Q 〉
G̅IF ⁄NF
(12)
< 07, 6 = 6F , G̅ = G̅IF
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M IF =
Q
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G̅IF = NF 〈−Q 〉
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Fiber tension 5 )
〉〈Q 〉 + G̅ TE ⁄NF
Matrix compression 5 )
(16) Q
(17)
< 07, 6 = 6F , G̅ = G̅ FE
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ACCEPTED MANUSCRIPT G̅ TF = NF O〈−Q 〉 + Q M TF =
〈−
〉〈−Q 〉 + G̅ TF ⁄NF
(18) Q
(19)
where NF represents the characteristic length of the element and is determined in accordance with
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the element geometry and formulation. For S4R element, it is the square root of element area [34].
The energy dissipated due to failure, C! , which corresponds to area of equivalent stress equivalent displacement curve given in Figure 2, has been specified for all failure modes to employ
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the damage evolution procedure. The value of C F affects the equivalent displacement, GI̅ .
The energies dissipated for four failure modes of woven fabric and uni-directional tape
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composite materials have been calculated by using FEA in this study. The 40
× 100
strips
with 0° and 90° material directions and one plied stacking sequence have been modeled to find the equivalent stress versus equivalent displacement curves for fiber and matrix directions. Most of the degradation models are based on two approaches; instantaneous unloading and gradual unloading [38]. In the analyses, the ply-discount theory has been applied to determine energies dissipated for the relevant failure modes, which is a common instantaneous unloading methodology used for
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degradation of material properties. Although in order to prevent convergence difficulties, the values of C! have been determined by allowing a deformation which corresponds a small fraction (5 %) of the initial equivalent displacement (Table 3), the stiffness degradation led to convergence
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difficulties during the progressive failure. Therefore, viscous regularization scheme has been employed to regulate tangent stiffness matrix during the progressive failure analyses [34]. Figure 3
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represents the energy dissipation of the matrix compression failure for UD Tape which has been obtained from numerical analysis with viscous regularization scheme. After the damage initiation was achieved in matrix due to the compression, the material stiffness started to degrade with respect to specified failure energy for matrix compression (Table 3) during damage evolution procedure. Therefore, the viscous regularization scheme started to regulate the tangent stiffness of the material after the sharp stiffness decrease as seen in matrix compression of UD laminate (Figure 3).
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ACCEPTED MANUSCRIPT 6. Results and Discussion
6.1
Comparison of FE results and experiments Figure 4 compares the experimental results with the finite element solutions obtained from two
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unidirectional laminates UD-1 and UD-2 for maximum out-of-plane displacement, respectively. Finite element analyses have been carried out by using 5 % imperfection. The first buckling mode shape of the laminates has one longitudinal half-wave and gives the maximum out-of-plane deflection at midpoint of the plates. The critical buckling load that has been obtained numerically
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agrees well with the experimental results for UD-1 whereas buckling of UD-2 was observed earlier in experiments than determined by the finite element solution (Table 4, Figure 4). Post-buckling
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behavior of UD-1 that has been obtained in the experiment is similar with the finite element results up to 45 U load level. However, after the load level of 45 U, the out-of-plane deflections obtained by the experiments have been found greater than the finite element result. For UD-2 the out ofplane displacement deviates with a constant amount compared to UD-1. The UD-1 specimen failed at the same load level that was observed in the experiments while the failure load of UD-2 is lower
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than the experimental value (Table 4).
The first ply failure was observed in the outer layer of UD laminate due to compression as the compressive stress in the matrix exceeded the allowable limit (Table 1). Figure 5 shows the damage
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variable field of matrix compression, ωW , for the mentioned ply. The same result was observed in the experiments; the damage initiation was first occurred at the upper end of the knife edges and
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then propagated towards to center of specimens (Figure 6). The experimental and numerical load vs. maximum out-of-plane curves obtained for two fabric specimens, FABRIC-1 and FABRIC-2 are given in Figure 7, respectively. For the first buckling mode shape of FABRIC laminates (one longitudinal half-wave), the maximum out-of-plane deflection is observed at center of plates in experiments. The critical buckling load, which is obtained from linear eigen value analysis, agreed well with the experimental results for FABRIC-1 and FABRIC-2 specimens (Table 5, Figure 7). Maximum out-of-plane deflections of FABRIC laminates which have been obtained from experiments are similar with the finite element results up to a load level of 50 U (Figure 7). Then, the out-of-plane deflections determined by the finite element analyses are
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ACCEPTED MANUSCRIPT found smaller than the experimental results. On the other hand, the failure loads obtained from the finite element analyses are found less than the experimental results for both FABRIC-1 and FABRIC2 (Table 5).
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6.2 Numerical Studies A number of numerical analyses have been realized to evaluate the effect of ply angle orientations on the composite laminates. Results are presented to investigate buckling loads, postbuckling behavior and failure characteristics of different laminates. The finite element models of UD
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and FABRIC laminates were modified for various ply angles and stacking sequences. Same boundary conditions, loading type and element types were used in all of the numerical analyses.
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The amplitude of initial geometric imperfections was assumed 1% of the panel thicknesses for these numerical examples.
6.2.1 Variation of critical buckling load with Ply Angle
The variation of critical buckling load with ply angle have been determined for the UD and
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FABRIC laminates having the same stacking sequence, thickness and boundary conditions that were used in the experiments. As shown in Figure 8, ply angle variation does not affect significantly the critical buckling load of FABRIC laminates due to quasi-isotropic material properties of AS4/8552
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5HS Fabric prepreg. However, the critical buckling loads of UD laminates decrease with increasing ply angle since the stiffness in the fiber direction of AS4/8552 UD tape is greater than transverse
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stiffness.
6.2.2 Effects of Ply Angle Variation on Buckling for UD Laminates Effects of ply angle on buckling and failure loads are presented in Tables 6-8 considering AS4/8552 Carbon Fiber Reinforced UD laminate with 0.184 mm ply thickness for the stacking sequences [θ/-θ]S, [θ/-θ]2S and [θ/-θ]S4, respectively. Table 6 shows that [0/-0]S UD laminate has highest buckling load and the buckling load decreases as ply angle increases. The highest failure load is obtained for [15/-15]S UD laminate, whereas the failure loads of the laminates with the stacking sequences [0/-0]S, [30/-30]S and [45/-45]S are close to each other. The failure load
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ACCEPTED MANUSCRIPT decreases significantly for ply angles greater than 60 . The decrease of the buckling load with
increasing ply angle is observed also for composite laminates with stacking sequence of [θ/-θ]2S and [θ/-θ]4S and almost a similar tendency is observed for the variation of failure load as shown in Tables 7 and 8. The relations between the load and out-of-plane displacement for different ply
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angles are presented in Figure 9 for AS4/8552 Carbon Fiber Reinforced UD laminate with the stacking sequence [θ/-θ]4S. The first buckling mode shapes of laminates with stacking sequences [60/-60]S, [75/-75]S, [90/-90]S, [60/-60]2S, [75/-75]2S, [90/-90]2S, [60/-60]4S, [75/-75]4S and [90/90]4S have two longitudinal half waves while the other laminates have one. The differences
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between mode shapes which have one longitudinal half wave and two longitudinal half waves are shown in Figure 10. On the other hand, for the [45/-45]S and [75/-75]2S laminates that were
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observed to buckle in one half wave mode-jump to two half-waves occurred at the loads of 2.9 kN and 11 kN, respectively. First buckling mode shapes of all 16 plied [θ/-θ]S4 UD laminates have one longitudinal half wave pattern and mode jump is not observed in any stacking sequence. Tables 9-11 show the effects of ply angle on buckling and failure loads using AS4/8552 Carbon Fiber Reinforced 5HS Fabric laminate with 0.28 mm ply thickness for the stacking sequences [θ/-
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θ]S, [θ /- θ / θ /- θ / θ]S and [θ/-θ]4S, respectively. It is seen from Tables 9-11 that, the buckling and failure loads are maximum for 0 and 90 ply angles, and decreases gradually and reaches the minimum for 45 ply angle. The maximum differences are about 10 % and 30 % for the bucking
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and failure loads, respectively. Figure 11 shows the load versus out-of-plane displacement for different ply angles for [45/-45]4S stacking sequence. Mode-jump occurred for [0/-0]S and [0/-
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0/0/-0/0]S laminates from one half wave to two half-waves at the load levels of 4.7 kN and 65 kN, respectively. No mode jump is observed for 16 plied [θ/-θ]S4 FABRIC laminates in any stacking sequence.
7.
Conclusions
This study presents the numerical analyses of buckling, post-buckling and failure of composite laminated plates built with different carbon-epoxy laminates and stacking sequences which are validated by experimental results. The buckling behavior of plates that has been determined experimentally are found quite similar with the FE results that were obtained by using 5 % initial
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ACCEPTED MANUSCRIPT geometric imperfection amplitude. The following conclusions have been acquired by the current study. 1. The buckling loads observed by the experiments are in good agreement with the numerically determined values for UD-1, FABRIC-1 and FABRIC-2 specimens. However, UD-2 specimen
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predicted a lower value than the experimental one. 2. Until about a load level 50 kN, the post-buckling behavior of the test specimens are quite similar with the analyses except the UD-2 specimen. For the same load level, out of plane displacements are found approximately 1 mm greater for the UD-2 specimen compared to UD-1 specimen.
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After 50 kN, the results deviate with small amounts from the experimental predictions.
3. Though the FABRIC specimens have failed at lower load levels than the values determined by FE
both UD and FABRIC laminates.
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analyses, failure prediction of the experiments and FE analyses are found close to each other for
4. In the experiments, after the damage initiation, a sudden decrease was observed in the stiffness of the specimens.
5. Analyses and experiments showed that the studied plates can withstand the loads which are
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about four times higher than their critical buckling loads during the post-buckling. 6. Numerical studies showed that the critical buckling loads of UD laminates decreases considerably with increasing ply angle. However, ply angle variation does not affect the critical
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buckling load values of FABRIC laminates significantly. 7. Highest post buckling stiffness values for UD laminates are observed at [15/-15] stacking sequence for [θ/-θ]S, [θ/-θ]2S and [θ/-θ]S4 laminates. Then stiffness decreases with increasing
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ply angle giving the minimum at 90 . For FABRIC laminates, it is observed that the post buckling
stiffness decreases with increasing ply angle giving the minimum at 45 .
8. Mode-jumps are observed for [45/-45]S UD, [75/-75]2S UD laminates and [0/-0]S FABRIC, [0/0/0/-0/0]S FABRIC laminates during the deformation. 9. Mechanical stiffness of UD and FABRIC laminates are found to increase at higher orders than the increase of laminate thickness. This study shows that the stacking sequence and the thickness of the laminates have significant influences on buckling, post buckling and failure performances of the composite laminated
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ACCEPTED MANUSCRIPT structures. Although the laminated structures withstand higher loads during post buckling, the sudden stiffness loses due to material failure and secondary instabilities must be taken into consideration.
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Q. J. Yang, B. Hayman and H. Osnes, "Simplified buckling and ultimate strength analysis of composite plates in compression," Composites: Part B, vol. 54, p. 343–352, 2013.
[23]
M. W. Hilburger and J. H. Starnes Jr., "Effects of Imperfections of the Buckling Response of Composite Shells," Thin-Walled Structures, vol. 42, p. 2004, 369-397. P. P. Camanho, P. Maimi and C. G. Davila, "Prediction of Size Effects in Notched Laminates
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[24]
using Continuum Damage Mechanics," Composites Science and Technology, vol. 67, p. 27152727, 2007. [25]
S. Mukhopadhyay, M. I. Jones and S. R. Hallett, "Compressive failure of laminates containing
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an embedded wrinkle; experimental and numerical study," Composites: Part A, vol. 73, p. 132–142, 2015.
P. Camanho and C. G. Davila, "Mixed-Mode Decohesion Finite Elements for the Simulation of
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[26]
Delamination in Composite Materials," NASA/TM-2002–211737, p. 1–37, 2002. [27]
P. Maimi, P. Camanho, J. Mayugo and C. Davila, "A continuum damage model for composite laminates: Part I – Constitutive model," Mechanics of Materials, vol. 39, p. 897–908, 2007.
[28]
P. Maimi, P. Camanho, J. Mayugo and C. Davila, "A continuum damage model for composite
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laminates: Part II – Computational implementation and validation", Mechanics of Materials, vol. 39, p. 909–919, 2007. [29]
R. Barretta, R. Luciano, "Exact solutions of isotropic viscoelastic functionally graded
[30]
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Kirchhoff plates", Composite Structures, vol. 118, p. 448-454, 2014. A. Apuzzo, R. Barretta, R. Luciano, "Some analytical solutions of functionally graded Kirchhoff
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plates", Composites: Part B, vol. 68, p. 266-269, 2015. [31]
R. Barretta, R. Luciano, "Analogies between Kirchhoff plates and functionally graded SaintVenant beams under torsion", Continuum Mechanics and Thermodynamics, vol. 27, p. 499505, 2015.
[32]
R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, R. Penna, "Functionally graded Timoshenko nanobeams: A novel nonlocal gradient formulation", Composites Part B, vol. 100, p. 208-219, 2016.
[33]
R. Barretta, L. Feo, R. Luciano, "Some closed-form solutions of functionally graded beams undergoing non-uniform torsion", Composite Structures, vol. 123, p. 132-136, 2015.
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ACCEPTED MANUSCRIPT [34]
Abaqus Analysis User's Manual Version 6.10, SIMULIA, 2010.
[35]
Z. Hashin, "Failure Criteria for Unidirectional Composites," Journal of Applied Mechanics, vol. 47, p. 329-334, 1980. "Composite Materials," in MIL-HDBK-17, US Department of Defense, 1990.
[37]
A. Matzenmiller, J. Lubliner and R. L. Taylor, "A Constituve Model for Anisotropic Damage in
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[36]
Fiber-composites," Mechanics of Materials, vol. 20, p. 125-152, 1995. [38]
D. W. Sleight, "Progressive Failure Analysis Methodology for Laminated Composite
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Structures," NASA/TP-1999-209107, p. 1-67, 1999.
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ACCEPTED MANUSCRIPT Table 1 Properties of each lamina.
Ply Thickness (mm) E11 (GPa) E22 (GPa) G12 (GPa) ν12 XT (MPa) Xc (MPa) YT (MPa) YC (MPa) S (MPa)
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Table 2 Lay-up and ply thicknesses of specimens Specimen Stacking Sequence UD-1, UD-2 [45/-45/0/45/90/90/-45/0]S FABRIC-1, FABRIC-2 [45/0/45/0/45]S
AS4/8552 5HS Fabric Elastic Properties 0.28 61.0 61.0 4.2 0.05 647.0 657.0 647.0 657.0 109.0
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AS4/8552 UD Tape Elastic Properties 0.184 130.0 8.5 4.2 0.35 1530.0 770.0 64.0 175.0 95.0
Table 3 Energies Dissipated due to Failure (N/mm) Gcft AS4/8552 UD Tape 49.5 AS4/8552 5HS Fabric 19.7
Gcfc 12.5 19.9
Thickness(mm) 2.94 2.80
Gcmt 1.3 19.7
Gcmc 9.5 19.9
Failure Load (kN) 82.9 84.4 78.0
Table 5 Buckling and Failure Loads of Fabric Laminates Laminate Buckling Load (kN) FABRIC-1 (FE) 16.3 FABRIC-1 (Experiment) 16.3 FABRIC-2 (Experiment) 16.3
Failure Load (kN) 91.8 77.8 71.0
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Table 4 Buckling and Failure Loads of UD Laminates Laminate Buckling Load (kN) UD-1(FE) 22.2 UD-1 (Experiment) 22.2 UD-2 (Experiment) 16.2
Table 6 Buckling and Failure Loads of [θ/-θ]S UD Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]S 0.41 5.3 [15/-15]S 0.36 6.2 [30/-30]S 0.26 5.6 [45/-45]S 0.22 5.0 [60/-60]S 0.21 4.3 [75/-75]S 0.14 3.5 [90/-90]S 0.12 2.9
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ACCEPTED MANUSCRIPT
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Table 7 Buckling and Failure Loads of [θ/-θ]2S UD Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]2S 3.3 18.7 [15/-15] 2S 3.1 26.0 [30/-30] 2S 2.4 23.2 [45/-45] 2S 2.2 18.5 [60/-60] 2S 2.1 14.6 [75/-75] 2S 1.3 12.4 [90/-90] 2S 0.9 9.6
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Table 8 Buckling and Failure Loads [θ/-θ]4S UD Laminates and Specimens for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]4S 26.0 73.0 [15/-15] 4S 24.7 78.9 [30/-30] 4S 19.7 83.0 18.0 68.0 [45/-45] 4S [60/-60] 4S 17.1 54.0 10.4 43.0 [75/-75] 4S [90/-90] 4S 7.4 35.6
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Table 9 Buckling and Failure Loads (kN) of [θ/-θ]S Fabric Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]S 0.95 18.6 [15/-15] S 1.00 18.5 [30/-30] S 1.02 16.4 0.92 14.3 [45/-45] S [60/-60] S 1.02 16.4 [75/-75] S 1.00 18.5 [90/-90] S 0.95 18.6
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Table 10 Buckling and Failure Loads (kN) of [θ/-θ/θ/-θ/θ]S Fabric Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0/0/-0/0]S 14.9 101.6 [15/-15/15/-15/15]S 15.6 100.8 [30/-30/30/-30/30]S 15.9 87.1 [45/-45/45/-45/45]S 14.2 75.9 [60/-60/60/-60/60]S 15.9 87.1 [75/-75/75/-75/75]S 15.6 99.3 [90/-90/90/-90/90]S 14.9 99.5 Table11 Buckling and Failure Loads (kN) of [θ/-θ]4S Fabric Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]4S 60.7 245.1 [15/-15]4S 63.6 238.2 [30/-30]4S 64.8 205.7 [45/-45]4S 57.6 182.6 [60/-60]4S 64.8 205.7 [75/-75]4S 63.6 238.2 [90/-90]4S 60.7 245.1
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ACCEPTED MANUSCRIPT
σ0
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Equivalent Stress (MPa)
Figure 1
0
δ0
δf
Equivalent Stress (MPa)
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Equivalent Displacement (mm)
Figure 2
200
160 Matrix Compression 120
80
Gcmc 40
0 0.0
0.1
0.2
0.3
Equivalent Displacement (mm)
Figure 3
17
0.4
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100 FEM (%5 imperfection) experiment UD-1 experiment UD-2
60
40
20
0 0
5
10
15
(b) Figure 5
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(a)
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Figure 4
20
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Out-of-Plane Displacement (mm)
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Load (kN)
80
Figure 6
18
(c)
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100 FEM (%5 imperfection) experiment FABRIC-1 experiment FABRIC-2
Load (kN)
80
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60
40
20
0 5
10
15
20
25
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0
Out-of-Plane Displacement (mm)
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Figure 7 30
16 ply UD 10 ply Fabric
Load (kN)
25
20
15
5
0
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10
0
15
30
45
60
75
90
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Ply Angle
Figure 8
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100
Load (kN)
80
60
Teta=0 Teta_15 Teta=30 Teta=45 Teta=60 Teta=75 Teta=90
40
20
0 0
5
10
15
Out-of-Plane Displacement (mm)
Figure 9
19
20
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(a)
(b)
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Figure 10 250
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Load (kN)
200
150
100
Teta=0 & 90 Teta=15 & 75 Teta=30 & 60 Teta=45
0 0
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50
5
10
15
20
25
Out-of-Plane Displacement (mm)
Figure 11
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Figure Legends
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Fig. 1 Experimental set-up used for UD and FABRIC test specimens Fig. 2 Eq. Stress versus Eq. Displacement Fig. 3 Energy Dissipation of Matrix Compression for AS4/8552 UD Tape with Viscous Regularization Scheme Fig. 4 Load vs. maximum out-of-plane displacement curves of UD specimens Fig. 5 Matrix Compression Damage Variable ( ) plot of UD Laminate. (a) First ply failure at maximum load level, P=82.9 kN. (b) Damage evolution of UD laminate of matrix compression at 13 mm out-of-plane displacement and (c) Damage evolution of UD laminate of matrix compression at 14 mm out-of-plane displacement Fig. 6 Failure in UD-1 Specimen Fig. 7 Load vs. maximum out-of-plane displacement curves for Fabric Specimens Fig. 8 Variation of buckling load for [θ/-θ]4S UD laminates and [θ/-θ/θ/-θ/θ]S fabric laminates Fig. 9 Load versus out-of-plane displacements of [θ/-θ]4S UD laminates Fig. 10 Buckling mode shapes of UD Laminates (out-of-plane displacement) for (a) [45/-45]4S (b) [60/-60]4S stacking sequences Fig. 11 Load versus out-of-plane displacements of [θ/-θ]4S FABRIC Laminates
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S1359-8368(17)30253-6
DOI:
10.1016/j.compositesb.2017.03.066
Reference:
JCOMB 5000
To appear in:
Composites Part B
Received Date: 20 January 2017 Revised Date:
25 March 2017
Accepted Date: 26 March 2017
Please cite this article as: Namdar E, Darendeliler H, Buckling, postbuckling and progressive failure analyses of composite laminated plates under compressive loading, Composites Part B (2017), doi: 10.1016/j.compositesb.2017.03.066. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Buckling, Postbuckling and Progressive Failure Analyses of Composite Laminated Plates under Compressive Loading Ömer Namdara,*, Haluk Darendelilerb
Structural Analysis Engineer, ELAN-AUSY GmbH, Channel 2, Harburger Schlossstrasse 24, 21079 Hamburg, Germany, E-mail: [email protected]
b
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a
Department of Mechanical Engineering, Middle East Technical University, 06800, Ankara Turkey, Email: [email protected]
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* Corresponding Author Tel: +49 (0) 40 28466660, E-mail address: [email protected]
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Abstract
In this study, buckling, post-buckling and progressive failure of composite laminated plates have been investigated numerically and experimentally. Buckling load, load-displacement relations for post buckling and maximum out-of-plane displacements of the plates are determined. Furthermore, the numerical results are compared with experimental findings for two different laminates made of
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woven fabric and uni-directional tapes. The comparisons show that there is a good agreement between numerical and experimental results obtained for buckling load and post-buckling behavior especially for the laminates with uni-directional tapes.
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Keywords: Buckling, Post-buckling, Progressive Failure, Composite Structures
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1. Introduction
Investigation of the buckling and post-buckling behavior of the composite laminated plates is
an important issue to observe their strength and stiffness characteristics since the structural performance of a composite material depends on its composition, orientation, fiber shape, matrix and fiber material properties and quality of bondings between fiber and matrix [1]. The critical buckling loads of the composite structures have been investigated extremely in the literature and there are many studies about post-buckling behavior of composite laminates [2-15]. The studied structures were generally subjected to mono-axial compression although other types of loading
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ACCEPTED MANUSCRIPT conditions, such as shear, were considered [19]. The shear contribution to the buckling response of composite structures was also investigated [20]. Analytical and numerical methods have been employed for predicting the buckling load and post-buckling behavior of the composite structures [3,10,18,21,22] and the outcomes such as critical buckling loads, post-buckling behavior and failure
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loads have been compared with the experimental results [6,7,9,12-18,22]. The effect of geometric imperfections on the buckling behavior has also been taken into consideration by carrying out numerical analyses and experiments [3,11,13-15,23]. Additionally, the investigation of secondary instability and mode-jumping of composite structures have been performed in several studies
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[3,4,17]. In some investigations, finite shell elements have been developed and used [6,12,15], whereas in most of the analyses commercial finite element programs have been employed by
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implementing user-developed subroutines [2,5,9,13,14].
In literature, the post-buckling investigations have been performed mostly for the composite structures which have been manufactured from uni-directional and woven fabric pre-pregs. In recent studies, the post-buckling behavior and progressive failure analyses of the variable-stiffness composite structures which were produced by tow placement technology have also been studied
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[9,13]. The progressive failure analyses have been performed using the continuum damage model and propagation of damage has been correlated with experiments [5,8,9,24-28]. Moreover, analytical and closed-form solutions of functionally graded plates and beams have been studied for
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various types of loading conditions [29-33].
In this paper, buckling, post-buckling and failure characteristics of composite laminated plates
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which made by uni-directional and woven fabric laminae have been investigated numerically and experimentally. The composite specimens have been manufactured by using prepreg forms which consist of thermoset epoxy and carbon reinforcements and the specimens have been configured as angle-ply symmetric and balanced laminates. Eigenvalue analysis, non-linear Riks and Newton Raphson methods have been employed to analyze buckling, post-buckling behavior and failure [34]. The progressive failure calculations have been carried out by using the Hashin’s Failure Criterion [35]. A test fixture was designed and manufactured to determine the large deformation of composite laminated plates during post-buckling under compressive in-plane loading. The results
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ACCEPTED MANUSCRIPT which had been obtained from numerical analyses and experiments were compared with each other.
2. Materials
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The laminated plates that used in this study had been manufactured by using two different prepreg laminae. These are AS4/8552 carbon fiber reinforced epoxy prepreg UD tape and AS4/8552 carbon fiber reinforced epoxy prepreg 5HS Fabric. Both prepreg laminae consist of same fiber and matrix materials. AS4/8552 UD Tape is a uni-directional fiber reinforced composite which
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all fibers are aligned in a single direction. AS4/8552 5HS is a fabric lamina, which the textile structure is formed by interlaced fibers that are 90° angle with each other [36]. The thickness
3.
Experimental Procedure
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values and mechanical properties of each lamina are given in Table 1.
A number of experiments were performed to evaluate the results of the numerical analyses by using four composite plates; UD-1 and UD-2 made from AS4/8552 UD Tape and FABRIC-1 and
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FABRIC-2 made from AS4/8552 5HS Fabric. The stacking sequences and ply thickness values of the manufactured laminated plates are given in Table 2. The stacking of the plies has been selected such that either of the plates are balanced or symmetric to prevent the secondary instabilities
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under compressive load.
The test fixture has been designed and manufactured to investigate the large displacement post-
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buckling behavior of composite laminated plates under compressive in-plane loading (Figure 1). The clamped boundary conditions were applied for the top and bottom edges of plates whereas the side edges of plates are simple supported. To provide the expected boundary condition in the tests, 30 mm length of the specimens from upper and lower edges were stuck in the metallic blocks of top and bottom fixtures. Furthermore, knife edge supports were located 10 mm inward from the side edges of the specimens. The supports were arranged such that the dimensions of the part of the plate under the effect of buckling was 400
× 330
. This configuration was also applied
exactly into the FE model in order to simulate the realistic behavior of the specimen.
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ACCEPTED MANUSCRIPT The composite laminates, which were tested in this study, had been cured at 175 ± 10
temperature and at 680 ± 50
pressure in an autoclave. Total duration of the autoclave
process was about 130-180 minutes. Furthermore, the vessel of autoclave had been purged of oxygen using an inert gas to prevent thermal combustion or charring of the materials which were
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cured [36]. After the curing process, specimens were cut with the help of a CNC milling machine.
4. Numerical Method
The buckling process of composite laminated plates has been analyzed by the commercial finite
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element software Abaqus [34]. The critical buckling loads and mode shapes for the first buckling mode of laminates have been obtained by employing the Linear Eigenvalue Extraction method.
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The first buckling mode shapes of the laminates with small amplitudes such as 0.1-5 % of the panel thicknesses have been used as the initial geometric imperfections and implemented into FE model for the non-linear analyses. Riks and Newton-Raphson methods have been employed for the non-linear analyses of composite laminated plates to investigate their buckling and post-buckling performances. The progressive failure analyses have been carried out to examine the damages on
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the laminates by using Newton-Raphson method since Riks method has long computation time and convergence difficulties with progressive failure option. To prevent divergence of the computation, to mitigate instabilities and to eliminate rigid body
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modes of the FE model, the adaptive automatic stabilization scheme has been employed for the buckling analyses which have been performed by using Newton-Raphson Method [34]. When the
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FE solution goes unstable due to the buckling behavior, large displacements occur and corresponding strain energy releases. Part of the strain energy is dissipated by the relevant scheme using the damping factor that is determined by means of the dissipated energy fraction. The convergence is controlled by comparing the energy dissipated by viscous damping and the total strain energy. The optimal dissipated energy fraction is determined through a trial and error procedure by evaluating the FE results obtained from Riks method. The analyses of progressive failure also create some convergence difficulties due to severe softening behavior and stiffness degradation. Abaqus software provides a scheme called as viscous regularization, which regulate the tangent stiffness matrix to be positive for small time increments
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ACCEPTED MANUSCRIPT and stabilizes damage evolution problems by introducing viscosity coefficients into the material model, to prevent divergence during the progressive failure analysis [34]. The built-in boundary condition was applied by restraining all degrees of freedom at the lower edge of finite element model. At the top edge, only the vertical displacement was allowed and an
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incremental vertical compressive load was applied. At both of side edges only the displacement in the out of plane direction is constrained and the rotation along the vertical edges was permitted to simulate the knife edge supports.
The composite laminates have been modeled by using quadrilateral shell element; S4R, reduced
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integration. The S4R is a finite-membrane-strain element which is defined as general-purpose shell with 4 nodes and six degrees of freedom [34]. The 460 mm by 350 mm finite element model
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consists of 6510 shell elements, and 6674 nodes. The elements are approximately 5 mm x 5 mm in size however the regions of the plates between knife edges and metallic blocks are modeled with a finer mesh by using an element size of 3 mm x 5 mm.
5.
Progressive Failure Analysis
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The damage analysis used in this study has been based on material stiffness degradation of model. The stress levels, which damage initiations were expected depending on the allowable strength values of materials in fiber, matrix and shear directions, have been determined by using
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Hashin’s Failure Criterion [35] defined as follows: Tensile Fiber Mode,
≥1
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+
>0
Compressive Fiber Mode, !
≥1
Tensile Matrix Mode, "
+
≥1
(1) <0 (2)
>0
Compressive Matrix Mode,
(3) <0
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ACCEPTED MANUSCRIPT "!
#
"! 2
where
− 1& +
+
2
≥1
(4)
is the normal stress in the fiber direction,
is the normal stress in the transverse to the
fiber direction. To determine the damage initiation, the stress level which indicates the initiation of material
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stiffness degradation, the effective stress tensor ' ( is calculated for all of the material points at each time step. The components of effective stress tensor are assumed to be stress values acting over the area of a section that still remains undamaged. Hence the effective stress components )*+ are used ' ( = -'
*+
and computed from the following relation [34, 37]
) ' ( = . ) 0 , ' = . ̂
1 4 351 − 6 7 3 0 0 , - = 3 3 3 0 2
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where ' ( , ' and the damage operator M are given as
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in the Hashin’s criterion for
0
1 51 − 6 7 0
: 9 9 0 9 9 1 9 51 − 6 78
(5)
0
(6)
; = <' ( = <-'
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The relationship between the strain tensor, ;, and effective stress tensor, ' ( , is given as [34, 37] (7)
' = => ;
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where < is the compliance matrix. Then taking the inverse of the strain stress equation (8)
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where => is the constitutive tensor for the damaged laminae and given as:
51 − 6 7A 1 51 − 6 751 − 6 7B A => = @ ? 0
51 − 6 751 − 6 7B A 51 − 6 7A 0
0 0 D 51 − 6 7C ?
(9)
In the above equation, ? = 1 − 51 − 6 751 − 6 7B B , A is the Young’s modulus in the fiber direction, A is the Young’s modulus in the matrix direction, C is the shear modulus and B
and
B are Poisson’s ratios. The variable of shear damage, 6 , is calculated using the damage variables
of fiber and matrix as follows [34].
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= 1 − 51 − 6E 751 − 6F 751 − 6E 751 − 6F 7
The damage variables, 6 , 6
and 6
(10)
are determined at each integration point for all plies based
on effective stress levels and take different values for fiber and matrix for both tension and compression [34,37]. The damage variables, 6E , 6F , 6E and 6F for a definite failure mode are
obtained by using following expression [26, 34]. GI̅ JG̅ − GK̅ L , G̅ ≥ GK̅ G̅JGI̅ − GK̅ L
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6=
(11)
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where GK̅ and GI̅ are the equivalent displacement levels which correspond to initiation of damage and completely damaged material, respectively, for the related failure mode.
The slope of the equivalent stress, M, versus equivalent displacement, G̅, curve is positive for the
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linear elastic material up to GK̅ , then a negative slope is achieved and the evaluation of the damage
variables is started for the particular failure mode (Figure 2). Equivalent stress and equivalent displacement values are calculated by using formulas given below [34]. ≥ 07 6 = 6E , G̅ = G̅IE
G̅IE = NF O〈Q 〉 + Q M IE =
〈
〉〈Q 〉 + G̅IE ⁄NF
Fiber compression 5 ) 〈−
Matrix tension 5 ) 〈
(15)
≥ 07, 6 = 6E , G̅ = G̅ TE
G̅ TE = NF O〈Q 〉 + Q M TE =
(13)
(14)
〉〈−Q 〉
G̅IF ⁄NF
(12)
< 07, 6 = 6F , G̅ = G̅IF
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M IF =
Q
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G̅IF = NF 〈−Q 〉
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Fiber tension 5 )
〉〈Q 〉 + G̅ TE ⁄NF
Matrix compression 5 )
(16) Q
(17)
< 07, 6 = 6F , G̅ = G̅ FE
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ACCEPTED MANUSCRIPT G̅ TF = NF O〈−Q 〉 + Q M TF =
〈−
〉〈−Q 〉 + G̅ TF ⁄NF
(18) Q
(19)
where NF represents the characteristic length of the element and is determined in accordance with
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the element geometry and formulation. For S4R element, it is the square root of element area [34].
The energy dissipated due to failure, C! , which corresponds to area of equivalent stress equivalent displacement curve given in Figure 2, has been specified for all failure modes to employ
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the damage evolution procedure. The value of C F affects the equivalent displacement, GI̅ .
The energies dissipated for four failure modes of woven fabric and uni-directional tape
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composite materials have been calculated by using FEA in this study. The 40
× 100
strips
with 0° and 90° material directions and one plied stacking sequence have been modeled to find the equivalent stress versus equivalent displacement curves for fiber and matrix directions. Most of the degradation models are based on two approaches; instantaneous unloading and gradual unloading [38]. In the analyses, the ply-discount theory has been applied to determine energies dissipated for the relevant failure modes, which is a common instantaneous unloading methodology used for
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degradation of material properties. Although in order to prevent convergence difficulties, the values of C! have been determined by allowing a deformation which corresponds a small fraction (5 %) of the initial equivalent displacement (Table 3), the stiffness degradation led to convergence
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difficulties during the progressive failure. Therefore, viscous regularization scheme has been employed to regulate tangent stiffness matrix during the progressive failure analyses [34]. Figure 3
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represents the energy dissipation of the matrix compression failure for UD Tape which has been obtained from numerical analysis with viscous regularization scheme. After the damage initiation was achieved in matrix due to the compression, the material stiffness started to degrade with respect to specified failure energy for matrix compression (Table 3) during damage evolution procedure. Therefore, the viscous regularization scheme started to regulate the tangent stiffness of the material after the sharp stiffness decrease as seen in matrix compression of UD laminate (Figure 3).
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ACCEPTED MANUSCRIPT 6. Results and Discussion
6.1
Comparison of FE results and experiments Figure 4 compares the experimental results with the finite element solutions obtained from two
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unidirectional laminates UD-1 and UD-2 for maximum out-of-plane displacement, respectively. Finite element analyses have been carried out by using 5 % imperfection. The first buckling mode shape of the laminates has one longitudinal half-wave and gives the maximum out-of-plane deflection at midpoint of the plates. The critical buckling load that has been obtained numerically
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agrees well with the experimental results for UD-1 whereas buckling of UD-2 was observed earlier in experiments than determined by the finite element solution (Table 4, Figure 4). Post-buckling
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behavior of UD-1 that has been obtained in the experiment is similar with the finite element results up to 45 U load level. However, after the load level of 45 U, the out-of-plane deflections obtained by the experiments have been found greater than the finite element result. For UD-2 the out ofplane displacement deviates with a constant amount compared to UD-1. The UD-1 specimen failed at the same load level that was observed in the experiments while the failure load of UD-2 is lower
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than the experimental value (Table 4).
The first ply failure was observed in the outer layer of UD laminate due to compression as the compressive stress in the matrix exceeded the allowable limit (Table 1). Figure 5 shows the damage
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variable field of matrix compression, ωW , for the mentioned ply. The same result was observed in the experiments; the damage initiation was first occurred at the upper end of the knife edges and
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then propagated towards to center of specimens (Figure 6). The experimental and numerical load vs. maximum out-of-plane curves obtained for two fabric specimens, FABRIC-1 and FABRIC-2 are given in Figure 7, respectively. For the first buckling mode shape of FABRIC laminates (one longitudinal half-wave), the maximum out-of-plane deflection is observed at center of plates in experiments. The critical buckling load, which is obtained from linear eigen value analysis, agreed well with the experimental results for FABRIC-1 and FABRIC-2 specimens (Table 5, Figure 7). Maximum out-of-plane deflections of FABRIC laminates which have been obtained from experiments are similar with the finite element results up to a load level of 50 U (Figure 7). Then, the out-of-plane deflections determined by the finite element analyses are
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6.2 Numerical Studies A number of numerical analyses have been realized to evaluate the effect of ply angle orientations on the composite laminates. Results are presented to investigate buckling loads, postbuckling behavior and failure characteristics of different laminates. The finite element models of UD
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and FABRIC laminates were modified for various ply angles and stacking sequences. Same boundary conditions, loading type and element types were used in all of the numerical analyses.
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The amplitude of initial geometric imperfections was assumed 1% of the panel thicknesses for these numerical examples.
6.2.1 Variation of critical buckling load with Ply Angle
The variation of critical buckling load with ply angle have been determined for the UD and
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FABRIC laminates having the same stacking sequence, thickness and boundary conditions that were used in the experiments. As shown in Figure 8, ply angle variation does not affect significantly the critical buckling load of FABRIC laminates due to quasi-isotropic material properties of AS4/8552
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5HS Fabric prepreg. However, the critical buckling loads of UD laminates decrease with increasing ply angle since the stiffness in the fiber direction of AS4/8552 UD tape is greater than transverse
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stiffness.
6.2.2 Effects of Ply Angle Variation on Buckling for UD Laminates Effects of ply angle on buckling and failure loads are presented in Tables 6-8 considering AS4/8552 Carbon Fiber Reinforced UD laminate with 0.184 mm ply thickness for the stacking sequences [θ/-θ]S, [θ/-θ]2S and [θ/-θ]S4, respectively. Table 6 shows that [0/-0]S UD laminate has highest buckling load and the buckling load decreases as ply angle increases. The highest failure load is obtained for [15/-15]S UD laminate, whereas the failure loads of the laminates with the stacking sequences [0/-0]S, [30/-30]S and [45/-45]S are close to each other. The failure load
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increasing ply angle is observed also for composite laminates with stacking sequence of [θ/-θ]2S and [θ/-θ]4S and almost a similar tendency is observed for the variation of failure load as shown in Tables 7 and 8. The relations between the load and out-of-plane displacement for different ply
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angles are presented in Figure 9 for AS4/8552 Carbon Fiber Reinforced UD laminate with the stacking sequence [θ/-θ]4S. The first buckling mode shapes of laminates with stacking sequences [60/-60]S, [75/-75]S, [90/-90]S, [60/-60]2S, [75/-75]2S, [90/-90]2S, [60/-60]4S, [75/-75]4S and [90/90]4S have two longitudinal half waves while the other laminates have one. The differences
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between mode shapes which have one longitudinal half wave and two longitudinal half waves are shown in Figure 10. On the other hand, for the [45/-45]S and [75/-75]2S laminates that were
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observed to buckle in one half wave mode-jump to two half-waves occurred at the loads of 2.9 kN and 11 kN, respectively. First buckling mode shapes of all 16 plied [θ/-θ]S4 UD laminates have one longitudinal half wave pattern and mode jump is not observed in any stacking sequence. Tables 9-11 show the effects of ply angle on buckling and failure loads using AS4/8552 Carbon Fiber Reinforced 5HS Fabric laminate with 0.28 mm ply thickness for the stacking sequences [θ/-
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θ]S, [θ /- θ / θ /- θ / θ]S and [θ/-θ]4S, respectively. It is seen from Tables 9-11 that, the buckling and failure loads are maximum for 0 and 90 ply angles, and decreases gradually and reaches the minimum for 45 ply angle. The maximum differences are about 10 % and 30 % for the bucking
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and failure loads, respectively. Figure 11 shows the load versus out-of-plane displacement for different ply angles for [45/-45]4S stacking sequence. Mode-jump occurred for [0/-0]S and [0/-
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0/0/-0/0]S laminates from one half wave to two half-waves at the load levels of 4.7 kN and 65 kN, respectively. No mode jump is observed for 16 plied [θ/-θ]S4 FABRIC laminates in any stacking sequence.
7.
Conclusions
This study presents the numerical analyses of buckling, post-buckling and failure of composite laminated plates built with different carbon-epoxy laminates and stacking sequences which are validated by experimental results. The buckling behavior of plates that has been determined experimentally are found quite similar with the FE results that were obtained by using 5 % initial
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ACCEPTED MANUSCRIPT geometric imperfection amplitude. The following conclusions have been acquired by the current study. 1. The buckling loads observed by the experiments are in good agreement with the numerically determined values for UD-1, FABRIC-1 and FABRIC-2 specimens. However, UD-2 specimen
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predicted a lower value than the experimental one. 2. Until about a load level 50 kN, the post-buckling behavior of the test specimens are quite similar with the analyses except the UD-2 specimen. For the same load level, out of plane displacements are found approximately 1 mm greater for the UD-2 specimen compared to UD-1 specimen.
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After 50 kN, the results deviate with small amounts from the experimental predictions.
3. Though the FABRIC specimens have failed at lower load levels than the values determined by FE
both UD and FABRIC laminates.
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analyses, failure prediction of the experiments and FE analyses are found close to each other for
4. In the experiments, after the damage initiation, a sudden decrease was observed in the stiffness of the specimens.
5. Analyses and experiments showed that the studied plates can withstand the loads which are
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about four times higher than their critical buckling loads during the post-buckling. 6. Numerical studies showed that the critical buckling loads of UD laminates decreases considerably with increasing ply angle. However, ply angle variation does not affect the critical
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buckling load values of FABRIC laminates significantly. 7. Highest post buckling stiffness values for UD laminates are observed at [15/-15] stacking sequence for [θ/-θ]S, [θ/-θ]2S and [θ/-θ]S4 laminates. Then stiffness decreases with increasing
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ply angle giving the minimum at 90 . For FABRIC laminates, it is observed that the post buckling
stiffness decreases with increasing ply angle giving the minimum at 45 .
8. Mode-jumps are observed for [45/-45]S UD, [75/-75]2S UD laminates and [0/-0]S FABRIC, [0/0/0/-0/0]S FABRIC laminates during the deformation. 9. Mechanical stiffness of UD and FABRIC laminates are found to increase at higher orders than the increase of laminate thickness. This study shows that the stacking sequence and the thickness of the laminates have significant influences on buckling, post buckling and failure performances of the composite laminated
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Structures," NASA/TP-1999-209107, p. 1-67, 1999.
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ACCEPTED MANUSCRIPT Table 1 Properties of each lamina.
Ply Thickness (mm) E11 (GPa) E22 (GPa) G12 (GPa) ν12 XT (MPa) Xc (MPa) YT (MPa) YC (MPa) S (MPa)
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Table 2 Lay-up and ply thicknesses of specimens Specimen Stacking Sequence UD-1, UD-2 [45/-45/0/45/90/90/-45/0]S FABRIC-1, FABRIC-2 [45/0/45/0/45]S
AS4/8552 5HS Fabric Elastic Properties 0.28 61.0 61.0 4.2 0.05 647.0 657.0 647.0 657.0 109.0
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AS4/8552 UD Tape Elastic Properties 0.184 130.0 8.5 4.2 0.35 1530.0 770.0 64.0 175.0 95.0
Table 3 Energies Dissipated due to Failure (N/mm) Gcft AS4/8552 UD Tape 49.5 AS4/8552 5HS Fabric 19.7
Gcfc 12.5 19.9
Thickness(mm) 2.94 2.80
Gcmt 1.3 19.7
Gcmc 9.5 19.9
Failure Load (kN) 82.9 84.4 78.0
Table 5 Buckling and Failure Loads of Fabric Laminates Laminate Buckling Load (kN) FABRIC-1 (FE) 16.3 FABRIC-1 (Experiment) 16.3 FABRIC-2 (Experiment) 16.3
Failure Load (kN) 91.8 77.8 71.0
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Table 4 Buckling and Failure Loads of UD Laminates Laminate Buckling Load (kN) UD-1(FE) 22.2 UD-1 (Experiment) 22.2 UD-2 (Experiment) 16.2
Table 6 Buckling and Failure Loads of [θ/-θ]S UD Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]S 0.41 5.3 [15/-15]S 0.36 6.2 [30/-30]S 0.26 5.6 [45/-45]S 0.22 5.0 [60/-60]S 0.21 4.3 [75/-75]S 0.14 3.5 [90/-90]S 0.12 2.9
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Table 7 Buckling and Failure Loads of [θ/-θ]2S UD Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]2S 3.3 18.7 [15/-15] 2S 3.1 26.0 [30/-30] 2S 2.4 23.2 [45/-45] 2S 2.2 18.5 [60/-60] 2S 2.1 14.6 [75/-75] 2S 1.3 12.4 [90/-90] 2S 0.9 9.6
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Table 8 Buckling and Failure Loads [θ/-θ]4S UD Laminates and Specimens for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]4S 26.0 73.0 [15/-15] 4S 24.7 78.9 [30/-30] 4S 19.7 83.0 18.0 68.0 [45/-45] 4S [60/-60] 4S 17.1 54.0 10.4 43.0 [75/-75] 4S [90/-90] 4S 7.4 35.6
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Table 9 Buckling and Failure Loads (kN) of [θ/-θ]S Fabric Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]S 0.95 18.6 [15/-15] S 1.00 18.5 [30/-30] S 1.02 16.4 0.92 14.3 [45/-45] S [60/-60] S 1.02 16.4 [75/-75] S 1.00 18.5 [90/-90] S 0.95 18.6
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Table 10 Buckling and Failure Loads (kN) of [θ/-θ/θ/-θ/θ]S Fabric Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0/0/-0/0]S 14.9 101.6 [15/-15/15/-15/15]S 15.6 100.8 [30/-30/30/-30/30]S 15.9 87.1 [45/-45/45/-45/45]S 14.2 75.9 [60/-60/60/-60/60]S 15.9 87.1 [75/-75/75/-75/75]S 15.6 99.3 [90/-90/90/-90/90]S 14.9 99.5 Table11 Buckling and Failure Loads (kN) of [θ/-θ]4S Fabric Laminates for different ply angles Laminate Buckling Load (kN) Failure Load (kN) [0/-0]4S 60.7 245.1 [15/-15]4S 63.6 238.2 [30/-30]4S 64.8 205.7 [45/-45]4S 57.6 182.6 [60/-60]4S 64.8 205.7 [75/-75]4S 63.6 238.2 [90/-90]4S 60.7 245.1
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Figure 1
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Figure 10 250
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Figure Legends
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Fig. 1 Experimental set-up used for UD and FABRIC test specimens Fig. 2 Eq. Stress versus Eq. Displacement Fig. 3 Energy Dissipation of Matrix Compression for AS4/8552 UD Tape with Viscous Regularization Scheme Fig. 4 Load vs. maximum out-of-plane displacement curves of UD specimens Fig. 5 Matrix Compression Damage Variable ( ) plot of UD Laminate. (a) First ply failure at maximum load level, P=82.9 kN. (b) Damage evolution of UD laminate of matrix compression at 13 mm out-of-plane displacement and (c) Damage evolution of UD laminate of matrix compression at 14 mm out-of-plane displacement Fig. 6 Failure in UD-1 Specimen Fig. 7 Load vs. maximum out-of-plane displacement curves for Fabric Specimens Fig. 8 Variation of buckling load for [θ/-θ]4S UD laminates and [θ/-θ/θ/-θ/θ]S fabric laminates Fig. 9 Load versus out-of-plane displacements of [θ/-θ]4S UD laminates Fig. 10 Buckling mode shapes of UD Laminates (out-of-plane displacement) for (a) [45/-45]4S (b) [60/-60]4S stacking sequences Fig. 11 Load versus out-of-plane displacements of [θ/-θ]4S FABRIC Laminates
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