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Composites Part B 116 (2017) 89e98

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Composites Part B

journal homepage: www.elsevier .com/locate/compositesb

Modeling framework for free edge effects in laminates under thermo-mechanical loading

M.S. Islam a, P. Prabhakar b, *

a Department of Mechanical Engineering, Khulna University of Engineering & Technology, Bangladeshb University of Wisconsin-Madison, Department of Civil & Environmental Engineering, WI 53706, United States

a r t i c l e i n f o

Article history:Received 20 June 2016Received in revised form14 November 2016Accepted 28 January 2017Available online 21 February 2017

Keywords:Free edge effectInterfaceLaminateFinite element analysis

* Corresponding author. Tel.: þ16082657834.E-mail address: pprabhakar4@wisc.edu (P. Prabha

http://dx.doi.org/10.1016/j.compositesb.2017.01.0721359-8368/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t

In this paper, a novel Quasi-2D (Q-2D) plane strain formulation for predicting interlaminar stresses inmulti-directional laminates is developed and implemented within the finite element method framework.In particular, analyses of free edge stresses in composite laminates subjected to uniform axial and/orthermal loading are conducted. The Q-2D modeling approach presented here is validated by comparingthe predicted interlaminar stresses with corresponding 3D models and previously published data.Computational time required for determining the interlaminar stresses using Q-2D model is approxi-mately 30 times lower than the 3D analysis of the same laminate. Also, the Q-2D model is implementedwithin a commercially available software by modifying a 3D model to behave like a 2D model. This isparticularly advantageous over developing an in-house finite element method code to capture thecomplex free edge stress states in multi-directional composites. Hence, the current framework canpotentially be used as a computational tool for efficiently predicting the interlaminar stresses in differentlaminates subjected to thermo-mechanical loading, which can assist in determining interlaminar regionssusceptible to free edge delamination.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Interlaminar regions in polymer based layered carbon fibercomposites are critical regions that are most susceptible todelamination under static and dynamic loading. Debonding ordelamination is observed to be a significant failure mechanism inlayered composites with considerable visible damage when sub-jected to load types like edge-wise and through-thicknesscompression, flexure, and dynamic impact. Delamination typefailure is directly influenced by the strength and toughness ofinterlaminar regions. This is due to significant localized stressesthat occur at the interlaminar regions, particularly at the free edgesdue to mismatch in property between plies, which commonlyreferred to as the “free edge effect” [1]. Hence, accurate determi-nation of stress distribution near the free edges is very importantdue to their significant impact on delamination or transversecracking in layered multi-directional laminates.

Stress state near the free edge is three dimensional in nature andclassical lamination theory (CLT) is unable to determine these

kar).

stresses accurately [1,2]. Therefore, various analytical and numeri-cal approaches such as closed form analytical solutions, boundarylayer theories, layer-wise theories, finite difference method andfinite element method have been used by earlier researchers tocalculate interlaminar stresses near the free edges. Puppo andEvensen [3] proposed the first analytical method to determineinterlaminar stresses in a composite laminate. Pagano [4] devel-oped a higher order plate theory to evaluate the interlaminarstresses, and Hsu and Herakovich [5] studied the free edge effectsusing perturbation and limiting free body approach for angle plylaminates. Tang [6], Davet and Destuynder [7], and Lin and Ko [8]studied similar problems using boundary layer theory. Pipes andPagano [9] used an approximate elasticity solution, while Pagano[10] and Lekhnitskii and Fern [11] used variational principle tostudy the free edge effects in laminates. Yin [12,13] used a varia-tional method that utilized Lekhnitskiis stress function for lami-nates under combined mechanical loading. Later, Yin [14] utilizedthe principle of complementary energy based on polynomial stressfunctions to evaluate the interlaminar stresses in laminates sub-jected non-uniform thermal loading. Andakhshideh and Tahani[15] used three dimensional multi-term extended Kantorovichmethod to investigate the interlaminar stresses near the free edges

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e9890

of general composite laminates under axial and shear loads.Amrutharaj et al. [16] investigated the free edge effects in com-posite laminates under uniaxial extension using the concept offracture process zone. D'Ottavio et al. [17] conducted a comparativestudy of different plate theories for free edge effects in laminatedplates and established that Layer-Wise (LW) theories are capable ofcapturing the free edge effects, while Equivalent Single Layer (ESL)theories fail. Even though the LW theories can capture the edgeeffects sufficiently well, they are computationally expensive [2] andthe results often depend on the number of sublayers consideredwithin each ply [18].

Pipes and Pagano [19] developed the first finite difference basednumerical method to solve the two dimensional governing elas-ticity equations for calculating the interlaminar stresses of longsymmetric laminate under uniform axial strain. Later, Atlus et al.[20] and Salamon [21] used three dimensional finite differencemethod to determine the interlaminar stresses in angle-ply lami-nates. Wang and Crossman [22,23] investigated edge effects insymmetric composite laminates subjected to uniform axial strainand thermal loading using finite element method. Herakovich et al.[24], Isakson and Levy [25], Rybicky [26], Kim and Hong [27], Icardiand Bertetto [28], Lessard et al. [29] and Yi and Hilton [30] also usedfinite element method to study free edge effects in laminates.Spilker and Chou [31] used a hybrid stress based finite elementmethod, Lee and Chen [32] used a layer reduction technique,Robbins and Reddy [33] used a displacement based variable kine-matic global local finite element method and Gaudenzi et al. [34]used a three dimensional multilayer higher order finite elementmethod to study similar problems. Lorriot et al. [35] investigatedthe onset of delamination in carbon/epoxy laminates by developinga model based on stress criterion.

From the previous studies mentioned above, it is well estab-lished that free edge effects are dominant in multidirectionallaminates and result in very high interlaminar stresses that pre-maturely initiate inter-layer delamination. Therefore, determininginterfaces with very large interlaminar stresses is critical for theassessment of delamination driven failure. Interfaces most sus-ceptible to delamination can be determined and strengthenedaccordingly during manufacturing to reduce their susceptibility tofailure. Towards that, a Quasi-2D formulation within the finiteelement method (FEM) framework is established in this paper fordetermining delamination prone interlaminar regions in multidi-rectional laminates subjected to thermo-mechanical loading.

In the current paper, a variational formulation presented byMartin et al. [36] is extended for combined thermal and axialloading on multi-directional laminates, referred to as “Quasi-2D”(Q-2D) formulation. Q-2D formulation is implemented within theFEM framework for accurately determining stress distribution nearthe free edges for cross-ply ð½0=90�sÞ and quasi-isotropic

Fig. 1. (a)3D Laminate; (b) Cros

ð½45=� 45=90=0�sÞ laminates. A novel technique is developed inthis paper for implementing the Q-2D model within the finiteelement framework, which involves modifying a thin slice of a 3Dlaminate to behave like a generalized 2Dmodel by enforcing multi-point constraints. The approach presented in this paper iscompared against corresponding 3D models and previously pub-lished data for establishing its validity and accuracy.

Other numerical approaches by earlier researchers have inves-tigated the full 3D model to capture the free edge effects. Forexample, Icardi and Bertetto [28] used a 3D modeling approachwith 20 noded quadratic isoparametric brick elements and 15noded quadratic singular wedge elements for capturing the stressstate at the free edges. Lessard et al. [29] also used 3D finite elementmethod with 20 noded quadratic brick elements for the same. Rajuand Crews [37] used a combined rectangular and polar mesh fortheir analysis, where the rectangular mesh was used to determinethe stress distributions and polar mesh was used to investigate thestress singularities. But in the current paper, accurate stress statesat the free edges are captured using a Quasi-2D reduced model,which takes about 30 times lesser computational time than a 3Dmodel. Another key advantage of the Q-2D model is the easyimplementation within any commercially available finite elementsoftware as opposed to the requirement of an in-house code.

This paper is divided into the following sections: Section 2 de-scribes the mathematical formulation for the Q-2D, followed by thedetails of the implementation within the finite element frameworkin Section 3. Discussion of results from several case studies is pre-sented in Section 4 followed by conclusions.

2. Mathematical formulation

Consider a laminate of length 2L and width 2b, which consists ofN layers (thickness h each) as shown in Fig. 1(a). Tensile load isapplied on the edges at

PþL and

P�L along the x1 direction, with

free edges atP

0 andP

2b. A temperature change of DT occursuniformly within the laminate.

The stress components are assumed to be independent of x1 inregions sufficiently far from the loading surface [19,36], such thatthe displacement field fUg can be defined as,

U1ðx1; x2; x3Þ ¼ ~U1ðx2; x3Þ þ ε11x1U2ðx1; x2; x3Þ ¼ ~U2ðx2; x3ÞU3ðx1; x2; x3Þ ¼ ~U3ðx2; x3Þ

(1)

where, ε11 is the uniform strain applied on the laminate along thex1 direction. Displacement field fUg and corresponding stress fieldfsg adhere to the following equations:

s-section of a 3D laminate.

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e98 91

vsijvxj

¼ 0; sij ¼ aijkhεmkh;ci ¼ 1;2;3 within U; (2)

εijðUÞ ¼12

vUi

vxjþ vUj

vxi

!within U; (3)

½Ui� ¼ 0;�sijnj

� ¼ 0 at the interfaces Gp;�sijnj

�¼ 0 on

X2b

andX0

(4)

�sijnj

� ¼ Fi onXþL

andXN h

2

;�sijnj

� ¼ �Fi onX�L

andX�N h

2

(5)

Introducing a trial field V with ε11 ¼ 0 and averaging equation(2) over the domain gives,

ZU

vsijvxj

VidU ¼ 0;cVi with i ¼ 1; 2; 3: (6)

Applying divergence theorem to the above equation yields,

ZU

sijvVi

vxjdU ¼

ZP

þL

FiVidS�ZP

�L

FiVidS (7)

Since, Vi does not depend on x1:

ZU

silvVi

vxldU ¼ 0;cVi with l ¼ 2;3: (8)

Using sij ¼ aijkhεmkh yields,

ZU

ailkh

�vUk

vxh� akhDT

�vVi

vxldU ¼ 0;cVi for i; k; h ¼ 1; 2; 3; l

¼ 2; 3

(9)

where, fεmg ¼ fεg � fagDT .Substituting the displacement field assumption given in equa-

tion (1) and (9) gives,

ZU

ailkh

v~Uk

vxh� akhDT

!vVi

vxldx1dx2dx3 þ ε11

ZU

ail11vVi

vxldx1dx2dx3

¼ 0

(10)

Splitting the volume integral into a surface integral in the planeof the cross-section (x2; x3) and line integral along the x1 directionyields,

ZþL

�L

dx1

ZS

ailkb

v~Ukvxb

� akbDT

!vVi

vxldx2dx3

�ZþL

�L

dx1

ZS

ailk1ak1DTvVi

vxldx2dx3

þ ε11

ZþL

�L

dx1

ZS

ail11vVi

vxldx2dx3

¼ 0; (11)

with l; b ¼ 2;3.

ZS

ailkb

v~Uk

vxb� akbDT

!vVi

vxldx2dx3

¼ DTZS

ailk1ak1vVi

vxldx2dx3 � ε11

ZS

ail11vVi

vxldx2dx3 (12)

Applying divergence theorem on the right hand side,

ZSp

ailkb

v~Uk

vxb� akbDT

!vVi

vxldx2dx3

¼ DTZvS

ailk1ak1Vinlds� ε11

ZvS

ail11Vinlds (13)

where, “s” represents a coordinate that denotes the boundary vS,starting at the origin of the x2 � x3 axes for the region S andtraversing in the counter-clockwise direction. Therefore, “s” iseither “x2” or “x3” depending on the edge on the boundary beingtraversed.

The above equation is modified to account for the through-thickness layers with different orientations in a multi-directionallaminate as:

XNp¼1

ZSp

apilkb

v~U

pk

vxb� apkbDT

!vVp

ivxl

dx2dx3

¼ DTXNp¼1

ZvSp

apilk1apk1V

pi n

pl ds� ε11

XNp¼1

ZvSp

apil11Vpi n

pl ds (14)

Upon expanding the right hand side of equation (14), contri-butions of the vertical edges is given by,

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e9892

DTXNp¼1

0B@ ZNh=2þph

�Nh=2þðp�1Þhapi2k1a

pk1V

pi ð2b; x3Þdx3

�ZNh=2þph

�Nh=2þðp�1Þhapi2k1a

pk1V

pi ð0; x3Þdx3

1CA

� ε11

XNp¼1

0B@ ZNh=2þph

�Nh=2þðp�1Þhapi211V

pi ð2b; x3Þdx3

�ZNh=2þph

�Nh=2þðp�1Þhapi211V

pi ð0; x3Þdx3

1CA (15)

Similarly, contribution of each interface is,

DTXN�1

p¼1

0B@�

Z2b0

�apþ1i3k1a

pþ1k1 � api3k1a

pk1

�Vpi

�x2;N

h2þ ph

�dx2

1CA

� ε11

XN�1

p¼1

0B@�

Z2b0

�apþ1i311 � api311

�Vpi

�x2;N

h2þ ph

�dx2

1CA

(16)

Contribution of top and bottom edges is,

DT

0B@�

Z2b0

a1i3k1a1k1V

1i

�x2;�N

h2

�dx2

þZ2b0

aNi3k1aNk1V

Ni

�x2;N

h2

�dx2

1CA� ε11

0B@

�Z2b0

a1i311V1i

�x2;�N

h2

�dx2 þ

Z2b0

aNi311VNi

�x2;N

h2

�dx2

1CA

(17)

Equation (14) is a generalized 2D formulationwith displacementfields in the x1; x2 and x3 directions. Inputs to the above formulationare the fourth order elasticity tensor of each layer of the laminatewithin the linear elastic limit, coefficient of thermal expansions,applied external strain and change in temperature. The effectiveloads calculated for a multi-directional laminate are applied to the

Fig. 2. 3D slice o

2D generalized representation of the laminate in the FEM modelexplained in the following section.

3. Implementation of the Quasi-2D formulation

The above formulation can be implemented in several wayswithin finite element method. However, a novel technique isdeveloped in this paper by modifying a thin slice of a 3D model tobehave like a generalized 2D model. A 3D model with a smallthickness (1mm in this case) along the x1-direction is considered asshown in Fig. 2(a) with only one element in the x1-direction. Multi-point constraints are applied between the front and back faces ofthe model such that the displacement fields are independent of thex1-direction. For illustration, consider a nodal point ‘m’ on the frontsurface and a corresponding nodal point on the back surface asshown in Fig. 2(b). These two points (m and n) are tied togethersuch that the displacements U1, U2 and U3 are identical for thenodal pair. Similarly, all the nodal points on the front surface aretied with their corresponding points on the back surface. Since,only one element exists in the x1-direction, the constraint enforcesthe requirement of displacement fields being independent of thatdirection, that is, U1, U2 and U3 are now functions of x2 and x3 only.This simulates the left hand side of equation (14). The externalloads established by the right hand side of equation (14) are appliedon the edges and interfaces between the layers of the model in thex2 � x3 plane.

Eight noded linear hexahedral elements (C3D8 in ABAQUS no-tation [38]) are used to mesh the Q-2D model as shown in Fig. 3(a).Very fine mesh is used near the interfaces and free edges to suffi-ciently capture the stress singularities within these regions. Theelement size in these regions is 1 mm and the total number of ele-ments in a single ply for ½45=� 45=90=0�s laminate is 5856. Theinterlaminar stresses near the free edges and interfaces appear tovary with the size of the elements. Hence, mesh sensitivity analysisis performed to determine the element sizes that minimize thedependency of stress profile on the mesh. Element sizes are variedlinearly from a very fine mesh at the free edges and interfaces tocoursemesh away in the interior regions of themodel in the x2 � x3plane. This mesh refinement is conducted until no significantchange in the interlaminar stress profile is observed. However, onlyone element is used along the x1 direction to enforce multi-pointconstraint described above. Due to relatively large number ofnodes on the front and back faces in the x2 � x3 plane, enforcingmulti-point constraints manually is cumbersome. Hence, a script toassign the multi-point constraints on a generic laminate is devel-oped, which is described in detail in the following paragraph.

A python™ script is written that reads an ABAQUS input file andincludes the multi-point constraints at the appropriate sections of

f a laminate.

Fig. 3. (a) Typical mesh in the Q-2D model; (b) Working principle of python™ script.

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e98 93

the file. Working procedure of the python™ script is shown inFig. 3(b). The input file obtained from ABAQUS without constraintsis read by the python™ script, which identifies the nodal pairsbetween the front and back faces. The script then writes theconstraint equations for each nodal pair in a different file. Then, themain input file is re-written with the equations file called using“*INCLUDE”. Fig. 4(a) shows a snapshot of the re-written input filewhich includes the equation file as an “input”, and Fig. 4(b) shows asnapshot of the equation file with constraint equations written.

For illustration, Fig. 5 shows the boundary conditions for½45=� 45=90=0�s laminate subjected to an uniform axial strain,which is derived from Equation (14). As mentioned in the previoussection, key inputs to the Q-2D model are the 4th order elasticitytensor of individual layers and the applied strain. Effective loadsthat depend on the elastic constants are applied as pressure alongthe normal direction and surface tractions parallel to the edges asshown in Fig. 5. Here, x1

�!, x2�! and x3

�! indicate the directions of theapplied effective loads on each layer. The loads corresponding tox3�! direction are not shown in the figure since the associated elasticconstants are equal to zero in the current example. Accordingly, theeffective interface loads are applied along the interfaces betweenthe layers.

For several cases considered in this paper, full 3D models arealso analyzed to validate the results from the corresponding Q-2Dmodels. Towards that, only 1=8th of a 3D model is analyzed due tosymmetry as shown in Fig. 6. Linear hexahedral (C3D8 asmentioned before) elements are used with a minimum elementsize at the interfaces and free edges equal to 1 mm. For example, thisyields a total number of elements in a single ply for½45=� 45=90=0�s laminate equal to 79696. Mesh sensitivity analysisfor the 3D models similar to Q-2D model is performed as well.

Fig. 4. Snapshot of (a) input fi

4. Results and discussions

The Q-2D model is applied to several laminates subjected tothree main types of loading: uniform axial extension (Section 4.1),uniform thermal loading (Section 4.2) and combined axial andthermal loading (Section 4.3). Results are compared with publisheddata (where available) and 3D model results to establish the reli-ability of the current modeling approach.

4.1. Case 1: axial loading

In this section, ½0=90�s and ½45=� 45=90=0�s laminates subjectedto a uniform axial (x1-direction) strain are considered. Thefollowing material properties are used: E11 ¼ 137:9 GPa,E22 ¼ E33 ¼ 14:48 GPa, G12 ¼ G13 ¼ G23 ¼ 5:86 GPa andn12 ¼ n13 ¼ n23 ¼ 0:21 , which correspond to a high modulusgraphite/epoxy laminate [18]. The ½0=90�s cross-ply laminate sub-jected to a uniform axial (x1-direction) strain has been previouslyinvestigated by Zhang et al. [1], Tahani and Nosier [2], Nguyen andCaron [18], Wang and Crossman [23], Carreira et al. [39] and Zhenet al. [40]. Fig. 7 shows the comparison of interlaminar stress dis-tribution at the 0/90 interface of ½0=90�s laminate subjected to auniform axial strain with few previously published articles. Eventhough, Fig. 7(a) shows that the present model predicts slightlyhigher interlaminar normal stress at the edges compared to pre-vious studies, the behavior is identical. The shear stress (s23) dis-tribution is comparable to other models (Fig. 7(b)).

For ½45=� 45=90=0�s laminate, seldom results are found in theliterature to compare the results from the current Q-2D model.Hence, a full 3D model is analyzed and the results from the Q-2Dmodel is compared with the corresponding 3D model. Fig. 8 shows

le and (b) equation file.

Fig. 5. Boundary conditions for uniaxial extension.

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e9894

the comparison of Q-2D and 3D models at different interfaces,which shows an excellent agreement between these two models.

4.2. Case 2: thermal loading

In this section, ½0=90�s and ½45=� 45=90=0�s laminates subjectedto a uniform temperature change DT throughout the modeling

Fig. 6. 3D model.

Fig. 7. Distribution of interlaminar stresses along the 0/90 interface of [0/90]s laminate s

domain are considered. For direct comparison with previouslypublished results, the following material properties are used: E11 ¼137:9 GPa, E22 ¼ E33 ¼ 14:48 GPa, G12 ¼ G13 ¼ G23 ¼ 5:86 GPa,n12 ¼ n13 ¼ n23 ¼ 0:21, a1 ¼ 0:36� 10�6 +C�1 anda2 ¼ a3 ¼ 28:8� 10�6 +C�1. Fig. 9 shows the distribution of s33(Fig. 9(a)) and s23 (Fig. 9(b)) for DT ¼ 1 +C. It is observed that thecurrent results are in excellent agreement with those determinedby previous researchers [2,14,18].

Fig. 10 shows the distribution of interlaminar stresses for½45=� 45=90=0�s laminate subjected to a uniform temperaturechange DT ¼ 1 +C along theþ45/-45 (Fig. 10(a)),�45/90 (Fig. 10(b))and 90/0 interfaces (Fig. 10(c)). It is observed that the current re-sults are in good agreement with the corresponding 3D modelresults.

4.3. Case 3: combined axial and thermal loading

In this section, ½0=90�s and ½45=� 45=90=0�s laminates subjectedto a uniform axial strain (ε11 ¼ 0:01) and uniform temperaturechange (DT ¼ 1 +C) are considered. The material properties usedhere are the same as those mentioned in Section 4.2. Fig. 11 and 12show the distribution of interlaminar stresses for ½0=90�s and½45=� 45=90=0�s laminates subjected to ε11 ¼ 0:01 and DT ¼ 25 +C.

ubjected to a uniform axial extension (a) normal stress s33 and (b) shear stress s23.

Fig. 8. Distribution of interlaminar stress of ½45=� 45=90=0�s laminate subjected to uniform axial extension along (a) 45/-45, (b) �45/90 and (c) 90/0 interface.

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e98 95

These results are compared with their corresponding 3D modelsand is observed (Fig. 11 and 12) that the current results are inexcellent agreement with the 3D results.

4.4. Comparison of Q-2D model with 3-D model

The Q-2D modeling framework developed in this paper utilizesa fairly straight forward derivation and an easy implementationwithin existing finite element method softwares. It is also highlyefficient with respect to computational cost when compared to the3D counterpart models. A direct comparison of the efficiency of Q-2D models with 3D models is performed due to insufficient datafound in the literature about the computational time, processorused, etc. for other publishedmethods. Table 1 shows a comparisonof the computational time required by the Q-2D and 3D models of½45=� 45=90=0�s laminate (it takes approximately the same timefor all loading cases). The total computational time required by theQ-2D model is approximately 2 min and that for the 3D model isapproximately 60 min using the same Intel® Core™ i7-3540 M3.00 GHz CPU. It should be noted that although 60 min seems like areasonable time for a 3D model, it is for an 8 layer laminate(½45=� 45=90=0�s) only. The time required will scale up withincreased number of layers in the case of thicker laminates (forexample 48 layers). Further, such a modeling approach will dras-tically reduce the computational cost associated with optimizationproblems, where large number of simulations are required. Hence,the Q-2D model can save the computational time significantlywhile predicting the interlaminar stress distributionwith high level

of accuracy as compared to 3D models.

5. Conclusions

In this study, a Quasi-2D modeling approach to determine freeedge effects in laminated composites was developed and imple-mented within the finite element method framework. The inter-laminar stresses determined using this approach are comparable tothat of a full 3D model. Also, the accuracy and effectiveness of thecurrent Q-2D model is validated with previously published results.In particular, free edge effects in ½0=90�s and ½45=� 45=90=0�slaminates subjected to uniform axial extension, uniform tempera-ture distribution and combined axial-thermal loading were inves-tigated with the Q-2D modeling approach. Primarily, normal (s33)and shear (s23) stresses at the interlaminar regions were examineddue to their propensity to create delaminations. The Q-2D modelwas capable of predicting the interlaminar stresses for differentlaminates subjected to uniform axial and/or thermal loading effi-ciently with high level of accuracy. The current formulation in-cludes thermal and uni-axial loading cases, however, furtherdevelopment is needed for capturing the effects of inhomogeneousmaterial/geometry or fully multi-axial loading. Key outcomes of theresearch presented in this paper are as follows:

� The computational time required to solve for the interlaminarstresses in multi-directional laminates using Q-2D model isapproximately 30 times lower than the corresponding 3Dmodelof the same laminate.

Fig. 9. Distribution of interlaminar stresses along the 0/90 interface of [0/90]s laminate due to a temperature change DT ¼ 1+C (a) normal stress s33 and (b) shear stress s23.

Fig. 10. Distribution of interlaminar stress of ½45=� 45=90=0�s laminate due to a temperature change DT ¼ 25 +C (a) 45/-45, (b) �45/90 and (c) 90/0 interface.

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e9896

Fig. 11. Distribution of interlaminar stress along the 0/90 interface of ½0=90�s laminate subjected to ε11 ¼ 0:01 and DT ¼ 25 +C (a) normal stress s33 and (b) shear stress s23.

Fig. 12. Distribution of interlaminar stress of ½45=� 45=90=0�s laminate subjected to ε11 ¼ 0:01 and DT ¼ 25 +C along (a) 45/-45, (b) �45/90 and (c) 90/0 interface.

Table 1Comparison of Q-2D and 3D model.

Model No. of Elements Approximate Time (min)

Q-2D 23424 23D 318786 60

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e98 97

� Existing finite element framework within a commerciallyavailable software is used to determine interlaminar stresses bymodifying a 3D model to behave like a 2D model. This isparticularly advantageous since developing an in-house FEMcode can be avoided for determining complex stress states atfree edges of multi-directional laminates.

M.S. Islam, P. Prabhakar / Composites Part B 116 (2017) 89e9898

Acknowledgements

The authors would like to acknowledge Dr. David Stargel fortheir support through the AFOSR Young Investigator Program(Grant No.: FA9550-15-1-0216) to conduct this research.

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