Composite Structures 119 (2015) 693–708

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Composite Structures

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3D FEA modelling of laminated composites in bending and their failuremechanisms

http://dx.doi.org/10.1016/j.compstruct.2014.09.0480263-8223/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Fax: +44 (0)1752 586101.E-mail address: maozhou.meng@plymouth.ac.uk (M. Meng).

M. Meng ⇑, H.R. Le, M.J. Rizvi, S.M. GroveSchool of Marine Science and Engineering, Plymouth University, United Kingdom

a r t i c l e i n f o

Article history:Available online 2 October 2014

Keywords:3DFEALaminateFlexuralCLT

a b s t r a c t

This paper developed three-dimensional (3D) Finite Element Analysis (FEA) to investigate the effect offibre lay-up on the initiation of failure of laminated composites in bending. Tsai-Hill failure criterionwas applied to identify the critical areas of failure in composite laminates. In accordance with the 3DFEA, unidirectional ([0]16), cross-ply ([0/90]4s) and angle-ply ([±45]4s) laminates made up of pre-preg Car-bon Fibre Reinforced Plastics (CFRP) composites were manufactured and tested under three-point bend-ing. The basic principles of Classical Laminate Theory (CLT) were extended to three-dimension, and theanalytical solution was critically compared with the FEA results. The 3D FEA results revealed significanttransverse normal stresses in the cross-ply laminate and in-plane shear stress in the angle-ply laminatenear free edge regions which are overlooked by conventional laminate model. The microscopic imagesshowed that these free edge effects were the main reason for stiffness reduction observed in the bendingtests. The study illustrated the significant effects of fibre lay-up on the flexural failure mechanisms incomposite laminates which lead to some suggestions to improve the design of composite laminates.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Laminated composites have been widely used in renewableenergy devices such as wind turbines [1] and underwater turbines[2] which are usually subjected to a combination of tension, bend-ing and twisting. Delamination plays an important role in the com-posites failure, while one of the main causes of compositesdelamination is the interlaminar shear stress [3]. Many theorieshave been developed to predict the distribution of interlaminarshear stress in composite laminates. It is well-known that ClassicalLaminate Theory (CLT) [4], First-order Shear Deformation Theory(FSDT) [5,6] and Refined Shear Deformation Theory (RSDT) [7] givea good prediction of mechanical behaviour with infinite compositeplates. However, these classical theories based on infinitely wideplates have experienced difficulties on regions near boundaries[8,9]. This is because these methods consider the laminated com-posites as shell elements which ignore the effects of the thicknessof the component. The shell method suffers from poor accuracy incase of high ratio of height to width.

Three-dimensional numerical analysis has been used to examinethe stress distribution in laminated composites. The pioneer workwas carried out by Pipes and Pagano using the Finite Difference

Method (FDM) [10,11]. They demonstrated the singularity of inter-laminar shear stress at the edge region in an angle-ply laminateunder tensile stress. Similar work investigated the interlaminarshear stress at free edges using FEA [12–14], Eigen-function expan-sions [15,16], Boundary Layer theory (BLT) [17,18], and Layer-wisetheory (LWT) [19]. A good review by Kant et al. [20] has covered theanalytical and numerical methods on free-edge problems of inter-laminar shear stress up to year 2000.

Previous work on 3D analysis has illustrated the increase ofinterlaminar shear stress at the edge region. Although the globalload may be lower than the composites strength, the interlaminarshear stress can induce the initial delamination at edge regionwhich will reduce the fatigue life of composites. This phenomenonhas been reported in composites design and manufacturing [21].In order to investigate the free edge effect on interlaminar shearstress, most of the previous works were focused on the uniformaxial loads. This type of loading condition does not induce somestress components, such as out-of-plane stresses, which neverthe-less have a significant effect on the bending failure behaviour. More-over, with the decrease of the support span in bending, these stresscomponents play an increasingly important role in composite fail-ure modes. Due to the nature of bending, laminates are subjectedto tension, compression and shear, so all of the six stress compo-nents should be considered when evaluating failure criteria. How-ever, there have been few reports on the free edge effect in bending.

Nomenclature

[a], [b], [d] block matrices of a bb d

� �matrix (inversed A B

B D

� �matrix)

r1, r2 radius of load cell and support rollersq mesh quality factorw, h, l width, height and length of laminate

[A], [B], [D] block matrices of A BB D

� �matrix

C off-axis stiffness matrix of laminaðCijÞk off-axis stiffness matrix of the kth plyDmax maximum deflectionEf

app apparent flexural modulus

EfCLT ; Ef

x flexural modulus evaluated by CLT

Ef1; Ef

2; Ef3 principal elastic moduli of fibre

E1, E2, E3 principal elastic moduli of laminaEm elastic modulus of matrixFmax maximum flexure forceGf

12; Gf23; Gf

13 principal shear moduli of fibreG12, G23, G13 principal shear moduli of laminaGm shear moduli of matrixI moment of inertiaK, Kf, Km bulk moduli of lamina, fibre and matrixL spanM, Mx momentNx;y;xy; Mx;y;xy force and moment per unit lengthP sinusoidal pressureQij; Qij extensional compliance matrix of unidirectional

and off-axis laminaQx force per unit length along the widthS compliance matrix of lamina

S off-axis compliance matrix of laminaTe, Tr transformation matrices of strain and stressVf fibre volume fractionV volume of a mesh elementWm matrix fraction in weightcxy in-plane shear straine normal strainf Halpin–Tsai adjusted parameter, for rectangle sec-

tion f = 1gxyx, gxyy interaction ratioh anglej curvaturem12, m23, m13 principal Poisson’s ratios of lamina

m f12; m f

23; m f13 principal Poisson’s ratios of fibre

mm Poisson’s ratio of matrixp circumference ratiok0 half-wavelength of fibres microbucklingr1, r2, r3 normal stress in local coordinate systemrx, ry, rz normal stress in global coordinate system

ðrult1 Þ

t; ðrult

2 Þt

principal tensile strength of lamina

ðrult1 Þ

clongitudinal compressive strength of lamina

ðrult2 Þ

ftransverse flexural strength of lamina

ðrultf Þ

tultimate tensile strength of fibre

ðrultm Þ

t; ðrult

m Þf

ultimate tensile and flexural strength of matrix

s12, s13, s23 shear stress in local coordinate system

sult12 ; sult

23 ; sult13 shear strength of lamina

sxy, sxz, syz shear stress in global coordinate system

Table 1Laminate configuration.

Laminate Lay-up Thickness (mm) Ply-thickness (mm)

Unidirectional [0]16 2.08 0.13Cross-ply [0/90]4s 1.92 0.12Angle-ply [±45]4s 1.92 0.12

694 M. Meng et al. / Composite Structures 119 (2015) 693–708

Due to the limitation of computing power, earlier works on 3Danalysis could only consider a few plies for the demonstration.When composite laminates are made of many plies with compli-cated orientation, the prediction of these models may lead to inac-curate results. Pipes and Pagano [10] illustrated the singularity ofinterlaminar shear stress at edge region of an angle-ply laminatewhich consisted of four plies, however, this singularity is unlikelyto occur in a laminate with many plies shown in the present work.Additionally, the FEA model for angle-ply laminate is unlikely to besimplified as symmetric in bending, due to the complicated ply lay-up pattern. This means that a full model need to be considered andsignificant computing resources are required for modelling.

The present work was intended to understand how the fibrelay-up affects the initiation of failure of laminated composites inbending. A series of composites with 3 common lay-up sequenceswere manufactured and tested in bending following ISO standards[22,23] to measure the critical failure loads and failure modes. Arobust 3D FEA and an extended CLT model were then applied toexamine the stress distribution under the measured failure loads.The stress distribution in critical areas of the laminated compositeswere examined and correlated with the observation of the initia-tion of failure in experiments.

2. Laminate preparation

Three different lay-ups of laminates were investigated for thedistributions of flexural and interlaminar shear stress. Table 1shows the laminate configuration. These three lay-up (unidirec-tional [0]16, cross-ply [0/90]4s and angle-ply [±45]4s) are the sim-plest examples of laminates which show a range of behaviour.The stress distributions and failure modes for a given laminate

lay-up could be extended in this direction. All of the laminateswere made up of 16 layers of high strength carbon fibre/epoxypre-preg (Cytec 977-2-35-12KHTS-134-150). The pre-preg plateswere placed on a mould and sealed in a vacuum bag, and then wereautoclave-cured at 0.6 MPa pressure. A heating rate of 3 �C/minfrom room temperature to 180 �C was applied, and then the pre-preg plates were held at 180 �C for 120 min and cooled at roomtemperature. For the cross-ply laminate, the bottom and top plieswere set as the longitudinal fibre orientation; therefore its flexuralmodulus was higher than tensile modulus. The angle-ply laminateand cross-ply laminate were cut from the same composite panel,with different cutting directions. In order to make the laminate‘self-balance’, the middle two plies were set at the same fibreorientation.

The final thicknesses of the three manufactured plates were notthe identical. There are probably two reasons: (a) the unidirec-tional laminate has a rougher surface than cross-ply laminate;(b) the void content in unidirectional laminate is slightly higherthan that in cross-ply laminate. Hypothesis (a) may lead to athicker laminate, since the dimension was measured by a digitalvernier calliper. Hypothesis (b) was confirmed by microscopeimage. Fig. 1 shows a void in microscope image of cross-section

Fig. 1. Optical microscope image of unidirectional laminate. A huge void was found,which was probably because of the manufacturing process.

M. Meng et al. / Composite Structures 119 (2015) 693–708 695

of unidirectional laminate. However, no voids were found in cross-ply and angle-ply laminates. In accordance with the actual thick-nesses of the laminates, the ply thicknesses of unidirectional andcross-ply laminates were adjusted in CLT and FEA models for con-sistency (shown in Table 1).

The flexural stress was measured by the long-beam method[24] with the dimension of length �width = 100 mm � 15 mm,while the interlaminar shear stress was measured by the short-beam method [25] with the dimension of length �width =20 mm � 10 mm.

3. Bending test results

The experiment was conducted according to ISO standards[24,25] using three-point bending. At least five samples in eachgroup were tested and the mean values were calculated. Theresults are shown in Table 2.

The value Dmax/L of unidirectional laminate was about 8%, asshown in Table 2, which was close to the ‘large-deflection criterion’(10%) [24]. Therefore, the flexural strength of unidirectional lami-nate was calculated by ‘large-deflection correction’,

rfmax

� �cor ¼

3FmaxL

2wh2 1þ 6Dmax

L

� �2

� 3Dmaxh

L2

� � !ð1Þ

Table 2Experimental results from three-point bending tests and the Standard Deviations (SDs).

Laminate groups Unidirectional [0]16

Long beam Short beam

Length (mm) 100 20Width (mm) 15.18 ± 0.03 10.14 ± 0.03Height (mm) 2.09 ± 0.06 2.13 ± 0.07Span (mm) 80 10Fmax (N) 853 ± 32 2933 ± 126Dmax (mm) 6.59 ± 0.27 –

Efapp (GPa) 120 ± 3.1 –

rappx (MPa) 1544 ± 49 –

ðrappx Þcor (MPa) 1598 ± 56 –

sappxz (MPa) – 101.9 ± 3.5

Fmax – maximum flexure load; Dmax – maximum deflection; Efapp – apparent flexural mo

‘large-deformation’ correction; sappxz – apparent interlaminar shear strength.

4. Methodologies

4.1. FEA approach

The technical term ‘symmetry’ includes symmetry in geometry,material and boundary condition. Although the lay-up sequence(as well as geometry) is symmetric for all specimens shown inTable 1, the angle-ply laminate has no through-thickness planeof symmetry in term of material orientation.

All of the specimens were ‘simply supported’, which presents alinear relationship between flexure load and deflection. The loadsapplied in the FEA models (Table 3) were taken as the maximummeasured loads in the three-point bending tests.

In the 3D FEA model, the boundary conditions are quite differ-ent from a 2D model, and some modelling techniques should beintroduced because the ‘simply supported’ and loading boundaryconditions might lead to inaccurate results due to the stress con-centration at these boundaries. Additionally, the ‘contact’ bound-ary condition is not appropriate in present work, so that thesehave been replaced by distributed loads with sinusoidal distribu-tion, which includes an downward (negative) distributed load (P)in the middle area of top surface (load-point) and half of an upward(positive) distributed load (P/2) at the left and right ends of bottomsurface, as shown in Fig. 2. For the long-beam specimens, the spans(L) were set as 80 mm (longitudinal laminate) and 79 mm (cross-ply laminate), while the short-beam specimens had a span of10 mm.

In order to avoid rigid movement, some assistant boundary con-ditions were applied to eliminate the six degree of freedom (DOF).With the natural symmetry of unidirectional laminate and cross-ply laminate, two symmetric planes were applied to eliminatethe DOFs of x, y, and the rotation about three axes. The two centralpoints at each end of the laminate (z = h/2) were restrained as z = 0to eliminate the last DOF. However, the ‘symmetric plane’ bound-ary conditions do not exist in angle-ply laminate, due to the asym-metric material properties. Two ‘edge displacement’ boundaryconditions were applied to replace the symmetric planes for elim-inating the DOFs. Fig. 3 shows the artificial boundary conditions forthe DOFs elimination in unidirectional and cross-ply laminates (a),and angle-ply laminate (b).

The properties of carbon fibre (HTS) and epoxy (977-2) wereused in FEA models. Table 4 gives the material properties of pre-preg CFRP composite from the manufacturers’ data sheets[26,27]. It can be seen from Table 4 that the flexural strength ofthe matrix is much higher than the tensile strength. This may affectthe transverse strength of composite lamina. The fibre volume frac-tion can be calculated from Wm,

Cross-ply [0/90]4s Angle-ply [±45]4s

Long beam Short beam Short beam

100 20 2015.08 ± 0.02 10.14 ± 0.10 10.10 ± 0.121.93 ± 0.01 1.94 ± 0.02 1.93 ± 0.0179 10 10630 ± 21 2257 ± 83 1395 ± 618.99 ± 0.31 – –79.7 ± 0.8 – –

1328 ± 39 – –1421 ± 48 – –– 86.1 ± 4.0 53.7 ± 2.8

dulus; rappx – apparent flexural strength; ðrapp

x Þcor – apparent flexural strength with

Table 3Loading forces in different groups of coupons.

Orientation groups Unidirectional Cross-ply Angle-ply

Long beam Short beam Long beam Short beam Short beam

Force (N) 853 2933 630 2257 1395

Fig. 2. Modelling conditions were equal to testing conditions. The two ‘simply supported’ boundary conditions at two ends were replaced by positive distributed loads (P/2).

Fig. 3. Boundary conditions applied in (a) symmetric laminates and (b) angle-ply laminate.

Table 4Material properties of fibre and matrix. The fibre transverse modulus is estimated as 10% of its longitudinal modulus, according to Refs. [28–30].

Symbol Ef1 Ef

2 ¼ Ef3 mf

12 ¼ mf13 mf

23qf qm

Value 238 GPa 23.8 GPa 0.2 0.4 1.77 g cm�3 1.31 g cm�3

Symbol Wma Em mm ðrult

f Þt ðrult

m Þt

ðrultm Þ

f

Value 35% 3.52 GPa 0.34 4.3 GPa 81.4 MPa 197 MPa

a Wm is the matrix fraction in weight; ðrultf Þ

tand rult

m

� �tare the tensile strength of fibre and matrix; rult

m

� �fis flexural strength of matrix.

696 M. Meng et al. / Composite Structures 119 (2015) 693–708

Vf ¼ð1�WmÞ=qf

ð1�WmÞ=qf þWm=qf¼ qmð1�WmÞ

qmð1�WmÞ þ qf Wmð2Þ

Substituting the values in Table 4 into Eq. (2), the fibre volumefraction can be estimated as Vf = 57.9%.

Employing the ‘rule of mixture’ [31] and ‘Halpin and Tsai’ [32]methods, the in-plane material properties of lamina can be calcu-lated. However, there is no agreed formula to calculate the trans-verse material properties (m23,G23). In the present work, thetransverse Poisson’s ratio was evaluated by the hydrostatic

Fig. 4. Mesh plot of 20 mm � 10 mm laminate with local refinement. The edge areawas refined to investigate the free edge effect.

M. Meng et al. / Composite Structures 119 (2015) 693–708 697

assumption as shown in Appendix A. For the orthotropic material,such as composite laminate, Tsai–Hill failure criterion [33,34] hasshown a good fit to experiments, and this is used in the presentwork:

ðG2þG3Þr21þðG1þG3Þr2

2þðG1þG2Þr23

�2ðG3r1r2þG2r1r3þG1r2r3�G4s223�G5s2

13�G6s212Þ<1 ð3Þ

G1 ¼1

rult2

� �2 �1

2 rult1

� �2 ; G2 ¼ G3 ¼1

2 rult1

� �2 ; G4

¼ 1

2 sult23

� �2 ; G5 ¼1

2 sult13

� �2 ; G6 ¼1

2 sult12

� �2 ð4Þ

There are six parameters of lamina strength in Eqs. (3) and (4), how-ever only four are independent rult

2 ¼ rult3 ; sult

12 ¼ sult13

� �. The longitu-

dinal tensile strength of lamina rult1

� �tcan be evaluated by the

‘maximum strain’ method [31]; rult2

� �tis assumed to be the trans-

verse flexural strength of the lamina, which was separatelyobtained by three-point bending test of [90]16 samples; the shearstrength sult

13 is estimated by the interlaminar shear test of [0]16

samples, which is shown in Table 2; sult23 is assumed to be the same

as the shear strength of matrix, which is in the order of 6% (ultimateshear strain) [35].

Substituting the material properties of fibre and matrix inTable 4, all the material properties of lamina required by FEA soft-ware were calculated, as shown in Table 5.

It is important to note that two coordinate systems areemployed: (a) the local coordinate system represents stress orstrain in the lamina level (subscripts 1, 2, and 3), and (b) the globalcoordinate system represents stress and strain at the laminatelevel (subscripts x, y, and z). The failure criterion must be appliedin local coordinate system. For example rx, ry and sxy are basedon global coordinates, which should be transformed to the localcoordinates (r1, r2 and s12) in accordance with the failurecriterion.

Orthotropic material properties were applied in the simulationand every off-axis ply used a rotated coordinate,

x

y

� �¼

cos h � sin h

sin h cos h

� �X

Y

� �ð5Þ

where X and Y are the transformed variables in the rotated (h) coor-dinate system. The elastic properties of lamina (Young’s modulus,Shear modulus and Poisson’s ratio) were transformed using Eq.(5) for the definition of FEA.

In FEA models, all of the 16 plies were built as 3D-solid element,and bonded together. Because the mesh quality could affect the 3DFEA results significantly, two methods for mesh quality controlwere employed: (a) distributed mesh was defined near edgeregion; (b) global elements were referred to ‘q’ factor, which wasevaluated by [36],

q ¼ 24ffiffiffi3p

VP12i¼1h2

i

3=2 ð6Þ

where V is the volume, and hi are the edge lengths. If q > 0.1, themesh size should not affect the solution quality.

Table 5Material properties of lamina for simulation.

Symbol E1 E2 = E3 G12 = G13

Value 139 GPa 8.8 GPa 4.7 GPa

Symbol ðrult1 Þ

t ðrult1 Þ

cðrult

2 Þt ¼ ð

Value 2.52 GPa 1.58 GPaa 123 MPa

a The longitudinal compressive strength of the lamina ðrult1 Þ

cis estimated by Ref. [26

The through-thickness mesh density has a weak influence onthe FEA results since the material properties within each ply areconsidered as homogeneous. However the mesh quality in thewidth direction has to be refined and the dimension of an individ-ual element at the edge should be comparable to the ply thickness.In the present work, geometry near the edge was refined to beapproximately one half-ply thickness along the width, and eachply was divided into 3 elements through-thickness, as shown inFig. 4. A finer mesh than this would not provide noticeableimprovement of the FEA solution, while demanding exponentiallyincreasing computing resources. Fig. 5 shows the relationshipbetween the mesh size (multiple of one-ply thickness) and thesolution. The 3D FEA models were solved by COMSOL Multiphysics[36], with approximate one million DOFs in each laminate.

4.2. 3D CLT formulae

A 3D formulation of CLT is required to extract the six stresscomponents in laminates. The formulae are shown in AppendixB. In the 2D version of CLT, the transverse shear stresses areneglected. It would be reasonable if the span is long enough andthe width is infinite. In the present work, these transverse stresscomponents are compared with 3D FEA models. With the 3D ver-sion of [a,b;b,d] matrix, the interlaminar shear stress sxz and trans-verse shear stress syz can be evaluated by the principle ofcontinuum mechanics [37],

sðkÞxz ¼Xk�1

j¼1

Z zj

zj�1

@sxz

@z

� �ðjÞ

dzþZ z

zk�1

@sxz

@z

� �ðkÞ

dz

¼ �Xk�1

j¼1

C11ðjÞb11 þ C12ðjÞb21 þ C16ðjÞb61

zj � zj�1� �n

þ 12

C11ðjÞd11 þ C12ðjÞd21 þ C16ðjÞd61

z2

j � z2j�1

�Q x

� C11ðkÞb11 þ C12ðkÞb21 þ C16ðkÞb61

z� zk�1ð Þ

nþ 1

2C11ðkÞd11 þ C12ðkÞd21 þ C16ðkÞd61

z2 � z2

k�1

� ��Qx ð7Þ

G23 m12 = m13 m23

3.0 GPa 0.26 0.48

rult3 Þ

tðrult

2 Þf sult

12 ¼ sult13 sult

23

123 MPa 102 MPa 79 MPa

], which used the same matrix and a similar fibre to the present work.

Fig. 5. The effect of mesh size near the edge region on the distribution of global interlaminar shear stress in short-beam angle-ply laminate. The results show that 0.5 wassufficient to get mesh independency.

Fig. 6. Distribution of tensile stress r1 on bottom surface of long-beam unidirec-tional laminate. The stress (FEA) shows a minor fluctuation about 2% between thefree edge and central areas.

698 M. Meng et al. / Composite Structures 119 (2015) 693–708

sðkÞyz ¼Xk�1

j¼1

Z zj

zj�1

@syz

@z

� �ðjÞ

dzþZ z

zk�1

@syz

@z

� �ðkÞ

dz

¼ �Xk�1

j¼1

C61ðjÞb11 þ C62ðjÞb21 þ C66ðjÞb61

zj � zj�1� �n

þ 12

C61ðjÞd11 þ C62ðjÞd21 þ C66ðjÞd61

z2

j � z2j�1

�Q x

� C61ðkÞb11 þ C62ðkÞb21 þ C66ðkÞb61

z� zk�1ð Þ

nþ 1

2C16ðkÞd11 þ C26ðkÞd21 þ C66ðkÞd61

z2 � z2

k�1

� ��Q x ð8Þ

The local interlaminar shear stress in the kth ply (along fibre orien-tation) is evaluated according to its orientation,

sðkÞ13 ¼ sðkÞxz cos hðkÞ þ sðkÞyz sin hðkÞ

sðkÞ23 ¼ �sðkÞxz sin hðkÞ þ sðkÞyz cos hðkÞð9Þ

Substituting the laminate dimension, flexure loads and deflectionsin Table 2 into 3D CLT formulae, the maximum ply normal stressand interlaminar shear stress can be obtained, as shown in Table 6.In order to process the data, a MATLAB program [38] wasdeveloped.

5. Results and discussion

5.1. Unidirectional laminates

For the long-beam method, the ISO standard considers the flex-ural stress in longitudinal direction by neglecting the other compo-nents. According to the 3D FEA model, the stress components r2

and r3 are very small compared with r1 (about 2%) because ofthe ‘simply supported’ boundary condition. The flexural stress r1

Table 6Maximum ply normal stress and interlaminar shear stress by 3D CLT.

Orientation groups Unidirectional [0]16 Cross-ply [0/90]4s

Long beam Short beam Long beam Sho

EfCLT (GPa) 139 – 86.5 –

rmax1 (MPa) 1598 ± 56 – 2157 ± 78 –

smax13 (MPa) – 101.9 ± 3.5 – 83.

shows a tiny increase (about 2%) near the free edge region, asshown in Fig. 6. Although the maximum tensile stress is muchlower than longitudinal tensile strength rult

1

� �t(2.52 GPa), the

compressive stress is very close to the compressive strengthrult

1

� �c(1.58 GPa), as shown in Table 5. Therefore, the long-beam

unidirectional laminate failed in compression, rather than tension.The microscope observation confirmed this hypothesis. Fig. 7

shows a typical failure image of long-beam unidirectionallaminate, and Fig. 8 shows the deflection-load curves of long-beamunidirectional laminate. Fig. 7(b) clearly shows the interfacebetween tensile and compressive failures within a unidirectionallaminate, while Fig. 7(c) indicates that fibres in the upper halffailed by compression. A survey of literature [39–42] shows that

Angle-ply [±45]4s Notes

rt beam Short beam

– Flexural modulus by CLT

– Maximum ply normal stress3 ± 2.6 40.6 ± 2.1 Maximum ply interlaminar shear stress

Fig. 7. Microscope image of failure mode in a long-beam unidirectional laminate under three-point bending (side-view). Approximate 70% of the plies failed by compression,and fibre microbuckling could be observed on the compressive side.

Fig. 8. Deflection–load curves of long-beam unidirectional laminate under three-point bending. Laminate failed rapidly after the first ‘stiffness losses’ appeared (progressivefailure is beyond the scope of this paper).

M. Meng et al. / Composite Structures 119 (2015) 693–708 699

the half-wavelength k0 of fibre microbuckling is typically 10–15times of fibre diameter, which is in accordance with the kinkingband shown in Fig. 7(b).

The observed stiffness dropped in small steps when the flexureload reached the peak, and each step of ‘stiffness losses’ representsthe failure of a single ply (compressive failure). The flexural stressre-distributed, and the lower plies withstood the maximum com-pressive stress but the tensile stress at bottom ply did not reachthe tensile strength. As a consequence, more and more plies failedby compressive stress, and then the sample broke into two partssuddenly when the last 1/3 of the plies failed. Previous literature[39–41,43] shows that the longitudinal compressive strength ofunidirectional laminate is about 60%�70% of its tensile strength.One possible reason is that fibre misalignment causes fibre micro-buckling. Fig. 9 is a schematic diagram to show the microbuckling

in long-beam unidirectional laminate. The carbon fibres areallowed to buckle into the weaker resin in lower plies and finallybreak under in-plane compressive stress.

For the short-beam laminate, the 3D FEA model shows a signif-icant increase (15%) of free edge effect on the interlaminar shearstress s13. However this value decays sharply inside the laminateand then converges to the CLT value (c.f. Table 6) in the centralarea, as shown in Fig. 10. This implies that the laminate failed ini-tially from edge area. Additionally, due to the short span, the out-of-plane normal stress r3, which is neglected in the ISO standard,shows a relatively high value in the FEA model. Similarly, this valuedecays inside the laminate, and is located at the loading area.Fig. 11 shows the distribution of out-of-plane normal stress r3

and typical failure images of the short-beam unidirectional lami-nate. The maximum out-of-plane normal stress r3 is very close

Fig. 9. Schematic diagram of fibre microbuckling of long-beam unidirectional laminate. With the same fibre orientation, the second ply is likely to ‘buckle’ following the firstply by the compressive stress, and then followed by the third ply, and so on.

Fig. 10. Distribution of interlaminar shear stress s13 on middle plane of short-beam unidirectional laminate. The higher value at free edge region implies the crack could beinitialized from this area.

700 M. Meng et al. / Composite Structures 119 (2015) 693–708

to the transverse tensile strength rult2

� �tin Table 5. It also indicates

that the ISO standard may underestimate the interlaminar shearstrength of short-beam unidirectional laminate.

5.2. Cross-ply laminates

Because of the bidirectional lay-up sequence, the flexural stres-ses are not continuous through thickness in cross-ply laminates.Fig. 12 shows these discontinuities in the long-beam cross-plylaminate in local coordinate system, while Fig. 13 shows thethrough-thickness stress distribution at the centre.

Fig. 13 shows that the maximum r1 and r2 at the centre of lam-inate are about ±2.2 GPa and ±140 MPa respectively. A comparisonof these values with the lamina strength shown in Table 5 illus-trates that the longitudinal compressive stress has exceeded thelamina compressive strength, while the longitudinal tensile stressis slightly lower than lamina tensile strength. In accordance with

the experimental condition, the top ply could withstand such highvalue of compressive stress, because the microbuckling was con-strained by the roller and supported by the transverse ply under-neath it. In this condition, the top ply would be more difficult to‘buckle’ compared to the situation in a unidirectional laminate(c.f. Fig. 9). Fig. 14 shows the schematics of fibre orientation inlong-beam cross-ply laminate. The out-of-plane buckling of fibresin the top ply is constraint by the roller and the transverse fibresin the adjacent ply. Therefore the compressive strength of thematerial is significantly improved.

On the other hand, it is widely recognised that the plasticmatrix could withstand higher compressive stress than tensilestress. Therefore, the 15th ply (90� orientation with low stiffness)in the tensile region was more likely to fail than the second ply(90�) in the compressive region. Indeed, the tensile stress in the15th ply had exceeded the transverse tensile strength of resinshown in Table 5. Therefore, failure sequence of long-beam

Fig. 11. Typical failure images of short-beam unidirectional laminate and the distribution of out-of-plane normal stress r3. The combination of interlaminar shear stress s13

and out-of-plane normal stress r3 leads to delamination at compressive (top) part of the laminate.

Fig. 12. Distributions of stress components r1 (left) and r2 (right) in long-beam cross-ply laminate and their side-views.

M. Meng et al. / Composite Structures 119 (2015) 693–708 701

cross-ply laminate can be explained as, (a) the 15th ply failed intension and the stiffness had a tiny drop (90� ply failed), (b) the16th ply (0� orientation with high stiffness) delaminated and failedin tension, and then the stiffness shown a huge decrease, (c) thedelamination propagated inside the laminate and it failed. Fig. 15shows a typical microscope failure image of long-beam cross-plylaminate, while Fig. 16 shows the deflection-load curves of long-beam cross-ply laminate.

Applying the Tsai–Hill failure criterion to the FEA results of thelong-beam cross-ply laminate, indicates that the interlaminarshear stress contributed about 4% to the criterion. However, thestress component r2 contributes much more due to the lowertransverse tensile strength rult

2

� �t. Fig. 17 shows the distribution

of the Tsai-Hill failure criterion in the long-beam cross-ply lami-nate. The maximum value appeared at the interfaces of the firstand second plies corresponding to the maximum transverse stressr2, as shown in Fig. 12. Delamination was also observed betweenthe 1st and the 2nd ply in the experiment as shown in Fig. 15.

For the short-beam cross-ply laminate, the interlaminar shearstress s13 is not continuous due to the bidirectional lay-upsequence, and the maximum value appears at the interfacebetween the 7th and 8th plies (z = 1.08 mm) rather than the mid-plane. This is different from the measured apparent interlaminar

shear stress (shown in Table 2). Fig. 18 shows the distribution ofinterlaminar shear stress s13 through-thickness at x = 13 mm. Thecoordinates are (13,0), (13,0.6) and (13,5) respectively.

It can be seen from Fig. 18 that the maximum value of s13 islower than the interlaminar shear strength shown in Table 5. Thetransverse and out-of-plane components of normal stress, r2 andr3 are much higher, compared with the short-beam unidirectionallaminate. Fig. 19 shows the distributions of these two normalstress components in the short-beam cross-ply laminate. The max-imum values of r2 and r3 are so high that they have exceeded thetransverse tensile strength ðrult

2 Þt. It indicates that the laminate

failed in transverse compression initializing at the second ply. Fol-lowing the ‘stiffness losses’ and stresses re-distribution, the maxi-mum interlaminar shear stress s13 exceeded the shear strength,and then the laminate failed. Fig. 20 shows a typical microscopeimage of interlaminar failure of short-beam cross-ply laminate.

5.3. Angle-ply laminate

For the angle-ply laminate, the distributions of these flexuralstresses are quite different from the symmetric laminates. More-over, the CLT and 3D FEA models present significantly differentresults. With the infinite plane hypothesis, the CLT method

Fig. 13. Through-thickness distributions of flexural stress r1 (s11) and r2 (s22) at central point of long-beam cross-ply laminate. The stresses jump rapidly at the interfacebetween longitudinal and transverse plies.

Fig. 14. Schematics of fibre microbuckling of long-beam cross-ply laminate. With the support of the second ply, the first ply is more difficult to fail by microbuckling.

Fig. 15. Typical microscope failure image of long-beam cross-ply laminate (left) and its tensile failure in 3-point bending (right).

702 M. Meng et al. / Composite Structures 119 (2015) 693–708

Fig. 16. Deflection-load curves of long-beam cross-ply laminate. The small and large ‘stiffness losses’ represent the failure of transverse and longitudinal plies.

Fig. 17. Distribution of Tsai–Hill ‘failure factor’ in long-beam cross-ply laminate.The transverse plies exceeded the failure criterion rather than the surface plies.

Fig. 18. Distribution of interlaminar shear stress s13 of short-beam cross-ply laminate.

M. Meng et al. / Composite Structures 119 (2015) 693–708 703

provides a relative smooth distribution of stresses. Fig. 21 showsthe distribution of interlaminar shear stress sxz (global) ands13 (local) through-thickness in short-beam angle-ply laminate,evaluated by CLT method. It can be seen that both of the maximumvalue of sxz and s13 appear at the mid-plane (z = 0.92 mm), and theshear stress s13 in local coordinate system is not continuousbecause of the complicated lay-up sequence.

The curves extracted from the CLT method show that the localinterlaminar shear stress s13 is lower than the global value.Furthermore, these curves are so uniform that they provide noinformation about the free edge effects.

The early works of Pipes and Pagano [10,11] had predicted thesingularity of interlaminar shear stress near the free edge regionof a [±45�]2 angle-ply laminate under axial load. 3D FEA modelsin the present work also show the increase of interlaminar shearstress in the short-beam angle-ply laminate under bending.Fig. 22 shows the through-thickness distribution of interlaminarshear stress of the short-beam angle-ply laminate. It can be seenthat both the sxz and s13 near free edge area fluctuate remarkably.The maximum values appear at the interface between the 4th and

The free edge effect is slight, compared with short-beam unidirectional laminate.

Fig. 19. Distributions of normal stress r2 (left) and r3 (right) in short-beam cross-ply laminate and their side-views. The maximum stresses appeared at the second ply (90�),and strong free edge effect on r3 is observed.

Fig. 20. Typical microscope failure image of short-beam cross-ply laminate. Theinitial delamination began from the 2nd ply, corresponding to the maximum r2 andr3 in Fig. 19.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

Inte

rlam

inar

she

ar s

tres

s τ 13

(M

Pa)

Through-thickness(mm)

Interlaminar shear stress through-thickness

Global τxz

Local τ13

Fig. 21. Interlaminar shear stress sxz and s13 distribution through-thicknesses inshort-beam angle-ply laminate (CLT). The discrete s13 represents the complicatedlamina orientation.

704 M. Meng et al. / Composite Structures 119 (2015) 693–708

5th plies, instead of the mid-plane (8th and 9th plies, as predictedby CLT, shown in Fig. 21). However, the distribution tends to beuniform inside the laminate. A small distance from the edge (2ply-thicknesses, 0.24 mm), the distribution of global shear stresssxz becomes a parabolic shape, while the maximum value of localshear stress s13 at the mid-plane drops approximate 20%. Finallyboth of the global and local shear stresses converge to the CLT atcentral area.

This extremely high global shear stress sxz at the free edgelocated at the interface of two plies, which may lead to delamina-tion, while the local shear stress s13 at the corresponding locationis very close to the shear strength sult

13 shown in Table 5. Moreover,quite a few points with these ‘extreme values’ can be found at theinterface of two plies, which are easier to induce the ‘multi-crack’at the edge area. Fig. 23 shows the surface plot and slice plot oflocal shear stress s13 of the short-beam angle-ply laminate, whileFig. 24 shows the diagram of the free edge effect.

It should be noted that the interaction ratio (gxyx) between nor-mal stress rx and in-plane shear stress sxy is too high to beneglected in angle-ply laminates. According to 3D CLT (shown inAppendix B), the transformed compliance matrix S of angle-plylaminate shows non-zero ‘interaction’ terms (S16 and S26), leadingto a definition of interaction ratios:

gxyx ¼ ExS16

gxyy ¼ ExS26

ð10Þ

The interaction ratio (gxyx) represents the ratio of the shear straincxy induced by normal stress rx, to the normal strain ex inducedby the same normal stress rx.

Fig. 25 shows the relationship between interaction ratio gxyx

and the off-axis angle (predicted by CLT). The interaction ratio(gxyx) evaluated by CLT predicted a value of about �0.7 in angle-ply lamina (45�). It illustrates the axial stress could induce ratherhigh in-plane shear stress, which is happening in the present caseof the short-beam angle-ply laminate. The authors found that formany commercial CFRP composites, the maximum value of inter-action ratio appears around 10–13� off-axis angle. Table 7 showsthe maximum interaction ratio of ten commercial CFRP compos-ites. In Table 7, there are ten different commercial CFRP compositesand their maximum interaction ratios are very close. In fact, thecoefficient of variation of off-axis angle is 1.3%.

Due to the complex structure in angle-ply laminate, the interac-tion ratio (g) strongly affects the distribution of the in-plane shearstress sxy in 3D. Indeed, the value of in-plane shear stress sxy is

Fig. 22. Distributions of interlaminar shear stress through-thicknesses in short-beam angle-ply (3D FEA model). The ‘stress peaks’ at edge area converge to CLT at centre, andthe maximum value appears at z = 1.44 mm (interface of 4–5 plies).

Fig. 23. 3D distribution of s13 in short-beam angle-ply laminate. The slice plotreveals the distribution of s13 in 3D scale, and the surface plot shows the variationof s13 in different plies with particular fibre orientation.

Fig. 24. Contour curves of interlaminar shear stress s13 in short-beam angle-plylaminate (z = 1.44 mm). The extremely high stress only appears near the edge area.

0 10 20 30 40 50 60 70 80 90-2.5

-2

-1.5

-1

-0.5

0

Angle in degree(θ)

Inte

ract

ion

ratio

Tensile ηxyx

Tensile ηxyy

Fig. 25. Interaction between axial stress and shear stress in off-axis laminate(according to CLT). The value of g represents the couple of normal stress to shearstress.

Table 7Engineering constants [44] and the interaction ratio (according to CLT).

E1 (GPa) E2 (GPa) m12 G12 (GPa) jgmaxxyx j h (�)

IM7/977-3 191 9.94 0.35 7.79 2.259 12T800/Cytec 162 9 0.4 5 2.622 10T700 C-Ply 55 121 8 0.3 4.7 2.301 11T700 C-Ply 64 141 9.3 0.3 5.8 2.224 12AS4/H3501 138 8.96 0.3 7.1 1.970 13IM6/epoxy 203 11.2 0.32 8.4 2.237 12AS4/F937 148 9.65 0.3 4.55 2.625 10T300/N5208 181 10.3 0.28 7.17 2.288 12IM7/8552 171 9.08 0.32 5.29 2.629 10IM7/MTM45 175 8.2 0.33 5.5 2.616 10Average 163.1 9.363 0.32 6.13 2.377 11.2SDs 25.85 0.96 0.03 1.37 0.23 1.14Coeff. var. 15.9% 10.2% 10.7% 22.4% 9.7% 1.3%a

a Divided by 90�.

M. Meng et al. / Composite Structures 119 (2015) 693–708 705

Fig. 26. Slice plot (upper) and surface plot (lower) of in-plane shear stress in short-beam angle-ply laminate. The values of s12 near the middle area of top and bottomsurfaces are so high that strong distortion was observed in the bending test.

Fig. 27. Typical failure image of angle-ply laminate under bending test condition.Cracks appeared at free edge area, but without penetrating inside the volume. Thepositions of cracks correspond to a peak of interlaminar shear stress, as shown inFig. 22. Specimen twisting induced by in-plane shear stress was observed.

706 M. Meng et al. / Composite Structures 119 (2015) 693–708

much higher than the other two shear stress components sxz andsyz. Because of the nature of three-point bending, the maximumnormal stress appears at the top and bottom plies. As a conse-quence, this ‘induced’ in-plane shear stress sxy may lead to strongtwisting at the two surfaces of the laminate. Fig. 26 shows the sliceplot and surface plot of in-plane shear stress s12 in short-beamangle-ply laminate.

Fig. 28. Deflection-load curves in angle-ply laminate from three-point be

The observation of microscope images confirmed the resultsfrom 3D FEA models. Instead of delamination failure (as likelyoccurred in unidirectional and cross-ply laminates), the failuremode in angle-ply laminate was the combination of in-plane shearstress s12 and interlaminar shear stress s13. Consequently, thecrack appeared near the two free edge sides of specimen, but with-out propagating through the whole width. Fig. 27 gives a typicalmicroscope failure image of short-beam angle-ply specimen underthree-point bending, while Fig. 28 shows the deflection-loadcurves.

5. Conclusions

In bending, composite laminates are subjected to both tensionand compression, which is fundamentally different from uniaxialloadings. This study has illustrated, by means of 3D FEA, 3D CLTand experiments, that all six stress components make contributionto the failure modes in the respective laminates. Three lay-upsequences were considered to cover the typical conditions of com-posite laminates. Compared with shell approximations, 3D FEA iscapable of modelling laminated composites with arbitrary lay-ups, and provides more accurate results. It has been shown thatthe combination of these three approaches can reveal the initiationof failure of composite laminates; used in isolation, the approachesare unlikely to be successful.

Study of the different failure mechanisms indicates ways inwhich laminate design might be improved. The unidirectional lam-inate failed in compression (fibre microbuckling); however withthe support of the lower transverse ply, the longitudinal ply inthe cross-ply laminate was more difficult to buckle, and as a con-sequence it could withstand a much higher compressive stress.This indicates that for practical composite structures, inserting atransverse ply into a unidirectional laminate (such as [0/90/0n])could significantly improve the bending performance.

It has been shown that the in-plane shear stress in angle-plylaminate is much higher than the interlaminar shear stress, andthis leads to laminate twisting under bending loads. However, thisstress component has been neglected by many previous studies.This study also shows that the maximum interaction ratio appearsat around 10–13� off-axis, therefore suggested that these orienta-tions should be avoided in the surface plies of practical compositelaminates.

It has also been demonstrated that the free edge effects arestrongly dependent on the laminate lay-up and loading span. Theasymmetric laminate (angle-ply) presents much more significant

nding test. Each crack represented a ‘stiffness losses’ in bending test.

M. Meng et al. / Composite Structures 119 (2015) 693–708 707

free edge effects than symmetric laminates (unidirectional andcross-ply), while these effects are more significant in short spanloading. Meanwhile, the free edge effects decay inside the laminatemuch more rapidly in the asymmetric laminate compared to sym-metric laminates. The stress components of all laminates convergeto the value predicted by CLT in central area.

Acknowledgements

The authors would like to thank Professor Long-yuan Li for hisadvice on FEA modelling, Dr Richard Cullen for his kind help withcomposites manufacturing, Terry Richards for his support of themechanical tests, and the financial support of the School of MarineScience and Engineering, Plymouth University.

Appendix A. Calculation of Poisson’s ratio m23

In the present work, the transverse Poisson’s ratio was derivedby hydrostatic assumption. Considering a bulk material underhydrostatic pressure, the change of volume is equal to the sum-mary of three strain vectors,

DV ¼ r=K ¼ e1 þ e2 þ e3 ð11Þ

Substituting the orthotropic properties (E2 = E3, G12 = G13, m12 = m13)into the relation of stress and strain,

e1

e2

e3

264

375 ¼

1=E1 �m21=E2 �m31=E3

�m12=E1 1=E2 �m32=E3

�m13=E1 �m23=E2 1=E3

264

375

r1

r2

r3

264

375 ð12-1Þ

e1 þ e2 þ e3 ¼ r 1� 2m12

E1þ 2ð1� m21 � m23Þ

E2

� �ð12-2Þ

The transverse Poisson’s ratio can be calculated by bulk modulus,while the bulk modulus is calculated by Halpin–Tsai empiricalequation,

m23 ¼ 1� m21 �E2

2Kþ E2ð1� 2m12Þ

2E1ð13Þ

K ¼ Kmð1þ fgVf Þ1� gVf

g ¼ Kf =Km � 1Kf =Km þ f

ð14Þ

Appendix B. 3D CLT formulae

With respect to the material symmetry, the compositecompliance matrix S is reduced to an orthotropic matrix. Applyingthe well-known stiffness transformation law [45], the off-axiscompliance matrix S and stiffness matrix C in 3D scale can beextended as,

S ¼ T�1e STr

C ¼ ½S��1 ð15Þ

S ¼

1=E1 �m12=E1 �m12=E1 0 0 0�m12=E1 1=E2 �m23=E2 0 0 0�m12=E1 �m23=E2 1=E2 0 0 0

0 0 0 1=G23 0 00 0 0 0 1=G12 00 0 0 0 0 1=G12

2666666664

3777777775ð16Þ

Te ¼

c2 s2 0 0 0 cs

s2 c2 0 0 0 �cs

0 0 1 0 0 00 0 0 c s 00 0 0 �s c 0�2cs 2cs 0 0 0 c2 � s2

2666666664

3777777775

ð17Þ

Tr ¼

c2 s2 0 0 0 2cs

s2 c2 0 0 0 �2cs

0 0 1 0 0 00 0 0 c s 00 0 0 �s c 0�cs cs 0 0 0 c2 � s2

2666666664

3777777775

ð18Þ

where S is the compliance matrix of lamina; c = cos(h) and s = sin(h).Substituting the three-dimensional version of composites com-

pliance matrix into CLT equations [4], the three-dimensional ver-sion of [A], [B] and [D] matrices can be written as,

½A� ¼XN

k¼1

ðCijÞkðzk � zk�1Þ

½B� ¼ 12

XN

k¼1

ðCijÞk z2k � z2

k�1

� �

½D� ¼ 13

XN

k¼1

ðCijÞk z3k � z3

k�1

� �ð19Þ

Assembling the [A], [B] and [D] matrices for A BB D

� �matrix, and its

inversed a bb d

� �matrix,

N

M

� �¼

A B

B D

� � ej

� �ð20-1Þ

½A;B; B;D� ¼A B

B D

� �ð20-2Þ

½a; b; b;d� ¼a b

b d

� �¼

A BB D

� ��1

ð20-3Þ

Once the three-dimensional [a,b;b,d] matrix is assembled, the flex-ural modulus of laminate can be evaluated by [4],

EfCLT ¼

12

h3d11

ð21Þ

Consider a composite laminate with symmetric lay-up patternunder three-point bending condition, the coupling matrix [B] = 0,the moment about x axes can be written as,

Mx ¼jx

d11¼ FL

4wð22Þ

If it is assumed that the curvature through-thickness is a constant,the strain and longitudinal stress are determined by,

ezx ¼ zjx ¼

zFLd11

4w

rz1 ¼ Ekez

x ¼ EkzFLd11

4w

ð23Þ

The maximum strain appears on the top and bottom surfacesz ¼ � h

2. However, the maximum stress is dependent on both thethrough-thickness coordinate and the ply modulus. The CLT formu-lae in the present work were solved by MATLAB.

708 M. Meng et al. / Composite Structures 119 (2015) 693–708

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