JOURNAL OFC O M P O S I T EM AT E R I A L SArticle

Study on stress concentration in notchedcross-ply laminates under tensile loading

Guangyan Liu and Kaili Tang

Abstract

This article presents a detailed analysis on stress concentration in notched unidirectional fiber-reinforced composites.

Due to the formation of longitudinal splitting at notch tips along the fiber direction, the extremely high stress concen-

trations ahead of the notch tips could be drastically reduced for composites under remote tension. The inability of the

widely used material property degradation method to accurately redistribute the local stresses at the notch tips is

examined. The notch blunting effect is investigated by modeling the longitudinal splitting as a thin plastic cohesive layer or

debonding, and results for the stress redistribution in the unnotched section directly ahead of the notch tips are

presented. By introducing the intra- and inter-ply damage modes, the failure of a double-notched cross-ply laminate is

predicted and compared with the experimental results from open literature.

Keywords

Splitting, strength, stress concentrations, finite element analysis

Introduction

Fiber-reinforced composites are widely used in aero-space, defense, and marine structures due to theiradvantages such as high specific stiffness, high specificstrength and corrosion-resistant properties. The struc-tural application of composite materials often requiresthe presence of holes or cut-outs. The behavior of com-posite materials with stress concentrations due to thesenotches is of great interest to designers because thedamage growth around these stress concentrationswill result in strength or life reduction of compositestructures. Over the past decades, several approacheshave been proposed to investigate the notched strengthand failure mechanisms of composite laminates. Oneapproach is based on concepts of linear elastic fracturemechanics (LEFM) carried over from homogeneousisotropic materials.1–3 LEFM-based models assumethat failure occurs if the notch, represented by anequivalent crack, reaches a critical size. The strengthof a laminate is related to the fracture toughness KIC

or the strain energy release rate GIC. These two quan-tities can be determined experimentally by crackedlaminates. A second approach is based on the elasticsolution of stress distribution in the vicinity of thenotch in an anisotropic plate, and makes use of simpli-fied stress fracture criteria. The two most popular

design criteria are the point stress criterion and averagestress criterion proposed by Whitney and Nuismer.4

The point stress criterion assumes that failure occurswhen the stress at a characteristic distance from thenotch boundary attains the strength of unnotchedmaterial, whereas the average stress criterion predictsfailure when the average stress over a characteristicdistance from the notch boundary reaches the strengthof unnotched material. The characteristic distance inthese two criteria must be determined experimentallyby notched laminates. Although the point and averagestress criteria provide reasonable solutions for some pre-liminary sizing design, the characteristic distance is nota material property and has no any physical meaning. Itdepends on the material type, lay-up, and geometry oflaminates.5

Department of Mechanics, School of Aerospace Engineering, Beijing

Institute of Technology, Beijing, China

Corresponding author:

Guangyan Liu, Department of Mechanics, School of Aerospace

Engineering, Beijing Institute of Technology, Beijing 100081, China.

Email: gliu@bit.edu.cn

Journal of Composite Materials

2016, Vol. 50(3) 283–296

! The Author(s) 2015

Reprints and permissions:

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DOI: 10.1177/0021998315573802

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For both the LEFM- and stress-based approachesmentioned earlier, the stress distribution in the vicinityof the notch is typically calculated by using anisotropicelasticity and the original notch geometry. In mostcases, however, complete failure is preceded by subcrit-ical damage which can alter the notch-tip stress distri-bution substantially. To provide accurate strengthprediction for notched composites, it is important totake into account the stress redistribution caused bydamage progression prior to complete failure. So far,a large body of research has been devoted to progres-sive failure analysis of notched composites based onfinite element methods combined with damage model-ing techniques. The material property degradationmethod (MPDM) is the most widely used damage mod-eling technique for strength prediction of notched com-posite laminates. This method assumes that thedamaged material can be replaced by an equivalentmaterial with degraded stiffness properties. Oncedamage is detected in finite elements based on aproper failure theory, the elastic moduli of the damagedelements are modified according to the failure modes.Tan et al.6–8 proposed a 2D progressive damage modelfor laminates containing central holes subjected to in-plane tensile or compressive loading. Internal state vari-ables are used to simulate the stiffness degradation ofdamaged elements. The predicted damage progressionpatterns agreed with the experimental results, but thepredicted ultimate strength values were very sensitive tothe selected values of the internal state variables. As aconsequence, investigation based on experiments isnecessary to validate these values. Camanho andMatthews9 extended the work of Tan et al.6–8 to 3Dto predict the damage progression and strength ofmechanically fastened joints. A good agreementbetween experimental results and numerical predictionscan be obtained if the first load drop-off in simulation istaken as the failure load. An alternative damage mod-eling technique is called the element-failure method(EFM).10–14 Instead of material properties, nodalforces of damaged elements are modified in EFM toreflect the changes in their load-bearing capability.The damage effect is achieved by applying a set of exter-nal nodal forces on the nodes of damaged elementswhich are equivalent to the internal nodal forces dueto the stresses in these elements before damage. Theadvantage of this method allows crack propagation inany direction without the need of remeshing and stiff-ness matrix modification. The EFM has been success-fully used to predict damage propagation in compositelaminates subjected to three-point bending,10 open-holetension,11–13 and filled-hole tension.14

Despite the extensive research on progressive failureanalyses of composite laminates, the fundamental

failure mechanism and behavior of composites arestill not fully understood. One main reason is that thestress concentrations in the notched composite lamin-ates are not accurately modeled. All of the above ana-lyses assumed composites as homogenized anisotropicmaterials, which introduces very severe stress concen-trations at notch tips and makes the finite element cal-culation mesh-dependent. However, compositematerials have fiber and matrix constituents, and thematrix is usually much weaker than the fiber. It hasbeen reported that notched laminates will exhibitmatrix splitting emanating from notch tips along thefiber direction.15–18 The growth of the splitting bluntsthe notches and reduces the stress concentrations cor-respondingly. Recently, Wisnom and co-workers19–23

carried out an elegant and systematic analysis fordamage progression in notched composite laminates.In-plane scaling, thickness scaling, stacking sequenceand lay-up effects are studied by considering both theintra-ply splitting in all plies and the inter-ply delamin-ation. The simulation results compare very well withthe experimental observations.

Chan et al.24 studied in depth the stress redistribu-tion due to plastic yielding or debonding of the inter-face in layered composites. The problem concerns acrack in one brittle layer impinging on the interfacewith the neighbouring brittle layer. In the presentpaper, we shall focus mostly on the stress concentrationrelief effect of the in-plane longitudinal splittingin fiber-reinforced composites. The paper is organizedas follows: the next section presents the stress concen-tration by homogenizing composite materials as aniso-tropic plates. In the Longitudinal splitting modeling byMPDM section, the ability of the commonlyused MPDM to predict the fiber stress relaxation result-ing from the longitudinal splitting in unidirectionallaminates is examined. In the Longitudinal splittingmodeling by cohesive elements section, an approachof modeling longitudinal splitting by cohesive elementswith plastic yielding or debonding is applied andthe stress concentration reduction at the notch tip isinvestigated. In the Progressive failure of double-notched composite laminate section, damage progres-sion in a double-notched tensile laminate is studiedby considering both intra- and inter-ply damagemodes. Some discussion and concluding remarks aregiven in the Discussion and Conclusion sections,respectively.

Stress concentration in homogeneousanisotropic plates

For a homogeneous anisotropic plate subjected toopen-hole tension (Figure 1), the tangential stress

284 Journal of Composite Materials 50(3)

concentration factor K on the circular hole boundarycan be expressed as25

K ¼���1¼

E�E1

�cos2’þ kþ nð Þsin2’� �

kcos2��

þ 1þ nð Þcos2’� ksin2’� �

sin2�

�n 1þ kþ nð Þsin’cos’sin�cos�� ð1Þ

where �� is the tangential stress at the position of angle� to the principal 1-axis, �1 is the remote tensile stress,’ is the angle between the principal 1-axis and the load-ing direction, E1 is the elastic modulus in the principal1-axis direction, and E� is the elastic modulus in the �direction given by

E� ¼ E1= sin4�þE1

E2cos4�þ

1

4

E1

G12� 2�12

� �sin22�

� ð2Þ

where E2 is the elastic modulus in the principal 2-axisdirection, and G12 and �12 are the shear modulus andPoisson’s ratio in the 1–2 plane, respectively. k and n inequation (1) are defined as

k ¼ ��1�2 ¼

ffiffiffiffiffiffiE1

E2

rð3Þ

n ¼ �i �1 þ �2ð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

E1

E2� �12

� �þ

E1

G12

sð4Þ

where i ¼ffiffiffiffiffiffiffi�1p

, and �1 and �2 are the complex roots ofthe anisotropic plate characteristic equation

�4 þE1

G12� 2�12

� ��2 þ

E1

E2¼ 0 ð5Þ

For isotropic materials (k¼ 1, n¼ 2), the tangentialstress concentration factor K (equation (1)) reduces to

K ¼ 1� 2cos2 ��’ð Þ ð6Þ

which gives K ¼ 3 at ��’ ¼ ��=2, and K ¼ �1 at��’ ¼ 0 or �.

For anisotropic materials loaded in the principal 1-axis direction (’ ¼ 0), the tangential stress concentra-tion factor K (equation (1)) reduces to

K ¼E�E1�kcos2�þ 1þ nð Þsin2�� �

ð7Þ

Figure 2 shows the tangential stress concentrationfactor around a circular hole of a typical unidirectionalcanbon fiber reinforced composite plate (IM7/8552,E1¼ 162GPa; E2¼ 8.96GPa; G12¼ 4.69GPa;�12¼ 0.316) based on equation (7)). The maximum tan-gential stress concentration factor Kmax(¼9.37) occursat � ¼ ��=2, and

Kmax ¼ 1þ n ¼ 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

E1

E2� �12

� �þ

E1

G12

sð8Þ

To study the stress concentration, four meshes inFigure 3 are used to model the unidirectional IM7/8552 composite plate subjected to open-hole tensionin the fiber direction. The numbers of elementsaround the circular hole are 28, 60, 84, and 116, respect-ively. Figure 4 shows the corresponding maximumstress concentration factors at � ¼ ��=2 position. Itcan be seen that the stress concentration factorincreases with refining the mesh. But even with thefinest mesh (mesh4), the stress concentration factor isonly 4.16, which is far more below the theoretical value(9.37). Therefore, to capture such a high stress concen-tration in finite element analyses, an extremely finemesh near the hole is needed. However, it is rare tosee models satisfying this requirement in open literatureavailable. In other words, most of the finite elementmodels with notches must be mesh-dependent.

-2

0

2

4

6

8

10

0 90 180 270 360Str

ess

conc

entr

atio

n fa

ctor

K

α (degree)

Figure 2. Tangential stress concentration factor around a cir-

cular hole of unidirectional IM7/8552 composite plate.

σ∞σ∞ φσα

σα

α

1

2

Figure 1. Open-hole tension of an anisotropic plate.

Liu and Tang 285

One direct question should be asked is whether thestress concentration factor in composites is really ashigh as that obtained by anisotropic elasticity theory,or whether composite materials can be idealized asanisotropic plates. Composites are composed of fiberand matrix constituents and the matrix is usuallymuch weaker than the fiber. At a very low tensile load-ing level, matrix cracking or longitudinal splitting mayinitiate at positions between strained and stress-relievedfibers near the hole edge due to a high local sheardeformation (Figure 5), and propagates along thefiber direction. The matrix cracking or longitudinalsplitting will blunt the notch tip to alleviate the stressconcentration. Therefore, it is questionable to modelcomposites as anisotropic plates without consideringthe matrix cracking or longitudinal splitting at thenotch tip, because the true stress concentration mightbe highly reduced. This also explains why most of thefinite element models in literature can only predict somefailure responses of certain types of composite struc-tures or loadings.

Longitudinal splitting modeling by MPDM

To perform progressive failure analyses of compositestructures by modeling the subcritical damages, theMPDM has been widely used by researchers. To theauthors’ knowledge, no attempts to verify the accuracyof the stress concentration prediction at notch tips dueto subcritical damages, modeled by the MPDM, havebeen reported in open literature except for the work ofIarve et al.26 Iarve et al.26 predicted the fiber stressrelaxation in unidirectional open-hole tension compos-ites with radial-type mesh configuration as shown inFigure 6. The material properties of elements in theshaded band including E22, G12, G23, and �12 weredegraded to model the matrix cracking-type damage.The width of the band is equal to the hole diameterD. The element size is chosen to be sufficiently small,so that the roughness of the edges of this band near thehole is much smaller than the hole diameter. Because ofthe degradation of transverse material properties of allelements in the band, which effectively isolatethe unnotched ligaments from the shaded band, it isexpected to get a stress concentration value ofW/(W-D), where W is the width of the composite.However, no obvious reduction in stress concentrationat the hole edge was observed compared with the resultwithout any material property degradation, and thestress concentration analysis showed some visiblemesh-dependency. To overcome this drawback of the

mesh1 mesh2

mesh3 mesh4

Figure 3. Four meshes of unidirectional composite plate under

open-hole tension.

σ∞σ∞

Stress relieved fiber (blue)

High local shear deformation

Strained fiber (black)

Figure 5. Open-hole 0� ply with stress-relieved fibers (in blue).

mesh1mesh2

mesh3mesh4

Figure 4. Stress concentration factor at the hole edge of uni-

directional composite plate under open-hole tension.

fiber direction

σ∞σ∞

Figure 6. Radial-type mesh configuration for predicting fiber

stress relaxation due to splitting in unidirectional open-hole

tension composite.

286 Journal of Composite Materials 50(3)

MPDM, Iarve and co-workers26–29 developed a meshindependent cracking (MIC) approach for progressivefailure analysis of composite laminates by combiningthe Regularized eXtended Finite Element Method(Rx-FEM) for matrix cracking and a cohesive interfacemodel for delamination. It was shown that the longitu-dinal splitting and other damage patterns can be accur-ately predicted by this approach.

To understand more about the inability of theMPDM to model matrix cracking and the consequentfiber stress relaxation effect in this analysis, a grid-typemesh is created by partitioning the hole section fromthe ligaments (Figure 7) in this paper. The two partitionlines (red line in Figure 7) are tangent to the hole edgeand align with the fiber direction. In physical open-holetension tests of unidirectional composites, the matrixcracking emanates at the hole edge due to shearing fail-ure, and propagates along the fiber direction to formlong and thin splitting. Therefore, instead of the wholeband of elements in the hole section, only materialproperties of two rows of elements adjacent to the par-tition lines are degraded to model the longitudinal split-ting. The row of degraded elements could be above orbelow the partition lines in Figure 7. The model with

the two rows of degraded elements lying in the liga-ments (outside the two partition lines) is referred toas Approach1, and the model with the two rows ofdegraded elements lying in the hole section (betweenthe two partition lines) is referred to as Approach2.Three material property degradation rules areemployed for the degraded elements: (i) G12 is multi-plied by 10�6; (ii) E2, E3, G12, and G13 are multiplied by10�6 or (iii) E2, E3, G12, G13, �12, and �13 are multipliedby 10�6. Figure 8 shows the distribution of stress �xxnormalized with respect to the applied far-field stress�1 in the cross section x¼ 0 as a function of y-coord-inate normalized with respect to the hole radius R. ForApproach1, stress oscillation was observed in the firstfew elements near the hole, and refining the mesh showsno improvement for the stress oscillation. Actually,even higher stress concentration than the virgin com-posite plate was predicted. Interestingly, Approach2predicted the function �xx/�1 to be nearly constantalong the cross-section and equal to W/(W-2R), whereW is the width of the composite plate. The three deg-radation rules showed almost identical stress distribu-tion. This is the expected result by implementing thematerial property degradation, i.e. the ligaments areeffectively isolated from the hole section by the longi-tudinal splitting.

It can be seen that the effectiveness of the MPDM tomodel the longitudinal splitting depends not only onthe mesh, but also the position of degraded elements.Only when the degraded elements lie in the hole sectionin Figure 7 can the splitting be reasonably modeled.For a unidirectional laminate subjected to open-holetension, splitting will initiate at positions betweenstrained and stress-relieved fibers near the hole edge,and propagate along the fiber direction (Figure 5).For this reason, a mesh by partitioning the hole sectionfrom the ligaments is preferred for composite failure

Figure 8. Stress distribution ahead of the notch tip in the unidirectional open-hole tension composite: (a) Approach1 and (b)

Approach2.

x

y

R

Figure 7. Grid-type mesh configuration for predicting fiber

stress relaxation due to splitting in unidirectional open-hole

tension composite.

Liu and Tang 287

analysis, not the typical radial-type ones widely seen inliterature.

For a general composite laminate with different fiberorientations, the stress distribution at the notch tip isvery complicated due to the constraining of one ply bythe neighboring plies. The initial element failure mightnot always occur in the hole section of the ply based onthe finite element calculation. Moreover, splitting mayform near the hole and propagate along different fiberdirections for different plies, which makes it difficult toconstruct a mesh satisfying the splitting requirement forall plies if the MPDM is used for element failure.Considering that the matrix between fibers is extremelythin, it is proper to use zero-thickness cohesive elem-ents30 to model the thin and straight layer of splitting.

Longitudinal splitting modeling by

cohesive elements

In this section, the ability of the cohesive elementmethod to model the stress relief at the notch tip dueto longitudinal splitting is studied. The cohesive elem-ent method combines a strength-based analysis to pre-dict the damage initiation, and a fracture mechanicsanalysis to predict the crack propagation. It has beendiscussed in the Stress concentration in homogeneousanisotropic plates section that the stress concentrationfactor from a finite element analysis is far below thetheoretical value by homogenizing composites as aniso-tropic plates. From this point of view, damage initi-ation may not be accurately predicted by using thestress states near the notch tips in a finite elementmodel. However, composites are composed of fiberand matrix constituents and the matrix is usuallymuch weaker than the fiber. Longitudinal splitting ini-tiates from the notch tip at an extremely low level ofexternal loading and the stress concentration drops. Tomore accurately model the stress concentration, prob-ably a micro-mechanical model should be created in thenotch area by explicitly modeling the fiber and matrixconstituents. To simplify the analysis, it is plausible tointroduce a line of cohesive elements at the notch tip toapproximate the matrix. It will be shown that by doingthis, the predicted longitudinal splitting can effectivelyblunt the notch. Although it is debatable, the stresses atthe notch tip are used to predict the damage initiationof cohesive elements in order to model the longitudinalsplitting. When this approach is employed for failureanalysis of composite laminates, a mesh dependencystudy should always be performed to make sure a con-verged failure load and damage pattern can beobtained.

Cohesive elements are frequently used at interfacesbetween individual plies of composite laminates fordelamination analysis,31–33 and seldom used for

intra-ply damage analysis.20–23 A cohesive element con-sists of an upper and a lower surface. Generally, thesetwo surfaces act as a single one before a prescribeddamage initiation criterion is satisfied. In contrast tocontinuum elements where stress–strain relations areused, cohesive elements are governed by the relationbetween tractions and separations of the two surfaces

�n�s�t

0@

1A ¼ k0n 0 0

0 k0s 00 0 k0t

24

35 �n

�s�t

0@

1A ð9Þ

where �n, �s, and �t are the one normal and two tan-gential traction components, �n, �s, and �t are the cor-responding separations. For intra-ply matrix damage,the stiffness components in equation (9) can beexpressed as

k0n ¼ E2=e

k0s ¼ 2G12=e

k0t ¼ 2G23=e

ð10Þ

where E2, G12, and G23 are the elastic moduli of thematrix. e is the thickness of the matrix between fibers.For delamination analysis, E2 and G12 in equation (10)should be replaced by E3 and G13, respectively, and ebecomes the thickness of the interface between plies.

For prediction of damage initiation, a strength-based quadratic criterion is selected. According to thiscriterion, the damage initiation is controlled by thethree tractions and the corresponding cohesivestrengths N, S, and T through the equation

�nh i

N

� �2

þ�sS

� �2þ�tT

� �2¼ 1 ð11Þ

where h i represents the Macaulay bracket of value zerowhen its argument is negative.

Damage propagation is defined based on fractureenergy. The dependence of the fracture energy on themode mix is defined based on criterion

Gn

GCn

� �2

þGs

GCs

� �2

þGt

GCt

� �2

¼ 1 ð12Þ

with the mixed-mode fracture energy GC¼GnþGsþGt

when the above condition is satisfied. In this expression,quantities Gn, Gs, and Gt refer to the work done by thetractions and their conjugate relative displacements inthe normal and shear directions, respectively.Quantities with superscript C denote the critical strainenergy release rates corresponding to each fracturemode.

288 Journal of Composite Materials 50(3)

After damage initiation occurs, the damage evolu-tion law is used to determine the rate at which the stiff-ness of cohesive elements is degraded. A scalar damagevariable D representing the state of damage is defined.It is assigned a value of 0 for undamaged elements, andacquires a value of 1 for completely damaged elements.The post-initiation traction versus separation relation isof the form

�n ¼1�Dð Þk0n�n �n 4 0

k0n�n otherwise

(

�s ¼ 1�Dð Þk0s �s

�t ¼ 1�Dð Þk0t �t

ð13Þ

Note that damage does not affect the cohesive rela-tion when �n � 0, i.e. in compression the interactionbetween the two surfaces of cohesive elements reducesto a penalty contact algorithm. The evolution law for Dis based upon the assumption that the tractionsdecrease linearly with increasing separation oncedamage has initiated. An effective separation isdefined by

�e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�nh i

2þ�2s þ �2t

qð14Þ

The work-conjugate effective traction is

�e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�nh i

2þ�2s þ �2t

qð15Þ

The damage variable D is then defined in terms ofthe fracture energy GC as

D ¼

2GC

�0e�maxe � �0e

��maxe

2GC

�0e� �0e

� � ð16Þ

where �maxe is the maximum value of �e attained during

the loading history, while �0e and �0e are the values of the

effective separation and effective traction at damageinitiation.

To examine the ability of the cohesive elementmethod to model the stress relief at the notch tip dueto longitudinal splitting, a central cracked unidirec-tional laminate subjected to remote tension in fiber dir-ection is studied (Figure 9). Material properties of thecomposite system IM7/5250-4 are listed in Table 1. Thelaminate has a length of l¼ 80mm and a width ofb¼ 40mm, and the central crack length is 2w¼ 10mm.The laminate is modeled by plane stress elements. Tomodel the longitudinal splitting, two rows of zero-thickness cohesive elements have been inserted in twolines (red lines in Figure 9) extending from the crack

tips along fiber direction. Here ‘‘zero-thickness’’ meansthe top and bottom surfaces of the cohesive elementshave coincident nodes. There is no doubt that a singu-lar stress state at the crack tip would be formed if nocohesive element is included, which makes the finiteelement calculation mesh dependent, and element fail-ure may occur at an extremely low loading level.

The initial stiffnesses of the cohesive elements arecalculated by assuming that the elastic constants inequation (10) are equal to those of the compositematerial in Table 1 and the cohesive elements have athickness of e¼ 0.005mm. The cohesive strengths of N,S, and T are assumed to be equal to the transversetensile strength Y, shear strengths S12 and S23 of thecomposite material, respectively. Since there is a lack offracture energies for this type of material, two sets ofmaterial parameters are selected in the finite elementanalysis: one with some typical critical strain energyrelease rates values from literature34 (GC

n ¼ 0.25N/mm, GC

s ¼ GCt ¼ 1.08N/mm); the other with these

values multiplied by 1000 (GCn ¼ 250N/mm,

GCs ¼ GC

t ¼ 1080N/mm). For the latter case, the cohe-sive elements will behave like a linear elastic-perfectlyplastic material with a yielding stress �. Finite elementcalculations show that both analyses gave quite similartrend for the curves in following Figures 10 and 11, andthe yielding stress is almost equal to the tangential

Table 1. Material properties of IM7/5250-4 composite.12

Modulus in fiber direction E1 (GPa) 172.4

Transverse moduli E2¼ E3 (GPa) 10.3

Shear moduli G12¼G13 (GPa) 5.52

Shear modulus G23 (GPa) 3.45

Poisson’s ratios �12¼ �13 0.32

Poisson’s ratio �23 0.4

Tensile strength in fiber direction X (MPa) 2826.5

Transverse tensile strengths Y¼Z (MPa) 65.5

Shear strengths S12¼ S13 (MPa) 122.0

Shear strength S23 (MPa) 122.0

Figure 9. Central cracked unidirectional laminate subjected to

remote tension.

Liu and Tang 289

cohesive strength, i.e. �S¼ 122.0MPa. Actually, thefailure of cohesive elements in this analysis is domi-nated by Mode II type failure, and Mode I and ModeIII type failures can be ignored. To verify this, a similarfinite element model was created by replacing the cohe-sive elements by springs with uncoupled three degreesof freedom. The springs behave in a linear elastic-per-fectly plastic manner with yielding stress �¼ 122.0MPain the longitudinal shear direction and in a linear elasticmanner in the other two directions. No substantial dif-ference was observed by comparing finite elementresults between the cohesive element method andspring element method. In this section, only resultswith perfectly plastic post yield cohesive elements arepresented.

Damage modeling by cohesive elements is usuallysensitive to element size. It has been demonstratedthat the energy regularization method can be used toalleviate the mesh dependency.35–39 A cohesive zone,where elements experience softening in the traction-separation response, develops ahead of the traction-free debonded crack. To accurately capture thecohesive zone stress distribution, the cohesive elementsize must be smaller than the cohesive zone length. Oneformula to estimate the numerical cohesive zone lengthfor bilinear cohesive laws has been proposed by Harperand Hallett,36 and later been successfully used by Chenet al.37 for delamination prediction. Since the longitu-dinal splitting in this study is mainly driven by shear,the formula containing only the Mode II-related termfor orthotropic materials is used to calculate the cohe-sive zone length

lch ¼ E0IIGC

s

2S2ð17Þ

1

2

3

4

5

0 0.5 1

yy(x

,0)/

∞

x/w

=1.8

=6.0

=9.4

Elastic solution(Eq.19)

σ∞/τ

σ∞/τ

σ∞/τ

ss

Figure 10. Stress distribution ahead of the crack tip in the

ligament section at several levels of applied stress to shear yield

stress of the cohesive layer.

1

2

3

4

5

0 1 2

yy(x

,0)/

∞

x/w

0.00.51.02.05.0

d/w

ss

Figure 12. Stress distribution ahead of the crack tip in the

ligament section for several debonding lengths.

}}

Plastic yielding

Debonding

Figure 13. Comparison of stress relief effect due to plastic

yielding and debonding.

02468

101214

0 1 2 3 4 5

∞/

d/w

ts

Figure 11. Relation between applied stress and length of

yielding zone in the cohesive layer.

290 Journal of Composite Materials 50(3)

where E0II is an equivalent elastic modulus and can bedetermined by36

1

E0II¼

ffiffiffiffiffiffiffiffi1

2E1

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

E1E3

� �1=2

��31E3þ

1

2G31

s: ð18Þ

Note that for delamination prediction, one more setof equations specialized for slender bodies should alsobe considered. Using equation (17) (GC

s ¼ 1.08N/mm,S¼ 122.0MPa), the cohesive zone length lch is calcu-lated to be equal to 2.01 mm. A uniform element sizeof 0.5mm� 0.5mm is used in this analysis and thereare about four elements in the cohesive zone length.

Figure 10 shows the distribution of normal stresssyy(x,0) (normalized by the applied far field stresss1) ahead of the crack tip across the ligament sectionfor three levels of s1/�. The curve shown for s1/�¼ 1.8 is only very slightly below the elasticdistribution25

�yy x, 0ð Þ

�1¼

x=wþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix=wð Þ

2þ2x=w

q ð19Þ

near the crack tip. Reduction of stress ahead of thecrack tip begins to be appreciable when s1/�¼ 6.0,

and becomes quite significant when s1/�¼ 9.4. Thedrop in stress just ahead of the crack tip for highers1/� is offset by a slight increase in stress relative tothe curve for lower s1/� in the region further from thecrack tip. This feature can be seen for all stress redis-tribution curves. With monotonic increase of theapplied far field stress s1, the yielding zone of half-length d (Figure 9) will propagate allowing splitting inthe form of tangential displacement discontinuity

Table 2. Material properties of E-glass/913composite used in

FE model.19–20

Modulus in fiber direction E1 (GPa) 43.9

Transverse moduli E2¼ E3 (GPa) 15.4

Shear moduli G12¼G13 (GPa) 4.34

Shear modulus G23 (GPa) 3.5

Poisson’s ratios �12¼ �13 0.3

Poisson’s ratio �23 0.5

Tensile strength in fiber direction X (MPa) 1540

Compressive strength in fiber direction X0 (MPa) 620

Transverse tensile strengths Y¼Z (MPa) 39

Transverse compressive strengths Y0 ¼Z0 (MPa) 128

Shear strengths S12¼ S13 (MPa) 89

Shear strength S23 (MPa) 89

25.8% 65% 0%

Final failure100%

Figure 14. Damage progression in cross-ply double notch tension specimen, from Hallett and Wisnom.19

Liu and Tang 291

across the cohesive line. Figure 11 shows the relation ofd/w to s1/�. When s1/� < 4, the yielding zone spreadsin an increasing speed with the monotonic increase ofs1. After that, the yielding zone spreads in an almoststeady state.

For composites with weaker matrix or lower cohe-sive yielding stress �, the stress relief effect ahead of thecrack tip shown in Figure 10 should be more pro-nounced. To illustrate this, an extreme case with fric-tionless debonding (i.e. �¼ 0) is studied. In the finite

Figure 17. Predicted damage pattern for double-notched cross-ply laminate. (a) Coarse mesh, (b) Base mesh and (c) Fine mesh.

Cohesive elements for splitting

Spring elements for fiber failure

(a) (b) (c)

Figure 15. Finite element meshes for double-notched cross-ply laminate.

Test Coarse Base Fine

0

50

100

150

200

250

300

350

App

lied

str

ess(

MP

a)

Figure 16. Failure stress for double-notched cross-ply laminate.

292 Journal of Composite Materials 50(3)

element model, the longitudinal splitting is explicitlymodeled as cracks and then meshed accordingly.Figure 12 shows the distribution of normal stresssyy(x,0) (normalized by the applied far field stresss1) ahead of the main crack tip across the ligamentsection for various lengths d of debonding. With theincreasing of debonding length, the normal stressahead of the crack tip is drastically reduced. Whenthe debonding length is larger than five times of thehalf crack length w, the normal stress is almost uni-formly distributed along the ligament and approachesto a value of b/(b-2w), where b is the width of the com-posite plate. Essentially tensile specimens are producedon either side of the crack with negligible stress concen-tration, which in turn reduces the strength of the lamin-ate to that of the ligament strength. Figure 13 shows thestress relief effect ahead of the crack tip by cohesiveplastic yielding and debonding for three levels of d/w,where d represents the yielding zone length or debond-ing length. We can conclude that debonding has a moresignificant effect on lowering the stress concentrationfactor at the crack tip than plastic yielding of a thincohesive layer.

Progressive failure of double-notched

composite laminate

Hallett and Wisnom19 have performed experiments ondouble-notched E-glass/913 composite laminates ([90/0]s) under remote tensile loading. Figure 14 showsthe damage history for a 20-mm wide specimen with a5-mm long notch on each side. Damage initiates at thenotch tips in the form of a transverse crack in the 90�

ply and short longitudinal splitting in the 0� ply. Withthe increasing of the applied remote load, transversecracks begin to diffuse and saturate the 90� ply, andthe longitudinal splitting propagates in the 0� ply mean-while. At 100% of the maximum load, a triangulardelamination shown as a darker region adjacent tothe splitting is observed. Thereafter, the damage propa-gates very rapidly and the specimen quickly reachesfinal failure due to fiber fracture across the width ofthe specimen. In this section, a progressive failure ana-lysis is carried out for this double-notched cross-plylaminate with the cohesive element method describedin the Longitudinal splitting modeling by MPDM sec-tion. The goal of this study is to compare the finiteelement prediction with the experimental results to seewhether the failure mechanisms of the laminate can beaccurately captured by this model. The material proper-ties of the composite system follow those of Hallett andWisnom19,20 and are given in Table 2.

Due to the in-plane and out-of-plane symmetry ofthe problem, only one-eighth of the laminate needs tobe analyzed. To investigate the mesh dependency, three

meshes are created as shown in Figure 15 by doublingor halving the element size from the bash mesh. Eachply is modeled with one 3D continuum shell element(SC8R in Abaqus notation) in the thickness direction.Different from conventional shell elements, continuumshell elements discretize the entire three-dimensionalbody and their thicknesses are determined from theelement nodal geometry. Continuum shell elementshave only displacement degrees of freedom.Therefore, from a modeling point of view, continuumshell elements look like three-dimensional continuumsolids.27 The Hashin failure criterion40 is employedfor matrix cracking in the 90� and 0� plies. To predictthe initiation and propagation of delamination, zero-thickness cohesive elements are inserted between thetwo plies, as used by Pham et al.41 The in-plane meshof the interface is the same as that of the ply. For lon-gitudinal splitting prediction, zero-thickness cohesiveelements are inserted in one line (green line in Figure15) emanating from the notch tip along the fiber direc-tion of the 0� ply. The behaviors of the cohesive elem-ents for delamination and longitudinal splitting areassumed to be the same, as described in theLongitudinal splitting modeling by cohesive elementssection. The initial stiffnesses of the cohesive elementsare determined by an assumed thickness ofe¼ 0.005mm in equation (10). For the critical strainenergy release rates, values of GC

n ¼ 0.25N/mm andGC

s ¼ GCt ¼ 0.9N/mm from literature34,42 are used.

Experimental results show that 0� fiber failure initiatesat the notch tip and rapidly progresses across the spe-cimen width but following an irregular path. This couldbe due to the influence of shear stress20 or positions ofimperfections in the fibers. Similar to matrix cracking,fiber failure can certainly be predicted based on thestresses of the continuum shell elements, but the stresseswill be mesh-dependent because of the stress concentra-tion at the notch tip. To alleviate the mesh dependency,one row of spring elements is put along the ligament(red line in Figure 15) ahead of the notch tip in the 0�

ply for fiber failure prediction. In this case, the influenceof shear stress is neglected and fiber failure can onlytake place on this horizontal line. The stiffness andstrength of the springs are calculated based on the elas-tic modulus and tensile strength in the fiber direction.

The maximum applied far field stresses at failure forthe three meshes are shown in Figure 16. It can be seenthat although there is some variation, all results arewithin 10% of the mean test ultimate strength.Analogous to the damage progression observed inexperiments, finite element simulation reveals thatdamage initiates at the notch tip in the form ofmatrix cracking in the 90� ply and longitudinal splittingin the 0� ply. On continuation of loading, the matrixcracking diffuses and the longitudinal splitting

Liu and Tang 293

propagates, and meanwhile delamination develops nearthe notch tip region. Ultimate failure occurs with fiberfracture at the notch tip in the 0� ply and it progressesrapidly across the width of the laminate. For conveni-ence and ease of comparison with the damaged speci-men, damage modes including matrix cracking,longitudinal splitting and delamination at 100% ofthe maximum load are superimposed to form a com-posite damage map as shown in Figure 17. The hori-zontal and vertical black lines represent the predictedmatrix cracking in the 90� and 0� ply, respectively.Although each damaged element may contain a greatnumber of microcracks, only one dominant matrixcrack is drawn for each damaged element. This explainsthe different crack densities for the three meshes. Thegreen line represents the predicted longitudinal splittingin the 0� ply, and the red shaded area denotes thedelamination. In comparison with the observeddamage from experiments in Figure 14, most of thefeatures of damage are replicated reasonably well bythe base and fine meshes. These features include theextensive transverse cracks in the 90� ply, the longitu-dinal splitting in the 0� ply, and the triangular delam-ination at the notch tip. The coarse mesh has adelamination area more like a rectangle than triangleand predicts more vertical matrix cracks in the 0� ply.This disagreement with the test is probably due to theeffect of mesh density. To verify this, the methodadopted in the literatures36–38 is used to calculate thenumerical characteristic element length. It is found thatonly in the coarse mesh, the element size is larger thanthe characteristic element length. In this case, the stressgradient cannot be accurately captured by the elements,which explains the discrepancy between the FE predic-tion and experimental results.

Discussion

By studying unidirectional and cross-ply compositelaminates, this paper demonstrated the necessity ofadopting a biased mesh and placing cohesive elementsalong potential splitting lines in the fiber direction inorder to model the failure mechanisms of compositesaccurately. It is of critical importance to extend thisidea to more general composite laminates with differentstacking sequences. Although it is possible to create afinite element mesh with the same configuration for allplies to accommodate the boundaries of the specimen,such as holes or cracks, and the potential splittingroutes for different plies, this pre-processing procedurecould be formidable and extremely time-consuming.One alternative approach is to use different mesh con-figurations for the individual plies, so that the elementedges align with the fiber direction for each ply. Insteadof coincident nodes, tie constraints can be applied to

connect the ply structural elements with the interfacecohesive elements for delamination prediction.43

Recently, based on the concept of eXtended FiniteElement Method (x-FEM),44 a regularized x-FEM(Rx-FEM) approach26–29 is developed for mesh-inde-pendent simulation of matrix cracking in each ply ofcomposite laminates. The Rx-FEM preserves the elem-ent Gauss integration schema for arbitrary crackingdirection and allows straightforward connectionsbetween neighboring plies with different fiber orienta-tions. By combining with interface cohesive model fordelamination and continuum damage mechanics modelfor fiber failure, the promising Rx-FEM approachshowed its robustness of progressive failure analysisof composite laminates.

Conclusion

The stress concentration in notched fiber-reinforcedcomposites is investigated. The study reveals that bysimply modeling composite laminates as anisotropicplates, the stress concentration at the notch tip will beextremely high and hence the finite element simulationwill be mesh dependent. Due to the weakness of thematrix phase compared with the fiber phase, matrixcracking in terms of longitudinal splitting may takeplace at a very early loading stage. The splitting hasan effect of blunting the notch, which in turn enhancesthe load-carrying capacity of the material. It is shownthat the commonly used MPDM might not be able toaccurately redistribute the stresses near the notch so asto capture the notch blunting effect. This article dem-onstrates that in order to characterize the stress concen-tration associated with the longitudinal splitting, it isnecessary to adopt a biased mesh and to introducecohesive elements along potential splitting routes.This is of paramount importance for the reliable pre-diction of failure modes and ultimate strengths ofnotched composite structures. It has also been demon-strated that debonding has a more significant effect onlowering the stress concentration factor at the notch tipthan plastic yielding of a cohesive layer.

A finite element model is developed to perform theprogressive failure analysis of a double-notched cross-ply laminate in tension. The model gives an accuratedescription of the main damage features prior to fiberfailure, including the longitudinal splitting in the 0� ply,extensive transverse cracks in the 90� ply, and triangu-lar delamination area at the interface. A good correl-ation of finite element simulation to experimentalresults in open literature is achieved.

Conflict of interest

None declared.

294 Journal of Composite Materials 50(3)

Funding

The project is supported by the National Natural ScienceFoundation of China (11242015) and the Excellent YoungScholars Research Fund of Beijing Institute of Technology.

References

1. Waddoups ME, Eisenmann JR and Kaminski BE.Macroscopic fracture mechanisms of advanced composite

materials. J Compos Mater 1971; 5: 446–454.2. Cruse TA. Tensile strength of notched laminates.

J Compos Mater 1973; 7: 218–229.3. Eriksson I and Aronsson CG. Strength of tensile loaded

graphite/epoxy laminates containing cracks, open and

filled holes. J Compos Mater 1990; 24: 456–482.4. Whitney JM and Nuismer RJ. Stress fracture criteria for

laminated composites containing stress concentrations.J Compos Mater 1974; 8: 253–265.

5. Camanho PP and Lambert M. A design methodology formechanically fastened joints in laminated composite

materials. Compos Sci Technol 2006; 66: 3004–3020.6. Tan SC and Nuismer PJ. A theory for progressive matrix

cracking in composite laminates. J Compos Mater 1989;23: 1029–1047.

7. Tan SC. A progressive failure model for composite lamin-ates containing openings. J Compos Mater 1991; 25:

556–577.8. Tan SC and Perez J. Progressive failure of laminated

composites with a hole under compressive loading.J Reinf Plast Compos 1993; 12: 1043–1057.

9. Camanho PP and Matthews FL. A progressive damagemodel for mechanically fastened joints in composite

laminates. J Compos Mater 1999; 33: 2248–2280.10. Tay TE, Tan VBC and Tan SHN. Element-failure: an

alternative to material property degradation method forprogressive damage in composite structures. J Compos

Mater 2005; 39: 1659–1675.11. Tay TE, Liu G and Tan VBC. Damage progression in

open-hole tension laminates by the SIFT-EFM approach.J Compos Mater 2006; 40: 971–992.

12. Liu G, Tay TE and Tan VBC. Failure progression and

mesh sensitivity analyses by the plate element-failure

method. J Compos Mater 2010; 44: 2363–2379.13. Tay TE, Liu G, Yudhanto A, et al. A micro-macro

approach to modeling progressive damage in composite

structures. Int J Damage Mech 2008; 17: 5–28.14. Tay TE, Liu G, Tan VBC, et al. Progressive failure ana-

lysis of composites. J Compos Mater 2008; 42: 1921–1966.15. Wolla JM and Goree JG. Experimental evaluation of

longitudinal splitting in unidirectional composites.J Compos Mater 1987; 21: 49–67.

16. Kortschot MT and Beaumont PW. Damage mechanics ofcomposite materials: I – Measurements of damage and

strength. Compos Sci Technol 1990; 39: 289–301.17. Kwon YW and Craugh LE. Progressive failure modelling

in notched cross-ply fibrous composites. Appl ComposMater 2001; 8: 63–74.

18. Tsai JL and Chen CW. Characterizing tensile strength ofnotched cross-ply and quasi-isotropic composite lamin-

ates. J Chin Inst Engineer 2009; 32: 327–332.

19. Hallett SR and Wisnom MR. Experimental investigation

of progressive damage and the effect of layup in notched

tensile tests. J Compos Mater 2006; 40: 119–141.20. Hallett SR and Wisnom MR. Numerical investigation of

progressive damage and the effect of layup in notched

tensile tests. J Compos Mater 2006; 40: 1229–1245.21. Hallett SR, Jiang WG, Khan B, et al. Modelling the inter-

action between matrix cracks and delamination damage

in scaled quasi-isotropic specimens. Compos Sci Technol

2008; 68: 80–89.22. Hallett SR, Jiang WG and Wisnom MR. Effect of stack-

ing sequence on open-hole tensile strength of composite

laminates. AIAA J 2009; 47: 1692–1699.23. Hallett SR, Green BG, Jiang WG, et al. The open hole

tensile tests: a challenge for virtual testing of composites.

Int J Fract 2009; 158: 169–181.24. Chan KS, He MY and Hutchinson JW. Cracking and

stress redistribution in ceramic layered composites.

Mater Sci Eng 1993; A167: 57–64.

25. Lekhnitskii SG, Tsai SW and Cheron T. Anisotropic

plates. New York: Gordon and Breach Science

Publishers, 1968.26. Iarve EV, Mollenhauer D and Kim R. Theoretical and

experimental investigation of stress redistribution in open

hole composite laminates due to damage accumulation.

Compos Part A 2005; 36: 163–171.

27. Iarve EV, Gurvich MR, Mollenhauer D, et al. Mesh-

independent matrix cracking and delamination modelling

in laminated composites. Int J Numer Meth Eng 2011; 88:

749–73.28. Mollenhauer D, Ward L, Iarve E, et al. Simulation of

discrete damage in composite overheight compact tension

specimens. Compos Part A 2012; 43: 1667–1679.

29. Swindeman MJ, Iarve EV, Brockman RA, et al. Strength

prediction in open hole composite laminates by using dis-

crete damage modeling. AIAA J 2009; 51: 936–945.30. Abaqus analysis user’s manual, Version 6.10. Chapter

29.5 Cohesive elements. Dassault Systemes, Providence,

RI, USA, 2010.31. Camanho PP, Davila CG, Ambur DR. Numerical simula-

tion of delamination growth in composite materials.

Technical Report NASA-TP-211041, National Aeronautics

and Space Administration, USA, 2001.

32. Camanho PP, Davila CG and de Moura MF. Numerical

simulation of mixed-mode progressive delamination in

composite materials. J Compos Mater 2003; 37:

1415–1438.33. Camanho PP, Davila CG and Pinho ST. Fracture ana-

lysis of composite co-cured structural joints using deco-

hesion elements. Fatig Frac Eng Mater Struct 2004; 27:

745–757.34. Petrossian Z and Wisnom MR. Prediction of delamin-

ation initiation and growth from discontinuous plies

using interface elements. Compos Part A 1998; 29:

503–515.35. Turon A, Davila CG, Camanho PP, et al. An engineering

solution for mesh size effects in the simulation of delam-

ination using cohesive zone models. Eng Fract Mech

2007; 74: 1665–1682.

Liu and Tang 295

36. Harper PW and Hallett SR. Cohesive zone length innumerical simulations of composite delamination. EngFract Mech 2008; 75: 4774–4792.

37. Chen BY, Tay TE, Baiz PM, et al. Numerical analysis ofsize effects on open-hole tensile composite laminates.Compos Part A 2013; 47: 52–62.

38. Ridha M, Wang CH, Chen BY, et al. Modelling complex

progressive failure in notched composite laminates withvarying sizes and stacking sequences. Compos Part A2014; 58: 16–23.

39. Ridha M, Tan VBC and Tay TE. Traction-separationlaws for progressive failure of bonded scarf repair ofcomposite panel. Compos Struct 2011; 93: 1239–1245.

40. Hashin Z. Failure criteria for unidirectional fiber com-posites. J Appl Mech 1980; 47: 58–80.

41. Pham DC, Sun XS, Tan VBC, et al. Progressive failureanalysis of scaled double-notched carbon-epoxy com-posite laminates. Int J of Damage Mech 2012; 21:

1154–1185.42. Shi YB, Hull D and Price JN. Mode II fracture of þ/-

angled laminate interfaces. Compos Sci Technol 1993; 47:173–184.

43. Song K, Li Y and Rose C. Continuum damage mechanicsmodels for the analysis of progressive failure in open-holetension laminates. In: 52nd AIAA structures, structural

dynamics and materials conference, NASA LangleyResearch Center, Denver, CO, 4–7 April 2011.

44. Moes N, Dolbow J and Belytschko T. A finite element

method for crack growth without remeshing. Int J NumerMeth Eng 1999; 46: 601–620.

296 Journal of Composite Materials 50(3)