An Improved Model for Fiber Kinking Analysis of Unidirectional Laminated Composites

An Improved Model for Fiber Kinking Analysisof Unidirectional Laminated Composites

Abdulreza Kabiri Ataabadi & Saeed Ziaei-Rad &

Hossein Hosseini-Toudeshky

Received: 9 January 2010 /Accepted: 14 April 2010 /Published online: 23 May 2010# Springer Science+Business Media B.V. 2010

Abstract This paper focuses on the fiber-kinking failure mode of unidirectional laminatedcomposites under the compressive loading. An available stress based fiber-kinking model isexplained and improved on the bases of strain concept. In the improved model, a newfracture surface is considered and the stresses are updated according to this new fracturesurface. By taking the advantage of damage variables, the models are implemented into afinite element code and the results of numerical analysis such as prediction of kink bandangles are discussed in details and compared with the available results in the literature. It isshown that the predicted kink band angles using the improved model are in good agreementwith the experimental results.

Keywords Composite . Modeling . Fiber-kinking . Failure . Unidirectional

1 Introduction

Design of composite structures requires the development of methods capable of predictingdifferent damage mechanisms and their evolution until the final failure. In addition, thesemodels should be applicable to industrial structures subjected to various loadings.

Fiber compressive failure is a complex failure mode in laminated composites.Depending on the material, different fiber compressive failure modes, such as micro

Appl Compos Mater (2011) 18:175–196DOI 10.1007/s10443-010-9145-z

A. Kabiri Ataabadi (*) : S. Ziaei-RadDepartment of Mechanical Engineering, Isfahan University of Technology, Isfahan, Irane-mail: [email protected]

S. Ziaei-Rade-mail: [email protected]

H. Hosseini-ToudeshkyDepartment of Aerospace Engineering, Amirkabir University of Technology, Tehran, Irane-mail: [email protected]

buckling, kinking and fiber failure are possible [1]. This article focuses on the fiber-kinkingdamage mode in the unidirectional composite laminates. Kinking can be defined as alocalized shear deformation of the matrix. Typically, the fibers break at the edges of theband and sometimes in the interior region [2]. In this failure mode which can be localized ina band across a specimen, the fibers are rotated by a large amount and the matrix is usuallysubjected to a large shearing deformation [2]. Some researchers considered the kinkingmode as a consequence of micro buckling, while others consider that as a separate failuremode [1].

Berbinau et al. [3] believes that the kink band in 0° unidirectional laminated composites,loaded in compression, forms in three steps of elastic kinking, plastic kinking and finalcollapse as a result of fiber failure at the kink boundaries. To study the fiber failure at thekink boundaries, they modeled the initial fiber waviness by a sine function and assumedthat the half wavelength is equal to the kink band width (50–80μm for T800/924C carbon/epoxy unidirectional laminated composites) and kink band width is about 10–15 times ofthe fiber diameter. He examined the shear strains developed in the matrix due to fiber microbuckling and concluded that criterion for fiber failure due to tensile stresses on its convexside is not acceptable and fiber failure is proved to be initiated on the compression side atthe points with maximum fiber curvature.

The mechanical model for the fiber kinking was also developed by Argon [2, 4]. Thismodel considered the fiber kinking as a separate failure mode. In this model and thosefollowed that, the elastic behavior of the matrix and the initial imperfections play animportant role in the damage analysis. The effects of matrix behavior on the compressivestrength of unidirectional laminated composites are examined by Soutis et al. [5]. Heperformed experiments on the T800/924C carbon/epoxy laminates for various temperaturesbetween the room temperature and 100°C. They founded that at 80°C the failure modeswitches from in-plane to out-of-plane fiber micro buckling. As the experiment temperatureincreases, the shear strength/stiffness of the resin are considerably decreased which causereduction of the amount of side support for the fibers and therefore reduce the strain level atwhich fiber buckling occurs.

The fiber kinking failure mode is usually observed in the high fiber-volume fractioncomposite materials when they are under the compressive loading [2]. A schematicrepresentation of a kink band is shown in Figs. 1 and 2.

In discussion on whether or not kinking is a consequence of micro buckling, the mainargument has to do with the orientation of the kink band boundary. Indeed, if kinking is a

Fig. 1 Micrograph fromexperiments in CFRP [6, 7] andkink band geometric parameters

176 Appl Compos Mater (2011) 18:175–196

consequence of micro buckling, then one would expect the kink band boundary to lienormal to the loading axis (original fiber direction) to lie in the plane of highest bendingstresses. This would mean the angle of β equal to zero in Fig. 2(a). However, it is found thatin most cases, β lies in the range of about 30°. On the other hand, the similarity between thekink bands and shear bands may suggests that the shear is the main factor at the onset ofkink band formation. In this case, it would be expected that kink bands would occur in theplanes of maximum shear stress, i.e., β=45°. Some thick kink bands have been found near45° direction [1], but this is not generally the case [2].

The kink band angle and the kink band width were studied by Hahn [8] for carbon fibercomposites (CFC), glass fiber composites (GFC) and aramid fiber composites (AFC). Thekink band angle β, and the band width w, were found to be the smallest for CFC at roomtemperature (β≈20° and w≈0.07–0.2mm). For GFC, no clear kink bands were observed atroom temperature. However, at 100°C, a kink band angle of β≈30° and a width of w≈1.2mmwere observed. For AFC tested at room temperature, the kink band angle was found to beabout β≈40° and width w≈0.45mm. Chaplin [9] noted that, the angle of rotation of the fibresin the kink band was twice of the β angle, so that no volumetric changes happened in thekinked region [2].

In an experimental study, Lee and Soutis [10] examined the effect of specimen size onthe axial compressive strength of unidirectional IM7/8552 carbon/epoxy laminates. Theyfound that in specimens which are broken within the gauge section or near the end-tabs, thefailure mechanism is micro buckling or fiber kinking. Optical microscopy images alsoshown that the kink bands propagated through the thickness of the specimen and across thespecimen width at an angle of 20–30° with a kink band width of approximately 90μm(about 12 fiber diameters).

Argon [4] assumed that the fiber shearing stresses are due to the existence of initial fibermisalignment. The shearing stresses cause further rotation of the fibres, which would in turnlead to further increase in the shear stresses. This close loop effect could then lead to thefailure of laminate. The relation between the compressive failure stress, Xc, the longitudinalshear failure stress, SL, and the initial fibre misalignment angle, θi(in radians) wasperformed from these analyses as [2]:

Xc ¼ SLqi

ð1Þ

Fig. 2 a Kink band, b Fibers misalignment frame

Appl Compos Mater (2011) 18:175–196 177

Several researchers reported the sensitivity of the compressive failure stress to the shearfailure stress [11–14]. Budiansky [13] extended the relation purposed by Argon’s analysisand suggested the following equation:

Xc ¼ SLqi þ g�

ð2Þ

Where, g° is the shear strain at the failure point and failure occurs when the shear failurestress criterion is satisfied in the material coordinate system.

Jumahat et al. [15] proposed a combined modes model based on Berbinau et al. [3] fibermicro buckling model and Budiansky et al. [13] fiber kinking model. This modelunderestimates the actual compressive strength value, because the predicted compressivestrength is the critical stress at which the fibers fail via micro buckling rather than the finalfailure stress of the whole laminate caused by both fiber micro buckling and plastic kinkingmechanisms. The plastic kinking failure mechanism is incorporated into the model asfollows:

Xc ¼ Xfiber microbuckling þ Xplastic kinking ð3Þ

In which the additional compressive strength caused by plastic deformation, Xplastic kinking

of the composite can be determined using the following equation:

Xplastic kinking ¼ tult � tygult � gy� �þ qi

ð4Þ

Where ty is the in-plan yield shear strength, tult is the in-plane ultimate shear strength, gyand gult are the corresponding shear strains. A comparison between the predictedcompressive strength of the fiber kinking model, fiber micro buckling model, combinedmodes model and experimental data were performed for UD HTS40/977-2 CFRP laminatedcomposite as shown in Fig. 3.

A 2D and 3D stress based kinking model has also been proposed by Pinho [2, 16]. Inthis model, the stresses are calculated from the strains in the global frame and thentransformed into the misalignment frame to determine the fiber shear stresses. Taking the

Fig. 3 Comparison of predictedcompressive strength withvarious models for UD HTS40/977-2 CFRP laminatedcomposite [15]


advantage of the shear constitutive law, increase of the misalignment angle can be easilydetermined. In the misalignment frame the matrix failure is checked and if the damageprocess is initiated, then the stresses in the kink band are degraded. This model has someobjections which will be noted in the following sections elaborately.

In this paper, the stress based model is explained firstly and then an improved strainbased model is performed. In the improved fiber kinking model proposed here, the fibersmisalignment angles are calculated based on the strains rather than the stresses and in theplanes parallel to the fibers directions, the matrix failure criterion is also checked and if thedamage process has been initiated, then the stresses in the fracture plane are degraded.The differences between this model and the stress based fiber kinking model are explainedin the following sections in details. Finally, the kinking analyses are performed for a fewlaminates and the obtained kink band results are compared with the available results in theliterature.

2 Stress Based Model

A unidirectional composite with misaligned region being compressed as depicted in Fig. 2is considered here. The stresses in the misalignment frame for a generic plane stress loadingare:

sam ¼ saþsb2 þ sa�sb

2 Cos 2qð Þ þ tab Sin 2qð Þsbm ¼ saþsb

2 � sa�sb2 Cos 2qð Þ � tab Sin 2qð Þ

tambm ¼ � sa�sb2 Sin 2qð Þ þ tab Cos 2qð Þ

ð5Þ

Where, σa, σb and tab are stress components in the global frame, sam ; sbm and tambm arestress components in the misalignment frame and θ is the fiber misalignment angle.

In the kinking model proposed by Pinho, kinking is the result of matrix failure in theplane parallel to the misaligned fibers. The proposed criteria by Puck and Schurmann [17, 18]for the fiber kinking resulted from matrix cracking are used as follows:

fmc ¼tT

ST�mTsn

� �2þ tL

SL�mL sn

� �2¼ 1 sn � 0

snYt

� �2þ tT

ST

� �2þ tL

SL

� �2¼ 1 sn > 0

8><>: ð6Þ

Where tT, tL and σn are the traction components on potential fracture planes as shown inFig. 4 and Yt, ST and SL are the transverse tensile strength, transverse and longitudinal shearstrengths respectively. μT and μL are friction coefficients in the transverse and longitudinaldirections and ST is the fracture plane resistance. The following relations between the aboveparameters and the fracture plane angle are also defined as [2, 19]:

tan 2f�ð Þ ¼ � 1

mT

ð7Þ

ST ¼ Yc2 tan f�ð Þ

ð8Þ

mL

SL¼ m

T

ST

ð9Þ


In which, fo is the fracture plane angle of the specimen under pure transverse loadingwhich is experimentally observed to be fo=53±2° for most of the composite materials[2, 17, 18] (see Fig. 5).

In the stress based model, the initial misalignment angle of fibers (θi) is obtained from apure axial compressive experiment. For this reason, the fibers misalignment angle (θc) andshear strain (gmc), at the failure condition for pure compressive loading are required. In thefollowing section, the calculation procedure of these parameters based on the pure axialcompression data is explained.

Fig. 4 Fracture surface angle for a general loading condition

Fig. 5 Fracture surface anglefor pure transverse compressivefailure [2, 20]


2.1 Determination of Model’s Parameters

For failure under a pure compression sa ¼ �Xc; sb ¼ tab ¼ 0ð Þ, Eq. 5 lead to:

sam ¼ �XcCos2 qð Þ

sbm ¼ �Xc Sin2 qð Þ

tambm ¼ tmc ¼ Xc Sin qð ÞCos qð Þð10Þ

These stresses can be employed in an appropriate matrix failure criterion, e.g. Eq. 6. Fora material with linear shear behavior, using an appropriate matrix failure criterion leadsdirectly to an expression for the specific value of the misalignment angle at failure point forpure compression (θc). For a material with nonlinear shear behavior, kinking can be resultedfrom either (i) matrix failure or (ii) instability mechanism due to the loss of shear stiffnessfor large shear strain values. The calculation procedure of model’s parameters for these twokinking modes is performed as follows [2, 19]:

Case 1 kinking due to the matrix failure.

In this case, by using Eq. 10, the stresses in the misalignment frame are calculated.These stresses are employed in Eq. 6 and the value of the misalignment angle at failurepoint, θc, is calculated from the following equation:

Xc Sin qcð ÞCos qcð Þ � mLSin2 qcð Þ

� � ¼ SL ð11ÞUsing this equation, the following expression is found for θc [2, 16, 19]:

qc ¼ arctan1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4 SL

Xcþ mL

� �SLXc

r

2 SLXcþ mL

� �0BB@

1CCA ð12Þ

Using the shear constitutive law, and having the shear stress tmc, the failure shear straingmc can be obtained and therefore, the initial misalignment angle, θi, can be calculated. Theshear stress at failure point and in the material axes is a function of the failure shear strainwith the same function as the constitutive law:

tmc ¼ fcl gmcð Þ ð13ÞFor 2D conditions under pure compression, the shear stress at the θc angle is:

tmc ¼ 1

2Sin 2qcð ÞXc ð14Þ

From the two above equations, the shear strain at failure point for a pure compressioncase, gmc, becomes:

gmc ¼ f �1cl

1

2Sin 2qcð ÞXc

� �ð15Þ

For instance, for a material linear behavior in shear deformation, Eq. 15 simply becomes:

gmc ¼Sin 2qcð ÞXc

2Gabð16Þ


The initial misalignment angle can be calculated by the following equation:

qi ¼ qc � gmc ð17ÞThe alternative form of Eq. 15 is:

fcl gmcð Þ ¼ 1

2Sin 2 qi þ gmcð Þ½ �Xc ð18Þ

The left hand side (LHS) of the above equation is a representation of the shearingconstitutive law and the right hand side (RHS) of that represents the shear stress resultingfrom the compressive longitudinal loading in a rotated coordinate system. If the shearbehavior of the material is linear, then Eq. 18 has only one solution at failure and kinkingresults from matrix cracking, as shown in Fig. 6(a). For a material with nonlinear shearbehavior, there could be more than one intersection point at each stress level for theparticular case of failure onset. In this situation, gmc corresponds to the first intersectionpoint (lower energy), as represented in Fig. 6(b). If the mentioned solution corresponds tothe second intersection, then the solution is not valid and failure occurs due to a differentdamage mechanism.

Case 2 kinking due to the instability.

If the composite laminate is progressively loaded in compression, the RHS curve inFig. 6(b) shifts up progressively. If the two LHS and RHS curves are tangent to each other,Fig. 6(c), and the matrix compressive failure criterion is not yet satisfied, For this case, asmall increase in the compressive load results in separation of two curves from each other.Physically, this means that there is no equilibrium position and catastrophic failure resultsdue to the unstable rotation of the fibers.

The values of θi and gmc corresponding to this type of failure can be obtained byconsidering Eq. 18 in conjunction with the condition that the left and right hand sides ofthis equation have the same slope at gmc [2, 19], as follows:

fcl gmcð Þ ¼ Xc

2Sin 2 qi þ gmcð Þ½ �

@ fcl@g

gð Þ gmc¼ Xc

Cos 2 qi þ gmcð Þ½ �

8>><>>: ð19Þ

Fig. 6 Left and right hand side of Eq. 18 for a material with (a) linear shear behavior, (b) nonlinear shearbehavior and failure by matrix cracking, and (c) nonlinear shear behavior and failure by instability


2.2 3D Stress Based Fiber Kinking Model

Most of the previously developed fiber kinking models assume that kinking happens in theplane of the lamina. Therefore, most of the experimental studies follow this in-planeapproach and constrain the specimens so that, out of plane movements are not allowed.However, many researchers agree with the 3D nature of fiber kinking failure.

The proposed 3D kinking model in [2, 16, 19] is explained in this section.A unidirectional lamina under a general compressive state of stress is considered as

shown in Fig. 7(a). The fiber kinking plane is assumed to be at an angle with the b axis, asshown in Fig. 7(b). Figure 7(c) shows the stresses on the (b,c) plane, while Fig. 7(d) showsthe stresses along the by and Cy directions (Fig. 7(d) assumes that by and Cy are the principaldirections in the (b,c) plane). The rotation to the misalignment plane is shown in Fig. 7(e). Thematrix fracture plane is represented in Fig. 7(f). The y angle in Fig. 7(b), (c) and (d) is equalto zero for the 2D fiber kinking model—in which through the thickness movements areconstrained. If the composite is constrained so that it cannot move in the b direction, then thefiber kinking plane will have an angle of y=90°. For a general loading condition, y will havea value between 0 and 180°. The following transformation equations can be used to find thestresses in a potential fiber kinking plane:

sb< ¼ sbþsc2 þ sb�sc

2 Cos 2yð Þ þ tbcSin 2yð Þsc< ¼ sb þ sc � sb<

taby ¼ tabCos yð Þ þ tcaSin yð Þtbycy ¼ � sbþsc

2 Sin 2yð Þ þ tbcCos 2yð Þtcya ¼ tcaCos yð Þ � tabSin yð Þ

ð20Þ

After defining the fiber kinking plane, the stresses in the rotated misalignment frame areobtained. The strain gm is obtained by solving the following equation iteratively:

fCL gmð Þ ¼ �sa � sby

2Sin 2 qi þ gmð Þð Þ þ tabyj jCos 2 qi þ gmð Þð Þ ð21Þ

Fig. 7 3D kinking model: a Solid under general loading, b Fiber kinking plane, c Stresses on the (a,b,c)coordinate system, d Stresses on the a; by ; cyð Þ coordinate system, e Stresses on the misalignment frame andf Matrix failure plane


And, therefore, the θ angle can be calculated as follow:

q ¼ taby

tabyj j qi þ gmð Þ ð22Þ

If Eq. 21 does not have a solution, then the failure occurs by instability mechanism. Theenvelope for failure by instability mechanism is defined by:

fCL gmð Þ ¼ �sa � sby

2Sin 2 qi þ gmð Þð Þ þ tabyj jCos 2 qi þ gmð Þð Þ

@fCL gmð Þ@gm

¼ � sa � sbyð ÞCos 2 qi þ gmð Þð Þ � 2 tabyj jSin 2 qi þ gmð Þð Þ

8>><>>: ð23Þ

Now, having the misalignment angle, θ, the stresses can be rotated to the misalignmentframe:

sam ¼ saþsby

2 þ sa�sby

2 Cos 2qð Þ þ tabySin 2qð Þsbm ¼ sa þ sby � sam

tambm ¼ � sa�sby

2 Sin 2qð Þ þ tabyCos 2qð Þtbmcy ¼ tbycyCos qð Þ � tcyaSin qð Þtcyam ¼ tcyayCos qð Þ þ tbycySin qð Þ

ð24Þ

It is also required to check the matrix failure using the following criteria:

fkink ¼ tTST � mTsn

� �2

þ tTSL � mLsn

� �2

¼ 1 ; sbm � � ð25Þ

fkink ¼ sn

Yt

� �2

þ tTST

� �2

þ tLSL

� �2

¼ 1 ; sbm > � ð26Þ

In the two above equations, the traction components in the fracture plane are given by:

sn ¼ sbmþscy

2 þ sbm�scy

2 Cos 2fð Þ þ tbmcSin 2fð ÞtT ¼ � sbmþscy

2 Sin 2fð Þ þ tbmcyCos 2fð ÞtL ¼ tambmCos fð Þ þ tcyamSin fð Þ

ð27Þ

Where the angle f is obtained by trying a small number of tentative angles in theinterval, 0≤f<π. As noted before, the angle y is corresponded to the principal directions ofb and c, and it can be obtained by:

tan 2yð Þ ¼ 2tbcsb � sc

ð28Þ

Pinho used a simple modification, which avoids iterating in each time step, consists ofdefining the misalignment frame orientation, θ, as the sum of an initial misalignment angle,θi, with the shear strain in the initial misalignment frame, gmi.

The strain in the initial misalignment frame is defined as:

gmi ¼ f �1CL � sa � sby

2Sin 2qið Þ þ tabyj jCos 2qið Þ

� �ð29Þ


The initial misalignment angle, θi, is deduced from experimental data by solving thefollowing equation iteratively:

qi ¼ qC � f �1cL

1

2Sin 2qið ÞXc

� �ð30Þ

And the misalignment frame angle, θ, becomes as:

q ¼ taby

tabyj j qi þ gmið Þ ð31Þ

2.3 Finite Element Implementation of Fiber Kinking Model

2.3.1 Material Behavior

For implementation of the failure model into the finite element code the material behaviormust be specified. Before damage initiation, the in-plane shear deformation is onlyconsidered to behave nonlinearly. To define the in-plane shear response, the maximumshear strain has to be defined as:

gmaxab ðtÞ ¼ max gab t0ð Þ

n o; t0 � t ð32Þ

And the inelastic shear strain as:

ginab ¼ gmaxab � fcL gmax

ab

� �=Gab ð33Þ

The function fcL (gab) represents the value of shear stress for gab≥0.The material behavior for shear deformations which has been shown in Fig. 8 can be

presented as:

tab ¼gabgabj j fCL gabj jð Þ gabj j ¼ gmax

abgabgabj j Gab < gabj j � ginab > gabj j < gmax

ab

(ð34Þ

Where the operator <•> is the Mc-Cauley bracket defined as xh i ¼ max o; xf g; x 2 R.

Fig. 8 a Experimental nonlinearshear behavior, b Irreversibility:loading, unloading and reloadingpaths


2.3.2 Damage Formulation

To explain the material behavior after damage initiation, a damage variable is used. Thisparameter depends on the material behavior on the fracture surface. In this research, inorder to obtain the damage variable, it is assumed that the material behave according toFig. 9(b) demonstrating a bilinear softening constitutive law.

According to Fig. 9, the maximum strain, εf, can be obtained as a function of the energyper unit area of the surfaces created by damage, Γ, the material strength, σ°, and theelement dimension, L2 [2, 20]:

"f ¼ 2*

s�L2ð35Þ

The damage variable, d, is defined to consider a linear degradation for the relevant stresscomponents, as shown in Fig. 9(b). The instantaneous value of the damage variable, dinst, isdefined as [2, 20]:

dinst ¼ max 0;min 1; "f"� "�

" "f � "�ð Þ � �

ð36Þ

In order to consider the irreversibility, the damage variable is defined as:

dðtÞ ¼ max dinst t0ð Þ� ; t0 � t ð37Þ

After the detection of the failure, the shear stresses in the kink band (tCyam ; tambm ) aswell as the stress normal to the kink band (depending on the sign of sam ) are degraded byusing the damage variable, dkink:

tambm 1� dkinkð Þtambm ; tcyam 1� dkinkð Þtcyamsam 1� dkink

samh isam

� �sam

ð38Þ

This failure process is associated with the rotation of fibers in the kink band, which is due tothe shear stress tambm . Therefore, in the scale of the kink band, failure propagation is controlled

by the driving stress skink ¼ tambm . The elastic component of gab geLab ¼ tabGab

� �, is considered

for the deriving strain, which is thus defined as "kink ¼ geLambm , where geLambm is obtained byrotation of the elastic strains.

Fig. 9 a Typical unidirectional composites loaded in transverse tension up to complete failure, b Materialconstitutive law with linear softening


The onset stress and strain are defined as:

s�kink ¼ skinkj jfkink¼1; "�kink ¼ "kinkj jfkink¼1 ð39ÞAnd the expression for the final strain "fkink is:

"fkink ¼2*kink

s�kinkLkinkð40Þ

The characteristic length Lkink in the above equation shown in Fig. 10 is defined as:

Lkink ¼ L1L2L3LamLcy

ð41Þ

Finally, the energy per unit area of the surface created in the damage process, Γkink, canbe obtained by the experiments [2, 20].

3 Strain Based Model

The previously proposed stress based model may be criticized as follow:

a) Before damage initiation, the effects of misalignment angle of the fibers are notconsidered in the stress calculation procedure.

b) The modification performed in the stress based model (Eq. 29), disregards theinstability mode. For example, when Eq. 30 is used, in the case which, XC=1,355MPa,θc=0.12 and the in-plane shear behavior of material is considered to be coincided withthe logarithmic function, t ¼ k1 ln k2g þ 1ð Þ with k1=46 and k2=100, θi is determinedto be about 0.064 radians. However, by using Eqs. 15 and 17, θi is determined to beabout −0.202 radians. This excessive difference means that, the previously performedmodification needs to be improved.

c) Matrix failure criterion (Eqs. 25 and 26) is used in the planes parallel to the fibers (seeFig. 7(f)) to investigate the fiber kinking failure. But after failure initiation, the stressesin the kink band are degraded (see Eq. 38). This means that matrix failure has

Fig. 10 Determination of the characteristic length within an element: a Rotation of an angle β, b Rotation ofan angle y, c Rotation of an angle θ, d Rotation of an angle f


happened in the plane parallel to the fibers but stresses are degraded in the planeperpendicular to the fibers.

d) The characteristic length defined in Eq. 41 does not coincide with the characteristiclength defined in Eq. 35 and Fig. 9 and those explained in the Section 2.3.2.

Therefore, the main aim of this investigation is the improvement of this stress basedmodel to alleviate some of these drawbacks. Therefore, a kinking failure model based onthe strains is presented here. The model is explained in the following sections and finally,the results are discussed and compared with the available experimental data.

3.1 Definition

– Stage-I

In this Stage, in order to calculate the misalignment angle at failure point, θc, the fiberkinking plane and the initial misalignment angle, θi, similar to the stress based model,Eqs. 12, 28 and 30 are used.

Now, the only important parameter which have to be determined is the misalignmentangle, θ, which can be calculated using the following equation instead of Eq. 31:

q ¼ qi þ gm ð42ÞIn this equation, θi is a material parameter and can have positive or negative values and

gm is the shear strain in the misalignment frame which can be determined using thefollowing equations instead of Eq. 29:

"by ¼ 1

2"b þ "c þ "b � "cð Þ cos 2yð Þ þ gbc sin 2yð Þð Þ ð43Þ

gaby ¼ gab cos yð Þ þ gca sin yð Þ ð44Þ

gm ¼ � "a � "byð Þ sin 2qið Þ þ gaby cos 2qið Þ ð45ÞAfter calculating the misalignment angle, θ, the stresses in the misalignment frame can be

determined using Eqs. 20 and 24. At this point, matrix failure is checked by Eqs. 25, 26 and 27.The angle f is obtained by trying a small number of tentative angles in the interval of 0≤f<π.

This modified model is implemented into an in-house finite element code for kinkinganalysis of compression specimens and the obtained results are compared with thepreviously developed stress based model in the next sections.

– Stage-II

In this stage after the calculation of fiber kinking plane angle and the initialmisalignment angle by Eqs. 28 and 30, the strains which are available in each incrementare transformed to the misalignment frame with the following transformations (see Fig. 7):

"a"b"c"ab"bc"ca

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;y Rotation about a axis

"a"by

"cy

"aby

"bycy

"cya

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;qi Rotation about cy axis

"am

"bm

"cy

"ambm

"bmcy

"cyam

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð46Þ


In which, "ij ¼ 12 gij. The misalignment angle can be calculated by:

q ¼ qi þ gambm ð47Þ

The stresses in the misalignment frame are:

sf gm ¼ E "f gm ð48Þ

Where E is the stiffness matrix and sf gm and "f gm are:

sf gm ¼

sam

sbm

scy

tambmtbmcytcyam

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;; "f gm ¼

"am

"bm

"cy

"ambm

"bmcy

"cyam

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð49Þ

At this point, similar to the stress based model, the matrix failure criterion is checkedusing Eqs. 25, 26 and 27.

3.2 Finite Element Implementation

In the implementation of the strain based failure model into the finite element code, the in-plane shear (ambm) has been considered to be fully nonlinear before damage initiation. Theother components of the stress tensor are considered to be elastic. To define the in-planeshear response, Eqs. 32, 33 and 34 are used. It should be noted that, instead of ab in thisequations ambm are used here.

Damage formulation for the strain based model (stage-I) is similar to the stress basedmodel. Therefore, in this section, the damage formulation for the strain based model instage-II is only explained.

After detection of damage, the shear stresses in the fracture surface which is parallel tothe fiber direction (misalignment axis, am) with an angle f and bm axis as shown in Fig. 7(f)(tT, tL(, are degraded as well as the stress normal to the fracture surface (depending on thesign of σn) using the damage variable, dkink as follows:

tT 1� dkinkð ÞtT ; tL 1� dkinkð ÞtLsn 1� dkink

snh isn

� �sn

ð50Þ

Parameters tT, tL and σn are determined using Eq. 27. Equation 50 tries to enhance theobjection (c) and has an agreement with Fig. 7(f).

To derive the damage variable, strain variable, εkink, magnitude of the traction at thefracture plane, σkink, and characteristic length, Lkink, must be calculated. The magnitude ofthe traction at the fracture plane is:

skink ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi< sn >2 þ t2kinkt

qð51Þ

With the shear component of the traction, tkink, defined as:

tkink ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2T þ t2L

qð52Þ


The elastic strain components on the fracture plane are:

"n ¼ 12 "bm þ "cyð Þ þ "bm � "cyð ÞCos 2fð Þ þ 2"bmcySin 2fð Þ½ �

gT ¼ � "bm � "cyð ÞSin 2fð Þ þ 2"bmcyCos 2fð ÞgeLL ¼ 2"eLambmCos fð Þ þ 2"cyamSin fð Þ

ð53Þ

Where the elastic component of the in-plane shear strain is defined as:

2"eLambm ¼tambm

Gambmð54Þ

Note that the elastic internal energy of the element at the onset of failure contributed tothe fracture process only. Therefore, the elastic part of "ambm is only considered in theanalyses. The strain component on the direction of σkink is defined as:

"kink ¼ < sn >

sn"nSin wð Þ þ gkinkCos wð Þ ð55Þ

The variables used in the above equation are defined as:

gkink ¼ gTGos lð Þ þ geLL Sin lð Þ l ¼ arctan tL

tT

� �; w ¼ arctan <sn>

tkink

� � ð56Þ

The onset stress and strain are determined from the value of σkink and εkink at the failureinitiation:

s�kink ¼ skink fkink¼1; "�kink ¼ "kink

fkink¼1 ð57Þ

Finally, the expression for "fkink is:

"fkink ¼2*kink

s�kinkLkinkð58Þ

The characteristic length, Lkink, is defined as:

Lkink ¼ L1L2L3LamLcfy

ð59Þ

Determination of Lam has been shown in Fig. 10(c) and Lcfy also can be determined by:

Lcfy ¼ minLcy

cos fð Þ ;Lbm

sin fð Þ �

ð60Þ

And Lbm defined as:

Lbm ¼ LaLby

Lamð61Þ

Lcy has been also defined in Fig. 10(b).The fracture toughness, Γkink, in Eq. 58 is the last term which needs to be determined.

For a mixed mode condition (tT, tL and σn acting concurrently) the following expressioncan be used [2]:

*kink ¼ 1

*b

"�ns�n

"�kinks�kink

� �2 !a

þ 1

*T

*�T t�T

"�kinks�kink

� �2 !a

þ"

1

*L

*eLL t�L

"�kinks�kink

� �2 !a#�1

a

ð62Þ

In which the parameter α can be obtained from experiments.


In a particular case, if the mixed mode data is unknown at all, a simple weighted averageof Γb, ΓT and ΓL might be appropriate to determine the fracture toughness, Γkink:

* ¼ *bs�ns�mat

� �2

þ *Tt�Ts�mat

� �2

þ *Lt�Ls�mat

� �2

ð63Þ

Where Γb is the mode-I interalaminar fracture toughness and *T ¼ *L ¼ GIIC [2].

3.3 Compressive Strength Prediction

In this section to evaluate the strain based kinking model, it is used to predict thecompressive strength of the laminates. The obtained results from the strain based model arecompared with those predicted by the combined modes model proposed by Jumahat et al.[15], for the laminate made of UD HTS40/977-2. Due to the lack of enough data, thematerial properties such as f0 for this composite material considered to be equal with thatbelong to T300/913 [2] (Table 1).

The nonlinear shear stress-strain behavior for UD HTS40/977-2 is shown in Fig. 11 [15].The analytical shear stress-strain equation for behavior shown in the figure is given by:

t gð Þ ¼ ty 1� exp �Ge12gty

� �� þ tult � ty� �

1� exp � GP12g

tult � ty

� �� ð64Þ

In the above equation, ty ¼ 52 MPa ; tult ¼ 101 MPa ; Ge12 ¼ 4:4 GPa ; and

GP12 ¼ 680 MPa.The effect of initial misalignment angle on the predicted compressive strength is

investigated and the obtain results are shown in Table 2. As pointed out in this table,compressive strength predicted by strain based model is due to instability mode and isapproximately 16% larger than the one predicted by combined modes model but thecompressive strength reduction as initial misalignment angle increases, is equal for twomodels.

The effect of shearing properties on the predicted compressive strength is alsoinvestigated and illustrated in Table 3. These compressive strengths are evaluated forsystems with one degree misalignment angles. As shown in this table, the compressivestrengths obtained from strain based model is approximately 10–22% larger than combinedmodes one.

4 Numerical Results

The capability of the fiber kinking models (stress based and strain based models) explainedin the previous sections, are investigated and the obtained results from the implemented

Table 1 Material properties for T300/913 [2] and HTS40/977-2 [15]

Materialproperties

Ea

(GPa)Eb

(GPa)Gab

(GPa)Yt(MPa)

Yc(Mpa)

Xc

(MPa)Xt

(MPa)Sab(MPa)

υbc υba f0 (°)

T300/913 132 8.8 4.6 68 198 1,355 2,005 150 0.4 0.021 53

HTS40/977-2 112 8.168 4.4 – – 1,396 – 101 – 0.022 –


models into a developed finite element code are discussed for composite specimens undercompression.

The model dimensions used for numerical analyses are 10*10*2 mm and created with 16layers and each layer is modeled with one element through the thickness of laminates.

4.1 Material Properties

The material properties for T300/913 unidirectional laminated composite used in numericalanalysis are shown in Table 1 [2]. The used experimental in-plane shear stress versus strainin the numerical analysis has been shown in Fig. 12. This curve has been estimated bylogarithmic function, k1 ln k2g þ 1ð Þ with k1=46 and k2=100. The fracture angle for puretransverse compression is considered as f0=53°. Regarding the through-the-thicknessdirection the composite is assumed to be transversely isotropic. The intralaminar toughnessis considered as *b ¼ 0:22 kj=m2. The parameter Γkink is set to 79.9kj/m2 (for strain basedmodel (stage-II) Γkink is also determined by Eq. 63) and *L ¼ *T ¼ 1:1 kj=m2.

4.2 Fiber Compression Failure Modelling

The formation of kink bands at a small value of β angle, normal to the loading direction ispredicted by both stress based and strain based failure models. For the stress based modeland the strain based model in stage-I, the formation of kink band results from the effects ofdamage variable on the shear stresses in the kink band. However, formation of the kink

Table 2 Predicted compressive strength using the combined modes model and stress based modelconsidering various initial misalignment angles

θi (°) 5 3 2 1

XC (combined modes model) 461 672 854 1,191

XC (strain based model) 565 (instability) 792 (instability) 1,004 (instability) 1,454 (instability)

Fig. 11 In-plane shearstress-strain response of �45½ �2SHTS40/977-2 laminatedcomposite [15]


band for the strain based model in stage-II results from the effects of damage variable onthe shear stresses on the fracture surface which results from matrix failure and is parallel tothe misaligned fibers.

In this section, the obtained results from the performed analyses using the stress basedmodel and the proposed strain based model (stage-I and stage-II), are discussed andcompared with each others.

4.2.1 Results from Stress Based Model

For T300/913 composite laminate, the kink band angle was observed to be 25±5° forstandard axial compression specimens with in-plane kinking as shown in Fig. 13(a).

The obtained results using the stress based failure model from our developed programhave been depicted in Fig. 13(b), (c) and (d). In this model, updating the misalignmentframe after damage initiation causes poor results for the kink band angle. Therefore, theanalysis has been carried out without updating the misalignment frame after damageinitiation. The predicted kink band angle for the developed stress based failure model isbetween 15° to 18°; however Pinho reported it about 15° [2].

4.2.2 Results from the Strain Based Model (Stage-I)

The obtained results from the implementation of the strain based model (stage-I) into thefinite element code are shown in Fig. 14 for the above example. Updating the misalignment

Table 3 Predicted compressive strength using combined modes model and stress based model consideringvarious shearing properties

Shear properties tY ¼ 45

tult ¼ 85

mL ¼ 0:57

f0 ¼ 53

tY ¼ 45

tult ¼ 85

mL ¼ 0:57

f0 ¼ 53

tY ¼ 60

tult ¼ 115

mL ¼ 0:57

f0 ¼ 53

tY ¼ 68

tult ¼ 130

mL ¼ 0:57

f0 ¼ 53

XC (combined modes model) 1,051 1,191 1,330 1,471

XC (strain based model) 1,351 (instability) 1,454 (instability) 1,556 (instability) 1,648 (instability)

Fig. 12 Experimental in-planeshear stress versus straincurve [2]


frame is an easier task here, because the formulation for the strain based model is simplerthan the stress based one. Therefore, for the numerical analysis carried out using this model,the misalignment frame has been updated after damage initiation. Figure 13 shows that thepredicted kink band angle using this new approach is about 20°.

Due to the differences between the formulations of these two kinking failure models, thenumerical results are slightly different. For example, the failure load in the stress basedmodel at the initiation of failure mode is smaller than that in the strain based one. In theother words; the displacement in the loading direction which caused the initiation ofkinking process in the strain based model (stage-I) is larger than that obtained from thestress based model.

4.2.3 Results from the Strain Based Model (Stage-II)

The obtained results from the implementation of the strain based model (stage-II) have beenshown in Fig. 15 for the previous example. The obtained load causing fiber kinkinginitiation in this modeling is lower than that obtained from the stress based modeling andthe obtained kink band angle is larger than that. This is due to the considering of themisalignment of fibers to derive the stresses in both before and after damage initiation. Theobtained damage variable contours are also depicted in Fig. 13 which clearly shows that

Fig. 13 a Kink band in the failed longitudinal compression test specimen [2], b The initial kink band angleabout 18° for stress based kinking model, c propagation of the kink band, d kink band broadening with anangle about 15°

Fig. 14 Strain based model: a Initial kink band angle about 23°, b Propagation of kink band with angle ofabout 20°


the kink band angle is about 30°. The misalignment angle contours are also illustrated inFig. 16. In this figure, sudden rotations of fibers in the kink band are also observable.

5 Conclusions

In this study, an available stress based fiber-kinking model was improved on the bases ofstrain concept. The modification used in the stress based model omits the instability modewhich may cause errors in the results. The strain based model proposed in this research(stage-I) can also predict the kink band angle approximately. However, comparing with theoriginal stress based model it is simpler. Also with updating the misalignment frame afterdamage initiation, the results are more accurate than the results obtained from the stressbased model for the kink band angle but more elements shows distortion. It was also shownthat the predicted kink band using the developed strain based model (stage-II), is in goodagreement with the experimental results. Furthermore the obtained damage variablecontours of two methods are not completely similar and the kink band angles predicted bythem are slightly different. More experiments must be carried out to evaluate these twomodels.

Fig. 15 Strain based model (stage-II): a Initial kink band angle, b and c Propagation of kink band with angleof about 30°

Fig. 16 Misalignment angle contours for strain based model (stage-II): a Initial kink band angle, b and cPropagation of kink band with angle of about 30°


References

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An Improved Model for Fiber Kinking Analysis of Unidirectional Laminated Composites

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