fracture analysis of frp-reinforced beams by orthotropic xfem

Author Query Form Journal Title : Journal of Composite Materials (JCM) Article Number : 418702 Dear Author/Editor, Greetings, and thank you for publishing with SAGE. Your article has been copyedited and typeset, and we have a few queries for you. Please address these queries when you return your proof corrections. Thank you for your time and effort. Please ensure that you have obtained and enclosed all necessary permissions for the reproduction of artistic works, (e.g. illustrations, photographs, charts, maps, other visual material, etc.) not owned by yourself, and ensure that the Contribution contains no unlawful statements and does not infringe any rights of others, and agree to indemnify the Publisher, SAGE Publications Ltd, against any claims in respect of the above warranties and that you agree that the Conditions of Publication form part of the Publishing Agreement. Any colour figures have been incorporated for the on-line version only. Colour printing in the journal must be arranged with the Production Editor, please refer to the figure colour policy outlined in the e-mail.

Please assist us by clarifying the following queries:

1. Under ‘FRP-reinforced concrete cantilever beam subjected to an edge moment ‘,Please check the edit made in the sentence “The value of the relative … present XFEM analysis”.

2. Figures 1, 2, 3, 4, 7, 10, 12, 13, 16, 17, 18, 20, 23, 25, 27, 28, 30 are poor quality. 3. Please update Ref. [42]. 4. Please check the renumbering of Appendix equations.

Administrator

Stamp

Administrator

Rectangle

XML Template (2011) [5.8.2011–5:09pm] [1–23]{APPLICATION}JCM/JCM 418702.3d (JCM) [PREPRINTER stage]

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

Fracture analysis of FRP-reinforced beamsby orthotropic XFEM

S. Esna Ashari and S. Mohammadi

Abstract

The extended finite element method is adopted for fracture analysis of delamination problems in fiber-reinforced

polymer (FRP) reinforced beams. In this method, the stress singularities near the debonding crack tip are modeled by

newly proposed orthotropic bimaterial enrichment functions, while crack faces and the material interface simulated by

the Heaviside function and the weak discontinuity enrichment function, respectively. The orthotropic crack tip enrich-

ments are utilized in orthotropic FRP plate and in a reduced isotropic form in the beam. The strain energy release rate is

numerically evaluated using mixed-mode stress intensity factors which are evaluated by the means of domain interaction

integral approach. The accuracy of this approach is examined through some numerical examples and the results are

compared with available reference results.

Keywords

extended finite element method, delamination, debonding, fiber-reinforced materials, orthotropic enrichment functions,

strengthened beams

Introduction

Fiber-reinforced polymers (FRPs) are widely used inmany kinds of engineering applications such as struc-tural strengthening, seismic retrofitting, and repairapplications.1–4 FRP materials, especially in the formof laminates and pultruded plates, are becoming amaterial of choice in order to strengthen existing con-crete or steel structures due to their high strength andstiffness-to-weight ratio, resistance to corrosion andenvironmental effects, and substantial fatiguedurability.

As a flexural reinforcement for beams, FRP platesare externally bonded to the tension face of a concreteor steel beam, by means of an adhesive layer, whichsignificantly improves the stiffness and load capacityof beams.5,6 The performance of the FRP–beam inter-face in providing an effective mean for stress transfer isso important and the success of the strengthening pro-cedure depends mainly on the quality of the FRP–beamjoint, the efficiency of the adhesive, and its vulnerabilityto debonding failures.7–11 Debonding failures are usu-ally caused by high concentrations of normal and shearstresses at the end of the bonded plate, bonding defectsin the application of the FRP plate, or the effect of

impact loadings.12,13 Debonding failures are associatedwith the initiation and growth of horizontal cracks at orvery close to the adhesive–beam interface. Any kind ofdebonding failure may reduce the expected lifetimeextension and significantly weaken the structural integ-rity and may also damage the structural performanceand safety due to reduced ductility. In many cases, thedebonding causes a brittle and sudden fracture mecha-nism of the component.

There are generally two approaches to the simulationof debonding failures in FRP-strengthened structures,recognized as stress- and fracture mechanics-basedapproaches.14 The first one involves evaluation of inter-facial stress distribution in strengthened members basedon an elastic assumption and comparison of the com-puted stresses with the adhesive strength to predict theload level of debonding failure. As an example,

School of Civil Engineering, University of Tehran, Tehran, Iran.

Corresponding author:

S. Mohammadi, High Performance Computing Lab, School of Civil

Engineering, University of Tehran, Tehran, Iran.

Email: [email protected]

Journal of Composite Materials

0(0) 1–23

! The Author(s) 2011

Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/0021998311418702

jcm.sagepub.com


Roberts15 presented a general formulation for the anal-ysis of composite beamswith partial interaction and pro-posed some models for the interface shear stressconcentration near the ends of the epoxy-bonded exter-nal plates. Taljsten16 used the linear elastic theory,derived for shear and peeling stresses, to calculate thecritical stress levels at the end of an outer reinforcementplate.

The second approach refers to fracture mechanicsand is more appropriate to predict debonding failuresdue to the presence of stress singularities at the end ofplate/beam interface and the crack nature of thedebonding failures. Within the category of fracturemechanics models, two main approaches are recog-nized. The first approach is based on cohesive interfacemodeling and the second approach on linear elasticfracture mechanics (LEFM) crack growth simulation.The cohesive interface approach employs a layer ofinterface elements or a general contact interfacebetween the FRP and the beam in which debonding issimulated as failure of the interface elements.12,13,17–22

In the second approach, fracture mechanics param-eters, typically the energy release rate (ERR or G) orthe J integral, are evaluated analytically or numericallyand compared with the relevant critical fracture energy.Based on this concept, the initiation and propagation ofthe horizontal edge delamination crack occur if theamount of energy released by the system due to infini-tesimal crack growth is larger than the specific fractureenergy. Existing LEFM debonding analyses mainlydiffer in the analytical and computational tools adoptedfor the evaluation of fracture mechanics parameters.For example, Yang et al.23 adopted the finite elementmethod (FEM) and the virtual crack extension methodwhile Greco et al.24 used a layered shear deformationmodel for the stress analysis and the virtual crack clo-sure method for the assessment of ERR. Rabinovitchand Frostig25 used a high-order theory for the stressanalysis of the strengthened beam and the J integralformulation for evaluation of ERR. An alternativeapproach, based on simplified stress analysis toolsand assessment of ERR through numerical differentia-tion of the total potential energy, was proposed byRabinovitch.26

The common numerical method used for analyzingthese kinds of problems27–30 is the FEM, which is capa-ble of analysis of non-linear systems and can be easilyadapted to general boundary conditions and complexgeometries. Besides the advantages of the FEM, thismethod is unable to represent the exact singular stressfields near the crack tip. Moreover, in crack problemssuch as debonding analysis, the elements associatedwith cracks must conform to crack faces and remeshingtechniques may be required to simulate the crackpropagation.

The extended finite element method (XFEM),enhances the conventional FEM capabilities by avoid-ing the requirement of mesh to conform to weak orstrong discontinuities and the necessity of adaptiveremeshing techniques in crack growth problems. Thismethod was first proposed by Belytschko and Black,31

combining the FEM with the concept of partition ofunity. In XFEM, elements containing a crack areenriched with a discontinuous function and near-tipasymptotic displacement fields. Sukumar et al.32 devel-oped partition of unity-based enrichment techniquesfor bimaterial interface cracks between two isotropicmedia, while Remmers et al.33 presented a partition ofunity finite element (FE) based on a solid-like shell ele-ment for simulation of delamination growth in thinlayered structures. Nagashima and Suemasu34,35

applied XFEM to stress analyses of structures contain-ing interface cracks between dissimilar isotropic mate-rials and also modeled interlaminar delaminations ofcomposite laminates by the near-tip functions forhomogeneous isotropic cracks in order to examine thebehavior of orthotropic composite materials.

Asadpoure et al.,36,37 Asadpoure andMohammadi,38 and Mohammadi39 extended themethod to orthotropic media by deriving a new set oforthotropic enrichment functions. Later, Motamediand Mohammadi40,41 further extended the method todynamic analysis of stationary and propagating cracksin orthotropic media and proposed new time-indepen-dent moving orthotropic enrichment functions fordynamic analysis of composites.42 Recently, EsnaAshari and Mohammadi43–45 proposed new orthotro-pic interlaminar crack tip enrichment functions, whichextend the capability of XFEM to full fracture analysisof two-dimensional crack problems in isotropic andorthotropic multilayer media. The interlaminar cracktip enrichment functions were derived from analyticalasymptotic displacement fields around a traction-freeinterfacial crack.

In this research, XFEM is adopted for fracture anal-ysis of debonding problems in FRP-strengthenedbeams by means of recently proposed bimaterial ortho-tropic enrichment functions.45 The near-tip enrichmentfunctions represent the stress singularities near thedebonding crack tip. In all numerical simulations, thebeams are assumed as an isotropic material, whilethe FRP plate as an orthotropic material. Combinedmodes I and II loading conditions are studied and thestrain ERR is numerically evaluated using the mixed-mode stress intensity factors (SIFs). SIFs are deter-mined by the use of domain form of the contourinteraction integral. In order to examine the perfor-mance of the proposed approach, various numericalexamples are simulated and the results compared withavailable reference solutions.

2 Journal of Composite Materials 0(0)


Mechanics of bimaterial interface cracks

For modeling an interface crack between FRP (ortho-tropic material) and concrete or metal (isotropic mate-rial) in the XFEM, proper enrichments, extracted fromthe near crack tip displacement fields, are required.Here, the analytical near crack tip displacement fieldsof a traction-free crack between two orthotropic mate-rials, derived by Lee,46 are used. These displacementfields can be utilized in both orthotropic and isotropicmaterials.

Figure 1 indicates a traction-free interfacecrack located in (x,y) plane between two orthotropicmaterials. The elastic principal directions areassumed normal to or parallel with the crack directionand the local polar coordinates r,�ð Þ defined on thecrack tip.

For each orthotropic material, a partial differentialequation in terms of a complex variable z ¼ xþmy canbe obtained with the following characteristicequation:47

m4 þ 2B12m2 þ K66 ¼ 0 ð1Þ

where

B12 ¼1

2

2a12 þ a66ð Þ

a11, K66 ¼

a22a11

ð2Þ

aij i,j ¼ 1,2,6ð Þ are the material parameters used in thestress–strain law:48

"� ¼ a�� ,� ¼ 1,2,6ð Þ ð3Þ

The characteristic roots of Equation (1) (m1 and m2)in most orthotropic materials, known as type 1, arepurely imaginary:

m1 ¼ ip, m2 ¼ iq ð4Þ

where

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB12 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB212 � K66

qrð5Þ

q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB12 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB212 � K66

qrð6Þ

The asymptotic displacement fields in the vicinity ofthe orthotropic biomaterial crack tip can be defined interms of p, q and the polar coordinate system r,�ð Þ.46

They are described comprehensively in Appendix.The functions that define the nature of the analytical

displacement fields will be added to the FEM

Figure 1. A typical interface crack located between two

orthotropic materials.

Figure 2. Node selection for enrichment. Nodes enriched with crack tip, Heaviside, and weak discontinuity functions are marked by

triangles, circles, and squares, respectively.

Ashari and Mohammadi 3

divya.t

2


approximation as the XFEM enrichment functions inorder to represent the singular crack tip stress fields.

In an isotropic material, values of B12 and K66 areequal to 1 and Equation (1) is reduced to:

m4 þ 2m2 þ 1 ¼ 0 ð7Þ

So, the roots are:

m1 ¼ i, m2 ¼ i ð8Þ

and

p ¼ q ¼ 1 ð9Þ

Consequently, all the displacement and stress fieldsand the enrichment functions are simplified in the iso-tropic material.

Extended finite element method

The XFEM is a numerical method which allows for themodeling the arbitrary strong and weak discontinuities,independent of the mesh discretization. XFEM uses thepartition of unity property to define an enriched solu-tion in order to reproduce discontinuous or singularfields. In this method, the mesh does not require toconform to the discontinuities nor significant meshrefinement is necessary near the singularities. A singleXFEM element, without the loss of accuracy, may con-tain two different materials and an interface in between.

XFEM for interface crack modeling

For an interface crack located in a domain within anFE mesh, shown in Figure 2, the XFEM displacementapproximation around the crack, uh, is enriched withthe discontinuous Heaviside function to model cracksurfaces, the crack tip asymptotic displacement fieldsto model crack tips, and the weak discontinuity enrich-ment function used to model the interface of two mate-rials. It can be expressed in th following form:

uhðxÞ¼XI

nI2N

NIðxÞuIþXJ

nJ2NH

aJNJðxÞ HðxÞ�H xj� ��

þXk

nk2NF

NkðxÞXl

blk Fl xð Þ�Fl xkð Þð Þ

!

þXr

nr¼N�

crNr xð Þ�r xð Þ

ð10Þ

The first term is the classical FE approximationwhere NI is the FE shape function and uI the vectorof regular degrees of nodal freedom.

The other terms are the enriched approximation,where NH, NF, and N� stand for the set of nodes thathave crack face, crack tip, and weak discontinuity intheir support domain, respectively.

aJ, blk and cr denote the vectors of additional degrees

of nodal freedom for modeling crack faces, crack tips,and weak discontinuity interfaces, respectively.

Crack face enrichment

H xð Þ is the Heaviside function that simulates the dis-continuity of displacement within an FE at the cracklocation:

HðxÞ ¼þ1 if ’ xð Þ4 0

�1 if ’ xð Þ5 0

(

where � xð Þ is the signed distance function computed ina normal direction to the crack side and takes the valueþ1 if the point x is above the crack and –1, otherwise.39

Material interface enrichment

� xð Þ is the enrichment function used for modeling weak(material) discontinuities within an FE and is defined interms of the signed distance function as:

�r xð Þ ¼ � xð Þ�� ð11Þ

This enrichment is used where a continuous displace-ment field exists but the strain fields are discontinuous,such as the interface of two different materials. Thisenrichment function is used only for the elements thatare cut by the interface, avoiding generation of exces-sive errors in surrounding elements.45 If the mesh con-forms to the material interface, no nodes are enrichedby the weak discontinuity function.

On an element cut by the crack or interface, the inte-gration of XFEM stiffness matrix and force vectorneeds to be performed carefully. An efficient way forthis procedure is to divide the element into sub-domainsmatching the crack faces or material interface. Then,the number of integration points over each sub-domainis chosen accordingly.39

Crack tip enrichment functions

The following basic crack tip enrichment functions thatspan the crack tip asymptotic displacement field



(Equation (A1)) are used in this study:45

Fl r,�ð Þ� �8

l¼1

¼ e�"�l cos " ln rlð Þþ�l2

� ffiffiffirlp

, e�"�l sin " ln rlð Þþ�l2

� ffiffiffirlp

,

e"�l cos " ln rlð Þ��l2

� ffiffiffirlp

, e"�l sin " ln rlð Þ��l2

� ffiffiffirlp

,

e�"�s cos " ln rsð Þþ�s2

� ffiffiffiffirsp

, e�"�s sin " ln rsð Þþ�s2

� ffiffiffiffirsp

,

e"�s cos " ln rsð Þ��s2

� ffiffiffiffirsp

, e"�s sin " ln rsð Þ��s2

� ffiffiffiffirsp

�ð12Þ

with

rj ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 � þ Z2

j sin2 �

q�j ¼ tan�1 Zj tan �

� �8><>: , j ¼ l,s,

Zl ¼ pZs ¼ q

�ð13Þ

where " (the oscillation index) and all parameters havebeen defined in Appendix. It should be noted that in theisotropic material rj ¼ r and �j ¼ �.

It is worth noting that these set of functions containcombined typical terms of

ffiffiffirjp

cos �j=2� �

orffiffirp

cos2 �j þ Z2 sin2 �j �1

4cos �j=2� �

. These include all gen-eralized terms of sin �j, sin �j sin �j=2

� �, sin �j cos �j=2

� �,. . .

and may generally span all various components oforthotropic material and isotropic bi-material enrich-ment functions, as proposed by Sukumar.32,45

Evaluation of ERR

The value of ERR is computed from the SIFs:49

G ¼1

4 cosh2 �"ð ÞD22K

21 þD11K

22 þ 2D12K

21K

22

�ð14Þ

D ¼ L�1#1 þ L�1#2 , where L is the Barnett and Lothetensor, defined in Equation (20).50 K1 and K2 are themodes I and II SIFs. These are evaluated using thedomain integral method, developed by Chow et al.51

Evaluation of different SIFs, in general, mixed-modeloadings is performed by the interaction integral M,

M ¼

ZA

�ijuauxi,1 þ �

auxij ui,1 �W 1,2ð Þ1j

h iq,jdA ð15Þ

where superscript aux stands for the auxiliary state, Athe region including the crack tip confined by � (Figure3), and q a smooth function that is unity inside thedomain A and zero outside the domain. Therefore,

the integrand of (15) vanishes for all the elements thatlie completely inside the domain (because of constant qand zero q,j), and it can simply be evaluated over theelements that are cut by the contour � (non-zero q,j).Wð1,2Þ for linear elastic condition is defined as

W 1,2ð Þ ¼1

2�ij"

auxij þ �

auxij "ij

� �ð16Þ

In the interaction integral method, auxiliary fieldsare superimposed onto the actual fields to satisfy equi-librium equation and traction-free boundary conditionon crack surfaces.52 One of the choices for the auxiliarystate is the displacement and stress fields in the vicinityof the interfacial crack tip.46

The SIFs are then determined from the M integral:51

ki ¼ 2X2m¼1

UimM u,uaux mð Þ� �

i ¼ 1,2ð Þ ð17Þ

where uaux 1ð Þ and uaux 2ð Þ are the auxiliary states associ-ated with K1 ¼ 1, K2 ¼ 0 and K1 ¼ 0, K2 ¼ 1, respec-tively, and

U ¼� �

L�1#1 þ L�1#2

��1 Iþ

�L�1#1 þ L�1#2

�2�S#1L

�1#1 � S#2L

�1#2

�2��1 ð18Þ

Components of S and L are computed for each indi-vidual layer, denoted by #1 and #2 for upper and lowermaterials, respectively. Explicit expressions of S and Lfor an orthotropic material have been derived byDongye and Ting53 and Deng,54

S21¼C66

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC11C22

p�C12

� �C22 C12þ2C66þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC11C22

p� �" #1

2

, S12¼�

ffiffiffiffiffiffiffiffiC22

C11

rS21

ð19Þ

Figure 3. The contour � containing the crack tip.



L11 ¼ C12 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC11C22

p� �S21, L22 ¼

ffiffiffiffiffiffiffiffiC22

C11

rL11 ð20Þ

and Cij is the contracted form of the fourth-order elas-tic constant tensor.

Numerical examples

In all numerical examples, the ERR is obtained withthe method mentioned in the previous section. For theintegration purposes, a 2� 2 Gauss quadrature rule isused over regular elements, while the quadrature parti-tioning approach adopted over enriched elements. Anyelement cut by the crack faces or the material interfaceis divided into four sub-triangles and the elements con-taining the crack tip are partitioned into six sub-trian-gles. Seven Gauss points are utilized for integrations ineach sub-triangle. Elements next to the main tip-enriched element need not any sub-division, and asimple 5� 5 Gauss quadrature rule is utilized.

Concrete beams strengthened with externallybonded glass fiber-reinforced polymer

In this example, the flexural behavior of a reinforcedconcrete beam strengthened by epoxy-bonded glass-fiber reinforced plate is investigated by the XFEM.The concrete beam is under the four-point bendingtest while reinforced by glass fiber-reinforced polymer(GFRP) bonded to the tension face, as shown inFigure 4. The plane stress state is assumed and thematerial properties are defined as:

Concrete.

E ¼ 21,000 MPa, ¼ 0:2:

GFRP.

E1 ¼ 37,230 MPa, E2 ¼ 1861 MPa,

G12 ¼ 1618MPa, 12 ¼ 0:27:

The beam dimensions are:

b ¼ 0:2 m, h ¼ 0:3 m, L ¼ 3 m

Two different thicknesses of the GFRP platetp ¼ 2,4 mm� �

are considered.A uniform structured FE model with 1386 quadri-

lateral four-noded elements and 1500 nodes isemployed. Figure 5 depicts the FEM mesh. It shouldbe noted that only one row of elements contains theGFRP laminate and part of the concrete at once.These elements are enriched by the weak discontinuityfunction.

The analysis is performed for different applied load-ings within the elastic condition and the mid-spandeflection is measured in each case and the results arecompared with the results reported in Hsu55 study.Hsu55 employed non-linear models of the FE codeABAQUS for simulating the behavior of strengthenedconcrete beams.

The variation of mid-span deflection as a function ofapplied load in elastic zone is plotted in Figure 6 fordifferent GFRP plate thicknesses in comparison withthe reference results. A very close agreement is observedbetween XFEM and reference curves in the linear elas-tic part, indicating the accuracy of XFEM approach inmodeling FRP, part of concrete, and the material inter-face in a single element.

FRP-reinforced concrete cantilever beam subjected toan edge moment

In this example, analysis of an edge debonding problemin concrete beams strengthened/repaired with

Figure 4. Geometry of the reinforced concrete beam strengthened by a GFRP plate.



externally bonded composite laminated plates, asshown in Figure 7, is considered. A bilayer specimencomposed of two homogeneous elastic layers with aninterface crack is simulated by present XFEM and theresults are compared with those obtained by Grecoet al.24 The first layer is isotropic and represents theconcrete beam to be strengthened and the other onethe orthotropic layer of typical carbon–epoxy unidirec-tional lamina. The beam is subjected to an edgemoment which is modeled by means of two oppositeuniform distributions of normal pressure acting at thex ¼ Lc edge with the plane stress condition. The mate-rial properties are defined as:

Concrete.

E ¼ 30,000 MPa, G12 ¼ 12,820:53 MPa:

CFRP.E1 ¼ 160,000 MPa, E2 ¼ 11,860 MPa,

G12 ¼ 5333:3 MPa, 12 ¼ 0:3:

The dimensions of the problem are:

Lc ¼ 1500 mm, Lp ¼ 1200 mm, tc ¼ 300 mm, tp ¼ 4 mm:

In order to evaluate the value of ERR, adaptivestructured FE models with different numbers of quad-rilateral four-noded elements are employed for differentcrack lengths. Element sizes are substantially smaller inthe vicinity of the crack, and the tip element sizes inx- and y-directions are 4 and 1.07mm, respectively,while the FRP laminate is discretized by 3.5 elementsin y-direction; so, the top element includes FRP, con-crete, and interface. The total number of rows and col-umns of FEs are 82 and 134, respectively, resulting in6834 elements and 7020 nodes. Figure 8 illustrates thetypical mesh utilized for a ¼ 200 mm.

The value of the relative J-integration domain size,rd (the distance between the crack tip and each edge ofthe integration domain), is assumed to be different forCFRP strip below the crack edge (about the thicknessof FRP plate) and about 27mm for other three edges(Figure 9) in the present XFEM analysis.

Greco et al.24 evaluated the strain ERR from a pro-posed analytical method and an FEM using a meshcharacterized with the crack tip element size of 1mmin the x-direction. The FE mesh, obtained by the com-mercial FE code LUSAS, is illustrated in Figure 10,where the mesh refinement in a region enclosing thecrack tip is also shown. It should be noted that theadhesive layer was modeled and delamination assumed

Figure 5. FEM discretization: (a) complete model and (b) discretization around the GFRP layer.

0

20

40

60

80

100

120

140

160

180(a) (b)

0 10 20 30 40 50 60 70

Loa

d (k

N)

Central deflection (mm)

XFEM

FEM [55]

Experiment [55]

0

50

100

150

200

250

0 10 20 30 40 50 60

Loa

d (k

N)

Central deflection (mm)

XFEM

FEM [55]

Experiment [55]

Figure 6. Central deflection of the concrete beam strengthened by GFRP plate: (a) tp ¼ 2 mm and (b) tp ¼ 4 mm.


divya.t

1


Figure 7. The geometry and boundary conditions of an interface crack in FRP-plated cantilever beam under bending.

Figure 8. FEM discretization of FRP-reinforced cantilever beam.

Figure 9. The rectangular J integral domain.



to occur at the interface between beam and the adhesivelayer. The FEs were 1mm long and 1mm high in theFRP and adhesive layers, and 160 elements were usedacross the thickness of concrete beam; including 80 ele-ments with a constant height in the upper half of thevertical section, and a graded mesh starting with anaspect ratio of 1 near the crack tip in the lower half.In the rest of the model, a uniform mesh was utilized.The total number of elements and nodes in Grecoet al.24 model were about 40,720 and 40,979,respectively.

The results of strain ERRs as a function of the cracklength, obtained by XFEM, are depicted in Figure 11and compared with reference results.24 This figure indi-cates that the total ERR remains practically constantfor most of the crack lengths and then decreases toward0 rapidly as the crack reaches the beam end.

Very close agreements are observed between theXFEM and available reference analytical and numeri-cal results, while the number of elements in the XFEMmesh is about 1=6th the number of elements of meshutilized in Greco et al.24 It should be noted that the

Figure 10. FEM discretization in Greco et al.24 report.

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000 1200

GE

ctc3 /

M2

a(mm)

XFEM

FEM [24]

Analytical [24]

Figure 11. Variation of strain ERRs with crack length.



XFEM results are closer to the analytical results thanthe reference two-layer FE model as the crack reachesthe beam end.

FRP-reinforced cantilever beam subjected to an edgetransverse force

A cantilever beam specimen subjected to an edge trans-verse force, as illustrated in Figure 12, is considered.

Again, the first layer is (isotropic) concrete and thesecond layer the (orthotropic) CFRP plate used tostrengthen the concrete beam. The geometry of the spe-cimen is the same as previous problem and the materialproperties are defined as:

Concrete.

E ¼ 30,000 MPa, G12 ¼ 12,820:53 MPa:

Figure 12. FRP-strengthened concrete cantilever beam subjected to an edge transverse force.

Figure 13. FE mesh employed for a ¼ 800 mm.



CFRP strips.

E1 ¼ 160,000 MPa, E2 ¼ 7610 MPa,

G12 ¼ 5333:3 MPa, 12 ¼ 0:3:

The analysis is performed for the bimaterial plate ina plane stress state, subjected to the edge transverseforce and for a range of crack lengths. Again, adaptivestructured FE models similar to example 2 are used fordifferent crack lengths and the solution around thecrack tip is enhanced with new orthotropic enrichmentfunctions. The mesh discretization used fora ¼ 800 mm is illustrated in Figure 13. The FRP lam-inate is discretized into 4.5 elements in y-direction andthe tip element sizes in x- and y-directions are 4 and0.91mm, respectively, with the total number of 8308elements and 8505 nodes.

The same relative J integral domain size as example2 is used to evaluate the strain ERRs. The variation ofstrain ERRs with respect to the crack length is depictedin Figure 14.

The results of XFEM simulations are compared withthe results reported by Greco et al.24 using an analyticalformulation and an FE model (similar to Figure 10) inFigure 14.

Clearly, very accurate results are obtained by newXFEM formulation. Similar to the case of the bending

moment, the presence of constraint at the left end pro-vides a mechanism for crack arrest with a decrease inthe ERR value (at about 0.92% of the strengthenedregion).24

The effect of partitioning in the enriched elements isnow investigated for the crack length of 400mm. Forthe sake of comparison, 8� 8, 6� 6, and 4� 4 Gaussquadrature rules are used in all enriched elements. Theresults are depicted in Table 1. It is observed that usingthe results of partitioning method is more accurate thanordinary Gauss quadrature rules.

A parametric study for the value of strain ERRs isnow presented by varying the CFRP plate thicknesswith all other parameters fixed. The results are illus-trated in Figure 15 and compared with the analyti-cal results obtained by Greco et al.,24 which show agood agreement. The analysis points out that anincrease in plate thickness leads to an increase instrain ERRs.

The effect of using tip enrichments is examined bycomparing the variations of stress components on thecrack tip element with and without tip enrichments.Figure 16 compares the stress fields around the cracktip when a¼ 800mm. Clearly, only using tip enrich-ment functions, the singular nature of stress field nearthe crack tip can be accurately represented.

Simply supported FRP-reinforced beam under three-point bending crack stability and propagation

This example is dedicated to the analysis of an interfa-cial crack between an FRP laminate and a concretebeam, previously studied by Greco et al.24 An interfa-cial crack at the edge of a simply supported beam,reinforced by FRP laminate and subjected to three-point bending is considered. The plane stress state ispresumed. The geometry and boundary conditions ofthe problem are indicated in Figure 17 and the materialproperties are:

Concrete.

E ¼ 30,000 MPa, G12 ¼ 12,820:53 MPa:

0

7.5

15

22.5

30

0 200 400 600 800 1000 1200

GE

ctc3 /

M2

a(mm)

XFEM

FEM [24]

Analytical [24]

Figure 14. Variations of the strain ERRs with respect to crack

length.

Table 1. Values of GEct3c=F

2 using different integration methods a ¼ 400 mmð Þ

a

XFEM

Reference valuesWithout partitioning

With partitioning 8� 8 6� 6 4� 4 FEM Analytical

400 6.00 6.31 6.27 6.34 6.33 5.97



CFRP.

E1 ¼ 160,000 MPa, E2 ¼ 9260 MPa,

G12 ¼ 5333:3 MPa, 12 ¼ 0:3:

The dimensions of the problem are:

Lc ¼ 3000 mm, Lp ¼ 2400 mm, tc ¼ 300 mm, tp ¼ 4 mm:

The FE mesh of Figure 18 is used to simulate theproblem and the element which contains the crack tip ismodeled with the new orthotropic enrichment func-tions. Again, the FRP laminate is discretized into 4.5

elements in y-direction. The total number of elementsand nodes are 10,988 and 11,205, respectively.

Again, the same relative J integral domain size asexample 2 is used to evaluate the strain ERRs of thepresent debonding problem.

Figure 19 illustrates the results of XFEM simula-tions for different crack lengths based on the neworthotropic enrichment functions.

Greco et al.24 performed a similar study on thisproblem. Figure 19 shows the results of analytical andnumerical values of strain ERRs obtained by Grecoet al.24 The numerical results were obtained by utilizinga mesh with the crack tip element size of 1mm in the

0

10

20

30

40

50

60

70

80

0 200 400 600 800 1000 1200

GE

ctc3 /

M2

a(mm)

Analytical-tp=4 [24]

Analytical-tp=16 [24]

XFEM-tp=4

XFEM-tp=16

Figure 15. Influence of the FRP plate thickness tp� �

on the strain ERR.

Figure 16. Variation of stress fields around the crack tip (along A–B): (a) with crack tip enrichments and (b) without crack tip

enrichments. The crack tip is located at (0,0.8) in the local curvilinear coordinates of the tip element.



x-direction and about 78,624 elements and 79,173nodes.24

It can be observed that the XFEM numerical esti-mates of strain ERRs are in close agreement with theanalytical solution reported by Greco et al.24 It shouldbe noted that these results are closer to analytical solu-tion than the reference FE results while the number ofelements in the XFEM mesh is about 1=7th of the ref-erence FE mesh.24

Moreover, a crack propagation problem is idealizedin this problem to illustrate the efficiency of the presentXFEM for simulation of several crack lengths on afixed FE mesh. As a result of having the interlaminarcrack to extend only along the material interface, noparticular propagation criterion is necessary in examin-ing this characteristic of present XFEM, and instead,different crack lengths are assumed to be generatedthrough the propagation process. The results of ERR,which are evaluated on the fixed FE model for differentcrack lengths, are shown in Table 2, and compared withthe previous results and the approximate reference solu-tions, showing very good quality results for all cases.

So far, the adhesive layer has been neglected andreasonably accurate results have been obtained. Butthis cannot be generalized to all problems. In someproblems, it may be more feasible to separately simu-late the adhesive layer (which can be performed by thepresent XFEM approach). In order to demonstratesuch a capability and to verify the mentioned

assumptions, this example is re-analyzed by a three-layer XFEM model, which includes FRP, the adhesivelayer (for instance with the thickness of ta¼ 2), and theconcrete beam. The results, shown in Figure 19, dem-onstrate the little effect of inclusion of adhesive layer asa separate layer on the overall response for differentcrack lengths.

It should be re-emphasized, however, that such aconclusion cannot be generalized to all problems, andrequires an independent study to assess the possibleeffects. It seems that a very thick adhesive layer isbetter to be simulated independent of the FRP lami-nate. This is the case for any potential adhesive layerwith sufficient bending stiffness. On the contrary, thinadhesives can be simulated in combination with FRP as

Figure 17. Simply supported beam reinforced by FRP laminate under three-point bending.

Figure 18. Near-tip structured FE discretization for a ¼ 400 mm.

0

1

2

3

4

5

6

7

0 200 400 600 800 1000 1200

GE

ctc3 /

M2

a (mm)

Analytical [24]

FEM [24]

2 layers XFEM

3 layers XFEM with ta=2

Figure 19. Results of XFEM simulations for debonding in

simply supported beam reinforced by FRP laminate.



a single layer or can even be neglected, while sufficientlyaccurate results are expected to be obtained for fractureparameters such as the ERR G.

FRP-strengthened concrete beam

To further assess the performance of the proposedapproach, an FRP-strengthened concrete beam,

previously studied by Rabinovitch26 is considered(Figure 20).

The strengthened beam is composed of concrete, anisotropic material, with the following properties:

Concrete.

E ¼ 30 GPa, ¼ 0:3:

Table 2. Comparison of GEct3c =F

2 values for different crack lengths on a fixed mesh

a (mm)

XFEM Reference

Adapted mesh Fixed mesh Analytical FEM

600 2.4295 2.4301 2.41 2.59

400 1.4654 1.5164 1.5 1.64

300 1.0835 1.1965 1.07 1.21

200 0.7928 0.8471 0.79 0.90

Figure 20. Geometry, loading, and boundary conditions.26

Figure 21. Near-tip adaptively structured XFEM mesh for a ¼ 150 mm.



and CFRP which is orthotropic with the followingcomponents of material modulus:

CFRP.

E1¼158GPa,E2¼32:7GPa,G12¼5:2GPa, 12¼0:27:

The crack is aligned along the interface of FRPstrips and concrete beam. The values of strain ERRsare evaluated based on the adaptively structured mesh,as shown in Figure 21. The FRP laminate is discretizedinto 3.5 elements through the thickness. Also, the totalnumber of elements and nodes in the FRP laminate areabout 536 and 675, respectively.

Again a rectangular integration domain is utilized inorder to evaluate the strain ERRs. The value of therelative J-integration domain size, rd (the distancebetween the crack tip and each edge of the integrationdomain), is assumed to be about the thickness of CFRPstrip below the crack edge and about 15mm for otherthree edges.

To determine the accuracy of the approach, compar-isons are made with the numerical investigation ofRabinovitch,26 which was based on the virtual crackextension method and using the high-order model and

Jintegral. The reference FE model, composed ofbi-linear four node elements, was utilized with about7000 quadrilateral elements in the CFRP strip with8008 nodes.

Figure 22 represents the results of XFEM simula-tions as a function of crack length, which are closelycomparable with the results of strain ERRs obtained bythe J integral method in Rabinovitch26 report.

Table 3 indicates the results of strain ERR by chang-ing the size of relative integration domain, rd (the dis-tance between the crack tip and each edge of theintegration domain). It is assumed that rd is aboutthe thickness of CFRP strip below the crack edge andthe other three edges differ.

The results indicate that the overall differencesbetween XFEM and the reference results vary between0.05% and 5.5%. Since the reference results cannot beregarded as the accurate and exact solution, while ageneral conclusion cannot be made to optimallyassign a particular rd for each crack length in this exam-ple, any values of rd in Table 3 can be used to obtain anacceptable engineering result. It should be noted thatthe path independency of the interaction integral is onlya pure analytical hypothesis, which may slightly be vio-lated in numerical analyses due to their own extraassumptions and approximations.

Delamination of metallic I beams strengthenedby FRP strips

In this example, a straight simply supported I shapedsteel beam with CFRP reinforcement subjected to apoint loading, as shown in Figure 23, is studied. Twodebonding cracks are located symmetrically at theinterface of the steel beam and the CFRP strips at thetwo ends (Figure 23). Due to the reinforcement symme-try, the analysis is performed for half of the beam in aplane stress state for a range of crack lengths.

0

50

100

150

200

250

300

350

400

450

0 50 100 150 200 250

G (J

/m2 )

a (mm)

XFEM

J integral [25]

Figure 22. Strain ERRs vs. crack length curves.

Table 3. The effect of different sizes of the relative J integral domain (rd) on the values of ERR for different crack lengths

a (mm)

The distance between the crack tip and three edges of the J integral domain (mm)Reference

results2610 12 15 17 20

50 89.13 90.21 90.96 90.95 91.50 90.91

100 141.70 143.32 144.42 144.41 145.20 145.45

200 309.40 312.67 314.84 314.83 316.37 300

Relative differences from the reference values (%)

2 0.76 0.05 0.05 0.6

2.6 1.5 0.7 0.7 0.17

3.1 4.2 5 5 5.5



The results of the present simulation are compared withthe results of Colombi.29

Material properties are defined as:

Steel.

E ¼ 210 GPa, 12 ¼ 0:3:

CFRP strips.

E1 ¼ 197 GPa, E2 ¼ 15:8 GPa,

G12 ¼ 8:18 GPa, 12 ¼ 0:3:

Adaptive structured FE meshes with differentnumber of quadrilateral four-noded elements for differ-ent crack lengths a ¼ 5� 160 mmð Þ are used in thisexample. The mesh is fixed and composed of 266 quad-rilateral elements and 402 nodes in the CFRP strip.

Figure 24 illustrates the typical mesh utilized fora ¼ 100 mm.

In these models, element sizes are substantially smal-ler in the vicinity of the crack, while the FRP laminateis discretized into 1.5 elements in y-direction. The totalnumber of rows and columns of FEs are 81 and 134,respectively.

The value of the relative J-integration domain size,rd (the distance between the crack tip and each edge ofthe integration domain), is assumed to be about thethickness of CFRP strip below the crack edge andabout 25mm for other three edges.

The results of the XFEM simulations for differentvalues of the initial delamination length are comparedwith the results of Colombi’s29 study. Colombi29 eval-uated the strain ERRs using both analytical andnumerical solutions. His results from the transformedsection approach were evaluated analytically, while the

Figure 23. Geometry and cross-section of the metallic beam.29

Figure 24. The near-tip FE mesh used for a ¼ 100 mm.



Figure 25. Details of the reference two-dimensional FE model of the reinforced beam.29

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140 160

G (J

/m2 )

a (mm)

XFEM

Trans. Sec. [29]

Elast. Found. [29]

FEM [29]

Figure 26. Strain ERR vs. the delamination crack length.



ERR from the two-parameter elastic foundation modelwas evaluated numerically using a virtual crack lengthequal to 1mm. The model was constructed by a one-dimensional FE for the beam and a two-dimensionalshell FE model was adopted for the composite stripsand adhesive layer, as depicted in Figure 25. Thenumber of elements and nodes in the CFRP stripwere about 3960 and 4955, respectively.

The XFEM results and reference results29 are illus-trated in Figure 26. Comparisons between the results of

all three methods indicate a very good agreementbetween the proposed model and the reference results.

Variable section beam reinforced by FRP

In this example, a cantilever, variable section beam issubjected to an edge transverse force F ¼ 30 Nð Þ. Thegeometry of the problem is shown in Figure 27. Thebeam is composed of concrete and reinforced withorthotropic CFRP plates. The material properties aredefined as:

Concrete.

E ¼ 30,000 MPa, G12 ¼ 12,820:53 MPa:

CFRP strips.

E1 ¼ 160,000 MPa, E2 ¼ 7610 MPa,

G12 ¼ 5333:3 MPa, 12 ¼ 0:3:

The XFEM analysis is performed for a range ofcrack lengths in a plane stress state in order to evaluatethe strain ERRs. An unstructured FE model, depictedin Figure 28, is used for different crack lengths which

Figure 27. A cantilever variable section beam subjected to an edge transverse force.

Figure 28. FEM discretization for the analysis of variable section beam a ¼ 900 mmð Þ.

0

1

2

3

4

5

6

7

8

500 700 900 1100 1300

GE

ctc3 /

F2

a(mm)

Figure 29. Variation of strain ERR as a function of crack length.



Figure 30. X and Y displacements and �xx , �yy , and �xy contours.



represent a crack propagation problem. The crack tip islocated in the finer mesh area and modeled with theproposed orthotropic enrichment functions. The meshdiscretization is composed of 1152 four-noded quadri-lateral elements and 1232 nodes. This is in fact a test todemonstrate the ability of the proposed approach withrather complex debonding problems.

Figure 29 depicts the results of GTE2t2=F2 obtained

by XFEM analysis for different crack lengths. A rapiddrop in GT is predicted at a ¼ 1100 mm, where theinterlaminar crack extends to the vicinity of clampededge.

Displacement and stress contours of the presentexample for a ¼ 900 mm are plotted in Figure 30.

Conclusions

An XFEM model for debonding failure analysis ofbeams strengthened with externally bonded FRPplates is proposed. Bimaterial orthotropic enrichmentfunctions, the Heaviside function, and weak disconti-nuity enrichment functions are utilized in XFEM anal-ysis to simulate the interface crack between two linearelastic isotropic (beam) and orthotropic (FRP plate)materials and fracture analysis of the domain.Combined modes I and II loading conditions are con-sidered and the mixed-mode SIFs numerically evalu-ated using the domain interaction integral approach.The strain ERR is evaluated based on the SIFs.Results obtained by the XFEM approach comparedwith predictions from various reference investigationsoutline the capability of the proposed model to predictthe debonding failure behaviour. The computed ERRsexhibit a satisfactory estimate with the available refer-ence results. In addition to simulation of several cracklengths (as a representation of crack propagation) on afixed FE mesh, effect of different types and sizes ofmesh and different relative J integral domain sizes hasbeen studied comprehensively to investigate the effi-ciency and robustness of the present method.

The present XFEM approach is capable of modelingany number of layers. The necessity of modeling theadhesive layer depends mainly on the general modelingconcepts, the required accuracy, and the computationalcosts. While we may choose to model the adhesive layeras a separate layer or an equivalent layer ofFRPþ adhesive, the obtained results demonstrate suf-ficient accuracy even if the adhesive layer is neglected.It should be emphasized, however, that such a conclu-sion cannot be generalized to all problems; as it may bemore feasible to separately simulate the adhesive layer(which can be performed by the present XFEMapproach). This requires an independent study toassess the possible effects. It seems that a very thickadhesive layer is better to be simulated independent

of the FRP laminate. This is the case for any potentialadhesive layer with sufficient bending stiffness. On thecontrary, thin adhesives can be simulated in combina-tion with FRP as a single layer or can even beneglected, while sufficiently accurate results areexpected to be obtained for fracture parameters suchas the ERR G.

Acknowledgments

The authors acknowledge the financial support of Universityof Tehran for this research under the grant number 8102051/1/02. Also, the technical support of the High Performance

Computing Lab, School of Civil Engineering, University ofTehran is gratefully acknowledged.

References

1. Meier U. Strengthening of structures using carbon fibre/

epoxy composites. Constr Build Mater 1995; 9(6):

341–351.2. Rizkalla S, Hassan T and Hassan N. Design recommen-

dations for the use of FRP for reinforcement and

strengthening of concrete structures. Prog Struct Eng

Mater 2003; 5(1): 16–28.

3. Lau K-T and Zhou L-M. Mechanical performance of

composite-strengthened concrete structures. Composites

Part B 2001; 32: 21–31.4. Teng JG, Chen JF, Smith ST and Lam L. FRP-strength-

ened RC structures. Chichester: John Wiley & Sons, 2002.5. Ferretti D and Savoia M. Cracking evolution in R/C ten-

sile members strengthened by FRP-plates. Eng Fract

Mech 2003; 70: 1069–1083.

6. Ramana VPV, Kant T, Morton SE, Dutta PK,

Mukherjee A and Desai YM. Behavior of CFRPC

strengthened reinforced concrete beams with varying

degrees of strengthening. Composites Part B 2000; 31:

461–470.7. Sebastian WM. Significance of mid-span debonding fail-

ure in FRP-plated RC beams. ASCE J Struct Eng 2001;

127(7): 792–798.

8. Mohamed Ali MS, Oehlers DJ and Bradford MA.

Interaction between flexure and shear on the debonding

of RC beams retrofitted with compression face plates.

Adv Struct Eng 2002; 5(4): 4223–4230.9. Chen JF and Teng JG. Shear capacity of FRP-strength-

ened RC beams: FRP debonding. Constr Build Mater

2003; 17(1): 27–41.10. Smith ST and Teng JG. Shear-bending interaction in

debonding failures of FRP-plated RC beams. Adv

Struct Eng 2003; 6(3): 183–199.

11. Yang ZJ, Chen JF and Proverbs D. Finite element model-

ling of concrete cover separation failure in FRP plated

RC beams. Constr Build Mater 2003; 17(1): 3–13.12. Mohammadi S and Moosavi AA. 3D multi delamination/

fracture analysis of composites subjected to impact load-

ings. J Compos Mater Part B 2007; 41(12): 1459–1475.13. Mohammadi S and Forouzan-Sepehr S. 3D adaptive

multi fracture analysis of composites. Mater Sci Forum

Part B 2003; 440–441: 145–152.



14. Rabinovitch O. Debonding analysis of fiber-reinforced-polymer strengthened beams: cohesive zone modelingversus a linear elastic fracture mechanics approach. Eng

Fract Mech 2008; 75: 2842–2859.15. Roberts TM. Shear and normal stresses in adhesive

joints. ASCE J Eng Mech 1989; 115(11): 2460–2476.16. Taljsten B. Strengthening of beams by plate bonding.

J Mater Civil Eng 1997; 9(4): 206–212.17. Wong RSY and Vecchio FJ. Towards modeling of rein-

forced concrete members with externally bonded fiber-

reinforced polymer composite. ACI Struct J 2003;100(1): 47–55.

18. Wu ZS, Yuan H and Niu HD. Stress transfer and frac-

ture propagation in different kinds of adhesive joints.ASCE J Eng Mech 2002; 128(5): 562–573.

19. Wu ZS and Yin J. Fracture behaviors of FRP-strength-

ened concrete structures. Eng Fract Mech 2003; 70:1339–1355.

20. Lu XZ, Teng JG, Ye LP and Jiang JJ. Bond-slip modelsfor FRP sheets/plates bonded to concrete. Eng Struct

2005; 27(6): 920–937.21. Wang J. Debonding of FRP plated reinforced concrete

beam, a bond-slip analysis. I: theoretical formulation. Int

J Solids Struct 2006; 43(21): 6649–6664.22. Teng JG, Yuan H and Chen JF. FRP-to-concrete inter-

faces between two adjacent cracks. Theoretical model for

debonding failure. Int J Solids Struct 2006; 43(18–19):5750–5778.

23. Yang QS, Peng XR and Kwan AKH. Strain energyrelease rate for interfacial cracks in hybrid beams. Mech

Res Commun 2006; 33(6): 796–803.24. Greco F, Lonetti P and Blasi N. An analytical investiga-

tion of debonding problems in beams strengthened using

composite plates. Eng Fract Mech 2007; 74(3): 346–372.25. Rabinovitch O and Frosting Y. Delamination failure of

RC beams strengthened with FRP strips: a closed form

high order fracture mechanics approach. J Eng Mech2001; 127(8): 852–861.

26. Rabinovitch O. Fracture-mechanics failure criteria for

RC beams strengthened with FRP strips – a simplifiedapproach. Compos Struct 2004; 64(3–4): 479–492.

27. Lu XZ, Jiang JJ, Teng JG and Ye LP. Finite elementsimulation of debonding in FRP-to-concrete bonded

joints. Constr Build Mater 2006; 20: 412–424.28. Bruno D, Carpino R and Greco F. Modelling of mixed

mode debonding in externally FRP reinforced beams.

Compos Sci Technol 2007; 67: 1459–1474.29. Colombi C. Reinforcement delamination of metallic

beams strengthened by FRP strips: fracture mechanics

based approach. Eng Fract Mech 2006; 73: 1980–1995.30. Pesic N and Pilakoutas K. Concrete beams with exter-

nally bonded flexural FRP-reinforcement: analyticalinvestigation of debonding failure. Composites Part B

2003; 34: 327–338.31. Belytschko T and Black T. Elastic crack growth in finite

elements with minimal remeshing. Int J Numer Methods

Eng 1999; 45: 601–620.32. Sukumar N, Huang ZY, Prevost JH and Suo Z. Partition

of unity enrichment for bimaterial interface cracks. Int J

Numer Methods Eng 2004; 59: 1075–1102.

33. Remmers JJC, Wells GN and de Borst R. A solid-like

shell element allowing for arbitrary delaminations. Int J

Numer Methods Eng 2003; 58: 2013–2040.34. Nagashima T and Suemasu H. Application of extended

finite element method to fracture of composite materials.

In: Proceedings of the European Congress on

Computational Methods in Applied Sciences and

Engineering (ECCOMAS), July 24–28 2004, Jyvaskyla,

Finland.35. Nagashima T and Suemasu H. Stress analysis of compos-

ite laminates with delamination using XFEM. Int J

Comput Methods 2006; 3: 521–543.36. Asadpoure A, Mohammadi S and Vafai A. Crack anal-

ysis in orthotropic media using the extended finite ele-

ment method. Thin Walled Struct 2007; 44(9): 1031–1038.

37. Asadpoure A, Mohammadi S and Vafai A. Modeling

crack in orthotropic media using coupled finite element

and partition of unity methods. Finite Elem Anal Des

2006; 42(13): 1165–1175.38. Asadpoure A and Mohammadi S. Developing new

enrichment functions for crack simulation in orthotropic

media by the extended finite element method. Int J Numer

Methods Eng 2007; 69: 2150–2172.39. Mohammadi S. Extended finite element method. UK:

Wiley/Blackwell Publishing, 2008.40. Motamedi D and Mohammadi S. Dynamic crack prop-

agation analysis of orthotropic media by the extended

finite element method. Int J Fract 2010; 161: 21–39.41. Motamedi D and Mohammadi S. Dynamic analysis of

fixed cracks in composites by the extended finite element

method. Eng Fract Mech 2010; 77: 3373–3393.

42. Motamedi D and Mohammadi S. Fracture analysis of

composites by time independent moving-crack orthotro-

pic XFEM. 2010 (Submitted for publication).43. Esna Ashari S and Mohammadi S. XFEM delamination

analysis of composite laminates by new orthotropic

enrichment functions. In: Proceedings of 1st International

Conference on Extended Finite Element Method

(XFEM2009), 28–30 September 2009, Aachen, Germany.

44. Esna Ashari S and Mohammadi S. Modeling delamina-

tion in composite laminates using XFEM by new ortho-

tropic enrichment functions. In: Proceedings of the

WCCM/APCOM 2010, 19–23 July 2010, Sydney,

Australia, IOP Conference Series: Materials Science and

Engineering, Vol. 10, paper 012240.45. Esna Ashari S and Mohammadi S. Delamination analysis

of composites by new orthotropic bimaterial extended

finite element method. Int J Numer Methods Eng 2011;

86: 1507–1543.

46. Lee KH. Stress and displacement fields for propagating

the crack along the interface of dissimilar orthotropic

materials under dynamic mode I and II load. J Appl

Mech 2000; 67: 223–228.47. Lee KH, Hawong JS and Choi SH. Dynamic stress inten-

sity factors KI and KII and dynamic crack propagating

characteristics of orthotropic materials. Eng Fract Mech

1996; 53: 119–140.

48. Lekhnitskii SG. Theory of an anisotropic elastic body. San

Francisco, CA: Holden-Day, 1963.


divya.t

3


49. Qian W and Sun CT. Methods for calculating stressintensity factors for interfacial cracks between two ortho-tropic solids. Int J Solids Struct 1998; 35: 3317–3330.

50. Barnett DM and Lothe J. Synthesis of the sextic and theintegral formalism for dislocation, Green’s function andsurface waves in anisotropic elastic solids. Phys Norv1973; 7: 13–19.

51. Chow WT, Boem HG and Alturi SN. Calculation ofstress intensity factors for interfacial crack between dis-similar anisotropic media using a hybrid element method

and the mutual integral. Comput Mech 1995; 15: 546–557.52. Sih GC, Paris P and Irwin G. On cracks in rectiline-

arly anisotropic bodies. Int J Fract Mech 1965; 1(3):

189–203.

53. Dongye C and Ting TCT. Explicit expressions of Barnett-

Lothe tensors and their associated tensors for orthotropic

materials. Q Appl Math 1989; 47: 732–734.54. Deng X. Mechanics of debonding and delamination in

composites: asymptotic studies. Compos Eng 1995; 5:

1299–1315.55. Hsu MH. Concrete beams strengthened with externally

bonded glass fiber reinforced plastic plates. Tamkang J

Sci Eng 2006; 9(3): 223–232.

56. Dundurs J. Edge-bonded dissimilar orthogonal elastic

wedges. J Appl Mech 1969; 36: 650–652.

Appendix

The general crack tip displacement fields for a crack along the interface of two dissimilar orthotropic materialsunder the mixed-mode loading have been obtained by Lee46 (also see Ref.45).

It should be noted that the following displacement fields are for the upper material and in order to obtain thedisplacement fields in the material below the interface, parameters "� and �"� are changed to �"� and "�,respectively.

ux ¼KIffiffiffiffiffiffi

2�p

1þ 4"2ð ÞD cosh "�ð Þe" ��lð Þpl �A cos " ln rlð Þ �

�l2

� þ 2" sin " ln rlð Þ þ

�l2

� ��

rlð Þ12þe�" ��lð ÞplA cos " ln rlð Þ �

�l2

� þ 2" sin " ln rlð Þ �

�l2

� �rlð Þ

12

�

� e" ��sð Þps �B cos " ln rsð Þ þ�s2

� þ 2" sin " ln rsð Þ þ

�s2

� �rsð Þ

12

�e�" ��sð ÞpsB cos " ln rsð Þ ��s2

� þ 2" sin " ln rsð Þ �

�s2

� �rsð Þ

12

��

þKIIffiffiffiffiffiffi

2�p

1þ 4"2ð ÞD cosh "�ð Þ�e" ��lð Þpl �A sin " ln rlð Þ þ

�l2

� � 2" cos " ln rlð Þ þ

�l2

� ��

rlð Þ12�e�" ��lð ÞplA sin " ln rlð Þ �

�l2

� � 2" cos " ln rlð Þ �

�l2

� �rlð Þ

12

�

þ e" ��sð Þps �B sin " ln rsð Þ þ�s2

� � 2" cos " ln rsð Þ þ

�s2

� �rsð Þ

12

þe�" ��sð ÞpsB sin " ln rsð Þ ��s2

� � 2" cos " ln rsð Þ �

�s2

� �rsð Þ

12

��

ðA1Þ


divya.t

4


uy ¼KIffiffiffiffiffiffi

2�p

1þ 4"2ð ÞD cosh "�ð Þe" ��lð Þql �A sin " ln rlð Þ þ

�l2

� � 2" cos " ln rlð Þ þ

�l2

� ��

rlð Þ12�e�" ��lð ÞqlA sin " ln rlð Þ �

�l2

� � 2" cos " ln rlð Þ �

�l2

� �rlð Þ

12

�

� e" ��sð Þqs �B sin " ln rsð Þ þ�s2

� � 2" cos " ln rsð Þ þ

�s2

� �rsð Þ

12

þe�" ��sð ÞqsB sin " ln rsð Þ ��s2

� � 2" cos " ln rsð Þ �

�s2

� �rsð Þ

12

��

þKIIffiffiffiffiffiffi

2�p

1þ 4"2ð ÞD cosh "�ð Þe" ��lð Þql �A cos " ln rlð Þ þ

�l2

� þ 2" sin " ln rlð Þ þ

�l2

� ��

rlð Þ12�e�" ��lð ÞqlA cos " ln rlð Þ �

�l2

� þ 2" sin " ln rlð Þ �

�l2

� �rlð Þ

12

�

� e" ��sð Þqs �B cos " ln rsð Þ þ�s2

� þ 2" sin " ln rsð Þ þ

�s2

� �rsð Þ

12

þe�" ��sð ÞqsB cos " ln rsð Þ ��s2

� þ 2" sin " ln rsð Þ �

�s2

� �rsð Þ

12

��

ðA2Þ

where

pl ¼ �p2a11 þ a12, ps ¼ �q

2a11 þ a12, ql ¼�p2a12 þ a22

p, qs ¼

�q2a12 þ a22q

ðA3Þ

and

A ¼ qþ �, �A ¼ q� �, B ¼ pþ �, �B ¼ p� �, Dk ¼ q� p½ �k, � ¼h21h12

� 12

ðA4Þ

h11 ¼ l11ð Þ1� l11ð Þ2h21 ¼ l21ð Þ1þ l21ð Þ2h12 ¼ l12ð Þ1þ l12ð Þ2

8<: ðA5Þ

l11ð Þk¼psp�plqq�p

n ok¼

qs�qlq�p

n ok

l21ð Þk¼qql�pqsq�p

n ok

l12ð Þk¼pl�psq�p

n ok

, k ¼ 1,2

8>>><>>>:

ðA6Þ

Subscripts k ¼ 1,2 denote the upper and lower materials, respectively. Also, variables rz,�z z ¼ l,sð Þ are related topolar coordinate system r,�ð Þ as

rz ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 � þ Z2

z sin2 �

q�z ¼ tan�1 Zz tan �ð Þ

(, z ¼ l,s,

Zl ¼ pZs ¼ q

�ðA7Þ

KI and KII are the modes 1 and 2 SIFs, respectively, and " the index of oscillation defined as

" ¼1

2�ln

1� �

1þ �

� ðA8Þ

where � is the second Dundurs parameter56 given by

� ¼h11ffiffiffiffiffiffiffiffiffiffiffiffih12h21p ðA9Þ


fracture analysis of frp-reinforced beams by orthotropic xfem

Documents