Cohesive Zone Parameters Selection for Mode-I Prediction of Interfacial Delamination

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1 Received for review: Jornal of Mechanical Engineering. All rights reserved. Received revised form: DOI: /sv-jme Original Scientific Paper Accepted for pblication: Cohesive Zone Parameters Selection for Mode-I Prediction of Interfacial Delamination Moslemi, M. Khoshravan, M. Mohsen Moslemi 1,* Mohammadreza Khoshravan 1 Yong Researchers and Elite Clb, Tabriz Branch Islamic Azad University, Tabriz, Iran University of Tabriz, Department of Mechanical Engineering, , Tabriz, Iran In order to determine the normal cohesive strength of composite laminates, a new test methodology was proposed. The vales of cohesive zone parameters (the cohesive strength and the separation energy) for mode I interlamiar fractre of E-glass/epoxy woven fabrication were compted from a series of experimental tests. Cohesive zone model simlation based on interface finite elements was condcted. A modified form of the Park-Palino-Roesler (PPR) traction-separation law together with a bilinear mixed-mode damage model was sed to simlate the damage processes, sing Abaqs cohesive elements. The nmerical reslts were compared with experimental tests and confirmed the adeqacy of normal cohesive strength. To ensre the sfficient dissipation of energy that sccessflly predicts delamination onset and propagation, cohesive zone length and minimm nmber of cohesive elements at cohesive zone length were determined. Interfacial penalty stiffness and the resistance crve of the composite specimen were also compted. The reslts show that the modified PPR model accrately simlates the fractre process zone ahead of the crack tip as compared to the bilinear model. Keywords: cohesive zone model, delamination, normal cohesive strength, finite element prediction Highlights Interlaminar Cohesive strength of composite material was determined. Cohesive zone modelling of delamination. Bilinear and modified PPR damage models. Experimental estimation of resistance crve. Cohesive zone length estimation. 0 INTRODUCTION Delamination is a defect type that freqently occrs in laminated composite materials, described as the separation of a layer or grop of layers from their adjacent ones, de to ot-of-plane shear loads. Delamination sally initiates from discontinities, sch as matrix cracks and free edges, or manfactring falts, sch as embedded defects and machining process sch as water-jet ctting [1]. Therefore, it is important to detect and analyse the progressive growth of delamination in order to predict the performance and to improve reliable and safe designs. Ultrasonic C-scan is one of the most efficient technics for tracing delamination in laminated composite materials []. Mode-I interface cracking is one of the most freqently modes of delamination in composite layered materials that is de to the loss of cohesion between layers of material. The delamination resistance of polymer matrix composites has been extensively investigated in the framework of fractre mechanics [3]. In order to model crack propagation, it was assmed that the delamination propagates when the available strain energy release rate is greater than or eqal to a critical vale, which is considered to be a mechanical parameter of the interface. Techniqes sch as the virtal crack closre (VCC), the J-integral, and the virtal crack extension are some of the most prevalent procedres that are sed to predict delamination growth. These techniqes are sed to compte the distribtion and components of the strain energy release rate. However, there are some difficlties when these techniqes are performed sing finite element codes. Another method to the nmerical modelling of the delamination growth can be developed within the framework of damage mechanics. Models formlated according to damage mechanics are based on the concept of the cohesive crack model, which considers a zone of vanishing thickness ahead of the crack front in order to describe more realistically the fractre natre withot the se of stress singlarity. The cohesive zone model was first developed by Dgdale [4], who demonstrated the concept that stresses in the material are confined by the yield stress and that a thin plastic zone is generated in front of the crack tip. Barenblatt [5] proposed a cohesive zone concept to stdy the fractre natre of brittle materials and to introdce a separation mechanism at the atomic scale in order to describe the real separation of materials, and to remove stress singlarity at the crack tip. Hillerborg et al. [6] sggested a model similar to the *Corr. Athor s Address: Yong Researchers And Elite Clb, Tabriz Branch Islamic Azad University, Tabriz, Iran, m.moslemi@tabriz.ac.ir 507

2 Barenblatt model. Their model allowed for existing cracks to grow as well as the initiation of new cracks. A cohesive zone model was freqently sed to the model fractre analysis of a different variety of materials sch as metals, polymers, concrete [7], ceramics, fnctionally graded materials [8], and fibrereinforced materials [9]; its range of applications will contine to expand. An important isse in conjnction with the se of the cohesive zone model is the specification of the traction-separation law. In particlar, the related fractre parameters, sch as cohesive strength and fractre toghness, as well as the shape of the traction-separation law, mst be determined. In the case of the traction-separation law, there are some models in the literatre that can be sed. For example, traction-separation based on an exponential form, a trapezoidal form and the bilinear form of Zhang and Palino [8] are traction-separation laws that have been widely adopted. Since there are no available standardized tests and de to the existence of some difficlties corresponding to the direct measrement of the theses parameters, in most cases they are determined by the comparison of a measred fractre property with nmerical predictions based on an idealized cohesive zone model, cf. [7], [10] and [11]. However, cohesive strength and fractre toghness are fond to have higher importance in comparison to the specific traction-separation shape that chosen for the cohesive zone modelling. Tron et al. [11] proposed a methodology to estimate the constittive parameters for the finite element simlation of progressive delamination sing a bilinear cohesive zone model. According to their methodology, the cohesive strength is proportional to the length of the cohesive zone, and this parameter shold be modified with respect to the cohesive zone length. It sggests that as the cohesive strength decreases, the length of the cohesive zone shold be artificially lengthened to ensre that it spans enogh elements of a given size. Yan and Li [1] investigated the effects of the cohesive law on dctile crack propagation and recommended obtaining realistic comptational reslts; the cohesive law mst be defined with proper parameters. The objective of this research is to provide a sitable methodology for the fractre characterization of delamination nder pre mode I loading. In this paper, a simple strctre test is proposed to compte the normal interlaminar cohesive strength of composite laminates. The vales for the critical mode I strain energy release rate (G Ic ) and mode-i cohesive strength (σ Ic ) were compted at room temperatre. These parameters and assmed damage models, inclding modified PPR model [13] and trianglar traction law, were sed with the Abaqs COH3D8 cohesive element to simlate bond line failre in strctres made from E-glass/Epoxy specimen. The resistance crve of the composite specimen compted sing the experimental method and a gideline methodology was proposed for selection of mode I cohesive zone length and the minimm reqired nmber of element in the cohesive zone length to obtain sccessfl prediction of the delamination onset and propagation. 1 COHESIVE ZONE MODEL THEORY Cohesive damage zone models relate traction to separation at an interface where a crack may initiate. Crack initiation is related to the cohesive strength, i.e., the maximm traction on the traction-separation law. When the area nder the traction-separation law reaches the fractre toghness, the traction declines to zero and new crack srfaces are generated. This phenomenon is shown in Fig. 1 for two different types of traction-separation law. a) b) Fig. 1. Traction-separation law a) trianglar model b) modified PPR model Fractre simlation of materials sing cohesive elements reqires sbstantial experience for determining the mesh reqirements and the accrate vales of parameters that characterize the tractionseparation law. In this work, in order to simlate Mode-I fractre of E-glass/epoxy woven composite laminate sing cohesive elements, two types of traction separation law, inclding mixed-mode Trianglar traction-separation response and modified PPR model, which is potential-based model, were sed and compared to experimental reslts. The objective is to determine an appropriate methodology to predict interlaminar crack growth in composite laminates. As shown in Fig 1, the sbscript m referred to mixed mode and sbscript c and referred to critical and ltimate (failre) vales, respectively, and i = I, II and III referred to pre modes of fractre. The area nder the crves represents the corresponding critical strain energy In order to determine the mixedmode traction-separation law, the properties reqired to be defined are the three critical strain energy release rates (G ic ), the penalty stiffnesses (K i ) and the 508 Moslemi, M. Khoshravan, M.

3 cohesive strengths (σ ic ). In the present work, Mode III is assmed to be identical to Mode II, so that the shear strengths in the two orthogonal directions, σ IIc and σ IIIc also critical strain energies G IIc, G IIIc are eqal. In the case of the simlation of Mode-I fractre, in order to predict delamination onset, analysis is more sensitive to the parameters σ Ic and G Ic than other interfacial properties. Therefore, in this work, the doble cantilever beam (DCB) test and the normal cohesive strength (NCS) test were carried ot to determine G Ic and σ Ic, respectively. Other fractre parameters were determined by the calibration of cohesive parameters. In other words, nmerical reslts are fit with experimental reslts according to the methodology described in [7], [10] and [11]. 1.1 Damage Initiation Criterion As shown in Fig. 1, the initial response of the cohesive elements at each damage model is based on linear elastic fractre mechanics (LEFM) and is assmed to be linear ntil a crack initiation criterion is satisfied. The penalty stiffness, K i, of each traction-separation response law that relates traction to the separation of cohesive elements before crack initiation is defined as below: K i i = σ c δ, (1) where i = I, II and III is fractre modes, σ ic and δ ic are the cohesive strength and critical separation of pre modes of fractre, respectively. In the present work, the penalty stiffness for all modes of fractre are considered to be the same, i.e., K I = K II = K III = K. Several methodologies have been proposed to determine the penalty stiffness of cohesive elements. The magnitde of the penalty stiffness mst be high enogh to avoid interpenetration of the crack srfaces and to avoid artificial compliance from being defined into the model by the cohesive elements. However, a more than enogh vale can lead to nmerical problems. Tron, et al. [11] assmed that whenever the throgh-the-thickness Yong s modls of the adjacent sb-laminate, E 3, is small enogh compared to K t, the effective elastic properties of the material will not be affected by the cohesive face. Where t is the thickness of adjacent sb-laminate of composite specimen. Therefore, an eqation to calclate the interface stiffness for Mode I is sggested, as mentioned below: ic E K = α 3, t () Cohesive Zone Parameters Selection for Mode-I Prediction of Interfacial Delamination where α is a parameter larger than 1. Tron et al. [11] recommended considering α = 50. This vale is sed for derivation of the penalty stiffness in the analysis presented in this work. After the linear part of the constittive law, traction at crack tip reaches its maximm vale and conseqently damage initiates. In order to simlate delamination onset in cohesive elements, there are for well-known crack initiation criteria: maximm stress, qadratic stress, maximm strain and qadratic strain criterion. In this analysis, the qadratic stress criterion that is expressed in Eq. (3) was sed. Under mixed-mode loading, an interaction between modes mst be taken into accont. This criterion model readily takes into accont the interaction of the traction components in the prediction of damage onset. Moreover, de to the high sensitivity of failre initiation to the strain and displacement, a stress-based criterion gives an accrate failre prediction as compared to other models, sch as strain-based or displacement-based criteria. The qadratic stress criterion is defined as: σ τ τ + + =1, τ τ z xz yz σ { } (3) Ic IIc IIIc where x, y, z refer to Cartesian coordinate as shown in Fig. 4a. The Macalay bracket, < >, states that the compressive stress does not contribte to crack onset. Whenever the damage initiation criterion of Eq. (3) for the cohesive element is satisfied, the stiffness of the element is declined according to the corresponding constittive law that is illstrated in Fig Damage Propagation Criterion After damage initiation, the softening procedre occrs. This procedre is governed by the corresponding traction separation law as below Bilinear Traction-Separation Model As shown in Fig. 1a, in the mixed-mode constittive law, σ mc and δ mc represent the cohesive strength and critical separation (separation at which crack initiates), respectively. Moreover, δ m refers to the separation at which failre occrs. The softening relation of cohesive elements that are governed by bilinear constittive law can be expressed as: σ i = dk i i ( 1 ) δ, (4) where d is a variable that relates damage condition, which has the magnitde d = 0 when the interface 509

4 is ndamaged, and the magnitde d = 1 when the interface is completely fractred. In nmerical analysis of damage evoltion, there are two crack evoltion criteria: energy- and displacement-based. In the analysis presented herein, the energy-based Benzeggagh and Kenane (BK) [14] damage evoltion criterion was sed, as expressed in Eq. (5). G G G G G shear c = Ic + ( IIc Ic ) G. T η (5) In Eq. (5), η refereed to the BK material parameter, G Shear = G II +G III and G T = G I + G II + G III. 1.. Modified PPR Model In this model, the traction force in the interface is obtained by differentiating a potential fnction with respect to the interface separation. Fig. 1b shows the typical modified PPR traction-separation law of cohesive zone modelling. Park et al. proposed this potential based model for mixed mode fractre [15]. This law can be sed for a wide variety of dctile and brittle fractres. Since it behaves nonlinearly prior to damage, it is reqired to develop a ser defined element in ABAQUS for this model. Bhattacharjee et al. [13] developed a modified version of the PPR model for analysis of tearing in thin soft materials. In their model, as with the trianglar constittive law, a linear elastic response was assmed before damage initiation (δ < δ c ). Therefore, this modified version of the PPR model allows for a straightforward implementation of the model in the commercial finite element code ABAQUS, sing tablar capability. After damage initiation, the softening relationship in this model can be expressed as [13]: δ α w δ w-1 σ= A[w( 1 ) ( + ) δ α δ δ α 1 w δ w α( 1 ) ( + ) ], δ α δ (6) where δ is normal displacement, and A, w, α, δ are PPR parameters that can be determined by satisfying all bondary conditions, inclding: σ= σ at δ = δ, σ =0 at δ = δ, c c σ lim, σ δ, δ δc δ = 0 G = d (7) where G is the total strain energy release rate. By applying the above-mentioned bondary conditions, it can be expressed as: δ 0 αα ( 1) λ σ w=, A= c, 1 αλ C α w w w w C = λ + + α λ 1 α λ α 1 [w( 1 ) ( ) ( 1 ) ( α λ ) ], (8) where λ δ. δ (9) = c δ can be determined sing the following eqation as: δ 1 G = Kδc + σ dδ. (10) In nmerical damage simlation, the corresponding the damage variables can be defined sing Eq. (4) and the corresponding traction in Eq. (6). After generating the damage variable at all the displacements, the modified PPR model can be directly implemented in ABAQUS sing the tablar capability as a fnction of relative displacement. 1.3 Cohesive Zone Length As shown in Fig. (), the length of the cohesive zone l cz is introdced as the distance between the crack tip and the point where the maximm cohesive traction is achieved. δc Fig.. Cohesive zone length of DCB specimen The length of the cohesive zone at the initiation of crack growth is independent of applied load and crack length. This means that the cohesive zone length at the crack initiation is a material property. There are several models in the literatre that estimated the length of the cohesive zone [4] to [6], [16] and [17]. All of the models for estimation of the cohesive zone length have similar forms as mentioned in Eq. (11). l cz = RE G Ic σ, (11) where G Ic is the critical strain energy release rate, σ Ic is the normal cohesive strength, and R is a parameter Ic 510 Moslemi, M. Khoshravan, M.

5 that depends on each cohesive zone model. For instance, Irwin s model [17] carried ot with R = 0.31 or in Dgdale [4] and Barenblatt [5] models, the vale for R is considered to be 0.4. minimm nmber of elements in the cohesive zone length. However, this nmber is not well established. FINITE ELEMENT SIMULATION Three-dimensional finite element models are developed in ABAQUS 6.1 [18] to model the delamination onset and growth, sing two different constittive laws. The DCB model was composed of two layers of eight-nodded solid elements, C3D8R (adherent layers), which were connected with a layer of zero thickness eight-node cohesive element, COH3D8, (cohesive layer). Adherent layers were connected to the cohesive layer by srface-to-srface tie constraints. Fig. 3 illstrates the deformed shape of the DCB specimen dring crack propagation. A more refined mesh near the crack tip, the oter srface of specimen and in the damage propagation region was sed. Bondary conditions inclded applying a vertical displacement and horizontally restraining the pper and lower edge node of the arms (Fig. 3). In order to predict an accrate propagation of delamination, it is necessary to have an adeqate fine finite element discretization in the cohesive zone length to achieve a good estimation of the interlaminar stress fields. When the cohesive zone is discretized by too few nmbers of elements, the distribtion of tractions ahead of the crack tip is not represented accrately. Ths, in order to achieve sccessfl FEM reslts, it is necessary to have a minimm nmber of elements in the cohesive zone length. The nmber of cohesive elements in the cohesive zone is: N e l = cz, (1) l where l e is the length of the cohesive elements in the direction of crack propagation and l cz is the cohesive zone length. There are several gidelines for the e Fig. 3. Deformed shape of simlated DCB with 3D elements 3 EXPERIMENTAL APPROACHES 3.1 Mode I Fractre Toghness Measrement The DCB Test Fig. 4a shows the geometry and dimensions of the DCB specimens. In the process of specimen fabrication, the E-glass/epoxy composite plate with a thickness of h = 4. mm was first prepared. The E-glass fibres were impregnated with a ML506 epoxy resin and HA11 hardener. The laminates were fabricated with the hand lay-p techniqe, and the pre-crack length was prodced by positioning a 13 μm thick Teflon insert at the mid-plane of the plate. In order to prodce plates with the desired fibre volme fraction, special pressre tool was applied in order to sqeeze ot excess resin. Then, the plate was left at room temperatre for 4 h to dry. After that, the plate was trimmed with a water jet machine along the longitdinal direction in order to obtain narrow specimens with the desired dimension and initial crack length. After trimming, the nominal length (l) and the nominal width (b) of the DCB specimen were 108 and 5 mm, respectively. The initial crack length (a) was 40 mm. Fibre volme fractions, V f, measred sing the resin brn-off method. Table 1 lists the mechanical property of E-glass/epoxy woven fabrication with V f = 49% that was sed in this research. All of the tests on DCB specimen were a) b) c) Fig. 4. a) DCB specimen, b) typical DCB test, c) crack length measrement dring propagation Cohesive Zone Parameters Selection for Mode-I Prediction of Interfacial Delamination 511

6 completed on a ZWICK electro-mechanical loading frame at room ambient temperatre. Fig. 4b shows a typical DCB test. All DCB tests were carried ot sing displacement control at the crosshead speed of 1 mm/min according to ASTM D558 standard [19]. Three specimens with a = 40 mm crack length were tested. A load-displacement response was recorded for each specimen dring the test. In this work, the crack length was monitored by bonding a strip of paper with the gradations printed on it to the specimen s edge and by taking photos dring the experiments with 5 s intervals sing a 5 megapixel digital camera. The experimental magnitdes of P δ a as a fnction of time were determined. The time of each P δ data point was compted sing the applied displacement and the corresponding loading rate. The time for each vale of crack length is the one at which the corresponding photo was taken. The photo in Fig. 4c was taken dring a test and shows the crack tip, allowing the crack length measrement. These experimental reslts were then sed to verify the adeqacy of the threedimensional finite element scheme tilized to obtain G Ic Data redction Method for G Ic In order to calclate the mode-i critical strain energy release rate, there are many commonly sed data redction methods, inclding compliance calibration method (CCM) that is based on the Irwin-Kies principle, direct beam theory (DBT) and the Corrected Beam Theory. In the present work, the corrected beam theory proposed by de Mora et al. [0] was sed. Only one specimen is sfficient to obtain the resistance crve (R-crve) of the specimen, which is the main advantage of the presented method. According to this theory, mode-i critical strain energy release rate can be compted as: P G = 3 δ Ic ba ( + ), (13) where P, a and b are load, crack length and specimen width, respectively. The parameter is the crack length correction to accont the crack tip rotation and deflection. According to the beam theory, the relationship between the compliance and the crack length is expressed by: C 13 / ( a + ) =. 13 / heb ( ) 1 (14) The crack length correction can be obtained sing the linear regression of C 1/3 verss crack length data. 3. Normal Cohesive Strength Measrement The objective of the NCS test is to determine the Mode I cohesive strength (σ Ic ) as an essential parameter for description of the traction separation law of cohesive elements. The fabrication of an NCS specimen is similar to that of a doble cantilever beam except for the delaminated area. In the process of NCS specimen fabrication, first a 14-layer woven rectanglar plate [0/90] 14 was prodced. After the 7 th layer of fabrication, a 13 μm thick Teflon layer was inserted at all sides of the plate so that a 10 mm 10 mm sqare area at the middle of plate was released. This area is the cohesive area of NCS specimen, as shown in Fig. 5a. After that, the plate was trimmed to a rectanglar dimension so that the cohesive area was located at the middle of rectanglar specimen. The specimen was bonded to fixtre srfaces. Prior to bonding, the srfaces of both fixtre and specimen, were lightly roghened with the sandpaper on the bonding face and cleaned with acetone. The area of bond is large enogh compared to the cohesive area so that debonding wold not occr between the specimen and fixtre srfaces the testing procedre. Fig. 5a shows the NCS specimen after testing, and Fig. 5b shows a typical NCS test. a) b) Fig. 5. a) NCS specimen after test, b) NCS test As shown in Fig. 5a, the dimensions of rectanglar plates B 1 and B are 80 mm and 50 mm, respectively, and the width of cohesive the sqare area (C) is 10 mm. All of the NCS tests were carried ot sing displacement control at the crosshead speed of 0.5 mm/min ntil fractre. In this process, three NCS failre tests were completed. In order to prevent slippage dring the test, the specimen was accrately balanced and a very little clamping force was reqired. Decohesion between the fibre and matrix in the NCS cohesive area is the dominant failre mode. Ths, it is obvios that the blk matrix behaves differently than the thin cohesive layer de to constraint effects indced by the adhesion between the fibre and matrix. 51 Moslemi, M. Khoshravan, M.

7 As a reslt, the blk matrix properties cold not be sed to determine normal cohesive strength, and this parameter shold be determined sing an NCS test method. For the prposes of data redction, all specimens were assmed to have failed instantly. The specimen failre is assmed at its maximm load vale. The normal cohesive strength vales (σ Ic ) were compted sing Eq. (15) as: σ max Ic = P, A (15) NCS where P max is the maximm load in which failre occrs and A NCS is the cohesive area of the NCS specimen. 4.1 Experimental Reslts 4 RESULTS AND DISCUSSION Fig. 6 shows the load-displacement response of three NCS experiments. The measred load is initially negligible, corresponding to the clearance of the fixtre. The test was preceded ntil a maximm load was achieved and followed by a sdden load drop, indicating specimen failre. Fig. 7. R-crve of E-glass/epoxy specimen Table 1. Mechanical properties of E-glass/epoxy E 1 E E 3 ν 1 G 1 G 13 G Table. Interfacial property of E-glass/epoxy σ Ic [MPa] τ IIc [MPa] τ IIIc [MPa] G Ic [J/m ] G IIc [J/m ] G IIIc [J/m ] Cohesive Zone Model Reslts FEM models of each specimen were carried ot sing the Ply elastic properties of adherent layers that are given in Table 1 and the interfacial properties obtained previosly and listed in Table. It shold be mentioned that when sing Eq. () for interfacial penalty stiffness, the vale of K = N/mm³ is sed for all DCB simlations. η 4..1 Determination of Cohesive Zone Length Fig. 6. Load-displacement response of NCS test The mean vale of maximm loads of Fig. 6 was considered as P max. Using Eq. (15) for data redction and sbstitting this vale for P max, the cohesive strength of composite material compted and is eqal to 1.4 MPa. To calclate the experimental resistance crve, the nmerical vales of P δ a parameters were recorded dring crack propagation and were sed to obtain the critical fractre energies verss crack length. Fig. 7 shows the experimentally obtained R-crve of the material. To simlate the crack propagation sing cohesive elements, the mean vale of G Ic was considered as the fractre toghness of the material. Table lists the cohesive properties of E-glass/epoxy woven composite laminate. Cohesive zone length was previosly introdced as the distance between the maximm traction and the crack front. Therefore, in order to calclate the distribtion of traction in the cohesive layer of model and the corresponding cohesive zone length, a very refined mesh sing element size of 0.15 mm was sed in the area near the crack tip of the DCB specimen. The distribtion of tractions ahead of the crack tip at the delamination onset of the DCB specimen is illstrated in Fig. 8. At the crack initiation point, traction at the crack tip vanished as expected from the cohesive zone theory. According to this analysis, the cohesive zone length of material is 3.76 mm, as shown in Fig. 8. Using the material property that shown in Table 1 and and Eq. (11), the parameter R is compted as 0.7. This vale is closest to the Irwin [17] (0.31) model. The cohesive zone length is a material property that has a high order of importance regarding obtaining a sccessfl prediction of delamination onset. This Cohesive Zone Parameters Selection for Mode-I Prediction of Interfacial Delamination 513

8 parameter was previosly introdced as a fnction of normal cohesive strength by Tron et al. [11]. Ths, sing an absolte vale for normal cohesive strength, this parameter as a material property is determined. de to the magnitde of tractions ahead of the crack tip. Fig. 8. Distribtion of traction ahead of crack tip 4.. Investigation of Mesh Refinement In order to investigate the effect of mesh refinement in the cohesive zone length on nmerical prediction of delamination onset, several DCB specimens were simlated with different sizes of cohesive elements in the cohesive zone length. The predicted loaddisplacement responses obtained sing DCB models are compared to the experimental soltion in Fig. 9. In this analysis, cohesive element sizes (in the direction of crack propagation) range from 0.15 mm to mm. The reslts illstrate that for all mesh sizes a converged soltion was obtained bt it is necessary to apply a mesh size, l e, less than 1 mm to accrately predict delamination initiation. The analysis sing coarser meshes significantly overpredicts the strength of the DCB specimen, and the response does not follow the experimental reslts. Cohesive zone length, l cz, for the material described in Tables 1 and was determined as 3.76 mm. Therefore, for a mesh size smaller than 1 mm, more than three elements wold span the cohesive zone length, which is enogh for an accrate prediction of the fractre onset. There are several gidelines for the minimm reqired element in cohesive zone length. For example, Moës and Belytschko [1] proposed sing more than 10 elements, while Falk et al. [16] sggested between and 5 elements. In the work of Dàvila et al. [], the minimm reqired element length to predict the delamination in a DCB model was 1 mm, and sing more than 3 elements in cohesive zone length of simlated DCB specimen was recommended. The difference in predictions from sing a coarse mesh in the modelling of delamination in a DCB specimen is Fig. 9. Damage simlation sing different mesh refinement 4..3 Comparison of Damage Models A stdy also was condcted to investigate the adeqacy of the two mentioned traction separation laws sed to simlate damage propagation. The objective was to determine how the sed models reprodce crack initiation and propagation. Fig. 10 shows the load-displacement of the cohesive zone model and experimental work on a DCB specimen. Fig. 10. Load displacement response of experimental and damage models In the crrent stdy, the vale of shape parameter α in the PPR model varied from 1 to 4; the reslt was that increasing the vale of the shape parameter α increased the rate at which material loses its stiffness once damage was initiated. In other words, increasing parameter α decreases the fractre process zone effect ahead of the crack tip. In the case of α <, there is a gradal fall in the load, bt in the case of α > a sdden drop in the load-displacement response is achieved, which means a large nmber of cohesive elements failed at the same time. For clarity, the reslts are not shown in this figre. In this stdy, a 514 Moslemi, M. Khoshravan, M.

9 vale of α = 1.7 was fond to be the optimm vale for the nmerical prediction of damage propagation. As shown in Fig. 10, the modified PPR model was fond to be adeqate to reprodce the experimentally observed behavior of the composite specimen, and reprodced approximate smooth crack initiation and propagation while the bilinear model depicted sdden damage propagation. The maximm difference between the experimental and bilinear models is 8.8 % while for PPR it is.6 %. This means the modified PPR model acconted fractre process zone which created ahead of crack tip. 5 CONCLUSION A methodology for the delamination characterization of composite laminates nder pre Mode I was proposed. An NCS test has been proposed to compte the Mode-I cohesive strength as a cohesive parameter. The Mode-I critical strain energy release rate verss the crack length of E-glass/epoxy composite laminate was compted sing corrected beam theory for data redction. A mixed-mode trianglar constittive relationship between stress (σ) and relative displacements (δ) of cohesive elements and modified PPR damage model were considered to simlate delamination onset and propagation. The reslts of the three-dimensional finite element analysis with cohesive parameters (σ Ic, G Ic ) enclosed the adeqacy of cohesive parameters. Accrate damage prediction was achieved sing the modified PPR model, and it was considered by the athors to be an accrate model for damage characterization of material. Modified PPR models accrately described the fractre process zone, which was created ahead of the crack tip as compared to bilinear model. To ensre the sfficient dissipation of energy, cohesive zone length as a material property was determined. Nmerical analysis with different discretizations of the cohesive zone length showed that nmerical predicted responses correlate well with the experimental soltions when at least 3 elements span the cohesive zone length. 6 ACKNOWLEDGMENT This work has been fnded by University of Tabriz, and its athors wold like to thank the University of Tabriz for the grant. 7 REFERENCES [1] Karpiliski, A. (006). An Introdction to the diagnosis of the delamination process for glass/epoxy com posites dring high-pressre abrasive water-jet ctting. Strojniški vestnik Jornal of Mechanical Engineering, vol. 5, no. 7, p [] Hasiotis, T., Badogiannis, E., Tsovalis, N.G. (011). Application of ltrasonic C-scan techniqes for tracing defects in laminated composite materials. Strojniški vestnik - Jornal of Mechanical Engineering, vol. 57, no. 3, p , DOI: /sv-jme [3] Brnner, A., Blackman, B., Davies, P. (008). A stats report on delamination resistance testing of polymer matrix composites. Engineering Fractre Mechanics, vol. 75, no. 9, , DOI: /j.engfracmech [4] Dgdale, D. (1960). Yielding of steel sheets containing slits. Jornal of the Mechanics and Physics of Solids, vol. 8, no., p , DOI: / (60) [5] Barenblatt, G. (196). The mathematical theory of eqilibrim cracks in brittle fractre. Advances in Applied Mechanics, vol. 7, no. 1, p , DOI: /j.ech [6] Hillerborg, A., Modéer, M., Petersson, P.-E. (1976). Analysis of crack formation and crack growth in concrete by means of fractre mechanics and finite elements. Cement and Concrete Research, vol. 6, no. 6, p , DOI: / (76) [7] Song, S.H., Palino, G.H., Bttlar, W.G. (006). Simlation of crack propagation in asphalt concrete sing an intrinsic cohesive zone model. Jornal of Engineering Mechanics, vol. 13, no. 11, p , DOI: /(ASCE) (006)13:11(115). [8] Zhang, Z.J., Palino, G.H. (005). Cohesive zone modelling of dynamic failre in homogeneos and fnctionally graded materials. International Jornal of Plasticity, vol. 1, no. 6, p , DOI: /j.ijplas [9] Khoshravan, M.R., Moslemi, M. (014). Investigation on mode III interlaminar fractre of glass/epoxy laminates sing a modified split cantilever beam test. Engineering Fractre Mechanics, vol. 17, p , DOI: /j. engfracmech [10] Song, S.H., Palino, G.H., Bttlar, W.G. (006). A bilinear cohesive zone model tailored for fractre of asphalt concrete considering viscoelastic blk material. Engineering Fractre Mechanics, vol. 73, no. 18, p , DOI: /j. engfracmech [11] Tron, A., Davila, C.G., Camanho, P.P., Costa, J. (007). An engineering soltion for mesh size effects in the simlation of delamination sing cohesive zone models. Engineering Fractre Mechanics, vol. 74, no. 10, p , DOI: /j.engfracmech [1] Yan, H., Li, X. (014). Effects of the cohesive law on dctile crack propagation simlation by sing cohesive zone models. Engineering Fractre Mechanics, vol. 16, p. 1-11, DOI: /j.engfracmech [13] Bhattacharjee, T., Barlingay, M., Tasneem, H., Roan, E., Vemaganti, K. (013). Cohesive zone modelling of mode I tearing in thin soft materials. Jornal of the Mechanical Cohesive Zone Parameters Selection for Mode-I Prediction of Interfacial Delamination 515

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