Experimental methods to determine modelparameters for failure modes of CFRP

DANIEL SVENSSON

Department of Applied MechanicsCHALMERS UNIVERSITY OF TECHNOLOGYGoteborg, Sweden 2013

THESIS FOR THE DEGREE OF LICENTIATE OF ENGINEERING IN SOLID ANDSTRUCTURAL MECHANICS

Experimental methods to determine modelparameters for failure modes of CFRP

DANIEL SVENSSON

Department of Applied MechanicsCHALMERS UNIVERSITY OF TECHNOLOGY

Goteborg, Sweden 2013

Experimental methods to determine modelparameters for failure modes of CFRPDANIEL SVENSSON

c DANIEL SVENSSON, 2013

Thesis for the degree of Licentiate of Engineering 2013:07ISSN 1652-8565Department of Applied MechanicsChalmers University of TechnologySE-412 96 GoteborgSwedenTelephone: +46 (0)31-772 1000

Chalmers ReproserviceGoteborg, Sweden 2013

Experimental methods to determine modelparameters for failure modes of CFRPThesis for the degree of Licentiate of Engineering in Solid and Structural MechanicsDANIEL SVENSSONDepartment of Applied MechanicsChalmers University of Technology

Abstract

The focus of this thesis is to develop methods to predict the damage response of CarbonFibre Reinforced Polymers (CFRP). In the pursuit of reducing the manufacturing costand weight of CFRP components, it is crucial to enable modelling of the non-linearresponse associated with various failure modes. Two failure modes are considered inthis thesis: fibre compressive failure and interlaminar delamination. Multidirectionallaminated composites are commonly used when a low weight is desired due to their highspecific strength and stiffness. In a carbon/epoxy composite, almost exclusively thefibres carry the load. However, along the fibre direction, the compressive strength isconsiderably lower than the tensile strength. With the same reasoning, the transversestrength is considerably lower than the in-plane strength. This makes delamination andfibre compressive failure two of the major concerns in structural design. Moreover, thepresence of delaminations severely reduces the compressive strength of a laminate. Thiscan cause catastrophic failure of the structure.

In Paper A, we suggest a test method for determining fracture properties associatedwith fibre compressive failure. A modified compact compression specimen is designedfor this purpose and compressive failure takes place in a region consisting exclusively offibres oriented parallel to the loading direction. The evaluation method is based on ageneralized J-integral and full field measurements of the strain field on the surface of thespecimen. Thus, the method is not restricted to small damage zones.

Paper B focuses on measuring cohesive laws for delamination in pure mode loading.The cohesive laws in mode I and mode II are measured with the DCB- and ENF-specimen,respectively. With a method based on the J-integral, the energy release rate associatedwith the crack tip separation is measured directly. From this, the cohesive laws are derived.It is concluded that the nonlinear response at the crack tip is crucial in the evaluation ofthe mode II fracture energy.

Keywords: composite, delamination, compressive failure, energy release rate, cohesivemodelling

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Preface

This work has been carried out during the years 2010-2013 at the Mechanics of Mate-rials research group at the University of Skovde. Funding from the Swedish NationalAeronautical Research Program (NFFP5) is gratefully acknowledged.

First of all, my deepest gratitude to my co-supervisor Associate Professor Svante Alfredssonfor the patient guidance, encouragement and advice he has provided me during this time.I am also grateful to my supervisor Professor Ulf Stigh for the generous sharing of hisknowledge and giving me the opportunity to do research.

Thanks to my present and former colleagues for their helpful attitude and creating suchan enjoyable environment.

Last but not certainly not least, a big thank you to my Anna for her never ending support,patience and love.

Daniel SvenssonSkovde, March 2013

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Thesis

This thesis consists of an extended summary and the following appended papers:

Paper A

D. Svensson, K.S. Alfredsson, U. Stigh and N.E Jansson. An exper-imental method to determine the fracture properties of compressivefibre failure in unidirectional CFRP. To be submitted for interna-tional publication (2013)

Paper BK.S. Alfredsson, D. Svensson, U. Stigh and A. Biel. Measurementof cohesive laws for initiation of delamination of CFRP. Submittedfor international publication (2013)

The appended papers were prepared in collaboration with the co-authors. Paper A:The author of the thesis was responsible for the major progress in planning the paper,developing the theory, performing the simulations and evaluating the experiments. Tookpart in designing the experimental setup and performing the experiments. Paper B: Theauthor took part in planning the paper and developing the theory. Responsible for thesimulations and the evaluation of the experiments.

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Contents

Abstract i

Preface iii

Thesis v

Contents vii

I Extended Summary 1

1 Introduction 1

2 Fibre compressive failure 2

3 Delamination 4

4 Summary of appended papers 64.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

References 7

II Appended Papers AB 11

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Part I

Extended Summary

1 Introduction

Long fibre composites such as Carbon Fibre Reinforced Polymers (CFRP) are commonlyused in advanced structural applications, e.g. in the aerospace and marine industry.This is mainly due to favourable mechanical properties such as high specific stiffness andstrength, low density and high resistance to corrosion. However, the wide range of possiblefailure modes and limited understanding of the material behaviour requires conservativedimensioning. This is further complicated by the fact that the material behaviour isdependent on e.g. lay-up, loading direction, specimen size and environmental effects suchas temperature and moisture. Thus, phenomenological determination of criteria to predictfailure on a structural level from coupon tests requires numerous testing. Thousands ofcoupon tests are not uncommon in the design of a safe advanced structure. Therefore, amajor challenge in the structural design is to remove excessive safety factors to e.g. reducethe manufacturing costs and environmental impact. It is well-known from experimentalexperience that assuming an ideal brittle behaviour is excessively conservative since theredistribution of stresses at high strain regions are not considered. Stresses around stressrisers such as holes or cut-outs are relaxed by damage evolution and further load canbe applied prior to failure. Therefore, a certain loss of integrity has to be allowed. Itis therefore desirable to model, experimentally determine and simulate the non-linearresponse that precedes failure of the laminate. Moreover, it has been observed thatspecimen size effects play an important role on the strength of a laminate, cf. e.g. [1].Thus, a fracture criterion should be associated with a length scale. This is introduced ina fracture mechanics based approach.

The scientific community is devoting substantial effort to develop test methods fordetermination of the fracture energy for various failure modes. It is also desirable toobtain accurate failure criteria of a laminate from the mechanical properties of thefibres, the matrix and the lay-up geometry. To successfully calibrate such models, animportant step is to determine the fracture energy for different failure modes in isolation.Several methods for each failure mode have been proposed and in [2] Laffan et al. give asummary. Standardized methods have been developed for interlaminar testing, e.g. [3],and for translaminar testing [4]. Furthermore, methods have been proposed for studyinglongitudinal [5] and transverse intralaminar matrix failure [6]. Moreover, test methods forfibre tensile failure have been proposed in e.g. [7]. Testing methods have been proposedfor fibre compressive failure in e.g. [8], [9] and [2].

In this thesis, experimental methods are developed for determining governing fractureproperties associated with fibre compressive failure and interlaminar failure. These failuremodes are considered to be two major design limiting fracture processes in structuraldesign. Thus, to accurately predict these failure modes is crucial in the design of improvedcomposite structures.

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2 Fibre compressive failure

Unidirectional (UD) CFRP is known for its high stiffness and strength along the fibredirection. However, the compressive strength is only about 60-70 % of the tensile strength[10]. The tensile strength is mainly governed by the tensile strength of the fibres whereasthe compressive strength, to a larger extent, depends on several material properties ofthe laminate. Multiple failure modes are associated with fibre compressive failure, e.g.elastic microbuckling, plastic microbuckling, longitudinal fibre splitting, fibre crushingand shear-driven failure. The distinction of elastic and plastic microbuckling refers to theamount of shear strain induced in the matrix. Experimental identification of the failuremechanisms preceding compressive failure is difficult due to the unstable and catastrophicnature of fibre compressive failure. However, plastic microbuckling is identified as thedominating failure mode, cf. e.g [11], [12]. Plastic microbuckling is the failure modewhere compressive loading introduces bending of fibres. This may start at a materialdiscontinuity such as fibre waviness or at a free edge where the fibres lack lateral support,cf. e.g. [13]. This places the matrix in a shearing mode. As the load increases, thefibres rotate further and yielding of the matrix is induced. Ultimately, the fibres fail attwo points due to a combination of axial compression and bending. A localized band ofbroken fibres denoted a kink-band is created and propagates into the intact region ofthe specimen, cf. Fig 2.1. According to the classical strength models by Budiansky [14]and Budiansky and Fleck [15], the governing parameters of the compressive strength of aUD laminate are the fibre misalignment and the shear yield strengths of the compositematerial.

In the formation of a kink-band, matrix/fibre splitting occurs in-between the rotatedfibres. This induces further rotation of the fibres prior to fibre failure. This mechanism isexplained by Fleck in [16]. It has also been experimentally observed in [17], where thefailure process at the micro scale has been observed while the specimen is kept underload. Thus, in compression the material properties of the matrix are important since thematrix provides lateral support to the fibres. Moreover, the bond between the matrixand the fibres is important for the compressive strength. It should be noted that, eventhough microbuckling leading to kink-band formation is recognized as the dominatingfailure mode, a shear-driven fibre failure mode is observed in e.g. [17]. In [17], Gutkin etal. highlights conditions for whether the fibre fails in a shear-driven mode or in a kinkingmode.

Unidirectional composites are rarely used in structural applications due to their poortransverse performance. Instead, a laminated composite is built-up by successive plieswith varying orientations. In [18], it is concluded that compressive failure occurs in amultidirectional centre cracked specimen due to microbuckling of the 0-plies and thefracture energy is increased when the portion of 0-plies is increased. It is thereforeimportant to determine the fracture energy associated with compressive failure of the0-plies in isolation. As yet there is no standardized test method for this purpose. A recentreview of earlier reported fracture energies associated with compressive failure is givenin [8]. From experiments with center crack specimens with a T800/924C laminate with(0,902,0)3s layup reported in [19], Pinho et al. derive the fracture energy for kink-band

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Figure 2.1: Kink band found in an experiment presented in Paper A. The scale barindicates 500 m

formation to about 76 kJ/m2. In [8], Pinho et al. use a compact compression (CC)specimen with a T300/913 laminate with cross ply layup and the fracture energy 79.9kJ/m2 associated with the 0-plies is reported. In the presented method, a normalizedenergy release rate is calculated from linear FE-simulations. The fracture energy is thendirectly determined from the maximum load. Later in [9], Catalanotti et al. use the samespecimen and layup geometry with a different material system (Hexcel IM7-8552) andthe fracture energy 47.5 kJ/m2 associated with the 0-plies is reported. The authors usedigital image correlation to calculate the fracture energy from the actual strain field onthe lateral surface of the specimen, i.e. the assumption of a small scaled damage zone isavoided. However, the use of cross ply laminates necessitates partitioning of the fractureenergy since the energy dissipated in the 90-plies needs to be deducted. Furthermore,interaction effects between the alternating 0-plies and 90-plies are neglected. Thereforeit is desired to determine the fracture energy by the use of a UD-layup. A test methodusing a four point bend specimen with a UD-layup is presented by Laffan et al. in [2]. Thematerial system is the same as in [9]. Also here, linear elastic FE-simulations are used todetermine how the energy release rate relates to the applied load. A lower fracture energyof 25.7 kJ/m2 is reported. The reason may be that the experiments are interrupted whendamage initiation is first detected. The fracture energy is then calculated from the load atonset of damage. It is reported that damage is initiated by a shear crack initiation at thenotch. The shear crack is propagating a small distance with the direction approximately45 to the mode I direction and then transforms into a kink band as the load increases.This transition phase is described in [17] by Gutkin et al.

In Paper A of this thesis, a modified CC-specimen is used to determine the fractureenergy associated with longitudinal compressive failure. Compressive failure takes placein a region with a UD-layup. Thus, partitioning of the fracture energy is not necessaryand interaction effects between the 0-plies and off-axis plies are avoided. The evaluation

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method is based on the concept of equilibrium of configurational forces [20] and full fieldmeasurement of the strain field. This method is therefore applicable also in the case of alarge damage zone.

3 Delamination

In general, the interlaminar strength of laminated composites is substantially lower thanthe in-plane strength of a multidirectional composite. The foremost reason for this is thatno fibres are oriented in the transverse direction. Moreover, the resin rich regions betweenthe lamina are zones of weakness. Thus, due to the comparatively poor performance intransverse loading, interlaminar delamination is a major concern in the design of structuralcomponents. In Fig 3.1, delamination in a cross ply laminate is shown. Delamination maystart at a stress concentration arising from an initial defect or damages occurring in theuse of the component. Moreover, a structural component often includes curved laminatesor laminates with a varying thickness. If the curved laminate is subjected to out-of-planebending, delamination may initiate due to the transverse stresses. Furthermore, plydrops are often used to progressively reduce the out of plane thickness along a compositelaminate. The axial load has to be transferred to the thinner section by interlaminarshear stresses. Thus, at the ply-dropping region, delamination can initiate in a shearingmode. Studies of the influence of ply-drops and out-of-plane curvatures can be foundin e.g. [22] and [23]. Moreover, delamination may also initiate and propagate due tolow-velocity impact [24]. Delaminations are difficult to detect and it can severely decreasethe structural compressive strength. Thus, the low compression strength after impactis a limiting design parameter in the industrial design of aero-structures. Since thestructural strength is severely decreased by the presence of delaminations, a conservativeprediction of the strength can be carried out by including delaminations in the model toe.g. determine the critical buckling load [25]. Two commonly used models for simulatingdelamination are, the Virtual-Crack-Closing-Technique (VCCT) based on linear elasticfracture mechanics and Cohesive Zone Modelling (CZM) that is not limited to smallprocess zones.

Fracture mechanics based methods have been successfully used for modelling onset andpropagation of delamination in laminated composites. In a fracture mechanics approach,propagation of delamination is assumed to initiate when the total energy release rate,G = GI + GII, is equal to the fracture energy, Gc, associated with the current modemix. Here, GI is the energy release rate in mode I (opening), GII is the energy releaserate in mode II (shearing) and the mode mix is given by e.g. the fraction GII/G. Thefracture energy is determined experimentally for various mode-mixes. Commonly usedtest methods for measuring the fracture energies in mode I and mode II are, the DoubleCantilever Beam (DCB)-test for mode I and the End Notch Flexure (ENF)-test for modeII. The most commonly used mixed-mode test is the Mixed Mode Bending (MMB)-test,cf. [26]. With this setup, the mode mix can be varied by adjusting the load point position.By determining Gc for a range of mode-mixes, a relation between Gc and the mode mixcan be obtained by a best curve fit. The energy release rate from these tests methods hasbeen calculated with different levels of sophistication. In the simplest case, the energy

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Figure 3.1: 2. Delamination, tensile fibre fracture and tensile/shear matrix cracks in across-ply laminate. Picture taken from [21]

release rate is calculated by assuming the Euler-Bernoulli beam theory and a rigid cracktip. The methods have later been extended to consider the flexibility ahead of the cracktip by adding a crack length correction to account for the influence of the anisotropiccomposite material. In [27], Juntti et al. give a comprehensive review of the developedevaluation methods. For the case of orthotropy, Bao et al. [28] present crack lengthcorrections based on an ingenious rescaling technique.

As a step toward a more complete model of the processes involved at the crack tip,the cohesive zone model is a strong candidate. A brief historical review on cohesivemodelling is given in Paper B. In a cohesive zone model, a planar damage zone is assumedwhere the behaviour is governed by a traction separation law. The onset of damage ismodelled by a stress criterion and the traction is assumed to decrease as the cohesiveseparation increases. At a large enough separation, the traction acting on the cohesivesurface drops to zero and crack propagation is initiated. For a given load history, thearea beneath the cohesive laws in mode I and mode II corresponds to the total fractureenergy. It is noted that not only the fracture energy needs to be accurately determined.The complete shape of the cohesive law has to be accurately modelled to predict criticalloads of structures suffering from delamination. For example, at regions subjected to highinterlaminar stresses such as ply-drop regions, the critical interlaminar stresses have agreat influence on the structural behaviour. Therefore, it is desirable to experimentallymeasure the cohesive law with high accuracy to model the interlaminar behaviour of alaminated composite.

Cohesive models of delamination are rather widely used in the scientific community.Several methodologies have been used for determining the cohesive law. One commonmethod is to use a best fit approach. A function by which the cohesive traction varies withthe cohesive separation is pre-selected. Then, the cohesive parameters that best reproducethe force-displacement curves are chosen. However, for many of the test specimen

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geometries, the shape of the cohesive law has a minor influence of the force-displacementrelation. Thus, the procedure is not sensitive to the parameter that it aims at measure.Another methodology is to experimentally measure the cohesive law with a method basedon evaluation of the path-independent J-integral, cf. [31]. Several test methods aredesigned so that the energy release rate, J , associated with the crack tip separation can bedirectly measured from external loads and displacements, cf. [29] and [30]. Subsequently,the cohesive laws in mode I and mode II are determined by differentiation of J withrespect to the opening separation and the shearing separation, respectively. This methoddoes not require any assumption of the material behaviour. Furthermore, the method isvalid for large damage zones. However, if the complete cohesive law for all mode-mixes isto be determined, it requires the existence of an associated potential. If a potential doesnot exist, then in mixed-mode loading, the cohesive law is path dependent. Thus, thecohesive law can only be determined for the specific load histories that were applied inthe experiments. However, this is not an issue when the measurement of the cohesivelaws is restricted to the pure mode cases, cf. Paper B.

For delamination, it is noted that two different fracture processes are involved. One isassociated with bridging of fibres in the wake of the growing crack and one is associatedwith the fracture process at the crack tip. The bridging stress behind the crack tip issmall compared to the initiation stress at the crack tip. However, in the presence of fibrebridging, the bridging fibres can contribute to substantially higher fracture energies. Thus,these two mechanisms act on two very different length scales. In Paper B, we focus onthe fracture process associated with the crack tip until initiation of delamination. This isthe important process when onset of delamination growth has to be avoided.

4 Summary of appended papers

4.1 Paper A

In this work a modified CC-specimen is designed to study longitudinal compressivefailure. Normally when using a CC-specimen a cross ply laminate is used and an in-planenotch is used to achieve a stress rising effect to nucleate compressive failure. Thus,interaction effects between the 0-plies and off-axis-plies are neglected and partitioningof the fracture energy is needed to determine the fracture energy associated with the0-plies. Furthermore, the data reduction scheme is often based on linear elastic fracturemechanics, i.e. the damage zone is assumed to be small.

In this work, localized high strains are achieved by decreasing the out-of-plane thick-ness towards the anticipated damage region that consists exclusively of 0-plies. Thus,compressive failure of 0-plies is obtained in isolation. The data reduction scheme isderived from Eshelbys concept of equilibrium of configurational forces [20]. The methodis similar to earlier work where the J-integral [31] is used to determine the energy releaserate associated with the damage zone. However, the J-integral is not applicable for thepresent geometry due to the varying out of plane thickness. Thus, a generalized form ofthe J-integral is used to determine the energy release rate associated with the damagezone. Full field strain measurement with the DIC-system Aramis is used in the evaluation.

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Thus, the assumption of a small damage zone is avoided. Numerical simulations are usedto verify the experimental results. The damage region is idealized as a cohesive zonemodel, i.e. the material behaviour within the damage zone is governed by a cohesive lawrelating the compressive stress and compressive separation. A cohesive law is proposedand the simulated results agree well with the main features of the experimental results.

4.2 Paper B

In this paper, methods are presented for measurement of the cohesive laws in modeI and mode II associated with interlaminar delamination. Cohesive laws for mode Iand mode II are measured with the DCB- and ENF-test, respectively. With a methodbased on the path-independent J-integral, the energy release rate, J , associated withthe crack tip separation can be measured directly from the applied load, load pointrotations and the separation at the crack tip. By differentiation of J with respect tothe separation at the crack tip, the cohesive laws are determined. FE-simulations areperformed with the cohesive laws implemented and the simulations show good agreementwith experimental results. The results indicate that the fracture energy in mode II canbe severely underestimated if the inelastic behaviour at the crack tip is ignored.

References

[1] J. Lee and C. Soutis. Measuring the notched compressive strength of compositelaminates: Specimen size effects. Composite Science and Technology 68 (2008),23592366.

[2] M. Laffan et al. Measurement of the fracture toughness associated with the longi-tudinal fibre mode of laminated composites. Composites Part A 43 (2012), 19301938.

[3] ASTM. D5528 Standard test method for mode I interlaminar fracture toughness ofunidirectional fiber-reinforced polymer matrix composites. 2007.

[4] ASTM. E1922-04 Standard test method for translaminar fracture toughness oflaminated polymer matrix composite materials. 2004.

[5] B. F. Sorensen and T. K. Jacobsen. Large-scale bridging in composites: R-curvesand bridging laws. Composites Part A: Applied Science and Manufacturing 29.11(1998), 14431451.

[6] S. Pinho, P. Robinson, and L. Iannucci. Developing a four point bend specimen tomeasure the mode I intralaminar fracture toughness of unidirectional laminatedcomposites. Composites Science and Technology 69 (2009), 13031309.

[7] M. Laffan et al. Measurement of the in situ ply fracture toughness associated withmode I fibre tensile failure in FRP. Part I: Data reduction. Composites Science andTechnology 70 (2010), 606613.

[8] S. Pinho, P. Robinson, and L. Iannucci. Fracture toughness of tensile and compressivefibre failure modes in laminated composites. Composites Science and Technology66 (2006), 20692079.

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[9] G. Catalanotti et al. Measurement of resistance curves in the longitudinal failure ofcomposites using digital image corroleation. Composites Science and Technology70.13 (2010), 198693.

[10] C. Soutis. Compressive strength of unidirectional composites; measurement andpredictions. ASTM STP 1242 (1997), 168176.

[11] P. Berbinau, C. Soutis, and I. Guz. Compressive failure of 0 unidirectional carbon-fibre-reinforced plastic (CFRP) laminates by fibre microbuckling. Composites Scienceand Technology 59 (1999), 14511455.

[12] W. Slaughter, N. Fleck, and B Budiansky. Microbuckling of Fibre Composites:The Roles of Multi-Axial Loading and Creep. J Eng. Mater. & Techn 115 (1993),308313.

[13] A. Jumahat et al. Fracture mechanisms and failure analysis of carbon fibre/toughenedepoxy composites subjected to compressive loading. Composites Structures 92 (2010),295305.

[14] B. Budiansky. Micromechanics. Computer and structures 16 (1983), 312.[15] B. Budiansky and N. Fleck. Compressive failure of fibre composites. J. Mechanics

and Physics of Solids 41.1 (1993), 183211.[16] N. Fleck. Compressive failure of fibre composites. Advances in applied mechanics

33 (1997), 43117.[17] R. Gutkin et al. On the transition from shear-driven fibre compressive failure to fibre

kinking in notched CFRP laminates under longitudinal compression. CompositesScience and Technology 70 (2010), 12231231.

[18] C. Soutis, P. Curtis, and N. Fleck. Compressive failure of notched carbon fibrecomposites. Proc R Soc 440.1909 (1993), 24156.

[19] C. Soutis and P. Curtis. A method for predicting the fracture toughness of CFRPlaminates failing by fibre microbuckling. Composites Part A 31 (2000), 733740.

[20] J. Eshelby. The force on an elastic singularity. Phil. Trans. R. Soc. London 244(1951), 87112.

[21] M. Alvarez E. Characterization of impact damage in composite laminates. FFA TN1998-24. Tech. rep. The Aeron Res Inst of Sweden, Bromma., 1998.

[22] Z. Petrossian and M. R. Wisnom. Prediction of delamination initiation and growthfrom discontinuous plies using interface elements. Composites Part A: AppliedScience and Manufacturing 29.56 (1998), 503 515.

[23] A. Weiss et al. Influence of ply-drop location on the fatigue behaviour of taperedcomposites laminates. Procedia Engineering 2.1 (2010), 1105 1114.

[24] A. Turon et al. Accurate simulation of delamination growth under mixed-modeloading using cohesive elements: Definition of interlaminar strength and elasticstiffness. Composite Structures 92 (2010), 18571864.

[25] E. Barbero and J. Reddy. Modeling of delamination in composite laminates using alayer-wise plate theory. International Journal of Solids and Structures 28.3 (1991),373 388.

[26] J. Reeder and J. Crews Jr. Mixed-Mode Bending Method for Delamination Testing.AIAA Journal 28 (1990), 12701276.

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[27] M. Juntti, L. Asp, and R. Olsson. Assessment of Evaluation Methods for the Mixed-Mode Bending Test. Journal of Composites Technology and Research 21 (1999),3748.

[28] G. Bao et al. The role of material orthotropy in fracture specimens for composites.International Journal of Solids and Structures 29 (1991), 11051116.

[29] B. F. Sorensen and P. Kirkegaard. Determination of mixed mode cohesive laws.Engineering Fracture Mechanics 73 (2008), 2006.

[30] U. Stigh et al. Some aspects of cohesive models and modelling with special applicationto strength of adhesive layers. International Journal of Fracture 165 (2010), 149162.

[31] J. Rice. A path independent integral and the approximative analysis of strainconcentration by notches and holes. Journal of applied mechanics 88 (1968), 379386.

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Part II

Appended Papers AB

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Paper A

An experimental method to determine the fracture prop-erties of compressive fibre failure in unidirectional CFRP

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An experimental method to determine the fracture properties of

compressive fibre failure in unidirectional CFRP

D. Svenssona, K.S. Alfredssona, U. Stigha, N.E. Janssonb

aUniversity of Skovde, S-541 28 Skovde, SwedenbGKN Aerospace Engine Systems Sweden, S-461 81 Trollhattan, Sweden

Abstract

A novel test specimen geometry for studies of longitudinal compressive failure of compos-ites is proposed. Damage is localized without using a premade in-plane notch. Instead,localized high strains are achieved by decreasing the out-of-plane thickness towards theanticipated damage region which contains exclusively unidirectional lamina. Thus, an iso-lation of the compressive fibre failure mode is permitted. Experiments performed showthat the test geometry produces fracture in the form of kink-band formation progressingalong the section with the smallest out-of-plane thickness. Fibre kinking takes place inthe direction with the least support from the surrounding material, i.e. in the out-of planedirection which is perpendicular to the direction of kink band progression. A method toextract the fracture energy associated with initiation of fibre-dominated compressive fail-ure is developed. The method is based on the concept of equilibrium of configurationalforces and full-field measurements of the strain field. Numerical simulations are used as atool for evaluating the local response in the most strained region of the test specimen. Theessential features of the response is captured by modelling the body of the test specimen asa continuum and the damage region as a cohesive zone. An important finding is that, forthe present test configuration, a cohesive law should, prior to softening, include a regionwhere the stress remains constant or is increasing with a reduced tangential stiffness as thecompression increases.

1. Introduction

Longitudinal compressive failure in Carbon Fibre Reinforced Polymer (CFRP) laminateshas been a topic for intensive research during the last decades. These materials are ingeneral brittle. However, stresses around e.g. holes are relaxed by progressive damagein a fracture process zone. If this process is ignored in an elastic stress analysis, theload capacity is underestimated since the high stresses at the boundary of the holes areredistributed. Thus, the process of progressive damage needs to be modelled. In [1],Guynn and Bradley analyze the zone of compressive failure based on the Dugdale model.Later Soutis et al. [2] analyses open hole compression specimens with a linear softeningcohesive zone model. The compressive strength and the size of the damage zone at failureare successfully predicted for various lay-ups and hole sizes. In [3], Budiansky and Fleck

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develop micromechanical models to predict the behaviour in the damage region. Pinho etal. gives an overview of the processes involved in a recent review [4].

Modelling of the structural integrity requires the fracture energy as input parameter.Standard tests to measure the fracture energy have been developed for certain failure modesbut no method has yet been successfully developed for measurement of fibre dominatedcompressive failure. In [5] Pinho et al. use a compact compression (CC) specimen with across ply laminate and a fracture energy of 79.8kJ/m2 is reported. Later Catalanotti et al.[6] use the same specimen geometry and layup with another material system and a fractureenergy of 45.7 kJ/m2 is reported. It is noted that partitioning of the fracture energy isneeded since the longitudinal fibre failure is accompanied by intralaminar cracking in the90-plies. The fracture energy of intralaminar cracking is however found to be of negligiblemagnitude compared to the magnitude associated with fibre failure. However, possibleinteraction effects are neglected. Thus, measurement of the fracture energy using a UD-layup is desirable. In [7], Laffan et al. use a 4-point bending specimen with a UD-layupand a fracture energy of 25.9 kJ/m2 is reported. It is reported that failure initiates by amode II crack that propagates a small distance prior to transition to in-plane kink-bandpropagation. This phenomenon is explained by Gutkin et al. in [8]. However, fibre splittingsometimes occur prior to fibre failure when using UD-layups, cf. e.g. [9]. This is avoidedwith cross ply laminates.

In this work, a modified CC-specimen is used to estimate the fracture energy associatedwith longitudinal compressive failure and a data reduction scheme is derived by use ofEshelbys concept of the equilibrium of configurational forces, [10]. How the concept ofequilibrium of configurational forces and cohesive modelling are related is presented in e.g.[11]. In CC-tests, the compressive strain is usually localized by the use of a premade notchthat achieves a stress rising effect. Here, the damage is localized without using a premadenotch. Instead, the out of plane thickness is decreased towards the anticipated damageregion and longitudinal compressive failure is obtained within a small region consisting ofa UD-layup. FE-simulations are performed to validate the experimental results where acohesive zone models the damage region. Cohesive zone models fit well into the structureof displacement based FE-programs and offers a convenient way to model the damage zone.To this end, not only the associated critical energy release rate has to be determined; theshape of the cohesive law must be defined. In this paper, governing parameters of thecohesive law are estimated from experimental results. These are implemented in Abaqus6.11 as a cohesive user material (UMAT). Simulations capture the main features of theexperimental behaviour.

The plan of the paper is as follows: First the method is presented in section 2. Insection 3 the experiments are presented. FE-simulations are presented in section 4. Thisis followed by a discussion of the results. The paper ends with a summary of the mainconclusions.

2

8

110

24.7

8.7

14 90

0.8

L1

L2 19 65

15.6

4

Figure 1: Sketch of specimen geometry. All dimensions in mm

2. Method

2.1. Design of specimen and test setup

The geometry of the specimen is shown in Fig. 1. A multidirectional (MD) laminate isused with nominal thickness, height and length of 4 mm, 65 mm and 110 mm, respectively.The material is manufactured using Resin Transfer Moulding (RTM) with a UD fabricreinforcement made from HTS carbon fibres and RTM6 resin. The nominal laminate layupindicated by L1 in Fig. 1 is [0/+60/-60/0/0/+60/-60/0]s with the 0-direction parallel tothe loading direction. At the mid-section, the thickness is 0.8 mm and all fibres are orientedparallel to the loading direction. To achieve this, the four inner 60-plies are dropped offcreating layup L2 Fig. 1. Closer to the mid-section, the outer plies including the 60sare removed with a milling operation. The region at the mid-section with a height andthickness of 1 mm and 0.8 mm, respectively, is denoted the waist, cf. Fig. 1.

The test-setup is visualized in Fig. 2. The specimen is mounted into a LLOYD loadingframe with a 10 kN load cell by inserting the free arms between two steel plates which en-sures that the load is applied symmetrically and also prevents global buckling deformation.Solid cylinders with a slightly smaller diameter are inserted into the holes in the platesand the specimen. Frictional forces on the contact surfaces of the specimen are reducedby lightly grinding them down. The lower cylinder is held fixed and the upper cylinder issubjected to quasi-static loading by applying a downwards prescribed displacement of 50m/s.

To capture relevant displacements, the specimen is prepared for digital image correla-tion (DIC) using an Aramis system by spraying a black-on-white speckle pattern on oneof the lateral surfaces of the specimen. A sufficiently large region is monitored with theAramis system so that displacements and rotations of the loading points are measured

3

Figure 2: Experimental setup

simultaneously with the strains along the notch of the specimen. Throughout the ex-periments, images are taken every five seconds by the Aramis system. For the presentexperimental series about 300 pictures are taken in each experiment.

2.2. Theoretical framework

In fracture mechanics, fracture energies are often evaluated by using a curve integral des-ignated the J-integral [12],

J =

C

(Wdy Tiui,1) dC =C

(Wn1 Tiui,1)dC (1)

where W is the strain energy density and Ti = ijnj is the traction vector acting onthe outside of the path C with the outward normal vector ni. Index notation is usedwhere i = 1, 2 indicate components along the x and ydirection; summation is indicatedby repeated indices and partial differentiation by a comma. The J-integral is zero forany closed path if the enclosed region does not include any singularity. With the notionsingularity, we here mean any object or feature of the body that, when moved in the elasticfield, changes the potential energy of the body.

For a test geometry with a constant out-of-plane thickness, evaluation of the J-integralalong the path indicated in Fig. 3 would yield an energetic balance in terms of energyper unit surface. However, the present specimen geometry has a varying out-of-planethickness. Thus, a generalized form of the J-integral has to be introduced in terms of thesurface integral,

P =

S

(W1j ijui,1)njdS =

S

(Wn1 Tiui,1)dS (2)

4

A B

CD

E

H

G

G

F

F x

y

r

r

b

b

s

waist

Figure 3: Integration path (left) and detail of the notch (right)

where ni is the outward normal vector on the surface S. In the following, indices i = 1,2,3indicate components along the x, y andzdirection, respectively. From the P-integral,an energetic balance can be formulated in terms of energy per unit length by exploitingthe fact that P is zero for a closed surface if the enclosed volume does not include anysingularity. The P-integral can also be interpreted as the sum of all configurational forcesthat act on the volume with outer boundary S. For the present geometry, the P-integralis evaluated over the closed surface that consists of the lateral surfaces and the surfacecreated when the dashed path indicated in Fig. 3 cuts through the thickness.

The resulting configurational force from each sub-surface is studied for the presentgeometry. Both terms in Eq. (2) are zero for the lateral surfaces and the surfaces thatincludes the paths A-B, C-D, E-F and G-H, since both n1 = 0 and T = 0 on thesesurfaces. Moreover, in the design of the specimen, special precaution is taken so thatsurfaces that include the vertical boundaries A-H, E-D and B-C are virtually undeformed,i.e. W = 0 andT = 0. That is, these surfaces are not associated with any configurationalforces. Thus, the only surfaces that provide non-zero configurational forces are the surfacesalong the notch G-F and along the holes where the loads are applied. The magnitudes of thecorresponding configurational forces are denoted Pnotch and Pload, respectively. Equilibriumof the configurational forces may then be formulated as

Pload = Pnotch (3)

The left hand side in Eq. (3) is dominated by the second term in Eq. (2) and can beformulated as

Pload = F (1 + 2) (4)

where F, 1 and 2 are the applied load and the rotations at the loading points, respectively[11]. No traction is acting on the surface that includes the boundary F-G, whereby theconfigurational force associated with this surface can be formulated as

5

Pnotch =

Snotch

Wdydz (5)

This configuational force can be separated into three terms corresponding to the configu-rational forces associated with the sub-surfaces that includes the boundaries indicated byr, b and waist, respectively, cf. Fig. 3. Thus we write

Pnotch = Pb + Pr + Pwaist (6)

Since the fracture process is assumed to be concentrated to the waist, the main focus is todetermine Pwaist. From Eqs. (3) and (6) we derive

Pwaist = Pload Pr Pb (7)In the experimental evaluation, the strains measured along the boundaries r and b are usedto calculate Pb and Pr. Since the strains are only measured on one of the lateral surfaces,assumptions have to be made regarding the variation of stress and strain componentsthrough the thickness. Since the tangential stress is the dominating stress componentalong the boundary, it is assumed that only this stress component contributes to the strainenergy. Furthermore, we assume that the state of strain is constant through the thicknessand that the deformation along b and r in Fig. 3 is governed by linear elasticity. Thus,the strain energy density is calculated as if each element along the boundary is subjectedto uniaxial stress, i.e.

W =1

2Et

2t (8)

where t is the tangential strain along the boundary and Et is the direct elastic modulusin the same direction. It is noted that Et varies along the notch due to the continuouslyvarying laminate layup and due to the change in direction along the radii.

Since the specimen has a constant out-of-plane thickness at the waist, its configurationalforce can be interpreted as the mean value of the energy release rate consumed in the waistmultiplied with its out-of plane width, b. A nominal energy release rate associated withthe waist is therefore given by

Jwaist = Pwaist/b (9)

The maximum value of Jwaist is identified as a measure of the fracture energy of the unidi-rectional material in the waist. It is denoted Jwaist,c.

2.3. Data reduction scheme

The configurational force Pload is determined with high precision by measuring the loadand the load point rotations, cf. Eq. (4). To determine Pr and Pb, the tangential strain,t, along the boundary G-F is measured with the Aramis system. Strains are recorded inpoints located every 0.15 mm along the boundary.

6

Given the strains along the boundary b, the strain energy density W is obtained from Eq.(8), where the elastic modulus, Et, is determined with laminate theory. Note that Et variesalong the boundary b due to the reduction of thickness by dropping of the inner 60-pliesand machining of the outer plies, as described earlier. Finally, Pb is formed by integrationsimilarily to Eq. (5).

The resolution of the strain measurements is not high enough to accurately measurethe strain along the radii and calculate Pr directly by integration. It is here assumed thatPr can be established based on measurement of the tangential strains at the points F andG, i.e. Fxx and

Gxx, cf. Fig 3. The relation between Pr,sim and the square of the tangential

strain at points F or G is determined from a linear elastic FE-simulation. A coefficient is first determined from the FE-simulation according to

=Pr,sim

(Fxx,sim)2=

Pr,sim(Gxx,sim)

2(10)

where it is noted that Fxx,sim = Gxx,sim due to symmetry. However, in the experiments

symmetry cannot be guaranteed, i.e. Fxx,exp and Gxx,exp are generally not equal. Thus,

the configurational force associated with the radii is evaluated from the experimental datathrough

Pr,exp =

2

[(Fxx,exp

)2+

(Gxx,exp

)2] (11)

where a correction factor is introduced to account for difficulties in measuring the tan-gential strains exactly at the positions F and G, at the start of the radii. To set thevalue of for a certain experiment it is assumed that, in the initial stage of the experi-ment, Pr,exp is related to Pb,exp with the same ratio as in a linear elastic FE-simulation,i.e. Pr,exp = Pb,exp where = Pr,sim/Pb,sim. For the present test specimen 0.46 andthe correction factor varies between 1.4-1.7 for the experiments presented in the nextsection. This suggests that the tangential strains, Fxx,exp and

Gxx,exp, are underestimated

by 15-23 %, cf. Eq. (11). Estimating the distance between the measuring points andpoints F/G based on the resolution in the DIC-measurement; this error seems reasonableconsidering the strain variation near the radii obtained with a FE-simulation. This typeof measurement error is considered neglectible in the evaluation of Pb. In FE-simulations,the strain gradient component normal to the loading direction is noticed to be small alongthe boundary b.

3. Experiments

Two experimental series, denoted series 1 and series 2, are carried out. Series 1 and 2 consistof five and four experiments, respectively. In series 2, measurement problems precludeevaluation according to the method discussed in the previous section. However, the loadsat failure are considered accurate and are presented below. Therefore, only experimentsfrom series 1 (denoted experiment 1-5 in the following) are fully evaluated in this paper.Experiments 1-4 are loaded beyond compressive failure whereas experiment 5 is unloaded

7

prior to failure and a microscopy study is performed in an attempt to detect damagemechanisms that eventually lead to compressive failure. These are difficult to identify in afailed specimen since compressive failure occurs in an unstable and catastrophic manner.

Composite specimens subjected to compressive loading are sensitive for variations ofgeometry and imperfections such as voids and fibre waviness. Therefore, the quality of thespecimens are studied. Specimen 2 is, after failure, cut and analysed in a microscopy study.It is observed that the present specimen is manufactured as intended with all lamellaspresent with the intended fibre orientation and ply drops are accurately positioned. Also,three tensile tests are performed on coupon specimens cut from tested specimens by cuttingten mm to the left, parallel with the boundary BC, cf. Fig 3. A FE-model of the tensiletest is analysed with Abaqus 6.11 using shell elements and the composite layup applicationto model the material. The elastic stiffnesses measured in the coupon tests are in goodagreement with the FE-simulations. A certain amount of fibre waviness is observed inthe UD-material at the waists of the specimens which probably reduces the compressivestrength, cf. e.g. [13].

The specimens fail due to compressive failure in the waist. In the microscopy studyof specimen 2, a kink-band is observed in the waist and the kink-band tip is found ap-proximately 47 mm behind the start of the waist. Magnified images of the kink-band tipare shown in Fig. 4. A kink-band height of about 200 m is observed. This correspondsto about 2-3 times larger than earlier reported kink-band heights, cf. e.g. [8] and [13].Possible reasons are briefly discussed in section 5.

Figure 4: Kink-band observed in the waist after failure. Two different magnificationsare shown and the scale bar in the left and right image indicates 1000 m and 500 m,respectively

In Fig. 5, load-displacement curves are shown; results from series 1 are presented in Fig. 5aand results from series 2 are presented in Fig. 5b. In series 1, the load point displacement,, is measured with high accuracy using the Aramis system. Experiments presented in Fig.5a are denoted experiment 1-5 from left to right. However, the displacements presented inFig. 5b do not correspond to the load point displacements. Due to measurement problems,

8

F(kN)

(mm)

00

1

10.2 0.4 0.6 0.8 1.2 1.4

2

3

4

5

6

F(kN)

(mm)1.6

00

1

10.2 0.4 0.6 0.8 1.2 1.4

2

3

4

5

6

Figure 5: Experimental results a) Load-displacement curves from experiment 1-5. Exper-iment 5, unloaded at F=4.4 kN, is indicated with +-signs b) Load versus displacement ofthe loading frame actuator from series 2

the applied load is plotted versus the position of the loading frame actuator denoted .At the critical load, all experiments fail in an unstable manner. The load drops rapidly.Afterwards, the load decreases with increasing displacement. The critical load varies withabout 20% which seems reasonable since failure is governed by an instability and smallvariations of the local geometry may lead to large variation of the stability load. The eventsat the unstable failure are not captured since the displacements are only recorded everyfive seconds and a jump of load and displacement is recorded at failure. The vertical dropshown in Fig. 5b is recorded since the displacement of the load actuator is prescribed ata rate of 50 m/s. All experiments show in principal an elastic brittle behaviour with aninitial stiffness of about 20 kN/m. At higher loads, the behaviour is slightly nonlinear andat 90 % of the critical load, the stiffness is reduced to about 85 % of the initial stiffness.

The applied load versus local compression in the waist (F-w) is shown in Fig. 6 forthe five experiments in series 1. The curves are only recorded to the point of failure sincemeasurement of w is only possible prior to failure. Here, w is measured from the relativedisplacement of two points located in the waist with an initial distance of 0.7 mm. Notethat values of w are presented with different scales in Fig. 6.

It is observed, that all experiments show similar initial behaviour. At higher loadsat about 2.5-4.0 kN, a substantial decrease of the slope is clearly visible. At this pointw 7 10 m, which corresponds to a nominal strain at the waist of 1.0-1.4 %. Thispoint is interpreted as damage initiation in the waist. As the load increases, the form ofthe F-w -curves differs a lot and a general local response is difficult to identify. A largevariation of w at failure is observed. This type of variation is expected at the local levelsince the behaviour is highly dependent on material imperfection and variations of the localgeometry. Experiment 5 was unloaded at F=4.4 kN and in Fig. 6 a noticable decreased

9

0 5 10 15 20 25 30 35 400

1

2

3

4

F(kN)

w (m)

Experiment 1

0 10 20 30 40 50 600

1

2

3

4

5

F(kN)

w (m)

Experiment 2

0 20 40 60 80 100 1200

1

2

3

4

5

6

F(kN)

w (m)

Experiment 3

0 5 10 15 20 25 300

1

2

3

4

5

F(kN)

w (m)

Experiment 4

0 5 10 15 200

1

2

3

4

F(kN)

w (m)

Experiment 5

Figure 6: Applied load vs local compression at the waist

10

2 4 6 80

0

1

2

3

4

5

6

710 12 14

S (mm)

t/F(1/MN)

Figure 7: Strain along S. Solid line: strain at F=1.4 kN. Dashed line: strain at F=4.5 kN

Table 1: In-plane mechanical properties in the material principal directions.

E11 (GPa) E22 (GPa) G12 (GPa) 12

120 10 3.5 0.25

slope is recorded prior to unloading. However, in the microscopy study, no obvious signs ofdamage are apparent that can explain the reduced slope in Fig. 6. One may note that Fig.6 indicates a remnant deformation after unloading of experiment 5 of about 2 m. Thiscorresponds to a strain of about 0.3 %. It is questionable if such a small variation in localgeometry is detectable in an optical microscopy study. Since the F-w -curves after initiationof local nonlinearity has a large variation, more experiments are needed to determine thelocal mechanisms that lead to compressive failure.

The tangential strain along the notch, i.e. the boundary indicated with b-waist-b inFig. 3, is shown in Fig. 7. The two curves correspond to two different stages in experiment4. The solid line corresponds to a low value of the load, F = 1.4 kN, and the dashed linecorresponds to a higher load level F = 4.5 kN. Both curves are normalized with respectto the currently applied load. It is observed that the strain is concentrated in the waistat the higher load level while strains elsewhere along the notch are relaxed as comparedto the case for the low load level. That is, at points along the notch located remote fromthe waist, strains do not increase linearly with the load. This indicates a redistribution ofstresses and strains when damage is progressing in the waist.

Mechanical stiffnesses used in the data reduction scheme are presented in Table 1. Theevaluation method discussed in the previous section is applied to experiments 1-4 and theresults are shown in Fig. 8. To the left, the evaluated configurational forces Pload, Pr andPb are shown versus the applied load. From these results, the energy release rate associated

11

P(N

)

F (kN)

PbPr

Pb + PrPload

100

2 3 4

10

20

30

40

50

60

70

F (kN)

Jwaist(kN/m) Experiment 1

100

2 3 4

5

10

15

25

30

20

P(N

)

F (kN)

PbPr

Pb + PrPload

10 50

2 3 4

10

20

30

40

50

60

70

F (kN)

Jwaist(kN/m) Experiment 2

10 50

2 3 4

5

10

15

25

30

35

20

P(N

)

F (kN)

PbPr

Pb + Pr

Pload

10 50

2 3 4 6

102030405060708090

100110

F (kN)

Jwaist(kN/m) Experiment 3

10 50

2 3 4

5

6

10

15

25

30

35

20

40

P(N

)

F (kN)

PbPr

Pb + Pr

Pload

10 52 3 40

10

20

30

40

50

60

70

80

505

F (kN)

Jwaist(kN/m) Experiment 4

10 52 3 40

10

15

25

20

Figure 8: Left: Configurational forces versus applied load. Right: Jwaist versus appliedload

12

with the waist, Jwaist, is calculated according to Eq. (7) and Eq. (9). To the right in Fig. 8,Jwaist is shown versus the applied load. The fracture energy is determined as the maximumvalue of Jwaist.

It is observed that Jwaist 0 in the early stages of the experiments. This is expectedprior to damage initiation in the waist. Near failure, the Jwaist-curves show varying be-haviour, arising from the measured strain along the boundary b. It is resonable to assumethat the assumption of linear elasticity results in conservative measurement of the fractureenergy since Pb is overestimated. This effect is particularly revealed in experiment 2, whereJwaist is decreasing due to a sudden increase of Pb. At this stage, it is observed in the DIC-measurement that the damage zone appears to expand into the boundary b and since linearelasticity is assumed along b, it results in a substantial increase of Pb. Of course, thoroughconclusions are difficult to establish since only strains on the lateral surface are avaliablefor measurement. The fracture energy, Jwaist,c, associated with compressive failure in thewaist for experiment 1-4 are: 25.5, 30.5, 39.8, and 23.8 kN/m respectively. In the followingsection, these values are compared with corresponding results from FE-simulations of thepresent experimental series.

4. Finite element simulations

Numerical simulations are performed in order to verify the experimental results and toincrease the understanding of the fracture process. For the present geometry, a globalinstability occurs at the critical load and the load drops instantaneously, cf. Fig. 5.Therefore, implicit dynamic simulations with numerical damping are performed to enablecapturing of the fracture process. This is done by using the quasi-static application inAbaqus.

A 2D-model of the specimen is created. The body of the test specimen is modelledas a continuum and the fractured region as a cohesive zone. To be more specific, theanticipated damage zone in the waist is modelled as a cohesive zone with the height 200m corresponding to the measured height of the kink-band, cf. Fig. 4.

Linear elastic plane stress elements are used for the continuum part of the FE-model.The mesh consists mainly of fully integrated 4-node elements, except near the radii wheretriangular elements occur. The varying thickness is modelled by dividing the specimeninto several sections as shown in Fig. 9. The height of the sections in the area of reducedthickness is typically 1 mm and the thickness of each section is determined as the meanthickness of the corresponding section of the specimen. Each section is assigned orthotropicproperties calculated with laminate theory and material properties according to Table 1.

The cohesive zone is modelled with 4-node cohesive elements. The deformation of thecohesive elements is governed by a cohesive law describing the normal compressive stress,, as a function of the compression, w. At this point it is important to note the differencebetween w and w; w is the compression measured as the relative displacement of twopoints/nodes at the waist with an initial distance of 0.7 mm while w is the compression inthe cohesive element with height 200 m.

13

Figure 9: Geometry of FE-model with indicated sections

Along the horizontal lines where the continuum elements and the cohesive elements sharenodes, the element sides are about 50 m long. This element size has proven to be smallenough to capture the global instability without numerical problems. Simulations havebeen performed with smaller elements, only to obtain the same results and longer compu-tation time. A 3D-model has also been used to validate that the 2D-model captures theglobal response satisfactorily. In the following, only results from the 2D-simulations arepresented and compared to the experimental results.

To model the loading, the boundary of each hole is subjected to a pin-joint constraintwith respect to the centre of the hole. The center of the hole is subjected to displacementboundary conditions that corresponds to the experimental loading.

Compressive failure of CFRP has earlier been modelled using cohesive laws. In e.g. [2]a cohesive zone model is used to model the fracture process in an open hole compressionspecimen. The damage evolution is often formulated assuming that, after onset of damage,the cohesive stress is decreasing linearly with the compression, cf. Fig. 10a. Such modelsare also available for use in conjunction with continuum elements in commercial FE-codes,e.g. Abaqus. However, simulations of the present geometry show that a linear softeningmodel incorrectly predicts that the load decreases at the onset of damage.

For the present experiments, the substantial decrease in the slope of the F-w -curvesin Fig. 6 is interpreted as initiation of damage. Beyond this stage the load increasesmonotonically with increased local compression until failure of the waist. Therefore, acohesive law should in this case, prior to softening, include a region where the stressremains constant or is increasing with a reduced tangential stiffness as the compressionincreases. Thus, we propose a cohesive law according to Fig. 10b. In this cohesive law,three values of the compressive stress 1, 2 and 3 and the corresponding compressionsw1, w2 and w3 are input parameters to be defined.

The first part of the cohesive law, i.e. stage 1 in Fig. 10, is assumed to be linearlyelastic. The corresponding initial stiffness is taken as the longitudinal stiffness of thepresent UD-laminate, i.e. w1 is given by w1 = 1hcz/E1, where E1 is the longitudinal

14

1

w1 wc w

21

3

w1 w2 w3

12

34

w

Figure 10: Left: Linear softening cohesive model Right: Proposed cohesive model

Table 2: Parameters derived from the simulations

Experiment 1 (MPa) 2 (MPa) 3 (MPa) w1 (m) w2 (m) w3 (m)

1 540 737 0 0.90 31 502 750 750 0 1.25 40 753 645 790 0 1.07 105 1404 645 925 0 1.07 23 68

elastic stiffness of the unidirectional composite in the waist and hcz is the height of thecohesive element. The stress 1 at the end of stage 1 is determined from a linear elasticsimulation by matching the load at damage initiation to the corresponding experimentalvalue. The tangent stiffness of stage 2 in the cohesive law is determined from nonlinearsimulations by matching the slopes in the experimental and simulated F-w -curves duringdamage growth. The values of 2 and w2 are determined by matching the critical loadfrom the simulations to the experimental value.

Table 2 gives the numerical values of the parameters obtained for the four experiments.The graphs in the left part of Fig. 11 show comparisons between the experimental F-w -curves and the corresponding curves obtained from simulation of the first two stages ofthe cohesive law, cf. Fig. 10. The experimental and simulated results are indicated withsolid lines and dashed lines, respectively. The numbered arrows indicate the correspondingstage in the cohesive law in the cohesive element at the notch, i.e. the most strainedcohesive element. These stages are indicated by the encircled numbers in Fig. 10. Theinitiation of stage 3 is shown in the simulated curves. At this point, the softening initiatesand the simulated compression exhibits a jump indicated by the horizontal ending of thesimulated F-w -curves. As described earlier, the experimental F-w -curves are shown untilfailure, since the measuring points the are lost at this stage. It is believed that the essentialfeatures of the cohesive law up to the maximum stress, 2, are well captured, even thoughthe agreement is far from perfect.

As mentioned previously, unstable failure occurs when proceeding the simulations into

15

w(m)

F(kN)

1

2

0

1

2

3

4

10 20 300 40 1

F(kN)

(mm)

1

23

40

1

2

3

4

0 0.2 0.4 0.6 0.8

w(m)

F(kN)

1

2

0

1

2

3

4

5

10 20 300 40 50 60 1

F(kN)

(mm)

1

23

4

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1.21.2

w(m)

F(kN)

1

2

0

1

2

3

4

5

6

200 40 60 80 100 120 1

F(kN)

(mm)

1

23

4

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1.2

w(m)

F(kN)

1

2

0

1

2

3

4

5

10 20 300 1

F(kN)

(mm)

1

23

40

1

2

3

4

5

0 0.2 0.4 0.6 0.8

Figure 11: Comparisons of simulated and experimental results. Experimental results areindicated with solid lines and simulated results with dashed lines

16

Table 3: Critical loads and fracture energies from experiments and simulations

Experiment F expc (kN) Jexpwaist,c (kN/m) F

simc (kN) J

simwaist,c (kN/m)

1 4.17 25.5 4.20 21.22 4.78 30.5 4.75 31.33 5.90 39.8 5.90 76.84 4.88 23.8 4.87 20.2

the softening part of the cohesive law, i.e. stage 3 in Fig. 10. Thus, the dynamic simulationpredicts a sudden drop of the load exactly as recorded in the experiments. During thesudden load drop, the deformation of the most strained cohesive element reaches the endof the softening part of the cohesive law. Thus, this part of the cohesive law cannotbe determined by comparison of the simulation with experimental results. Instead, thenegative slope in stage 3 of Fig. 10 is chosen small enough so that convergence is obtainedin the simulation increments immediately after the load drop. To determine the stresslevel 3 in stage 4 of the cohesive law, the experimental load-displacement curves to theright in Fig. 11 are studied. In the present case 3 = 0 gives the best agreement with theexperimental load-displacement curves after failure. It seems unreasonable that no stress isacting between the fractured surfaces. However, it can be expected that significant damage,which is not modelled here, is introduced at neighbouring areas where compressive failurehas occurred.

The graphs in the right part of Fig. 11 show comparisons of simulated and experimentalload-displacement (F ) curves. Note that experimental values of displacement are onlyrecorded every five seconds and therefore the experimental response during the sudden loaddrop is not recorded. With this in mind, the comparison shows that the essential featuresof the global behaviour are well captured.

The simulated and experimental values of the fracture energies, Jwaist,c, and criticalloads, Fc, are presented in Table 3 and the results are also visualized in Fig. 12. Thepresented values of J simwaist,c are determined by using Eqs. (5), (8) and (9). Since failureoccurs in the simulation when stage 3 of the cohesive law is reached, J simwaist,c is given by

J simwaist,c =

waist

Wdy =

w20

dw +22(hwaist hcz)

2Et(12)

where the first term is the area beneath the cohesive law until initiation of stage 3. Thesecond term is the energy release rate consumed by the linear elastic elements along thewaist at failure. The parenthesis in the second term equals 0.8 mm since the height of thewaist and cohesive zone is 1 mm and 0.2 mm, respectively.

The fracture energies in the FE-models agree well with the experimental results forthree experiments. However, in the simulation of experiment 3, the simulated fractureenergy is about twice the experimentally obtained fracture energy. This may have severalexplainations but one that appears reasonable is that the large difference arise from the

17

Jwaist,c(kN/m)

Fc (kN)

SimulationExperiment

0 1 2 3 4 5 6 7

10

20

30

40

50

60

70

80

0

Figure 12: Comparison of fracture energies from simulations and experiments

experimental measurement of w. It is noted that w at failure is about 110 m in experiment3 while the corresponding values in experiment 1,2 and 4 are about 20-40 m. In the F-w -curve of experiment 3, three jumps of 10, 5 and 17 m, are noticed at F 3, 4.2and 5.6 kN without visibly affecting the global behaviour. It is believed that some eventoccured at the outer ply, where the compression is measured, which is not representativefor the material in the waist. Assuming that the specimen fails due to an unstable kinkingprocess, this magnitude of compression seems unlikely prior of failure. This could be anexplanation to the poor correlation in experiment 3 between the experimental fractureenergy Jexpwaist,c and the fracture energy used in the simulation J

simwaist,c. Thus, the simulated

fracture energy appears precarious for experiment 3.Clearly, numerical simulation of compressive failure is a complicated task. All aspects

of the complicated fracture process can not be expected to be captured by the idealized FE-model used in the present paper. Moreover, uncertainties associated with the experimentalmeasurements cannot be overlooked. However, the comparison between experimental andsimulated results indicates a relevance of modelling compressive failure as a cohesive zone.

5. Discussion

In the present paper a novel test specimen geometry for studies of compressive failure ofcomposites is proposed. Localized high strains are achieved by decreasing the out-of-planethickness towards the anticipated damage region which contains exclusively unidirectionallamina. Experiments performed show that the test geometry produces fracture in the formof kink-band formation progressing along the waist, i.e. along the section with the smallestout-of-plane thickness. Kinking takes place in the direction with the least support from thesurrounding material, i.e in the out-of plane direction, cf. Fig. 4, which is perpendicularto the direction of kink band progression. In [14], Berbinau et al. concludes that laminateshaving 0 outer layers tend to fail due to out-of-plane microbuckling, whereas outer off-axisplies permits in-plane microbuckling failure.

The fact that the damage region contains exclusively unidirectional lamina has severalimplications. On the one hand, the unidirectional plies do not have any support fromsurrounding plies as is the case when compression failure is tested using a cross ply laminate.

18

This might give a different failure mechanism as compared to the cross ply situation. Thisthought is supported by the fact that the height of the kink-band is 200 m in the presentstudy, which is about 2-3 times larger than kink-band heights reported in the literature,cf. e.g. [8] and [13]. Part of this difference may also be attributed to differences in thematerials studied. For example, the ply thickness in the composite studied here is 0.250 mmwhile the composite studied in [8] has a ply thickness of 0.125 mm. On the other hand,the present approach with exclusively unidirectional lamina in the damage region doesnot require the use of a special procedure to identify the contribution of the 0-directedlamina from the measured properties for the cross ply laminate. Indeed, the fractureenergy detected with the proposed method can be attributed to longitudinal compression.The proposed method yields fracture energies for the specific composite of about 20-40kN/m. In [7], the critical energy release rate is measured on a UD-specimen and a valueof 25.7 kN/m is reported. This value corresponds to the energy release rate at the onsetof damage by the formation of a shear crack initiating at the notch. It is reasonable toassume that the fracture process is different in the present work due to the substantiallysmaller thickness used in this work. This promotes out-of-plane microbuckling rather thanshear crack initiation due to longitudinal compression. Moreover, the failures observedin this work does not initiate at a premade in-plane stress riser. Using cross ply compactcompression specimens, fracture energies of 47.5 kN/m and 79.9 kN/m are reported for twodifferent materials in [5] and [6]. In [15] a fracture energy of 38.8 kN/m is reported for a(0/902/0)3S laminate from which the fracture energy associated with kink-band formationwas calculated as 76 kN/m [5].

Experiments reported in section 3 show that failure takes place in an unstable man-ner, i.e. the load drops rapidly to about half the value prior to fracture as the kink-bandformation propagates rapidly. The fact that the sudden load drop takes place under pre-scribed displacements, indicates that we are dealing with a global instability inherent to thegeometry of the test specimen. That is, the instability is analogous to similar problems en-countered for test specimens for delamination of composites, cf. e.g [16]. This conclusionis supported by the fact that the FE-simulations display a similar force/displacement-response, i.e. the load drops rapidly as soon as the most strained cohesive element entersthe softening part of the cohesive law. The obvious practical consequence of this is thatthe softening part of the cohesive law cannot be captured with the present test geometry.It is, however, believed that this can be circumvented in future developments of the testgeometry. By increasing the distance between the loading points and the most strained re-gion, a geometry more favourable from stability aspects is achieved. This kind of geometrychanges also reduces the length of the damage zone along the waist, i.e. the redistributionof stresses during an experiment is less pronounced. In an ideal situation, this would enableextraction of the fracture energy using methods not relying on full field measurement ofstrains.

The numerical simulations performed in the present paper are used as a tool for eval-uating the local response in the most strained region. The objective is to capture theessential features of the response with a model that is as simple as possible. Thus, in theFE-model, the body of the test specimen is modelled as a linear elastic continuum and the

19

fractured region as a cohesive zone with and idealized cohesive law. With a more refinedmodel, better agreement with the experimental results can be obtained. For instance, anon-linear elastic model for the continuum would be necessary to capture the nonlinearresponse initiating at about 1 kN in the experiments, cf. Fig. 11. In order to capture thecontinuous change of slope in the experimental F-w -curves, a more complex form of thecohesive law is needed. However, it is believed that the present level of model complexityis sufficient for the present purpose, i.e. to verify the experimental results and to increasethe understanding of the fracture process. In this context, an important finding is thatthe cohesive law, prior to softening, includes a region where the stress remains constant oris increasing with a reduced tangential stiffness as the compression increases. The presentevaluation method results in a maximum stress of the cohesive law, i.e. 2 in Fig. 10 inthe interval 740-925 MPa. These values are relatively low as compared to what might beexpected from previous results from the literature [17]. It is possible that the fibre wavinessthat is observed in the waist of the specimens may introduce microbuckling at a fairly lowlongitudinal stress, cf. e.g. [13]. Moreover, it is reasonable to assume that out-of-planebuckling initiates at a fairly low compressive stress in the UD material since no off-axisplies provide lateral support to prevent out-of-plane kinking, cf. e.g. [13].

6. Conclusions

A method is proposed for measurement of the fracture energy associated with longitudinalcompressive failure. Failure occurs in a region consisting of a unidirectional layup. Thus,a compressive fibre failure process is isolated and reduction of dissipated energy associatedwith non-0-plies and interaction effects are not necessary. The fracture energy is evaluatedby using a generalized J-integral and full field measurement of the strains with a DIC-system. Thus, the assumption of a small damage zone is avoided. Experimental resultsshow large scatter and a fracture energy of 20-40 kN/m is reported. FE-simulations of thereported experiments are performed. Comparing the experimental and simulated resultsindicates a relevance of modelling compressive failure as a cohesive zone. The simulationscapture the main features of the experimental behaviour.

Acknowledgements

The authors would like to acknowledge funding from the Swedish National AeronauticalResearch Program (NFFP5) in support of this work through the joint project FIKOMwith GKN Aerospace Sweden. The authors are grateful to Dr. Anders Biel for his help inconnection with the experiments and to Dr. Fredrik Edgren at GKN Aerospace Swedenfor performing the microscopy study.

20

References

[1] E.G. Guynn and W.L. Bradley. Micromechanics of compression failures in open holecomposite laminates. Annual Progress Report, Apr. 1986-Auq. 1987 for NASA re-search grant NAG-1-659. 227p, 1987.

[2] C. Soutis, N.A. Fleck, and P.A. Smith. Failure prediction technique for compressionloaded carbon fiber-epoxy laminate with open holes. Journal of Composite Materials,25:147698, 1991.

[3] B. Budiansky and N.A. Fleck. Compressive failure of fibre composites. J. Mechanicsand Physics of Solids, 41(1):183211, 1993.

[4] S.T. Pinho, R. Gutkin, S. Pimenta, N.V. De Carvalho, and P. Robinson. On longi-tudinal compressive failure of carbon-fibre-reinforced polymer: from unidirectional towoven, and from virgin to recycled. Phil. Trans. R. Soc. A, 370(165):18711895, 2012.

[5] S.T. Pinho, P. Robinson, and L. Iannucci. Fracture toughness of tensile and compres-sive fibre failure modes in laminated composites. Composites Science and Technology,66:20692079, 2006.

[6] G. Catalanotti, P.P. Camanho, J. Xavier, C.G. Davila, and A.T. Marques. Measure-ment of resistance curves in the longitudinal failure of composites using digital imagecorroleation. Composites Science and Technology, 70(13):198693, 2010.

[7] M.J. Laffan, S.T Pinho, P. Robinson, L. Iannucci, and A.J. McMillan. Measurementof the fracture toughness associated with the longitudinal fibre mode of laminatedcomposites. Composites Part A, 43:19301938, 2012.

[8] R. Gutkin, S.T. Pinho, P. Robinson, and P.T. Curtis. On the transition from shear-driven fibre compressive failure to fibre kinking in notched cfrp laminates under lon-gitudinal compression. Composites Science and Technology, 70:12231231, 2010.

[9] C. Soutis, P.T. Curtis, and N.A. Fleck. Compressive failure of notched carbon fibrecomposites. Proc R Soc, 440(1909):24156, 1993.

[10] J.D. Eshelby. The force on an elastic singularity. Phil. Trans. R. Soc. London, 244:87112, 1951.

[11] U. Stigh and T. Andersson. An experimental method to determine the completestress-elongation relation for a structural adhesive layer loaded in peel. Fracture ofPolymers, Composites and Adhesives (Eds. Williams J.G. and Pavan A.), 27:297306,2000.

[12] J.R. Rice. A path independent integral and the approximative analysis of strainconcentration by notches and holes. Journal of applied mechanics, 88:379386, 1968.

21

[13] C. Soutis. Compressive strength of unidirectional composites; measurement and pre-dictions. ASTM STP, 1242:168176, 1997.

[14] P. Berbinau, C. Soutis, P. Goutas, and P.T. Curtis. Effect of off-axis ply orientationon 0-fibre microbuckling. Composites Part A, 30:11971207, 1999.

[15] C. Soutis and P.T. Curtis. A method for predicting the fracture toughness of cfrplaminates failing by fibre microbuckling. Composites Part A, 31:733740, 2000.

[16] K.S. Alfredsson and U. Stigh. Stability of beam-like fracture mechanics specimens.Engineering Fracture Mechanics, 89:98113, 2012.

[17] J. Lee and C. Soutis. A study on the compressive strength of thick carbon fibre-epoxylaminates. Composites Science and Technology, 67(10):20152026, 2007.

22

Paper B

Measurement of cohesive laws for initiation of delami-nation of CFRP

38

Measurement of cohesive laws for initiation of

delamination of CFRP

K.S. Alfredsson, D. Svensson, U. Stigh, A. Biel

University of Skovde, S-541 28 Skovde, Sweden

Abstract

With a cohesive zone, delamination of laminated material can be analysed using non-linearfinite element codes. For carbon fibre reinforced composites, two different delaminationprocesses are identified; one at the crack-tip with a sub millimetre sized process zoneand one with a large-scale process zone associated with fibre bridging. In the presentpaper we focus on the crack-tip process that is important when crack growth has to beavoided. Methods to measure the cohesive relations associated with the fracture processesare developed. These methods are used to measure the cohesive relations in mode I andII, respectively. The material is a carbon fibre reinforced composite.

Keywords: CFRP, Cohesive modelling, Experimental, Cohesive law

1. Introduction

Delamination of Carbon Fibre Reinforced Polymer composites (CFRP) is one of the majorconcerns in the design and use of advanced composite structures. Delamination may startat unidentified defects originating from the production process or damages occurring in theuse of the component. Two different mechanisms and corresponding length scales can beidentified in the process of delamination. At the close proximity of a crack tip, a processregion can be identified. With epoxy resins, the associated fracture energy is in the range of102 N/m and the yield strength is in the range of 101 MPa. A simple estimate predicts thesize of the process zone to about 101 mm. That is too small to indicate interaction withouter boundaries of structures, but large enough to imply interaction with the fibres. Froman experimental point of view, this shows that the fracture properties should be measuredin the relevant composite and one cannot rely on bulk properties for the resin. At thelarger length scale and in the wake of a growing crack, crack bridging may occur. Thisprocess often contributes significantly and increases the total fracture energy to about 103

N/m. The bridging stress is however small, in the range of 100 MPa. That is, the processzone is very large, in the range of 101 mm, [1]. Thus, the two fracture processes areassociated with two very different length scales. In some applications, the enhancement ofthe strength due to crack bridging can be considered. However, in the aeronautic industry,no defects are allowed to grow during the use of a composite structure. Moreover, defectsfrom the production stage are likely to lack bridging fibres. Therefore, if no defects from

1

Tcrack tip

s

st

t

v

w dx

y

Figure 1: Left : Cohesive zone heading a crack tip. Traction T holds the cohesive surfacestogether. Right : Traction and separation separated in orthogonal components relative themiddle surface of the cohesive surfaces.

the productions stage are allowed to grow during use, fibre bridging cannot be consideredin aeronautical applications.

In the present paper, we study delamination at the smaller length scale, i.e. with-out considering fibre bridging. The study is performed within the framework of cohesivemodelling. As compared to linear elastic fracture mechanics, this can be viewed as a steptoward a more complete model of the actual damage process. With cohesive modelling weassume the existence of a planar process zone heading the crack tip. All inelastic materialprocesses in the real process zone are modelled by a cohesive law acting on the cohesivesurface. Figure 1 illustrates a cohesive model. The traction T is assumed to decreaseas the separation of the cohesive surfaces increases. At large enough separation, thetraction is zero indicating the formation of new crack surfaces. Historically, Barenblattintroduced the cohesive model to increase the understanding of brittle fracture in his sem-inal paper 1962, [2]. Later, a number of researchers showed the usefulness of the conceptto model fracture in a large variety of applications: e.g. strength of structures of concrete,[3], in-plane strength of composites, [4], creep crack growth, [5], and fracture of adhesives,[6]. A major step forward was the realization that cohesive models fits well within thestructure of deformation based finite element analysis, [6], [7]. That is, strength analysisof structures can be performed as non-linear stress analyses using conventional FE-codes.Today, cohesive models are included in many commercially available FE-codes.

In the present paper, we develop methods to measure the cohesive laws in mode Iand II, respectively. The theory is developed in section 2 and the experiments and theirevaluation are reported in section 3. In section 4 one of the methods developed in section2.2 is used to re-evaluate mode II experiments previously presented in [8]. The paper endswith some conclusions.

2

2. Theory and methods

Methods to measure cohesive laws have been relatively sparingly reported. The embryo tosuch methods can be traced to [9]. The J-integral gives the release of potential energy ofan elastic body per unit created crack area, associated with the propagation of the crackfront. It can be calculated from the integral

J =

C(W dy Tiui,x dC) . (1)

where C is a counter-clockwise integration path, and W , Ti and ui are the strain energydensity, the traction vector and the displacement vector, respectively, cf. Fig. 1. In-dex notation is employed with index i = 1, 2 indicating components along the x andycoordinates, respectively; summation is indicated by repeated indices and partial dif-ferentiation by a comma. The crack is assumed to lie in a plane with y = constant. IfW does not contain any explicit dependence of the xcoordinate, the integration path Cstarting at the lower crack surface and ending at the upper crack surface can be chosenfreely as long as C does not encircle objects that change the energy of the body if theobjects positions are changed. Thus, by choosing C close to the crack tip,1 we get

J =

CW dy =

w0

dw +

v0

dv (2)

where , , w, and v are the cohesive normal stress, shear stress, opening and shear atthe crack tip, respectively, cf. Fig. 1.2 That is, if we are able to continuously measure Jfrom the external loads acting on a specimen during an experiment, and at the same timemeasure v and w at the crack tip, we would be able to differentiate the measured J(v, w)data to derive the cohesive laws (v, w) and (v, w), cf. Eq. (2). Two facts complicatethis idea. Firstly, the cohesive law is not likely to be elastic in nature. That is, it is notlikely that the strain energy density W exists. However, if the loading within the cohesivezone can be regarded as monotonically increasing, a pseudo-potential A can replace W ,[10]. In this case the mathematical difference between an elastic and inelastic material isimmaterial. It is only when un-loading from an inelastically deformed state occurs that thedifference between elasticity and inelasticity reveals itself. The second problem originatesfrom the fact that most expressions for J in terms of external loads implicitly or explicitlydepend on an assumption of the material behaviour. In [9], it is however shown that somespecimen geometries allow for a direct measurement of J from the applied load withoutthe need for a too restricted assumption of the behaviour of the material. This idea is

1In the present paper, the crack tip is considered to be situated at the left end of the process zone,cf. Fig. 1. It should be noted that the definition of the position of crack tip differs among authors. Forinstance, in studies of crack bridging, the right end of the process zone is usually considered as the cracktip and the process zone is referred to as a bridging zone.

2Note that the cohesive shear stress is defined as = xy in order to get a positive shear stress nearthe crack tip for the ENF-specimen, cf. Fig. 2.

3

H/2

P, D

a b

L

aL

x

y

H/2H/2

a

L

x

y

H/2

P, D/2

P, D/2

a b

Figure 2: a) Double Cantilever Beam-specimen; b) End Notched Flexure-specimen. Bothloaded with pres