Experimental methods to determine model parameters for failure modes of CFRP

Transcription

1 Experimental methods to determine model parameters for failure modes of CFRP DANIEL SVENSSON Department of Applied Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 213

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3 THESIS FOR THE DEGREE OF LICENTIATE OF ENGINEERING IN SOLID AND STRUCTURAL MECHANICS Experimental methods to determine model parameters for failure modes of CFRP DANIEL SVENSSON Department of Applied Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 213

4 Experimental methods to determine model parameters for failure modes of CFRP DANIEL SVENSSON c DANIEL SVENSSON, 213 Thesis for the degree of Licentiate of Engineering 213:7 ISSN Department of Applied Mechanics Chalmers University of Technology SE Göteborg Sweden Telephone: +46 () Chalmers Reproservice Göteborg, Sweden 213

5 Experimental methods to determine model parameters for failure modes of CFRP Thesis for the degree of Licentiate of Engineering in Solid and Structural Mechanics DANIEL SVENSSON Department of Applied Mechanics Chalmers University of Technology Abstract The focus of this thesis is to develop methods to predict the damage response of Carbon Fibre Reinforced Polymers (CFRP). In the pursuit of reducing the manufacturing cost and weight of CFRP components, it is crucial to enable modelling of the non-linear response associated with various failure modes. Two failure modes are considered in this thesis: fibre compressive failure and interlaminar delamination. Multidirectional laminated composites are commonly used when a low weight is desired due to their high specific strength and stiffness. In a carbon/epoxy composite, almost exclusively the fibres carry the load. However, along the fibre direction, the compressive strength is considerably lower than the tensile strength. With the same reasoning, the transverse strength is considerably lower than the in-plane strength. This makes delamination and fibre compressive failure two of the major concerns in structural design. Moreover, the presence of delaminations severely reduces the compressive strength of a laminate. This can cause catastrophic failure of the structure. In Paper A, we suggest a test method for determining fracture properties associated with fibre compressive failure. A modified compact compression specimen is designed for this purpose and compressive failure takes place in a region consisting exclusively of fibres oriented parallel to the loading direction. The evaluation method is based on a generalized J-integral and full field measurements of the strain field on the surface of the specimen. 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 associated with 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 of the mode II fracture energy. Keywords: composite, delamination, compressive failure, energy release rate, cohesive modelling i

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7 Preface This work has been carried out during the years at the Mechanics of Materials research group at the University of Skövde. Funding from the Swedish National Aeronautical Research Program (NFFP5) is gratefully acknowledged. First of all, my deepest gratitude to my co-supervisor Associate Professor Svante Alfredsson for 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 his knowledge and giving me the opportunity to do research. Thanks to my present and former colleagues for their helpful attitude and creating such an enjoyable environment. Last but not certainly not least, a big thank you to my Anna for her never ending support, patience and love. Daniel Svensson Skövde, March 213 iii

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9 Thesis This thesis consists of an extended summary and the following appended papers: Paper A Paper B D. Svensson, K.S. Alfredsson, U. Stigh and N.E Jansson. An experimental method to determine the fracture properties of compressive fibre failure in unidirectional CFRP. To be submitted for international publication (213) K.S. Alfredsson, D. Svensson, U. Stigh and A. Biel. Measurement of cohesive laws for initiation of delamination of CFRP. Submitted for international publication (213) 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. Took part in designing the experimental setup and performing the experiments. Paper B: The author took part in planning the paper and developing the theory. Responsible for the simulations and the evaluation of the experiments. v

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11 Contents Abstract Preface Thesis Contents i iii v vii I Extended Summary 1 1 Introduction 1 2 Fibre compressive failure 2 3 Delamination 4 4 Summary of appended papers Paper A Paper B References 7 II Appended Papers A B 11 vii

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13 1 Introduction Part I Extended Summary Long fibre composites such as Carbon Fibre Reinforced Polymers (CFRP) are commonly used 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 and strength, low density and high resistance to corrosion. However, the wide range of possible failure modes and limited understanding of the material behaviour requires conservative dimensioning. This is further complicated by the fact that the material behaviour is dependent on e.g. lay-up, loading direction, specimen size and environmental effects such as temperature and moisture. Thus, phenomenological determination of criteria to predict failure on a structural level from coupon tests requires numerous testing. Thousands of coupon tests are not uncommon in the design of a safe advanced structure. Therefore, a major challenge in the structural design is to remove excessive safety factors to e.g. reduce the manufacturing costs and environmental impact. It is well-known from experimental experience that assuming an ideal brittle behaviour is excessively conservative since the redistribution of stresses at high strain regions are not considered. Stresses around stress risers such as holes or cut-outs are relaxed by damage evolution and further load can be applied prior to failure. Therefore, a certain loss of integrity has to be allowed. It is therefore desirable to model, experimentally determine and simulate the non-linear response that precedes failure of the laminate. Moreover, it has been observed that specimen 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 in a fracture mechanics based approach. The scientific community is devoting substantial effort to develop test methods for determination of the fracture energy for various failure modes. It is also desirable to obtain accurate failure criteria of a laminate from the mechanical properties of the fibres, the matrix and the lay-up geometry. To successfully calibrate such models, an important 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 a summary. Standardized methods have been developed for interlaminar testing, e.g. [3], and for translaminar testing [4]. Furthermore, methods have been proposed for studying longitudinal [5] and transverse intralaminar matrix failure [6]. Moreover, test methods for fibre tensile failure have been proposed in e.g. [7]. Testing methods have been proposed for fibre compressive failure in e.g. [8], [9] and [2]. In this thesis, experimental methods are developed for determining governing fracture properties associated with fibre compressive failure and interlaminar failure. These failure modes are considered to be two major design limiting fracture processes in structural design. Thus, to accurately predict these failure modes is crucial in the design of improved composite structures. 1

14 2 Fibre compressive failure Unidirectional (UD) CFRP is known for its high stiffness and strength along the fibre direction. However, the compressive strength is only about 6-7 % of the tensile strength [1]. The tensile strength is mainly governed by the tensile strength of the fibres whereas the compressive strength, to a larger extent, depends on several material properties of the laminate. Multiple failure modes are associated with fibre compressive failure, e.g. elastic microbuckling, plastic microbuckling, longitudinal fibre splitting, fibre crushing and shear-driven failure. The distinction of elastic and plastic microbuckling refers to the amount of shear strain induced in the matrix. Experimental identification of the failure mechanisms preceding compressive failure is difficult due to the unstable and catastrophic nature of fibre compressive failure. However, plastic microbuckling is identified as the dominating failure mode, cf. e.g [11], [12]. Plastic microbuckling is the failure mode where compressive loading introduces bending of fibres. This may start at a material discontinuity 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, the fibres rotate further and yielding of the matrix is induced. Ultimately, the fibres fail at two points due to a combination of axial compression and bending. A localized band of broken fibres denoted a kink-band is created and propagates into the intact region of the 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 a UD laminate are the fibre misalignment and the shear yield strengths of the composite material. In the formation of a kink-band, matrix/fibre splitting occurs in-between the rotated fibres. This induces further rotation of the fibres prior to fibre failure. This mechanism is explained by Fleck in [16]. It has also been experimentally observed in [17], where the failure process at the micro scale has been observed while the specimen is kept under load. Thus, in compression the material properties of the matrix are important since the matrix provides lateral support to the fibres. Moreover, the bond between the matrix and the fibres is important for the compressive strength. It should be noted that, even though microbuckling leading to kink-band formation is recognized as the dominating failure mode, a shear-driven fibre failure mode is observed in e.g. [17]. In [17], Gutkin et al. highlights conditions for whether the fibre fails in a shear-driven mode or in a kinking mode. Unidirectional composites are rarely used in structural applications due to their poor transverse performance. Instead, a laminated composite is built-up by successive plies with varying orientations. In [18], it is concluded that compressive failure occurs in a multidirectional centre cracked specimen due to microbuckling of the -plies and the fracture energy is increased when the portion of -plies is increased. It is therefore important to determine the fracture energy associated with compressive failure of the -plies in isolation. As yet there is no standardized test method for this purpose. A recent review of earlier reported fracture energies associated with compressive failure is given in [8]. From experiments with center crack specimens with a T8/924C laminate with (,9 2,) 3s layup reported in [19], Pinho et al. derive the fracture energy for kink-band 2

15 Figure 2.1: Kink band found in an experiment presented in Paper A. The scale bar indicates 5 µm formation to about 76 kj/m 2. In [8], Pinho et al. use a compact compression (CC) specimen with a T3/913 laminate with cross ply layup and the fracture energy 79.9 kj/m 2 associated with the -plies is reported. In the presented method, a normalized energy release rate is calculated from linear FE-simulations. The fracture energy is then directly determined from the maximum load. Later in [9], Catalanotti et al. use the same specimen and layup geometry with a different material system (Hexcel IM7-8552) and the fracture energy 47.5 kj/m 2 associated with the -plies is reported. The authors use digital image correlation to calculate the fracture energy from the actual strain field on the lateral surface of the specimen, i.e. the assumption of a small scaled damage zone is avoided. However, the use of cross ply laminates necessitates partitioning of the fracture energy since the energy dissipated in the 9 -plies needs to be deducted. Furthermore, interaction effects between the alternating -plies and 9 -plies are neglected. Therefore it is desired to determine the fracture energy by the use of a UD-layup. A test method using a four point bend specimen with a UD-layup is presented by Laffan et al. in [2]. The material system is the same as in [9]. Also here, linear elastic FE-simulations are used to determine how the energy release rate relates to the applied load. A lower fracture energy of 25.7 kj/m 2 is reported. The reason may be that the experiments are interrupted when damage initiation is first detected. The fracture energy is then calculated from the load at onset of damage. It is reported that damage is initiated by a shear crack initiation at the notch. The shear crack is propagating a small distance with the direction approximately 45 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 fracture energy associated with longitudinal compressive failure. Compressive failure takes place in a region with a UD-layup. Thus, partitioning of the fracture energy is not necessary and interaction effects between the -plies and off-axis plies are avoided. The evaluation 3

16 method is based on the concept of equilibrium of configurational forces [2] and full field measurement of the strain field. This method is therefore applicable also in the case of a large damage zone. 3 Delamination In general, the interlaminar strength of laminated composites is substantially lower than the in-plane strength of a multidirectional composite. The foremost reason for this is that no fibres are oriented in the transverse direction. Moreover, the resin rich regions between the lamina are zones of weakness. Thus, due to the comparatively poor performance in transverse loading, interlaminar delamination is a major concern in the design of structural components. In Fig 3.1, delamination in a cross ply laminate is shown. Delamination may start at a stress concentration arising from an initial defect or damages occurring in the use of the component. Moreover, a structural component often includes curved laminates or laminates with a varying thickness. If the curved laminate is subjected to out-of-plane bending, delamination may initiate due to the transverse stresses. Furthermore, ply drops are often used to progressively reduce the out of plane thickness along a composite laminate. The axial load has to be transferred to the thinner section by interlaminar shear stresses. Thus, at the ply-dropping region, delamination can initiate in a shearing mode. Studies of the influence of ply-drops and out-of-plane curvatures can be found in e.g. [22] and [23]. Moreover, delamination may also initiate and propagate due to low-velocity impact [24]. Delaminations are difficult to detect and it can severely decrease the structural compressive strength. Thus, the low compression strength after impact is a limiting design parameter in the industrial design of aero-structures. Since the structural strength is severely decreased by the presence of delaminations, a conservative prediction of the strength can be carried out by including delaminations in the model to e.g. determine the critical buckling load [25]. Two commonly used models for simulating delamination are, the Virtual-Crack-Closing-Technique (VCCT) based on linear elastic fracture mechanics and Cohesive Zone Modelling (CZM) that is not limited to small process zones. Fracture mechanics based methods have been successfully used for modelling onset and propagation of delamination in laminated composites. In a fracture mechanics approach, propagation of delamination is assumed to initiate when the total energy release rate, G = G I + G II, is equal to the fracture energy, G c, associated with the current mode mix. Here, G I is the energy release rate in mode I (opening), G II is the energy release rate in mode II (shearing) and the mode mix is given by e.g. the fraction G II /G. The fracture energy is determined experimentally for various mode-mixes. Commonly used test methods for measuring the fracture energies in mode I and mode II are, the Double Cantilever Beam (DCB)-test for mode I and the End Notch Flexure (ENF)-test for mode II. 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 G c for a range of mode-mixes, a relation between G c and the mode mix can be obtained by a best curve fit. The energy release rate from these tests methods has been calculated with different levels of sophistication. In the simplest case, the energy 4

17 Figure 3.1: 2. Delamination, tensile fibre fracture and tensile/shear matrix cracks in a cross-ply laminate. Picture taken from [21] release rate is calculated by assuming the Euler-Bernoulli beam theory and a rigid crack tip. The methods have later been extended to consider the flexibility ahead of the crack tip by adding a crack length correction to account for the influence of the anisotropic composite material. In [27], Juntti et al. give a comprehensive review of the developed evaluation methods. For the case of orthotropy, Bao et al. [28] present crack length corrections 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 cohesive modelling is given in Paper B. In a cohesive zone model, a planar damage zone is assumed where the behaviour is governed by a traction separation law. The onset of damage is modelled by a stress criterion and the traction is assumed to decrease as the cohesive separation increases. At a large enough separation, the traction acting on the cohesive surface drops to zero and crack propagation is initiated. For a given load history, the area beneath the cohesive laws in mode I and mode II corresponds to the total fracture energy. 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 critical loads of structures suffering from delamination. For example, at regions subjected to high interlaminar stresses such as ply-drop regions, the critical interlaminar stresses have a great influence on the structural behaviour. Therefore, it is desirable to experimentally measure the cohesive law with high accuracy to model the interlaminar behaviour of a laminated composite. Cohesive models of delamination are rather widely used in the scientific community. Several methodologies have been used for determining the cohesive law. One common method is to use a best fit approach. A function by which the cohesive traction varies with the cohesive separation is pre-selected. Then, the cohesive parameters that best reproduce the force-displacement curves are chosen. However, for many of the test specimen 5

18 geometries, the shape of the cohesive law has a minor influence of the force-displacement relation. 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 based on evaluation of the path-independent J-integral, cf. [31]. Several test methods are designed so that the energy release rate, J, associated with the crack tip separation can be directly measured from external loads and displacements, cf. [29] and [3]. Subsequently, the cohesive laws in mode I and mode II are determined by differentiation of J with respect to the opening separation and the shearing separation, respectively. This method does not require any assumption of the material behaviour. Furthermore, the method is valid for large damage zones. However, if the complete cohesive law for all mode-mixes is to be determined, it requires the existence of an associated potential. If a potential does not exist, then in mixed-mode loading, the cohesive law is path dependent. Thus, the cohesive law can only be determined for the specific load histories that were applied in the experiments. However, this is not an issue when the measurement of the cohesive laws is restricted to the pure mode cases, cf. Paper B. For delamination, it is noted that two different fracture processes are involved. One is associated with bridging of fibres in the wake of the growing crack and one is associated with the fracture process at the crack tip. The bridging stress behind the crack tip is small compared to the initiation stress at the crack tip. However, in the presence of fibre bridging, 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 on the fracture process associated with the crack tip until initiation of delamination. This is the 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 compressive failure. Normally when using a CC-specimen a cross ply laminate is used and an in-plane notch is used to achieve a stress rising effect to nucleate compressive failure. Thus, interaction effects between the -plies and off-axis-plies are neglected and partitioning of the fracture energy is needed to determine the fracture energy associated with the -plies. Furthermore, the data reduction scheme is often based on linear elastic fracture mechanics, i.e. the damage zone is assumed to be small. In this work, localized high strains are achieved by decreasing the out-of-plane thickness towards the anticipated damage region that consists exclusively of -plies. Thus, compressive failure of -plies is obtained in isolation. The data reduction scheme is derived from Eshelby s concept of equilibrium of configurational forces [2]. The method is similar to earlier work where the J-integral [31] is used to determine the energy release rate associated with the damage zone. However, the J-integral is not applicable for the present geometry due to the varying out of plane thickness. Thus, a generalized form of the J-integral is used to determine the energy release rate associated with the damage zone. Full field strain measurement with the DIC-system Aramis is used in the evaluation. 6

19 Thus, the assumption of a small damage zone is avoided. Numerical simulations are used to verify the experimental results. The damage region is idealized as a cohesive zone model, i.e. the material behaviour within the damage zone is governed by a cohesive law relating the compressive stress and compressive separation. A cohesive law is proposed and 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 mode I and mode II associated with interlaminar delamination. Cohesive laws for mode I and mode II are measured with the DCB- and ENF-test, respectively. With a method based on the path-independent J-integral, the energy release rate, J, associated with the crack tip separation can be measured directly from the applied load, load point rotations and the separation at the crack tip. By differentiation of J with respect to the separation at the crack tip, the cohesive laws are determined. FE-simulations are performed with the cohesive laws implemented and the simulations show good agreement with experimental results. The results indicate that the fracture energy in mode II can be 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 composite laminates: Specimen size effects. Composite Science and Technology 68 (28), [2] M. Laffan et al. Measurement of the fracture toughness associated with the longitudinal fibre mode of laminated composites. Composites Part A 43 (212), [3] ASTM. D5528 Standard test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites. 27. [4] ASTM. E Standard test method for translaminar fracture toughness of laminated polymer matrix composite materials. 24. [5] B. F. Sörensen and T. K. Jacobsen. Large-scale bridging in composites: R-curves and bridging laws. Composites Part A: Applied Science and Manufacturing (1998), [6] S. Pinho, P. Robinson, and L. Iannucci. Developing a four point bend specimen to measure the mode I intralaminar fracture toughness of unidirectional laminated composites. Composites Science and Technology 69 (29), [7] M. Laffan et al. Measurement of the in situ ply fracture toughness associated with mode I fibre tensile failure in FRP. Part I: Data reduction. Composites Science and Technology 7 (21), [8] S. Pinho, P. Robinson, and L. Iannucci. Fracture toughness of tensile and compressive fibre failure modes in laminated composites. Composites Science and Technology 66 (26),

20 [9] G. Catalanotti et al. Measurement of resistance curves in the longitudinal failure of composites using digital image corroleation. Composites Science and Technology 7.13 (21), [1] C. Soutis. Compressive strength of unidirectional composites; measurement and predictions. ASTM STP 1242 (1997), [11] P. Berbinau, C. Soutis, and I. Guz. Compressive failure of unidirectional carbonfibre-reinforced plastic (CFRP) laminates by fibre microbuckling. Composites Science and Technology 59 (1999), [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), [13] A. Jumahat et al. Fracture mechanisms and failure analysis of carbon fibre/toughened epoxy composites subjected to compressive loading. Composites Structures 92 (21), [14] B. Budiansky. Micromechanics. Computer and structures 16 (1983), [15] B. Budiansky and N. Fleck. Compressive failure of fibre composites. J. Mechanics and Physics of Solids 41.1 (1993), [16] N. Fleck. Compressive failure of fibre composites. Advances in applied mechanics 33 (1997), [17] R. Gutkin et al. On the transition from shear-driven fibre compressive failure to fibre kinking in notched CFRP laminates under longitudinal compression. Composites Science and Technology 7 (21), [18] C. Soutis, P. Curtis, and N. Fleck. Compressive failure of notched carbon fibre composites. Proc R Soc (1993), [19] C. Soutis and P. Curtis. A method for predicting the fracture toughness of CFRP laminates failing by fibre microbuckling. Composites Part A 31 (2), [2] J. Eshelby. The force on an elastic singularity. Phil. Trans. R. Soc. London 244 (1951), [21] M. Álvarez E. Characterization of impact damage in composite laminates. FFA TN Tech. rep. The Aeron Res Inst of Sweden, Bromma., [22] Z. Petrossian and M. R. Wisnom. Prediction of delamination initiation and growth from discontinuous plies using interface elements. Composites Part A: Applied Science and Manufacturing (1998), [23] A. Weiss et al. Influence of ply-drop location on the fatigue behaviour of tapered composites laminates. Procedia Engineering 2.1 (21), [24] A. Turon et al. Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: Definition of interlaminar strength and elastic stiffness. Composite Structures 92 (21), [25] E. Barbero and J. Reddy. Modeling of delamination in composite laminates using a layer-wise plate theory. International Journal of Solids and Structures 28.3 (1991), [26] J. Reeder and J. Crews Jr. Mixed-Mode Bending Method for Delamination Testing. AIAA Journal 28 (199),

21 [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), [28] G. Bao et al. The role of material orthotropy in fracture specimens for composites. International Journal of Solids and Structures 29 (1991), [29] B. F. Sörensen and P. Kirkegaard. Determination of mixed mode cohesive laws. Engineering Fracture Mechanics 73 (28), 26. [3] U. Stigh et al. Some aspects of cohesive models and modelling with special application to strength of adhesive layers. International Journal of Fracture 165 (21), [31] J. Rice. A path independent integral and the approximative analysis of strain concentration by notches and holes. Journal of applied mechanics 88 (1968),

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23 Part II Appended Papers A B 11

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25 Paper A An experimental method to determine the fracture properties of compressive fibre failure in unidirectional CFRP

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27 An experimental method to determine the fracture properties of compressive fibre failure in unidirectional CFRP D. Svensson a, K.S. Alfredsson a, U. Stigh a, N.E. Jansson b a University of Skövde, S Skövde, Sweden b GKN Aerospace Engine Systems Sweden, S Trollhättan, Sweden Abstract A novel test specimen geometry for studies of longitudinal compressive failure of composites 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 the anticipated damage region which contains exclusively unidirectional lamina. Thus, an isolation of the compressive fibre failure mode is permitted. Experiments performed show that the test geometry produces fracture in the form of kink-band formation progressing along the section with the smallest out-of-plane thickness. Fibre kinking takes place in the direction with the least support from the surrounding material, i.e. in the out-of plane direction which is perpendicular to the direction of kink band progression. A method to extract the fracture energy associated with initiation of fibre-dominated compressive failure is developed. The method is based on the concept of equilibrium of configurational forces and full-field measurements of the strain field. Numerical simulations are used as a tool for evaluating the local response in the most strained region of the test specimen. The essential features of the response is captured by modelling the body of the test specimen as a continuum and the damage region as a cohesive zone. An important finding is that, for the present test configuration, a cohesive law should, prior to softening, include a region where the stress remains constant or is increasing with a reduced tangential stiffness as the compression increases. 1. Introduction Longitudinal compressive failure in Carbon Fibre Reinforced Polymer (CFRP) laminates has been a topic for intensive research during the last decades. These materials are in general brittle. However, stresses around e.g. holes are relaxed by progressive damage in a fracture process zone. If this process is ignored in an elastic stress analysis, the load capacity is underestimated since the high stresses at the boundary of the holes are redistributed. 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 softening cohesive zone model. The compressive strength and the size of the damage zone at failure are successfully predicted for various lay-ups and hole sizes. In [3], Budiansky and Fleck 1

28 develop micromechanical models to predict the behaviour in the damage region. Pinho et al. 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 modes but no method has yet been successfully developed for measurement of fibre dominated compressive failure. In [5] Pinho et al. use a compact compression (CC) specimen with a cross ply laminate and a fracture energy of 79.8 kj/m 2 is reported. Later Catalanotti et al. [6] use the same specimen geometry and layup with another material system and a fracture energy of 45.7 kj/m 2 is reported. It is noted that partitioning of the fracture energy is needed since the longitudinal fibre failure is accompanied by intralaminar cracking in the 9 -plies. The fracture energy of intralaminar cracking is however found to be of negligible magnitude compared to the magnitude associated with fibre failure. However, possible interaction effects are neglected. Thus, measurement of the fracture energy using a UDlayup is desirable. In [7], Laffan et al. use a 4-point bending specimen with a UD-layup and a fracture energy of 25.9 kj/m 2 is reported. It is reported that failure initiates by a mode II crack that propagates a small distance prior to transition to in-plane kink-band propagation. This phenomenon is explained by Gutkin et al. in [8]. However, fibre splitting sometimes occur prior to fibre failure when using UD-layups, cf. e.g. [9]. This is avoided with cross ply laminates. In this work, a modified CC-specimen is used to estimate the fracture energy associated with longitudinal compressive failure and a data reduction scheme is derived by use of Eshelby s concept of the equilibrium of configurational forces, [1]. How the concept of equilibrium 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 notch that achieves a stress rising effect. Here, the damage is localized without using a premade notch. Instead, the out of plane thickness is decreased towards the anticipated damage region and longitudinal compressive failure is obtained within a small region consisting of a UD-layup. FE-simulations are performed to validate the experimental results where a cohesive zone models the damage region. Cohesive zone models fit well into the structure of 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; the shape of the cohesive law must be defined. In this paper, governing parameters of the cohesive law are estimated from experimental results. These are implemented in Abaqus 6.11 as a cohesive user material (UMAT). Simulations capture the main features of the experimental behaviour. The plan of the paper is as follows: First the method is presented in section 2. In section 3 the experiments are presented. FE-simulations are presented in section 4. This is followed by a discussion of the results. The paper ends with a summary of the main conclusions. 2

29 Ø8 11 L L 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 is used with nominal thickness, height and length of 4 mm, 65 mm and 11 mm, respectively. The material is manufactured using Resin Transfer Moulding (RTM) with a UD fabric reinforcement made from HTS carbon fibres and RTM6 resin. The nominal laminate layup indicated by L1 in Fig. 1 is [/+6/-6///+6/-6/]s with the -direction parallel to the loading direction. At the mid-section, the thickness is.8 mm and all fibres are oriented parallel to the loading direction. To achieve this, the four inner 6-plies are dropped off creating layup L2 Fig. 1. Closer to the mid-section, the outer plies including the ±6s are removed with a milling operation. The region at the mid-section with a height and thickness of 1 mm and.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 loading frame with a 1 kn load cell by inserting the free arms between two steel plates which ensures 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 plates and the specimen. Frictional forces on the contact surfaces of the specimen are reduced by lightly grinding them down. The lower cylinder is held fixed and the upper cylinder is subjected to quasi-static loading by applying a downwards prescribed displacement of 5 µm/s. To capture relevant displacements, the specimen is prepared for digital image correlation (DIC) using an Aramis system by spraying a black-on-white speckle pattern on one of the lateral surfaces of the specimen. A sufficiently large region is monitored with the Aramis system so that displacements and rotations of the loading points are measured 3

30 Figure 2: Experimental setup simultaneously with the strains along the notch of the specimen. Throughout the experiments, images are taken every five seconds by the Aramis system. For the present experimental series about 3 pictures are taken in each experiment Theoretical framework In fracture mechanics, fracture energies are often evaluated by using a curve integral designated the J-integral [12], Z Z J= (W dy Ti ui,1 ) dc = (W n1 Ti ui,1 )dc (1) C C where W is the strain energy density and Ti = σij nj is the traction vector acting on the outside of the path C with the outward normal vector ni. Index notation is used where i = 1, 2 indicate components along the x and y direction; summation is indicated by repeated indices and partial differentiation by a comma. The J-integral is zero for any closed path if the enclosed region does not include any singularity. With the notion singularity, we here mean any object or feature of the body that, when moved in the elastic field, changes the potential energy of the body. For a test geometry with a constant out-of-plane thickness, evaluation of the J-integral along the path indicated in Fig. 3 would yield an energetic balance in terms of energy per unit surface. However, the present specimen geometry has a varying out-of-plane thickness. Thus, a generalized form of the J-integral has to be introduced in terms of the surface integral, ZZ ZZ P = (W δ1j σij ui,1 ) nj ds = (W n1 Ti ui,1 )ds (2) S S 4

31 A B r H E G F y x G F s b waist b D C r Figure 3: Integration path (left) and detail of the notch (right) where n i is the outward normal vector on the surface S. In the following, indices i = 1,2,3 indicate components along the x, y and z direction, respectively. From the P-integral, an energetic balance can be formulated in terms of energy per unit length by exploiting the fact that P is zero for a closed surface if the enclosed volume does not include any singularity. The P-integral can also be interpreted as the sum of all configurational forces that act on the volume with outer boundary S. For the present geometry, the P-integral is evaluated over the closed surface that consists of the lateral surfaces and the surface created when the dashed path indicated in Fig. 3 cuts through the thickness. The resulting configurational force from each sub-surface is studied for the present geometry. Both terms in Eq. (2) are zero for the lateral surfaces and the surfaces that includes the paths A-B, C-D, E-F and G-H, since both n 1 = and T = on these surfaces. Moreover, in the design of the specimen, special precaution is taken so that surfaces that include the vertical boundaries A-H, E-D and B-C are virtually undeformed, i.e. W = and T =. That is, these surfaces are not associated with any configurational forces. Thus, the only surfaces that provide non-zero configurational forces are the surfaces along the notch G-F and along the holes where the loads are applied. The magnitudes of the corresponding configurational forces are denoted P notch and P load, respectively. Equilibrium of the configurational forces may then be formulated as P load = P notch (3) The left hand side in Eq. (3) is dominated by the second term in Eq. (2) and can be formulated as P load = F (θ 1 + θ 2 ) (4) where F, θ 1 and θ 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 the configurational force associated with this surface can be formulated as 5

32 P notch = W dydz (5) S notch This configuational force can be separated into three terms corresponding to the configurational forces associated with the sub-surfaces that includes the boundaries indicated by r, b and waist, respectively, cf. Fig. 3. Thus we write P notch = P b + P r + P waist (6) Since the fracture process is assumed to be concentrated to the waist, the main focus is to determine P waist. From Eqs. (3) and (6) we derive P waist = P load P r P b (7) In the experimental evaluation, the strains measured along the boundaries r and b are used to calculate P b and P r. Since the strains are only measured on one of the lateral surfaces, assumptions have to be made regarding the variation of stress and strain components through the thickness. Since the tangential stress is the dominating stress component along the boundary, it is assumed that only this stress component contributes to the strain energy. Furthermore, we assume that the state of strain is constant through the thickness and 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 subjected to uniaxial stress, i.e. W = 1 2 E tǫ 2 t (8) where ǫ t is the tangential strain along the boundary and E t is the direct elastic modulus in the same direction. It is noted that E t varies along the notch due to the continuously varying 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 configurational force can be interpreted as the mean value of the energy release rate consumed in the waist multiplied with its out-of plane width, b. A nominal energy release rate associated with the waist is therefore given by J waist = P waist /b (9) The maximum value of J waist is identified as a measure of the fracture energy of the unidirectional material in the waist. It is denoted J waist,c Data reduction scheme The configurational force P load is determined with high precision by measuring the load and the load point rotations, cf. Eq. (4). To determine P r and P b, the tangential strain, ǫ t, along the boundary G-F is measured with the Aramis system. Strains are recorded in points located every.15 mm along the boundary. 6

33 Given the strains along the boundary b, the strain energy density W is obtained from Eq. (8), where the elastic modulus, E t, is determined with laminate theory. Note that E t varies along the boundary b due to the reduction of thickness by dropping of the inner 6 -plies and machining of the outer plies, as described earlier. Finally, P b is formed by integration similarily to Eq. (5). The resolution of the strain measurements is not high enough to accurately measure the strain along the radii and calculate P r directly by integration. It is here assumed that P r can be established based on measurement of the tangential strains at the points F and G, i.e. ǫ F xx and ǫ G xx, cf. Fig 3. The relation between P r,sim and the square of the tangential strain at points F or G is determined from a linear elastic FE-simulation. A coefficient α is first determined from the FE-simulation according to α = P r,sim (ǫ F xx,sim )2 = P r,sim (ǫ G xx,sim )2 (1) where it is noted that ǫ F xx,sim = ǫ G xx,sim due to symmetry. However, in the experiments symmetry cannot be guaranteed, i.e. ǫ F xx,exp and ǫ G xx,exp are generally not equal. Thus, the configurational force associated with the radii is evaluated from the experimental data through P r,exp = α [ (ǫ ) F 2 ( ) ] xx,exp + ǫ G 2 2 xx,exp γ (11) where a correction factor γ is introduced to account for difficulties in measuring the tangential strains exactly at the positions F and G, at the start of the radii. To set the value of γ for a certain experiment it is assumed that, in the initial stage of the experiment, P r,exp is related to P b,exp with the same ratio as in a linear elastic FE-simulation, i.e. P r,exp = βp b,exp where β = P r,sim /P b,sim. For the present test specimen β.46 and the correction factor γ varies between for the experiments presented in the next section. This suggests that the tangential strains, ǫ F xx,exp and ǫ G xx,exp, are underestimated by %, cf. Eq. (11). Estimating the distance between the measuring points and points F/G based on the resolution in the DIC-measurement; this error seems reasonable considering the strain variation near the radii obtained with a FE-simulation. This type of measurement error is considered neglectible in the evaluation of P b. In FE-simulations, the strain gradient component normal to the loading direction is noticed to be small along the boundary b. 3. Experiments Two experimental series, denoted series 1 and series 2, are carried out. Series 1 and 2 consist of five and four experiments, respectively. In series 2, measurement problems preclude evaluation according to the method discussed in the previous section. However, the loads at failure are considered accurate and are presented below. Therefore, only experiments from 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

34 prior to failure and a microscopy study is performed in an attempt to detect damage mechanisms that eventually lead to compressive failure. These are difficult to identify in a failed specimen since compressive failure occurs in an unstable and catastrophic manner. Composite specimens subjected to compressive loading are sensitive for variations of geometry and imperfections such as voids and fibre waviness. Therefore, the quality of the specimens 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 lamellas present 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 cutting ten mm to the left, parallel with the boundary BC, cf. Fig 3. A FE-model of the tensile test is analysed with Abaqus 6.11 using shell elements and the composite layup application to model the material. The elastic stiffnesses measured in the coupon tests are in good agreement with the FE-simulations. A certain amount of fibre waviness is observed in the UD-material at the waists of the specimens which probably reduces the compressive strength, cf. e.g. [13]. The specimens fail due to compressive failure in the waist. In the microscopy study of specimen 2, a kink-band is observed in the waist and the kink-band tip is found approximately 47 mm behind the start of the waist. Magnified images of the kink-band tip are shown in Fig. 4. A kink-band height of about 2 µm is observed. This corresponds to 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 magnifications are shown and the scale bar in the left and right image indicates 1 µm and 5 µm, respectively In Fig. 5, load-displacement curves are shown; results from series 1 are presented in Fig. 5a and 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 in Fig. 5b do not correspond to the load point displacements. Due to measurement problems, 8

35 F (kn) 3 F (kn) (mm) (mm) 1.6 Figure 5: Experimental results a) Load-displacement curves from experiment 1-5. Experiment 5, unloaded at F =4.4 kn, is indicated with +-signs b) Load versus displacement of the 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 with about ±2% which seems reasonable since failure is governed by an instability and small variations of the local geometry may lead to large variation of the stability load. The events at the unstable failure are not captured since the displacements are only recorded every five seconds and a jump of load and displacement is recorded at failure. The vertical drop shown in Fig. 5b is recorded since the displacement of the load actuator is prescribed at a rate of 5 µm/s. All experiments show in principal an elastic brittle behaviour with an initial stiffness of about 2 kn/m. At higher loads, the behaviour is slightly nonlinear and at 9 % 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 for the five experiments in series 1. The curves are only recorded to the point of failure since measurement of w is only possible prior to failure. Here, w is measured from the relative displacement of two points located in the waist with an initial distance of.7 mm. Note that values of w are presented with different scales in Fig. 6. It is observed, that all experiments show similar initial behaviour. At higher loads at about kn, a substantial decrease of the slope is clearly visible. At this point w 7 1 µm, which corresponds to a nominal strain at the waist of %. This point is interpreted as damage initiation in the waist. As the load increases, the form of the F-w-curves differs a lot and a general local response is difficult to identify. A large variation of w at failure is observed. This type of variation is expected at the local level since the behaviour is highly dependent on material imperfection and variations of the local geometry. Experiment 5 was unloaded at F =4.4 kn and in Fig. 6 a noticable decreased 9

36 4 Experiment Experiment 2 F (kn) 3 2 F (kn) w (µm) w (µm) 6 5 Experiment Experiment 4 F (kn) F (kn) w (µm) w (µm) 4 Experiment 5 F (kn) w (µm) Figure 6: Applied load vs local compression at the waist 1

37 ǫt/f (1/MN) S (mm) 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. E 11 (GPa) E 22 (GPa) G 12 (GPa) ν slope is recorded prior to unloading. However, in the microscopy study, no obvious signs of damage 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 µm. This corresponds to a strain of about.3 %. It is questionable if such a small variation in local geometry is detectable in an optical microscopy study. Since the F-w-curves after initiation of local nonlinearity has a large variation, more experiments are needed to determine the local mechanisms that lead to compressive failure. The tangential strain along the notch, i.e. the boundary indicated with b-waist-b in Fig. 3, is shown in Fig. 7. The two curves correspond to two different stages in experiment 4. The solid line corresponds to a low value of the load, F = 1.4 kn, and the dashed line corresponds to a higher load level F = 4.5 kn. Both curves are normalized with respect to the currently applied load. It is observed that the strain is concentrated in the waist at the higher load level while strains elsewhere along the notch are relaxed as compared to the case for the low load level. That is, at points along the notch located remote from the waist, strains do not increase linearly with the load. This indicates a redistribution of stresses and strains when damage is progressing in the waist. Mechanical stiffnesses used in the data reduction scheme are presented in Table 1. The evaluation method discussed in the previous section is applied to experiments 1-4 and the results are shown in Fig. 8. To the left, the evaluated configurational forces P load, P r and P b are shown versus the applied load. From these results, the energy release rate associated 11

38 P (N) P (N) P (N) P (N) P load P b + P r P b P r 1 P load P b + P r P b P r F (kn) F (kn) P load P b + P r P b P r F (kn) P load P b + P r P b P r F (kn) Jwaist (kn/m) Jwaist (kn/m) Jwaist (kn/m) Jwaist (kn/m) Experiment 1 1 Experiment F (kn) F (kn) Experiment F (kn) 55 Experiment F (kn) 6 Figure 8: Left: Configurational forces versus applied load. Right: load Jwaist versus applied 12

39 with the waist, J waist, is calculated according to Eq. (7) and Eq. (9). To the right in Fig. 8, J waist is shown versus the applied load. The fracture energy is determined as the maximum value of J waist. It is observed that J waist in the early stages of the experiments. This is expected prior to damage initiation in the waist. Near failure, the J waist -curves show varying behaviour, arising from the measured strain along the boundary b. It is resonable to assume that the assumption of linear elasticity results in conservative measurement of the fracture energy since P b is overestimated. This effect is particularly revealed in experiment 2, where J waist is decreasing due to a sudden increase of P b. At this stage, it is observed in the DICmeasurement that the damage zone appears to expand into the boundary b and since linear elasticity is assumed along b, it results in a substantial increase of P b. Of course, thorough conclusions are difficult to establish since only strains on the lateral surface are avaliable for measurement. The fracture energy, Jwaist,c, associated with compressive failure in the waist for experiment 1-4 are: 25.5, 3.5, 39.8, and 23.8 kn/m respectively. In the following section, these values are compared with corresponding results from FE-simulations of the present experimental series. 4. Finite element simulations Numerical simulations are performed in order to verify the experimental results and to increase the understanding of the fracture process. For the present geometry, a global instability occurs at the critical load and the load drops instantaneously, cf. Fig. 5. Therefore, implicit dynamic simulations with numerical damping are performed to enable capturing of the fracture process. This is done by using the quasi-static application in Abaqus. A 2D-model of the specimen is created. The body of the test specimen is modelled as a continuum and the fractured region as a cohesive zone. To be more specific, the anticipated damage zone in the waist is modelled as a cohesive zone with the height 2 µm 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 where triangular elements occur. The varying thickness is modelled by dividing the specimen into several sections as shown in Fig. 9. The height of the sections in the area of reduced thickness is typically 1 mm and the thickness of each section is determined as the mean thickness of the corresponding section of the specimen. Each section is assigned orthotropic properties calculated with laminate theory and material properties according to Table 1. The cohesive zone is modelled with 4-node cohesive elements. The deformation of the cohesive elements is governed by a cohesive law describing the normal compressive stress, σ, as a function of the compression, w. At this point it is important to note the difference between w and w; w is the compression measured as the relative displacement of two points/nodes at the waist with an initial distance of.7 mm while w is the compression in the cohesive element with height 2 µm. 13

40 Figure 9: Geometry of FE-model with indicated sections Along the horizontal lines where the continuum elements and the cohesive elements share nodes, the element sides are about 5 µm long. This element size has proven to be small enough to capture the global instability without numerical problems. Simulations have been performed with smaller elements, only to obtain the same results and longer computation time. A 3D-model has also been used to validate that the 2D-model captures the global response satisfactorily. In the following, only results from the 2D-simulations are presented and compared to the experimental results. To model the loading, the boundary of each hole is subjected to a pin-joint constraint with respect to the centre of the hole. The center of the hole is subjected to displacement boundary 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 compression specimen. The damage evolution is often formulated assuming that, after onset of damage, the cohesive stress is decreasing linearly with the compression, cf. Fig. 1a. Such models are 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 softening model 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-curves in Fig. 6 is interpreted as initiation of damage. Beyond this stage the load increases monotonically with increased local compression until failure of the waist. Therefore, a cohesive law should in this case, prior to softening, include a region where the stress remains constant or is increasing with a reduced tangential stiffness as the compression increases. Thus, we propose a cohesive law according to Fig. 1b. In this cohesive law, three values of the compressive stress σ 1, σ 2 and σ 3 and the corresponding compressions w 1, w 2 and w 3 are input parameters to be defined. The first part of the cohesive law, i.e. stage 1 in Fig. 1, is assumed to be linearly elastic. The corresponding initial stiffness is taken as the longitudinal stiffness of the present UD-laminate, i.e. w 1 is given by w 1 = σ 1 h cz /E 1, where E 1 is the longitudinal 14

41 σ σ σ 1 σ 2 σ 1 σ w 1 w c w w 1 w 2 w 3 w Figure 1: Left: Linear softening cohesive model Right: Proposed cohesive model Table 2: Parameters derived from the simulations Experiment σ 1 (MPa) σ 2 (MPa) σ 3 (MPa) w 1 (µm) w 2 (µm) w 3 (µm) elastic stiffness of the unidirectional composite in the waist and h cz is the height of the cohesive element. The stress σ 1 at the end of stage 1 is determined from a linear elastic simulation by matching the load at damage initiation to the corresponding experimental value. The tangent stiffness of stage 2 in the cohesive law is determined from nonlinear simulations by matching the slopes in the experimental and simulated F-w-curves during damage growth. The values of σ 2 and w 2 are determined by matching the critical load from 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-wcurves and the corresponding curves obtained from simulation of the first two stages of the cohesive law, cf. Fig. 1. The experimental and simulated results are indicated with solid lines and dashed lines, respectively. The numbered arrows indicate the corresponding stage in the cohesive law in the cohesive element at the notch, i.e. the most strained cohesive element. These stages are indicated by the encircled numbers in Fig. 1. The initiation of stage 3 is shown in the simulated curves. At this point, the softening initiates and the simulated compression exhibits a jump indicated by the horizontal ending of the simulated F-w-curves. As described earlier, the experimental F-w-curves are shown until failure, since the measuring points the are lost at this stage. It is believed that the essential features of the cohesive law up to the maximum stress, σ 2, are well captured, even though the agreement is far from perfect. As mentioned previously, unstable failure occurs when proceeding the simulations into 15

42 F (kn) F (kn) F (kn) F (kn) F (kn) w(µm) (mm) w(µm) (mm) F (kn) w(µm) (mm) F (kn) w(µm) (mm) F (kn) Figure 11: Comparisons of simulated and experimental results. Experimental results are indicated with solid lines and simulated results with dashed lines 16

43 Table 3: Critical loads and fracture energies from experiments and simulations Experiment F exp c (kn) J exp waist,c (kn/m) F sim c (kn) J sim waist,c (kn/m) the softening part of the cohesive law, i.e. stage 3 in Fig. 1. Thus, the dynamic simulation predicts a sudden drop of the load exactly as recorded in the experiments. During the sudden load drop, the deformation of the most strained cohesive element reaches the end of the softening part of the cohesive law. Thus, this part of the cohesive law cannot be determined by comparison of the simulation with experimental results. Instead, the negative slope in stage 3 of Fig. 1 is chosen small enough so that convergence is obtained in the simulation increments immediately after the load drop. To determine the stress level σ 3 in stage 4 of the cohesive law, the experimental load-displacement curves to the right in Fig. 11 are studied. In the present case σ 3 = gives the best agreement with the experimental load-displacement curves after failure. It seems unreasonable that no stress is acting between the fractured surfaces. However, it can be expected that significant damage, which is not modelled here, is introduced at neighbouring areas where compressive failure has occurred. The graphs in the right part of Fig. 11 show comparisons of simulated and experimental load-displacement (F ) curves. Note that experimental values of displacement are only recorded every five seconds and therefore the experimental response during the sudden load drop is not recorded. With this in mind, the comparison shows that the essential features of the global behaviour are well captured. The simulated and experimental values of the fracture energies, Jwaist,c, and critical loads, F c, are presented in Table 3 and the results are also visualized in Fig. 12. The sim presented values of J waist,c are determined by using Eqs. (5), (8) and (9). Since failure sim occurs in the simulation when stage 3 of the cohesive law is reached, J waist,c is given by J waist,c sim = W dy = waist w2 σd w + σ2 2(h waist h cz ) 2E t (12) where the first term is the area beneath the cohesive law until initiation of stage 3. The second term is the energy release rate consumed by the linear elastic elements along the waist at failure. The parenthesis in the second term equals.8 mm since the height of the waist and cohesive zone is 1 mm and.2 mm, respectively. The fracture energies in the FE-models agree well with the experimental results for three experiments. However, in the simulation of experiment 3, the simulated fracture energy is about twice the experimentally obtained fracture energy. This may have several explainations but one that appears reasonable is that the large difference arise from the 17

44 Jwaist,c (kn/m) Simulation Experiment F c (kn) Figure 12: Comparison of fracture energies from simulations and experiments experimental measurement of w. It is noted that w at failure is about 11 µm in experiment 3 while the corresponding values in experiment 1,2 and 4 are about 2-4 µm. In the F- w-curve of experiment 3, three jumps of 1, 5 and 17 µm, are noticed at F 3, 4.2 and 5.6 kn without visibly affecting the global behaviour. It is believed that some event occured at the outer ply, where the compression is measured, which is not representative for the material in the waist. Assuming that the specimen fails due to an unstable kinking process, this magnitude of compression seems unlikely prior of failure. This could be an explanation to the poor correlation in experiment 3 between the experimental fracture J exp waist,c sim energy and the fracture energy used in the simulation J waist,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 FEmodel used in the present paper. Moreover, uncertainties associated with the experimental measurements cannot be overlooked. However, the comparison between experimental and simulated 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 of composites is proposed. Localized high strains are achieved by decreasing the out-of-plane thickness towards the anticipated damage region which contains exclusively unidirectional lamina. Experiments performed show that the test geometry produces fracture in the form of kink-band formation progressing along the waist, i.e. along the section with the smallest out-of-plane thickness. Kinking takes place in the direction with the least support from the surrounding material, i.e in the out-of plane direction, cf. Fig. 4, which is perpendicular to the direction of kink band progression. In [14], Berbinau et al. concludes that laminates having outer layers tend to fail due to out-of-plane microbuckling, whereas outer off-axis plies permits in-plane microbuckling failure. The fact that the damage region contains exclusively unidirectional lamina has several implications. On the one hand, the unidirectional plies do not have any support from surrounding plies as is the case when compression failure is tested using a cross ply laminate. 18

45 This might give a different failure mechanism as compared to the cross ply situation. This thought is supported by the fact that the height of the kink-band is 2 µm in the present study, 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 the materials studied. For example, the ply thickness in the composite studied here is.25 mm while the composite studied in [8] has a ply thickness of.125 mm. On the other hand, the present approach with exclusively unidirectional lamina in the damage region does not require the use of a special procedure to identify the contribution of the -directed lamina from the measured properties for the cross ply laminate. Indeed, the fracture energy detected with the proposed method can be attributed to longitudinal compression. The proposed method yields fracture energies for the specific composite of about 2-4 kn/m. In [7], the critical energy release rate is measured on a UD-specimen and a value of 25.7 kn/m is reported. This value corresponds to the energy release rate at the onset of damage by the formation of a shear crack initiating at the notch. It is reasonable to assume that the fracture process is different in the present work due to the substantially smaller thickness used in this work. This promotes out-of-plane microbuckling rather than shear crack initiation due to longitudinal compression. Moreover, the failures observed in this work does not initiate at a premade in-plane stress riser. Using cross ply compact compression specimens, fracture energies of 47.5 kn/m and 79.9 kn/m are reported for two different materials in [5] and [6]. In [15] a fracture energy of 38.8 kn/m is reported for a (/9 2 /) 3S laminate from which the fracture energy associated with kink-band formation was calculated as 76 kn/m [5]. Experiments reported in section 3 show that failure takes place in an unstable manner, i.e. the load drops rapidly to about half the value prior to fracture as the kink-band formation propagates rapidly. The fact that the sudden load drop takes place under prescribed displacements, indicates that we are dealing with a global instability inherent to the geometry of the test specimen. That is, the instability is analogous to similar problems encountered for test specimens for delamination of composites, cf. e.g [16]. This conclusion is supported by the fact that the FE-simulations display a similar force/displacementresponse, i.e. the load drops rapidly as soon as the most strained cohesive element enters the softening part of the cohesive law. The obvious practical consequence of this is that the 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 test geometry. By increasing the distance between the loading points and the most strained region, a geometry more favourable from stability aspects is achieved. This kind of geometry changes also reduces the length of the damage zone along the waist, i.e. the redistribution of stresses during an experiment is less pronounced. In an ideal situation, this would enable extraction of the fracture energy using methods not relying on full field measurement of strains. The numerical simulations performed in the present paper are used as a tool for evaluating the local response in the most strained region. The objective is to capture the essential features of the response with a model that is as simple as possible. Thus, in the FE-model, the body of the test specimen is modelled as a linear elastic continuum and the 19

46 fractured region as a cohesive zone with and idealized cohesive law. With a more refined model, better agreement with the experimental results can be obtained. For instance, a non-linear elastic model for the continuum would be necessary to capture the nonlinear response initiating at about 1 kn in the experiments, cf. Fig. 11. In order to capture the continuous change of slope in the experimental F-w-curves, a more complex form of the cohesive law is needed. However, it is believed that the present level of model complexity is sufficient for the present purpose, i.e. to verify the experimental results and to increase the understanding of the fracture process. In this context, an important finding is that the cohesive law, prior to softening, includes a region where the stress remains constant or is increasing with a reduced tangential stiffness as the compression increases. The present evaluation method results in a maximum stress of the cohesive law, i.e. σ 2 in Fig. 1 in the interval MPa. These values are relatively low as compared to what might be expected from previous results from the literature [17]. It is possible that the fibre waviness that is observed in the waist of the specimens may introduce microbuckling at a fairly low longitudinal stress, cf. e.g. [13]. Moreover, it is reasonable to assume that out-of-plane buckling initiates at a fairly low compressive stress in the UD material since no off-axis plies 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 longitudinal compressive failure. Failure occurs in a region consisting of a unidirectional layup. Thus, a compressive fibre failure process is isolated and reduction of dissipated energy associated with non- -plies and interaction effects are not necessary. The fracture energy is evaluated by using a generalized J-integral and full field measurement of the strains with a DICsystem. Thus, the assumption of a small damage zone is avoided. Experimental results show large scatter and a fracture energy of 2-4 kn/m is reported. FE-simulations of the reported experiments are performed. Comparing the experimental and simulated results indicates a relevance of modelling compressive failure as a cohesive zone. The simulations capture the main features of the experimental behaviour. Acknowledgements The authors would like to acknowledge funding from the Swedish National Aeronautical Research Program (NFFP5) in support of this work through the joint project FIKOM with GKN Aerospace Sweden. The authors are grateful to Dr. Anders Biel for his help in connection with the experiments and to Dr. Fredrik Edgren at GKN Aerospace Sweden for performing the microscopy study. 2

47 References [1] E.G. Guynn and W.L. Bradley. Micromechanics of compression failures in open hole composite laminates. Annual Progress Report, Apr Auq for NASA research grant NAG p, [2] C. Soutis, N.A. Fleck, and P.A. Smith. Failure prediction technique for compression loaded carbon fiber-epoxy laminate with open holes. Journal of Composite Materials, 25: , [3] B. Budiansky and N.A. Fleck. Compressive failure of fibre composites. J. Mechanics and Physics of Solids, 41(1): , [4] S.T. Pinho, R. Gutkin, S. Pimenta, N.V. De Carvalho, and P. Robinson. On longitudinal compressive failure of carbon-fibre-reinforced polymer: from unidirectional to woven, and from virgin to recycled. Phil. Trans. R. Soc. A, 37(165): , 212. [5] S.T. Pinho, P. Robinson, and L. Iannucci. Fracture toughness of tensile and compressive fibre failure modes in laminated composites. Composites Science and Technology, 66: , 26. [6] G. Catalanotti, P.P. Camanho, J. Xavier, C.G. D avila, and A.T. Marques. Measurement of resistance curves in the longitudinal failure of composites using digital image corroleation. Composites Science and Technology, 7(13): , 21. [7] M.J. Laffan, S.T Pinho, P. Robinson, L. Iannucci, and A.J. McMillan. Measurement of the fracture toughness associated with the longitudinal fibre mode of laminated composites. Composites Part A, 43: , 212. [8] R. Gutkin, S.T. Pinho, P. Robinson, and P.T. Curtis. On the transition from sheardriven fibre compressive failure to fibre kinking in notched cfrp laminates under longitudinal compression. Composites Science and Technology, 7: , 21. [9] C. Soutis, P.T. Curtis, and N.A. Fleck. Compressive failure of notched carbon fibre composites. Proc R Soc, 44(199):241 56, [1] J.D. Eshelby. The force on an elastic singularity. Phil. Trans. R. Soc. London, 244:87 112, [11] U. Stigh and T. Andersson. An experimental method to determine the complete stress-elongation relation for a structural adhesive layer loaded in peel. Fracture of Polymers, Composites and Adhesives (Eds. Williams J.G. and Pavan A.), 27:297 36, 2. [12] J.R. Rice. A path independent integral and the approximative analysis of strain concentration by notches and holes. Journal of applied mechanics, 88: ,

48 [13] C. Soutis. Compressive strength of unidirectional composites; measurement and predictions. ASTM STP, 1242: , [14] P. Berbinau, C. Soutis, P. Goutas, and P.T. Curtis. Effect of off-axis ply orientation on -fibre microbuckling. Composites Part A, 3: , [15] C. Soutis and P.T. Curtis. A method for predicting the fracture toughness of cfrp laminates failing by fibre microbuckling. Composites Part A, 31:733 74, 2. [16] K.S. Alfredsson and U. Stigh. Stability of beam-like fracture mechanics specimens. Engineering Fracture Mechanics, 89:98 113, 212. [17] J. Lee and C. Soutis. A study on the compressive strength of thick carbon fibre-epoxy laminates. Composites Science and Technology, 67(1): ,

49 Paper B Measurement of cohesive laws for initiation of delamination of CFRP

50 38

51 Measurement of cohesive laws for initiation of delamination of CFRP K.S. Alfredsson, D. Svensson, U. Stigh, A. Biel University of Skövde, S Skövde, Sweden Abstract With a cohesive zone, delamination of laminated material can be analysed using non-linear finite element codes. For carbon fibre reinforced composites, two different delamination processes are identified; one at the crack-tip with a sub millimetre sized process zone and one with a large-scale process zone associated with fibre bridging. In the present paper we focus on the crack-tip process that is important when crack growth has to be avoided. Methods to measure the cohesive relations associated with the fracture processes are developed. These methods are used to measure the cohesive relations in mode I and II, 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 major concerns in the design and use of advanced composite structures. Delamination may start at unidentified defects originating from the production process or damages occurring in the use of the component. Two different mechanisms and corresponding length scales can be identified in the process of delamination. At the close proximity of a crack tip, a process region can be identified. With epoxy resins, the associated fracture energy is in the range of 1 2 N/m and the yield strength is in the range of 1 1 MPa. A simple estimate predicts the size of the process zone to about 1 1 mm. That is too small to indicate interaction with outer boundaries of structures, but large enough to imply interaction with the fibres. From an experimental point of view, this shows that the fracture properties should be measured in the relevant composite and one cannot rely on bulk properties for the resin. At the larger length scale and in the wake of a growing crack, crack bridging may occur. This process often contributes significantly and increases the total fracture energy to about 1 3 N/m. The bridging stress is however small, in the range of 1 MPa. That is, the process zone is very large, in the range of 1 1 mm, [1]. Thus, the two fracture processes are associated with two very different length scales. In some applications, the enhancement of the 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, defects from the production stage are likely to lack bridging fibres. Therefore, if no defects from 1

52 crack tip T s t y w d x v s t Figure 1: Left: Cohesive zone heading a crack tip. Traction T holds the cohesive surfaces together. Right: Traction and separation separated in orthogonal components relative the middle surface of the cohesive surfaces. the productions stage are allowed to grow during use, fibre bridging cannot be considered in aeronautical applications. In the present paper, we study delamination at the smaller length scale, i.e. without considering fibre bridging. The study is performed within the framework of cohesive modelling. As compared to linear elastic fracture mechanics, this can be viewed as a step toward a more complete model of the actual damage process. With cohesive modelling we assume the existence of a planar process zone heading the crack tip. All inelastic material processes in the real process zone are modelled by a cohesive law acting on the cohesive surface. Figure 1 illustrates a cohesive model. The traction T is assumed to decrease as the separation δ of the cohesive surfaces increases. At large enough separation, the traction is zero indicating the formation of new crack surfaces. Historically, Barenblatt introduced the cohesive model to increase the understanding of brittle fracture in his seminal paper 1962, [2]. Later, a number of researchers showed the usefulness of the concept to 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 the structure of deformation based finite element analysis, [6], [7]. That is, strength analysis of 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 I and II, respectively. The theory is developed in section 2 and the experiments and their evaluation are reported in section 3. In section 4 one of the methods developed in section 2.2 is used to re-evaluate mode II experiments previously presented in [8]. The paper ends with some conclusions. 2

53 2. Theory and methods Methods to measure cohesive laws have been relatively sparingly reported. The embryo to such methods can be traced to [9]. The J-integral gives the release of potential energy of an elastic body per unit created crack area, associated with the propagation of the crack front. It can be calculated from the integral J = (W dy T i u i,x dc). (1) C where C is a counter-clockwise integration path, and W, T i and u i are the strain energy density, the traction vector and the displacement vector, respectively, cf. Fig. 1. Index notation is employed with index i = 1, 2 indicating components along the x and y coordinates, respectively; summation is indicated by repeated indices and partial differentiation by a comma. The crack is assumed to lie in a plane with y = constant. If W does not contain any explicit dependence of the x coordinate, the integration path C starting at the lower crack surface and ending at the upper crack surface can be chosen freely as long as C does not encircle objects that change the energy of the body if the objects positions are changed. Thus, by choosing C close to the crack tip, 1 we get w v J = W dy = σdŵ + τdˆv (2) C where σ, τ, w, and v are the cohesive normal stress, shear stress, opening and shear at the crack tip, respectively, cf. Fig That is, if we are able to continuously measure J from the external loads acting on a specimen during an experiment, and at the same time measure v and w at the crack tip, we would be able to differentiate the measured J(v, w) data to derive the cohesive laws σ(v, w) and τ(v, w), cf. Eq. (2). Two facts complicate this idea. Firstly, the cohesive law is not likely to be elastic in nature. That is, it is not likely that the strain energy density W exists. However, if the loading within the cohesive zone can be regarded as monotonically increasing, a pseudo-potential A can replace W, [1]. In this case the mathematical difference between an elastic and inelastic material is immaterial. It is only when un-loading from an inelastically deformed state occurs that the difference between elasticity and inelasticity reveals itself. The second problem originates from the fact that most expressions for J in terms of external loads implicitly or explicitly depend on an assumption of the material behaviour. In [9], it is however shown that some specimen geometries allow for a direct measurement of J from the applied load without the need for a too restricted assumption of the behaviour of the material. This idea is 1 In 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. For instance, in studies of crack bridging, the right end of the process zone is usually considered as the crack tip and the process zone is referred to as a bridging zone. 2 Note that the cohesive shear stress is defined as τ = τ xy in order to get a positive shear stress near the crack tip for the ENF-specimen, cf. Fig. 2. 3

54 P, D/2 H/2 H/2 P, D/2 a y x a L H/2 H/2 b a y x b L P, D al Figure 2: a) Double Cantilever Beam-specimen; b) End Notched Flexure-specimen. Both loaded with prescribed load point displacements. Fibre orientation is indicated at the right part of the specimens. developed in [11]. In [12] a different path of derivation is taken. Starting from the basic equations of Euler-Bernoulli beam theory, the authors show how the cohesive law can be measured from the external loads. It was later shown that the same result can be derived using the J-integral, [13] or by a direct application of the concept of energetic forces, [14]. These methods have previously been used to measure cohesive laws for adhesives and for fibre bridging, cf. e.g. [13], [15]. In this section, two methods to measure the cohesive laws for delamination are presented. With a Double Cantilever Beam-specimen (DCB) the cohesive law in mode I is measured; with an End Notched Flexure specimen (ENF) the cohesive law in mode II is measured, cf. Fig. 2. The basic assumption in the development of the experimental methods is the existence of a cohesive law associated with a potential Φ. Thus, we assume { σ = Φ w τ = Φ (3) v As stressed above, if Φ does not exist, it suffices that a pseudo-potential can replace Φ in the specific loading history. In the following sections, methods to measure cohesive laws in mode I (DCB) and II (ENF) are presented Mode I With an inner integration path, J is given from Eq. (2) as J = w σdŵ (4) Taking an outer integration path according to Fig. 3 yields contributions to J only from the left boundaries where the forces P act. As shown in Eq. (1), contributions to the first term only appear at vertical boundaries. At the left boundary, the material is stressed and W is non-zero. However, by using a long enough specimen, the right boundary is unstressed and W =. The second term in Eq. (1) is only non-zero where traction acts, i.e. only at the left boundary. 4

55 P, D/2 H/2 H/2 P, D/2 a y L x q 2 q 1 P upper beam, undeformed Figure 3: Outer integration path for Double Cantilever Beam-specimen and rotations associated with the Timoshenko beam theory. Using the Timoshenko beam theory, we distinguish the rotation of the cross-section of the beam, θ 1, from the rotation of the neutral axis, θ 2, cf. Fig. 3. These rotations are given by the displacements, u 1 and u 2, by { θ1 = u1 y (5) θ 2 = u2 x From these, we identify the shear of the beam as γ = θ 2 θ 1. The shear results in strain energy which contributes to the first part of Eq. (1), W dy = P (θ 1 θ 2 )/2B, where B is the out-of-plane width of the specimen. The second part of Eq. (1) gives Ti u i,x dc = P θ 2 /B. Thus, for the upper beam of the DCB-specimen we collect the contribution P (θ 1 + θ 2 )/2B to J. By doing the same exercise for the lower beam, we arrive at J = 2P θ m (6) B where θ m (θ 1 + θ 2 )/2 is the average of the cross-section and neutral axis rotations. 3 By equating the two expressions for J, Eq. (4) and Eq. (6), we get an expression from which the cohesive stress can be calculated, σ = 2 (P θ m ) (7) B w As shown in [14], this expression is not limited to beam theory and it can be derived by direct application of the concept of energetic forces for 3D-elasticity. It is also readily extended to large deformations by replacing θ with sin θ, cf. [17]. 3 As demonstrated in [16], the error is often small in using the cross-section rotation instead of the average rotation in FE-analyses. 5

56 B A C H/2 H/2 D y P a b x L al Figure 4: Integration path in ENF-specimen. Paths A and C are taken just above and below the cohesive zone. Path D is taken below the loading point Mode II Figure 4 shows an outer integration path used to derive a relation between the external load P and J. The contributions to J are denoted J A, J B, J C, and J D, respectively. If b is large enough, it is safe to neglect J B and J D as compared to J A and J C, cf. [18] and [19]. In this case, the shear stress in the cohesive zone switches sign below the loading point. With a too small b, the shear stress switches sign at a point further to the right. The contributions J A and J C to J originate from the second term in Eq. (1), viz. J A + J C = A T i u i,x dx T i u i,x dx = C b τv dx (8) where v u C x u A x and ( ) denotes ( )/ x. A careful re-derivation of the theory in [18] and extending to the Timoshenko beam theory and an asymmetrically loaded ENF-specimen gives τ = E 1H 16 v + τ (9) where τ = 3αP/2BH for < x < b and τ = 3(1 α)p/2bh for b < x < b + αl is identified as the shear stress according to Jouravski s theory in the plane y = ; E 1 denotes the Young s modulus in the fibre direction and H is the thickness of the ENF-specimen, cf. Fig. 2. Multiplying both sides of Eq. (9) with v and integrating from x = to x = b gives b τv dx = E 1H 16 b v v dx + τ b v dx (1) 6

57 whereby we have constructed the negative of the right-hand-side of Eq. (8). The righthand-side of Eq. (1) can be integrated to form b τv dx = E 1H 32 [ v (b) 2 v () 2] + τ [v(b) v()] (11) Following [18], we now neglect v(b) as compared to v() and v (b) as compared to v () = 24αP a/e 1 BH 2. The latter expresses the boundary conditions that the bending moments in each leg at x = equal αp a/2 and the horizontal stress resultants equal zero at this point. Collecting all terms in Eq. (1) yields J 18α2 P 2 a 2 E 1 B 2 H 3 3αP v() + 2BH where it should be noted that a is the initial crack length and v() is the shear deformation at the initial position of the crack tip, cf. [18]. Hence, the experimental method to measure J does not require monitoring of the position of the crack tip during crack propagation. By comparing with the corresponding expression used in e.g. [18] and [19], we note that the only difference appears due to the unsymmetrically positioned load. 4 That is, the extension to Timoshenko s beam theory does not influence J. The reason is that the elastic energy due to shear deformation is inversely proportional to the height of a Timoshenko beam. That is, this part of the energy is unaltered if the crack propagates infinitesimally under prescribed loading; the shear energy per unit length at the crack tip before and after crack propagation are the same. Thus, shear deformation of the beams does not contribute in mode II. This is obviously not the case in mode I, where no energy is stored in the region heading the crack tip, cf. Eq. (6). We also note that the first term in Eq. (12) corresponds to the energy release rate in an unsymmetrically loaded and elastic ENF-specimen where the influence of a cohesive zone is neglected, i.e. the expression to be used when the conditions of LEFM are met. With Eqs. (2) and (12) we arrive at τ = [ ] 18α 2 P 2 a 2 3αP v() + (13) v E 1 B 2 H 3 2BH Thus, we measure the applied force P and the shear deformation v at the crack tip. 3. Material, specimens, experiments and evaluation The material studied is a CFRP-laminate with all fibres in the longitudinal direction of the test specimens. In order to minimize effects of fibre bridging, one of the laminas at the crack is oriented slightly off-axis with a nominal direction 5. The crack is formed by cutting one of the lamina to a shorter length and replacing it by an equally thick Teflon film in the crack area. In the ENF-experiments, the Teflon film remains between the crack surfaces in order (12) 4 Note that the beam height is defined differently in [18] and [19]. 7

58 Figure 5: Vertical DCB-specimen in test machine. The legs of the specimen are separated with a prescribed rate during an experiment. Two LVDTs are fixed at the crack tip region and measures the expansion of the specimen at this point. to minimize friction. After manufacturing in an autoclave, the specimens are thoroughly examined for defects by NDE-techniques. Before conducting the experiments, a wedge is used to propagate the crack beyond the resin filled crack-tip-area formed during the curing process. The new crack tip and crack length are carefully identified and measured. Finite element models of the test specimens are analysed using Abaqus [2] in order to verify the experimentally obtained cohesive laws. Since B < H, 2D plane stress FE-models are used. In the FE-models, the composite material is modelled as a continuum using the fully integrated element CPS4. The four-node cohesive element COH2D4 is used to model a cohesive zone heading the crack tip Mode I A total of four successful DCB-experiments are conducted. The nominal dimensions of the specimens are H = 16 mm, B = 8.3 mm, a = 155 mm and L = 275 mm, cf. Fig. 2a. Each specimen is measured individually and the data are used in the evaluation. A custom-made test machine is used to conduct the experiments, cf. [21]. In the DCBexperiments, the loading points are separated,, with a prescribed rate, cf. Fig. 2a. During an experiment, the reaction forces P, the rotation at the loading points θ and the separation, w exp ext, of the two points on the outside of the specimen at the position of the crack tip are measured, cf. Fig. 5. A semi-inverse method, described below, is used to derive the cohesive law σ(w) in pure mode I using Eq. (7). In experiments, all data suffer from measurement errors. To limit the influence, we first adapt a Prony-series to the J(w exp ext )-data using a suitable number of terms, and afterwards differentiate the series. The procedure is described in some detail in [21]. From Eq. (2) 8