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14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 ON COHESIVE LAWS FOR DELAMINATION OF COMPOSITES U. Stigh1*, D. Svensson1 1 University of Skövde, PO Box 408, SE 541 28 Skövde, Sweden *ulf.stigh@his.se Abstract Analysis of delamination of carbon fibre reinforced composite using cohesive models is studied. A method to measure the cohesive law associated with delamination is presented. The method allows for identification of a cohesive law fit to model the fracture process at the crack tip, i.e. not considering fibre bridging. Due to the small size of the cohesive zone, an elaborated method involving simulations of the fracture process is developed. The results show larger scatter in the parameters of the cohesive law than in the fracture energy. 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 102 N/m and the yield strength in the range of 101 MPa. A simple estimate predicts the size of the process zone to about 10-1 mm. That is 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 should not rely on bulk properties for the resin. In the wake of a growing crack, crack bridging may occur involving a longer length scale. This process often contributes significantly and increases the total fracture energy to about 103 N/m. The bridging stress is however small, in the range of 100 MPa. That is, the process zone is very large, in the range of 101 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 are likely to lack bridging fibres. Therefore, if no defects from 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 consideration of 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 complex model of the actual damage process. This is done by assuming the existence of a planar process zone heading the crack tip. All inelastic material processes in the real process zone are modelled by the 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 1 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 σ crack tip T τ δ w v τ σ Figure 1. Left: Cohesive zone heading a crack tip. Traction T holds the cohesive surfaces together. In the present paper, the crack tip is considered to be situated at the left end of the process zone. It should be noted that the definition of the position of crack tip differs among authors. For instance, 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 in studies of crack bridging. Right: Traction and separation separated in orthogonal components relative the middle surface of the cohesive surfaces 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], and fracture of adhesives, [5]. 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 FE-codes. Today, cohesive models are included in many commercially available FE-codes. Methods to measure cohesive laws have been relatively sparingly reported. The embryo to such methods can be traced to [8]. 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 − Ti ui , x dS ) (1) S Where, W, Ti, ui, and S are the strain energy density, the traction vector, the displacement vector and a counter-clockwise integration path, respectively. Index notation is employed with index i = 1, 2 indicating components along the x- and y-coordinates, respectively; summation is indicated by repeated indexes and partial differentiation by a comma. The crack is assumed to lie in a plane y = constant. If W does not contain any explicit dependence of the y- coordinate and no loads act inside S, the integration path S starting at the lower crack surface and ending at the upper crack surface, can be chosen freely. Thus, by choosing S close to the crack tip, we get w v J = ∫ W dy = ∫ σ dw + ∫ τ dv ˆ (2) S 0 0 where σ, τ , w, and v are the cohesive normal stress, shear stress, opening and shear at the crack tip, respectively. 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. 2 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 However, if the loading within the cohesive zone can be regarded as proportional and monotonically increasing, a pseudo-potential A can replace W, [9]. In this case the 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 [8], 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 developed in [10]. In [11] 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, [12], or by a direct application of the concept of energetic forces, [13]. These methods have previously been used to measure cohesive laws for adhesives and for fibre bridging; a recent overview is given in [14]. In the present paper, a method to measure the cohesive law for mode I delamination of a carbon fibre reinforced composite (CFRP) is presented. 2 Experimental The double cantilever beam (DCB) specimen is used to measure the cohesive law in mode I. A brief introduction to the theory is given here, cf. [12] for a more detailed derivation. P, Δ/2 P, Δ/2 y y H/2 x H/2 x H/2 H/2 a a P, Δ/2 P, Δ/2 L L Figure 2. Double cantilever beam specimen subjected to prescribed displacements, Δ, of the loading points. The fibre orientation is indicated at the right part of the left illustration of the specimen. Outer integration path is indicated in right illustration. The out of plane width is denoted B. In the DCB-experiments, the loading points are separated ∆ with a prescribed rate, cf. Fig. 2. During an experiment, the reaction forces, P, the rotations of the loading points, θ, and the crack tip opening, w, are measured continuously. From these data the cohesive law in mode I can be determined as explained below. 2.1 Theoretical background Taking advantage of the path independence of Eq. (1), J is evaluated along two alternative integration paths encircling the crack tip. The paths have common start and end points and due to path independence, the expressions can be equated. With an integration path close to the crack tip, J is given by w J ( w ) = ∫ σ ( w ) dw ˆ ˆ (3) 0 With an integration path along the outer boundary an alternative expression is derived, cf. Fig. 3 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 2b. Evaluation of the terms in Eq. (1) along the path yields non-zero contributions to J only from the left boundaries where P is applied. The first term in Eq. (1) is zero at horizontal boundaries and the specimen is assumed to be long enough to consider the right vertical exp Figure 3. Experimental setup in the DCB-experiments. The two LVDT measures the separation wext on the outsides of the specimen over the crack tip. boundary unstressed, i.e. W = 0. The second term is only non-zero if there is a traction acting on the current boundary, thus only at the left boundary. The contributions to J can be calculated using beam theory though the result is not dependent on this assumption, [13]. Evaluation gives 2Pθ J= (4) B where B is the out of plane width of the specimen and θ the rotation of the loading points. Equation (4) is valid for large deformations if θ is replaced by sin θ, cf. [15]. By equating the expressions for J and differentiating the resulting expression, the cohesive law is given by 2 d( Pθ ) σ ( w) = (5) B dw The cohesive law in mode I can therefore be determined if P, θ and w are accurately measured during the experiment. To determine the cohesive law from the experimental J-w data we adapt a Prony-series to the experimental results and the series is differentiated. This procedure minimises unavoidable defects in experimental data, cf. [16] for details. 2.2 Experiments The material studied is a CFRP-laminate with all fibres in the longitudinal direction of the test specimens. The longitudinal and transversal elastic modulus are E1 = 26G12 and E2 = 1.9G12 respectively where G12 is the shear modulus. Poisson´s ratio is ν12 = 0.3. Directions 1 and 2 correspond to the longitudinal and transversal direction in Fig. 2, respectively. Four successful experiments are conducted on the studied lamina. The initial crack of the specimen is formed by cutting one lamina to a shorter length and replacing it by an equally thick Teflon film at the crack. After manufacturing in an autoclave, the specimens are thoroughly examined for defects by NDE-technique. To sharpen the crack tip a wedge is used to propagate the crack 4 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 beyond the resin filled crack-tip area formed during the curing process. The crack lengths are measured before the experiments. A custom made test machine is used to conduct the experiments, cf. [12] and Fig. 3. The force P is measured with a force transducer. A shaft encoder is used to measure the rotation at the loading points and two LVDT are positioned on exp the outsides of the specimen to measure the separation above the crack tip, wext . The nominal dimensions of the specimens are L = 270 mm, B = 8.3 mm, H = 16 mm and a = 155 mm. In the evaluation of the experimental data the individual dimensions of the specimens are used. Figures 4a,b show P-∆ and J − wext graphs from the experiments. The data is normalized in exp relation to the average critical value in all experiments. Subscript c denotes critical value, i.e. the value at the moment of crack propagation when the cohesive stress is zero. A bar over the parameters denotes the average value from the experiments. Figure 4. Left: Reaction force versus separation of the loading points. Right: J versus the separation measured at the outer boundary of the specimen. In Figure 4a it is observed that the reaction forces increases virtually linearly with the separation of the loading points, almost until the maximum is reached. This indicates that linear elastic fracture mechanics (LEFM) would work well with the current geometry. That is, the inelastic zone at the crack tip is small compared to size of the specimen. Furthermore, Fig. 4b shows that the cracks propagate with almost constant energy release rate. Since Eq. (4) does not explicitly depend on the crack length, the fracture energy is considered to be measured with high accuracy, cf. [17] for an analysis of different methods to evaluate the fracture energy using the DCB-specimen. The fracture energy varies within about ±6 %, cf. Table 1. 2.3 Evaluation of experiments To determine the complete cohesive law in mode I, the separation at the crack tip, w, must be measured with high accuracy. Initial simulations indicate that the separation, wA, measured at the outside of the specimen does not equal w, cf. Fig. 5. That is, the specimen expands in the transversal direction. The transversal expansion is largest when the cohesive stress is at its peak value. However, at the moment of crack propagation the expansion is small. If these effects are not accounted for when evaluating an experiment, the separation at the crack tip is overestimated and the shape of the derived cohesive law is inaccurate. That is, if we assume w = wext the derived cohesive law will have a lower peak stress and stiffness. It should exp be noted that the fracture energy, equal to the area under σ(w) curve, is unaffected. A reasonable value of the critical crack tip opening wc, corresponding to zero σ and crack 5 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 growth, can be obtained from measurements at point A since the transversal expansion is small at the moment of crack propagation. That is wc ≈ wext,c can be used as a first estimate of exp,A the critical separation of the cohesive law. Furthermore, the horizontal positioning of the LVDT is critical. Moving the measuring point less than one millimetre towards the loading point, i.e. from point A to point B in Fig. 5, results in a substantially larger separation, i.e. wext > wext . The horizontal position of the measuring point also has a large effect on the sim,B sim,A measured separation at the moment of fracture. Figure 6 illustrates the effects of the position of the measuring points. In the example, point B is offset one millimetre towards the loading point. The influence rapidly increases with the elastic stiffness of the cohesive law. As expected, the w/wc ratio is almost constant in the elastic part of the cohesive law, cf. Fig. 6. P B A Crack tip Figure 5. Crack tip area at the upper beam with typical measuring points A and B. Figure 6. Separation measured at point A and B in relation to the crack tip opening w. Left: Separation at point A in relation to w. Right: Separation at point B in relation to w. The values w are normalized in relation to the critical separation wc. Note the different magnitudes of the vertical axes. The effects discussed above must be addressed when evaluating the experiments. Inspection of the specimens after conducting the experiments reveals that the initial crack length was exp underestimated and therefore the two LVDT that measures the separation wext was not positioned exactly above the crack tip in all experiments. The initial crack tip is easier to identify after the specimens is completely cracked into two halves. Moreover, the positions of the LVDT are easily identified since they leave marks on the specimen. The two LVDT were typically positioned about one millimetre towards the loading point. That is, the effects discussed above have influenced the experimental data. A simulation model of each experiment is created to back out the cohesive law for each 6 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 individual experiment. The initial crack length in the simulation models is set to the crack length measured after the experiment. Since B < H, a 2D plane stress model is considered adequate. The individual dimensions of each experiment are used when creating the models. The fully integrated element CPS4 in Abaqus is used for the laminate. To model the anisotropic behaviour, the Abaqus command Lamina is used to assign the elastic properties. The element mesh consists of 60 by 800 equally sized continuum elements in the vertical and longitudinal direction, respectively. The horizontal length of all elements is 0.3375 mm. This corresponds to about one seventh of the fully developed process zone, thus the elements are sufficiently small to capture the stress distribution ahead of the crack tip. In the vertical direction, the continuum element size varies between 0.252 to 0.269 mm depending on which experiment to simulate. Convergence studies are performed to ensure that the results do not depend on the element size. The model is validated to elementary beam theory to assure that the model provides the correct bending stiffness. The cohesive zone heading the crack tip is modelled with the four node cohesive element COH2D4. To validate the FE-model, the simulated and the experimental P-∆ plots are compared, cf. Fig. 7. This comparison can be made without prior knowledge of the complete cohesive law since the P-∆ relation for the present specimens is essentially within the realm of LEFM. That is, only the fracture energy, the geometry and the elastic properties determine the P-∆ relation with only minor influences of the cohesive stress. Figure 8 indicates that the FE-model is accurate and the maximum force in the simulation differs less than three percent from the experimental result. Figure 7. Reaction force versus the separation of the loading points. Results from simulation model of experiment 1 (solid curve) is compared with experimental results from the same experiment (crosses). A triangularly shaped σ (w) relation is assumed. Thus, three parameters govern the law: the peak strength, σ , the corresponding separation, w0, and the critical separation, wc, cf. Fig. 8. ˆ The fracture energies of the experiments are considered to be measured with high accuracy. Thus, we constrain the parameters of the cohesive law to give the fracture energy of the experiment. The parameters wc and w0 can be chosen freely while σ = 2 J c wc , cf. Fig. 8. ˆ Simulations that accounts for the individual initial crack lengths and the position of the two LVDT are performed. With a trial and error approach, iteratively more suitable values of wc sim are found. That is, the parameter wc is determined to give the same value of wext, c as the measured wext, c . With wc determined, σ = 2 J c wc is also determined. Thus, only w0 is left to exp ˆ be determined. The method used to find a suitable w0 is to study the elastic parts of the exp J ( wext ) curves. It is observed that the specimen is linearly elastic until w0 is reached, thus the 7 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 exp J ( wext ) curves are parabolically increasing until the peak stress is reached. σ σ σ σwc = 2JIc JIc w0 wc w w0 wexp ext,c wc Figure 8. Left: Cohesive law used in the simulation model to back out the separation at the crack tip. Right: The parameters are constrained by the measured fracture energy. A suitable parabolic function, J(w), that captures the first part of the curve is determined. The exp parabolic function is plotted together with J ( wext ) and the value of J is determined at which the curves deviate. Together with the previously determined parameters this gives a good estimate of w0. Figure 9. Plots from the top left to the bottom right correspond to experiment 1-4, respectively. Experimental exp exp J ( wext ) curves are indicated by cross signs. Dashes indicates the optimal J(w) relation that gives the J ( wext ) curve indicated by solid lines. This procedure is employed to analyze all experiments and gives good agreement with the experimental results, cf. Fig 9. The parameters in the adapted cohesive laws are summarized in Table 1. As noted, the cohesive properties vary within ±20 % between individual experiments. 8 14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary Paper ID: 159-ECCM14 Table 1 Resulting parameters in the cohesive law and the corresponding fracture energy. All values are normalized in relation to the average values in the adapted cohesive laws. Experiment σ σ ˆ ˆ w0 w 0 wc w c Jc J c 1 0.80 0.91 1.16 0.96 2 1.20 1.09 0.86 1.06 3 0.80 0.91 1.16 0.96 4 1.20 1.09 0.82 1.01 3 Summary and conclusions In this paper DCB-experiments are evaluated to derive the cohesive law for delamination in mode I for a CFRP. It is shown that the substantial difference between the longitudinal and transversal stiffness of the composite, together with the short cohesive zone lead to a substantial transversal expansion of the DCB-specimen. Moreover, it is noted that the position of the initial crack tip is difficult to identify prior to the experiment. Therefore, the positioning of the measuring device is not as accurate as desired. Similar effects have not been observed when measuring cohesive laws for adhesives using metal adherends. In this case, it suffices to measure the expansion of the cohesive zone by LVDT positioned on the outside of the specimen. A novel technique to back-out the cohesive law is presented. The method is based on accurate measurement of the initial position of the crack tip and simulations of the experiments. In these simulations, we assume a triangularly shaped cohesive law. This assumption might prove too restricted to capture the cohesive law for delamination. An indication of this is the substantial variation of cohesive data between different experiments. The fracture energy, i.e. the area beneath the cohesive law is however accurately measured in the experiments. The adapted J(w) relation substantially differs from the experimental data indicated by crosses in Fig. 9. That is, ignoring the effects discussed here results in a cohesive law with as low as half the peak stress derived here. Acknowledgement The authors acknowledge from financial support from NFFP. Dr Anders Biel, University of Skövde, generously shared experimental data. 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