ON COHESIVE LAWS FOR DELAMINATION OF COMPOSITES

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1 ON COHESIVE LAWS FOR DELAMINATION OF COMPOSITES U. Stigh 1*, D. Svensson 1 1 University of Skövde, PO Box 408, SE Skövde, Seden *ulf.stigh@his.se Abstract Analysis of delamination of carbon fibre reinforced composite using cohesive models is studied. A method to measure the cohesive la associated ith delamination is presented. The method allos for identification of a cohesive la 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 sho larger scatter in the parameters of the cohesive la 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. To 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 10 2 N/m and the yield strength in the range of 10 1 MPa. A simple estimate predicts the size of the process zone to about 10-1 mm. That is large enough to imply interaction ith the fibres. From an erimental point of vie, this shos that the fracture properties should be measured in the relevant composite and one should not rely on bulk properties for the resin. In the ake of a groing crack, crack bridging may occur involving a longer length scale. This process often contributes significantly and increases the total fracture energy to about 10 3 N/m. The bridging stress is hoever small, in the range of 10 0 MPa. That is, the process zone is very large, in the range of 10 1 mm, [1]. Thus, the to fracture processes are associated ith to very different length scales. In some applications, the enhancement of the strength due to crack bridging can be considered. Hoever, in the aeronautic industry, no defects are alloed to gro 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 alloed to gro during use, fibre bridging cannot be considered in aeronautical applications. In the present paper, e study delamination at the smaller length scale, i.e. ithout consideration of fibre bridging. The study is performed ithin the frameork of cohesive modelling. As compared to linear elastic fracture mechanics, this can be vieed as a step toard 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 la 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 ne crack surfaces. Historically, Barenblatt introduced the cohesive model to 1

2 crack tip T δ σ τ τ σ v 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 shoed 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 forard as the realization that cohesive models fits ell ithin 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 las 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 ith the propagation of the crack front. It can be calculated from the integral S ( d i i, xd ) J = W y Tu S (1) Where, W, T i, u i, and S are the strain energy density, the traction vector, the displacement vector and a counter-clockise integration path, respectively. Index notation is employed ith 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 licit dependence of the y- coordinate and no loads act inside S, the integration path S starting at the loer crack surface and ending at the upper crack surface, can be chosen freely. Thus, by choosing S close to the crack tip, e get J = W dy = σ dˆ + τ dv S v (2) 0 0 here σ, τ,, and v are the cohesive normal stress, shear stress, opening and shear at the crack tip, respectively. That is, if e are able to continuously measure J from the external loads acting on a specimen during an eriment, and at the same time measure v and at the crack tip, e ould be able to differentiate the measured J(v,) data to derive the cohesive las σ(v,) and τ(v,), cf. Eq. (2). To facts complicate this idea. Firstly, the cohesive la is not likely to be elastic in nature. That is, it is not likely that the strain energy density W exists. 2

3 Hoever, if the loading ithin the cohesive zone can be regarded as proportional and monotonically increasing, a pseudo-potential A can replace W, [9]. In this case the difference beteen an elastic and inelastic material is immaterial. It is only hen un-loading from an inelastically deformed state occurs that the difference beteen elasticity and inelasticity reveals itself. The second problem originates from the fact that most ressions for J in terms of external loads implicitly or licitly depend on an assumption of the material behaviour. In [8], it is hoever shon that some specimen geometries allo for a direct measurement of J from the applied load ithout 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 sho ho the cohesive la can be measured from the external loads. It as later shon 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 las for adhesives and for fibre bridging; a recent overvie is given in [14]. In the present paper, a method to measure the cohesive la 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 la in mode I. A brief introduction to the theory is given here, cf. [12] for a more detailed derivation. P, Δ/2 H/2 H/2 P, Δ/2 a y x L P, Δ/2 H/2 H/2 P, Δ/2 a y L x 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 idth is denoted B. In the DCB-eriments, the loading points are separated ith a prescribed rate, cf. Fig. 2. During an eriment, the reaction forces, P, the rotations of the loading points, θ, and the crack tip opening,, are measured continuously. From these data the cohesive la in mode I can be determined as lained belo. 2.1 Theoretical background Taking advantage of the path independence of Eq. (1), J is evaluated along to alternative integration paths encircling the crack tip. The paths have common start and end points and due to path independence, the ressions can be equated. With an integration path close to the crack tip, J is given by ( ) σ ( ˆ) J = dˆ (3) 0 With an integration path along the outer boundary an alternative ression is derived, cf. Fig. 3

4 2b. Evaluation of the terms in Eq. (1) along the path yields non-zero contributions to J only from the left boundaries here 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 Figure 3. Experimental setup in the DCB-eriments. The to LVDT measures the separation outsides of the specimen over the crack tip. ext on the 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 J 2Pθ = (4) B here B is the out of plane idth of the specimen and θ the rotation of the loading points. Equation (4) is valid for large deformations if θ is replaced by sinθ, cf. [15]. By equating the ressions for J and differentiating the resulting ression, the cohesive la is given by 2d( Pθ ) σ ( ) = (5) B d The cohesive la in mode I can therefore be determined if P, θ and are accurately measured during the eriment. To determine the cohesive la from the erimental J- data e adapt a Prony-series to the erimental results and the series is differentiated. This procedure minimises unavoidable defects in erimental data, cf. [16] for details. 2.2 Experiments The material studied is a CFRP-laminate ith all fibres in the longitudinal direction of the test specimens. The longitudinal and transversal elastic modulus are E 1 = 26G 12 and E 2 = 1.9G 12 respectively here G 12 is the shear modulus. Poisson s ratio is ν 12 = 0.3. Directions 1 and 2 correspond to the longitudinal and transversal direction in Fig. 2, respectively. Four successful eriments 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 edge is used to propagate the crack 4

5 beyond the resin filled crack-tip area formed during the curing process. The crack lengths are measured before the eriments. A custom made test machine is used to conduct the eriments, cf. [12] and Fig. 3. The force P is measured ith a force transducer. A shaft encoder is used to measure the rotation at the loading points and to LVDT are positioned on the outsides of the specimen to measure the separation above the crack tip, ext. 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 erimental data the individual dimensions of the specimens are used. Figures 4a,b sho P- and J ext graphs from the eriments. The data is normalized in relation to the average critical value in all eriments. Subscript c denotes critical value, i.e. the value at the moment of crack propagation hen the cohesive stress is zero. A bar over the parameters denotes the average value from the eriments. 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 ith the separation of the loading points, almost until the maximum is reached. This indicates that linear elastic fracture mechanics (LEFM) ould ork ell ith the current geometry. That is, the inelastic zone at the crack tip is small compared to size of the specimen. Furthermore, Fig. 4b shos that the cracks propagate ith almost constant energy release rate. Since Eq. (4) does not licitly depend on the crack length, the fracture energy is considered to be measured ith high accuracy, cf. [17] for an analysis of different methods to evaluate the fracture energy using the DCB-specimen. The fracture energy varies ithin about ±6 %, cf. Table Evaluation of eriments To determine the complete cohesive la in mode I, the separation at the crack tip,, must be measured ith high accuracy. Initial simulations indicate that the separation, A, measured at the outside of the specimen does not equal, cf. Fig. 5. That is, the specimen ands in the transversal direction. The transversal ansion is largest hen the cohesive stress is at its peak value. Hoever, at the moment of crack propagation the ansion is small. If these effects are not accounted for hen evaluating an eriment, the separation at the crack tip is overestimated and the shape of the derived cohesive la is inaccurate. That is, if e assume = ext the derived cohesive la ill have a loer peak stress and stiffness. It should be noted that the fracture energy, equal to the area under σ() curve, is unaffected. A reasonable value of the critical crack tip opening c, corresponding to zero σ and crack 5

6 groth, can be obtained from measurements at point A since the transversal ansion is,a small at the moment of crack propagation. That is c ext,c can be used as a first estimate of the critical separation of the cohesive la. Furthermore, the horizontal positioning of the LVDT is critical. Moving the measuring point less than one millimetre toards the loading point, i.e. from point A to point B in Fig. 5, results in a substantially larger separation, sim,b sim,a i.e. ext > ext. The horizontal position of the measuring point also has a large effect on the 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 toards the loading point. The influence rapidly increases ith the elastic stiffness of the cohesive la. As ected, the / c ratio is almost constant in the elastic part of the cohesive la, cf. Fig. 6. P B A Crack tip Figure 5. Crack tip area at the upper beam ith typical measuring points A and B. Figure 6. Separation measured at point A and B in relation to the crack tip opening. Left: Separation at point A in relation to. Right: Separation at point B in relation to. The values are normalized in relation to the critical separation c. Note the different magnitudes of the vertical axes. The effects discussed above must be addressed hen evaluating the eriments. Inspection of the specimens after conducting the eriments reveals that the initial crack length as underestimated and therefore the to LVDT that measures the separation as not positioned exactly above the crack tip in all eriments. The initial crack tip is easier to identify after the specimens is completely cracked into to halves. Moreover, the positions of the LVDT are easily identified since they leave marks on the specimen. The to LVDT ere typically positioned about one millimetre toards the loading point. That is, the effects discussed above have influenced the erimental data. A simulation model of each eriment is created to back out the cohesive la for each ext 6

7 individual eriment. The initial crack length in the simulation models is set to the crack length measured after the eriment. Since B < H, a 2D plane stress model is considered adequate. The individual dimensions of each eriment are used hen 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 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 beteen to mm depending on hich eriment 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 ith the four node cohesive element COH2D4. To validate the FE-model, the simulated and the erimental P- plots are compared, cf. Fig. 7. This comparison can be made ithout prior knoledge of the complete cohesive la since the P- relation for the present specimens is essentially ithin the realm of LEFM. That is, only the fracture energy, the geometry and the elastic properties determine the P- relation ith 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 erimental result. Figure 7. Reaction force versus the separation of the loading points. Results from simulation model of eriment 1 (solid curve) is compared ith erimental results from the same eriment (crosses). A triangularly shaped σ () relation is assumed. Thus, three parameters govern the la: the peak strength, σˆ, the corresponding separation, 0, and the critical separation, c, cf. Fig. 8. The fracture energies of the eriments are considered to be measured ith high accuracy. Thus, e constrain the parameters of the cohesive la to give the fracture energy of the eriment. The parameters c and 0 can be chosen freely hile ˆ σ = 2J c c, cf. Fig. 8. Simulations that accounts for the individual initial crack lengths and the position of the to LVDT are performed. With a trial and error approach, iteratively more suitable values of c sim are found. That is, the parameter c is determined to give the same value of ext,c as the measured ext, c. With c determined, ˆ σ = 2J c c is also determined. Thus, only 0 is left to be determined. The method used to find a suitable 0 is to study the elastic parts of the J ( ) curves. It is observed that the specimen is linearly elastic until 0 is reached, thus the ext 7

8 J ( ) curves are parabolically increasing until the peak stress is reached. ext σ σ σ σ c = 2J Ic J Ic 0 c 0 ext,c c Figure 8. Left: Cohesive la 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(), that captures the first part of the curve is determined. The parabolic function is plotted together ith J ( ext ) and the value of J is determined at hich the curves deviate. Together ith the previously determined parameters this gives a good estimate of 0. Figure 9. Plots from the top left to the bottom right correspond to eriment 1-4, respectively. Experimental J ( ext ) curves are indicated by cross signs. Dashes indicates the optimal J() relation that gives the J ( ext ) curve indicated by solid lines. This procedure is employed to analyze all eriments and gives good agreement ith the erimental results, cf. Fig 9. The parameters in the adapted cohesive las are summarized in Table 1. As noted, the cohesive properties vary ithin ±20 % beteen individual eriments. 8

9 Table 1 Resulting parameters in the cohesive la and the corresponding fracture energy. All values are normalized in relation to the average values in the adapted cohesive las. Experiment ˆ σ ˆ σ 0 0 c c J c J c Summary and conclusions In this paper DCB-eriments are evaluated to derive the cohesive la for delamination in mode I for a CFRP. It is shon that the substantial difference beteen the longitudinal and transversal stiffness of the composite, together ith the short cohesive zone lead to a substantial transversal ansion of the DCB-specimen. Moreover, it is noted that the position of the initial crack tip is difficult to identify prior to the eriment. Therefore, the positioning of the measuring device is not as accurate as desired. Similar effects have not been observed hen measuring cohesive las for adhesives using metal adherends. In this case, it suffices to measure the ansion of the cohesive zone by LVDT positioned on the outside of the specimen. A novel technique to back-out the cohesive la is presented. The method is based on accurate measurement of the initial position of the crack tip and simulations of the eriments. In these simulations, e assume a triangularly shaped cohesive la. This assumption might prove too restricted to capture the cohesive la for delamination. An indication of this is the substantial variation of cohesive data beteen different eriments. The fracture energy, i.e. the area beneath the cohesive la is hoever accurately measured in the eriments. The adapted J() relation substantially differs from the erimental data indicated by crosses in Fig. 9. That is, ignoring the effects discussed here results in a cohesive la ith as lo as half the peak stress derived here. Acknoledgement The authors acknoledge from financial support from NFFP. Dr Anders Biel, University of Skövde, generously shared erimental data. References [1] Sorensen L, Botsis J, Gmür Th, Humbert L: Bridging tractions in mode I delamination: Measurements and simulations, Composites Science and Technology, 68, (2008) [2] Barenblatt G: The mathematical theory of equilibrium cracks in brittle fracture, in Advances in Applied Mechanics 7, (1962) 9

10 [3] Hillerborg A, Modeer M, Petersson P E: Analysis of crack formation in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research 6, (1976) [4] Bäcklund J: Fracture analysis of notched composites. Computer Structures 13, (1981) [5] Stigh U: Damage and crack groth analysis of the double cantilever beam specimen. Int J Fract 37, R13 18 (1988) [6] Needleman A: A continuum model for void nucleation by inclusion debonding. ASME J Appl Mech 54, (1987) [7] Stigh U: Initiation and groth of an interface crack. In: Mechanical behaviour of adhesive joints, Bordeaux, France, Pluralis, Paris, France (1987) [8] Rice JR: A path independent integral and the approximative analysis of strain concentration by notches and cracks, ASME J Appl Mech 88, (1968) [9] Nilsson F: Fracture mechanics from theory to applications. Department of Solid Mechanics, KTH, Stockholm (2001) [10] Suo Z, Bao G, Fan B: Delamination R-curve phenomena due to damage, J Mech Phys Solids 40, 1-16 (1992) [11] Olsson P, Stigh U: On the determination of the constitutive properties of thin interphase layers An exact inverse solution, Int J Fract 41, R71 R76 (1989) [12] Andersson T, Stigh U: The stress-elongation relation for an adhesive layer loaded in peel using equilibrium of energetic forces, Int J Solids Struct 41, (2004) [13] Stigh U, Andersson T: An erimental method to determine the complete stresselongation relation for a structural adhesive layer loaded in peel, In: Proceedings of the 2nd ESIS TC4 Conference on Polymers and Composites, Eds. J.G. Williams and A. Pavan, ESIS Publication 27, Elsevier, Amsterdam, (2000) [14] Stigh U, Alfredsson K S, Andersson T, Biel A, Carlberger T, Salomonsson K, Some aspects of cohesive models and modelling ith special application to strength of adhesive layers, International Journal of Fracture, published online, DOI: /s (2010) [15] Nilsson F: Large displacement aspects on fracture testing ith double cantilever beam specimen, Int J Fract 139, (2006) [16] Andersson T, Biel A: On the effective constitutive properties of a thin adhesive layer loaded in peel. Int J Fract 141, (2006) [17] Biel A, Stigh U: Effects of constitutive parameters on the accuracy of measured fracture energy using the DCB-specimen, Engineering Fracture Mechanics 75, (2008) 10