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How Similitude Can Improve Our Understanding of Fatigue Delamination Growth

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How Similitude Can Improve Our Understanding of Fatigue Delamination Growth Misinterpreting the Results: How Similitude can Improve our Understanding of Fatigue Delamination Growth *Submitted for publishing in Composites Science and Technology on March 25 2010 Calvin Rans, René Alderliesten, Rinze Benedictus Structural Integrity Group, Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands Abstract: Use of the strain energy release rate for characterizing delamination growth in composite and bonded structures is now commonplace. Although its use is common, a consensus on the best formulation of strain energy release rate has not been reached for describing fatigue delamination growth. Most commonly selected formulations are the maximum strain energy release rate and the strain energy release rate range. This paper examines the use strain energy release rate range and how a misconception of its definition is misleading our understanding of fatigue delamination growth. Keywords: Delamination, fatigue, strain energy release rate, similitude 1. INTRODUCTION The study and characterization of delamination growth behaviour has gained significant attention in the research community due to the growing use of composite structures in a wide range of structural applications. A widely utilized approach is to apply a linear elastic fracture mechanics (LEFM) description using the strain energy release rate (G) in a manner analogous to metal crack growth in terms of the stress intensity factor (K). The primary reason for the use of G rather than K is that the local crack-tip stresses used to determine K are difficult to obtain in inhomogeneous composite laminates [1, 2]. Use of G avoids the need to determine crack-tip stresses. Based on an analogy with the Paris relation for metal crack growth, delamination growth rate has been experimentally observed to have a linear relationship on a log- log scale when plotted vs. different formulations of the strain energy release rate, G. What has resulted is a consensus that for fatigue delamination growth (db/dN): db n = CfG⎡ ()⎤ (1) dN ⎣ ⎦ with C and exponent n being material constants. No consensus, however, has been reached on the formulation of f(G), resulting in it is simply being viewed as a curve fitting parameter that can be manipulated to best fit the experimental data. By far the most prevalent formulations used in the literature are f(G) = Gmax [3, 4] based on the importance of this parameter for static delamination growth, and f(G) = ΔG (where ΔG = Gmax - Gmin) [1, 5-14] based on an analogy with the use of ΔK for fatigue crack growth in metals. Both of these parameters can result in misleading results for fatigue delamination growth. Examples of potentially misleading results can be illustrated with data from the literature. Figure 1 shows an example of Mode I fatigue delamination growth data for Preprint submitted to Composite Science and Technology (25/03/2010) Page 1 of 18 various R-ratios obtained by Hojo et al. [7] for a unidirectional carbon epoxy laminate plotted vs. Gmax and ΔG = Gmax - Gmin. The variation in behaviour for Figure 1a is not unexpected as fatigue is a result of cyclic loading, and use of Gmax does not distinguish between the different minimum load cycles for the various R-ratios. The use of ΔG = Gmax – Gmin in Figure 1b attempts to remedy this by using the applied strain energy release rate range. A reduction in the variation between the various R- ratios is indeed observed, however, there is still a clear R-ratio dependency. Furthermore, this dependency seems counter-intuitive. For the same ΔG, a higher delamination growth rate is observed for a lower R-ratio, and thus lower mean stress. In other words, a reduction in load results in increased fatigue damage! 10-6 10-6 R=0.2 P305 R=0.2 P305 R=0.5 R=0.5 10-7 R=0.7 10-7 R=0.7 10-8 10-8 10-9 10-9 10-10 10-10 da/dN [m/cycle] da/dN [m/cycle] 10-11 10-11 10-12 10-12 50 100 150 200 40 60 80 100 120 140 G [N/m] ΔG =G -G [N/m] (a) Imax (b) I Imax Imin Figure 1: R-ratio dependency observed in Mode I delamination growth in unidirectional carbon- epoxy laminates (reproduced using data from [7]) Figure 2 shows Mode I fatigue delamination growth results for a single Double Canteliver Beam (DCB) test of a unidirectional carbon-epoxy laminate carried out by Khan et al. [11]. Khan performed the fatigue test using a constant cyclic displacement at an R-ratio of 0.35. Once the delamination stopped growing (due to a decrease in G with delamination length) Khan increased the opening displacement and resumed testing under the same R-ratio. The two tests performed on the same specimen under the same R-ratio resulted in two clearly distinct curves, suggesting that the delamination growth resistance for the single specimen changed between tests! Preprint submitted to Composite Science and Technology (25/03/2010) Page 2 of 18 opening displacement 1 10-3 opening displacement 2 10-4 db/dN [mm/cycle] 50 100 150 200 250 G -G [J/m2] max min Figure 2: Variation in delamination resistance for Mode I delamination growth within the same test specimen [11, 15] This paper will examine how the use of f(G) = Gmax - Gmin can lead to misleading interpretations of experimental data, similar to those illustrated in the above two examples. The source of these misinterpretations stems from the fact that the definition of ΔG = Gmax - Gmin violates the underlying principle used to describe fatigue crack growth: the principle of similitude. A similitude based definition of ΔG will be presented and how the use of this similitude based definition can improve our understanding of delamination growth will be discussed. 2. SIMILITUDE AS APPLIED TO METAL CRACK GROWTH The fundamental idea behind the principle of similitude is related to the fact that the size of an object is relative and physical phenomena within an object can be reduced to size independent relations [16, 17]. This principle is used regularly in engineering to describe physical phenomena. A simple example is the Hooke’s Law relation which relates force-deflection behaviour in a structure based upon the size independent similitude parameters of stress and strain. Using these similitude parameters, a size independent material constant, elastic modulus, can be defined. The principle of similitude can also be applied to fatigue crack growth in which the crack growth behaviour is reduced to material behaviour independent of crack length. To do so, a similitude parameter must be defined which relates the conditions for similitude between the various crack geometries. The driving force for growth of a crack (either static or fatigue) is related to the stress state ahead of the crack tip; thus similitude implies that for a similar stress state ahead of a crack tip (regardless of crack length) similar crack growth behaviour should be observed. In metals, we represent this stress state by the linear elastic stress intensity factor, K, which becomes the basis for similitude in crack growth analysis. It should be noted that the use of K is a simplification as the stress state in front of a crack is not linear elastic. Defining K as a similitude parameter for crack growth, we can now differentiate between static and fatigue crack growth. Static crack propagation is related to a critical stress state in front of the crack tip and is thus propagation occurs when a limit or critical K value is reached. Fatigue is related to the cyclic stress state in front of the Preprint submitted to Composite Science and Technology (25/03/2010) Page 3 of 18 crack tip, thus the similitude parameter for fatigue becomes ∆K = Kmax – Kmin. Experimental observation has shown that for fatigue, the crack growth rate follows a linear trend, when plotted on a log-log scale vs. ΔK, for the majority of the crack growth life (the so-called Paris region). This has led to the commonly applied Paris relation for describing fatigue crack growth: da n =ΔCK()cg (2) dN cg where the material constants C and n have been given the subscript cg to distinguish them from C and n for delamination growth. Numerous other formulations have been proposed to describe crack growth that includes the two asymptotes bounding the Paris region (defined by the threshold ΔK for delamination growth and the critical K for static fracture), however these unnecessarily complicate the objective of this paper. Only the Paris region of growth will be discussed. The reality for metals is that Ccg and ncg are not simple material constants. The most notable case is the variation in crack growth behaviour (and thus Ccg and ncg) for fatigue loading under various R-ratios. This variation is not a result of a failure of the similitude principle, rather it arises due to the simplification in characterizing the crack growth driving force by a linear elastic description (K) of the stress state ahead of the crack tip. Knowing this has allowed mechanisms that alter the crack driving force which are not captured by ∆K to be identified. In the case of the R-ratio dependency in metals, that effect is related to variations in the amount of crack tip plasticity with varying R-ratio, and the reduction in crack growth driving force related to plasticity induced crack closure in the wake of the crack. Identifying this mechanism has permitted the continued use of a linear elastic stress intensity factor description of crack driving force with a correction related to the crack closure mechanism. Recognizing that for closure, there is no crack growth driving force when the crack tip is closed, similitude can be redefined based on the stress intensity factor range between maximum crack opening and crack closure: da n n =ΔCK()eff = CK()max − K close (3) dN where Kclose is the linear elastic stress intensity factor which is calculated for the tensile stress at which the crack tip closes.
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