An efficient methodology for the experimental characterization of mode II delamination growth under fatigue loading

Crack growth rate curves provide information about the delamination resistance of composite materials under cyclic loading. The existing methodologies for mode II fatigue testing using three-point bending end-notched flexure (3-ENF) under constant cyclic displacement conditions yield discontinuous delamination growth rate curves, therefore requiring a batch of several specimens to be tested under different severity conditions in order to fully characterize the crack growth. This work describes a variable cyclic displacement test procedure that, in combination with the real time monitoring of the specimen’s compliance, allows the crack growth rate to be measured for the desired range of severities with a single specimen, thus avoiding any human intervention during the test.


Introduction
1 Emerging delaminations and their growth under repeated or cyclic loads can reduce the load carry-2 ing capacity of composite structures. In consequence, a reliable design should account for this damage 3 mechanism. The experimental characterization of interlaminar fracture properties under fatigue load-4 ing assesses the damage tolerance of the composite materials in service. 5 The no-growth criterion and the damage tolerance approach are two alternatives to deal with fatigue. However, while the 4-ENF test is not preferable because of friction effects [14], only further 23 research will show whether 3-ENF or C-ELS is better suited for cyclic mode II fatigue delamination 24 characterization [10]. 25 Figure 1: a) Sinusoidal shaped loading cycles with constant displacement. The sign convention used in this work is negative for displacements which result in compressive reaction forces. b) 3-ENF test configuration, where L is the mid-span length, a is the crack length, a0 is the initial crack length and 2h is the specimen's total thickness of the specimen.
The selection of the testing parameters for mode I fatigue experiments is not critical. For a test 26 conducted under displacement control and constant displacement amplitude, the energy release rate 27 (the severity of the load) decreases as the crack grows. That is, the crack growth rate vs. load severity 28 curve sweeps from left to right until the crack growth rate becomes unnoticeable (the threshold, the 29 severity for which the crack growth rate tends to zero). Nevertheless, even in mode I, the determination 30 of the threshold value remains elusive due to the need of high sensitivity measurement devices to capture 31 the actual growth rates [15]. 32 For mode II 3-ENF experiments the contrary applies as the range of the crack growth rate curve 33 swept in a single test under constant cyclic displacement is very narrow. This results from the de-   usually by means of the specimen's compliance. Indeed, the use of the compliance to control the machine, or any other behavioral-based control technology, can lead to unexpected load setpoints. Figure 3: Chosen relation between the maximum cyclic energy release rate, Gmax, normalized to the quasistatic fracture toughness, Gc, and the crack length. Gmax,0 and a0 are the initital energy release rate and crack length, while G max,f and a f are the energy release rate and crack length at the end of the test.
where m G and n are the slope and the y-intercept, respectively: Assuming a linear elastic behavior of the specimen, the energy release rate reads [24]: thus, the minimum cyclic displacement (δ min ) is related to G max by where B is the width of the specimen and C is the specimen's compliance (δ/P , where P is the load).

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The compliance of the specimen increases as the crack length grows.
where m cc and C 0 are fitting parameters.

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Equations 1 to 6 can be used to write the dependence of (δ min ) with the specimen compliance 99 measured in real time: variable, as the number of cycles, N , is.

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To deduce the δ min (N ) function, a relation between the crack length (or, equivalently, the compli-107 ance, equation 6) and the number of cycles is necessary. Here, we assume that the crack growth rate 108 follows the Paris' law based expression [26]: Although the simplest expression of the Paris like power law has been used to relate crack growth 110 to the energy release rate, it is worth noting that the methodology described here can also be used with  The estimated A and p parameters are used to find the expression that relates the crack length to 122 the number of cycles, by integrating equation 8: and Finally, by substituting equations 1, 9, and 6 and its derivative, into equation 5, the functions of 126 the minimum cyclic displacement with the number of cycles reads: (13) Hence, the minimum cyclic displacement is a function of the number of cycles (N ), the specimen 128 width (B), the initial conditions (a 0 , G max,0 /G c ), the user-defined gradient of energy release rate (m G ), 129 the static compliance calibration parameters obtained prior to the fatigue test (m cc , C 0 ) and, finally,  In this work it is assumed that the fatigue delamination growth, under pure mode II loading 134 conditions, depends only on the peak energy release rate, G max , and the load ratio (equivalent to 135 the minimum to maximum cyclic displacement ratio, R = δ min /δ max , for small deflections that the behavior of the specimen was linear elastic, the selection of the initial minimum displacement 171 (δ min,0 ) was related to G max,0 using equations 5, and 6 and its derivative, and the initial crack length 172 (a 0 ): We established a maximum initial severity, G max,0 /G c , of 0.55 to avoid the horizontal movement 174 of the sample occurring for displacements below -3.5 mm. Other authors use mechanical restraints to 175 avoid this [10] although this could affect the results.

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The TestStar v3.5C control software for the MTS servohydraulic testing machine includes a "Calcu-177 lated Channels" option that allows for internal variables, either external inputs or calculated through 178 simple arithmetic operations, to be generated. It was used to compute the dynamic compliance, C * , by 179 processing the instantaneous signals of load and displacement in line with the methodology described 180 in [11]. Next, the crack length was derived from the dynamic compliance using equation 6: One set of data (the dynamic compliance, C * , the minimum cyclic load, P min , and the number of  where the crack length, a, is taken at the midpoint of the cycle.

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Constant displacement tests were performed using eight different initial severities in order to cover   the third region the crack tends to arrest with a higher slope than that seen in the second region.

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For these reasons discussed in the next section, the curve was truncated, neglecting the first and third 231 regions. Only the mid-region was considered in the modified Paris' law calculation.       da/dN [mm/cycle] 04 C Initial severity 55% 05 C Initial severity 40% 06 C Initial severity 30% 07 C Initial severity 25% 08 C Initial severity 20% 09 C Initial severity 18% 10 C Initial severity 14% 11 C Initial severity 12% 01 V Severity from 50% to 10% (m G = − 0.030) 02 V Severity from 50% to 10% (m G = − 0.090) 03 V Severity from 50% to 10% (m G = − 0.225) Modified Paris' Law fitting of the constant displacement tests Range of severities covered by constant displacement tests Figure 6: Relation between crack propagation rate and peak energy release rate for R=0.3. The results from both variable displacement ("V" labelled) and constant displacement ("C" labelled) tests are presented for comparison proposes.

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The proposed experimental methodology for mode II testing in the 3-ENF configuration enables a 250 chosen range of load severities to be swept while the crack grows in a predefined crack length increment. to sweep the desired severity range. In any event, the severity range achieved would be much larger 258 than that attained using a constant displacement test.

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The variable cyclic displacement method leads to crack growth rate data practically indistinguish-260 able from that resulting from the complete set of eight constant displacement tests. In particular, the  severity being applied. This is more likely to happen as the shedding rate increases and that deviation 283 would also be more important the stronger the variation of the FPZ with the load severity is.
284 Figure 6 illustrates the crack growth rate curves obtained from the variable cyclic displacement tests for the three shedding rates, m G , explored in this study. of the specimen's compliance, avoids the need of human intervention during the test.

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The usefulness of this methodology has been exemplified with an experimental testing campaign 339 in which the crack growth rate curve obtained is compared with the modified Paris' law fitting data 340 from a batch of constant cyclic displacement tests.

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The range of severities covered by a single test using the developed methodology spans from 0.45 342 to 0.1. Due to the specimen movement, the initial severity could not be higher than 0.50.

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The time saved employing the methodology developed has been demonstrated (the example per-344 formed shows a reduction of 1/80) and how the duration of the test, which is determined by the 345 shedding rate and the range of severities explored, is limited by the requirement of forming the com-346 plete failure process zone corresponding to the actual load severity is discussed.