A stress-relaxation approach to determine onset of delamination in angle ply laminates

Abstract

A new test method, named multi-relaxation test, is proposed for detecting on-set of delamination in fibre-reinforced polymers. Multi-relaxation test is based on the principle that uses change of stress relaxation behaviour of fibre-reinforced polymer to detect the occurrence of delamination. In this study, angle-ply laminated fibre-reinforced polymer (APL-FRP) is used to demonstrate and evaluate multi-relaxation test for detection of the delamination occurrence. The stress relaxation behaviour is characterized using a standard, three-element viscoelastic model in which the Eyring’s law is used to govern the time-dependent stress response to deformation. Results suggest a high possibility of using the trend line change of viscous stress at the beginning of stress relaxation to determine the critical stroke for the onset of delamination. The results also suggest that value for the corresponding static stress is very close to the value reported in the literature for APL-FRP of the same fibre lay-up. The major advantage of multi-relaxation test over other tests for the same purpose is that multi-relaxation test is able to detect delamination without relying on ancillary information such as acoustic signals. Therefore, multi-relaxation test can be used to characterize critical loading and deformation in fibre-reinforced polymer structures of any size and geometry, even when subjected to a loading mode that mimics the in-service loading.

Keywords

Delamination fibre-reinforced polymers stress relaxation

Introduction

Delamination has long been a major damage mode of fibre-reinforced polymers (FRP). Its occurrence causes significant loss in structural integrity and can lead to catastrophic failure without much warning in advance. In view of such a problem, many test methods have been developed to quantify FRP’s resistance to delamination.^1–3 Most of the test methods are based on classical fracture mechanics, using coupon specimens that contain a specific starting defect to initiate crack growth in the inter-laminar region, from which critical energy release rate for delamination growth is determined. Since calculation of the energy release rate is based on linear-elastic fracture mechanics, specimens for this type of test methods need to satisfy conditions such as small deformation and small-scaled yielding for crack growth. As a result, these test methods need to use specimens with certain fibre orientation, stiffness, and starting defect size, which limit usefulness of the test data, which is mainly for materials evaluation and comparison. Alternative test methods that do not have such specimen requirements are needed to quantify delamination resistance for FRP that is used in load-carrying structures.

Early work on the alternative test methods (e.g., see Wang and Crossman⁴) has shown that the use of notch-free specimens to characterize FRP’s delamination resistance has involved matrix cracking that may not lead to the delamination development.⁵ For angle-ply laminated FRP (APL-FRP), the issue is further complicated by localized delamination initiation which often occurs along free edges of the specimens, commonly known as edge delamination, due to the change of constraint condition and presence of defects along the specimen edges. As a result, use of this type of specimens for characterization of delamination resistance requires either idealization of stress state to extract the delamination criteria,^6,7 or delicate instrumentation to obtain ancillary signals to identify the occurrence of delamination.^8,9 Both approaches are limited to the use of small coupon specimens for the testing, and thus not suitable for large FRP structures of various geometry and dimensions.

Objective of this study is to develop a new test method to detect critical loading for delamination initiation in FRP without the above limitations on size and geometry of the test piece. This paper describes an approach for the new test method and validates the approach by comparing results from the new test method with those reported in literature for specimens of the same fibre lay-up and similar mechanical properties for the matrix.¹⁰ It should be noted that use of small coupon specimens is simply because results for such a type of specimens are available in literature for comparison. However, as to be shown in this paper, since the new approach uses only results from the mechanical testing to characterize delamination, it can be applied to FRP structures of any size and geometry, even under the loading modes that mimic the in-service conditions.

Concept of the new approach

The new approach is to use stress relaxation behaviour of FRP to detect delamination. As a polymer-based material, FRP shows stress drop at a constant deformation level after loading. Profile for this stress drop is expected to change with the change of stress distribution in FRP. Since delamination in FRP leads to stress concentration in front of the inter-laminar cracks and stress relief in the cracked inter-laminar region, the occurrence of delamination should be detectable by monitoring the change of stress relaxation behaviour under loading.

In this study, the stress relaxation behaviour is characterized using a standard, linear viscoelastic model, as shown in Figure 1, to mimic the time function of stress drop profile during the stress relaxation. The viscoelastic model consists of a static branch and a viscous branch that are connected in parallel. Spring constants for the static and viscous branches, and reference stress and reference strain rate for the damper in the viscous branch are represented by E_st, E_v, $σ_{0}$ and ${\overset{\cdot}{ɛ}}_{0}$ , respectively. Since the two branches in Figure 1 are connected in parallel, the applied strain ( $ɛ_{A}$ ) is equal to the total strain in each branch, that is, $ɛ_{st}$ for the static branch and $ɛ_{v}$ for the viscous branch. Furthermore, since spring and damper in the viscous branch are connected in serious, $ɛ_{v}$ is equal to summation of their strains, $ɛ_{v, sp}$ and $ɛ_{v, D}$ , respectively.

The applied stress ( $σ_{A}$ ) is summation of stresses applied to the two branches, which are $σ_{st}$ for the static branch and $σ_{v}$ for the viscous branch, each of which can be determined from strain generated in its spring element, that is

σ_{st} = E_{st} ɛ_{st}

(1)

σ_{v} = E_{v} ɛ_{v, sp}

(2)

During stress relaxation, value for $ɛ_{v, sp}$ decreases with time, but the corresponding strain in the damper ( $ɛ_{v, D}$ ) increases accordingly. In this study, $ɛ_{v, D}$ is assumed to be zero at the beginning of stress relaxation and $ɛ_{v, sp}$ to approach zero when time goes to infinity.

In addition to equation (1), $σ_{st}$ can also be determined from the difference between $σ_{A}$ and $σ_{v}$ at the beginning of stress relaxation, i.e.

σ_{st} = σ_{A} (0) - σ_{v} (0)

(3)

To determine $σ_{v} (0)$ value for the model in Figure 1, it is assumed that relationship between $σ_{v}$ and strain rate of the damper, ${\overset{\cdot}{ɛ}}_{v, D}$ , is governed by the Eyring’s law¹¹

(\frac{{\overset{\cdot}{ɛ}}_{v, D}}{{\overset{\cdot}{ɛ}}_{0}}) = \sinh (\frac{σ_{v}}{σ_{0}})

(4)

where

σ_{0}

and

{\overset{\cdot}{ɛ}}_{0}

are reference stress and reference strain rate, respectively, with the following expressions

σ_{0} = KT / V

(5)

{\overset{\cdot}{ɛ}}_{0} = \frac{σ_{0}}{τ_{r} E_{v}}

(6)

where K being the Boltzmann constant, T absolute temperature, V activation volume for the deformation, and

τ_{r}

relaxation time.

Based on the model in Figure 1, with variation of $σ_{v}$ governed by equation (4), the following expression can be derived to represent the time function of the applied stress drop ( $Δ σ_{A}$ ), equivalent to stress drop in the viscous branch ( $Δ σ_{v}$ ), during the stress relaxation^12,13

Δ σ_{A} = Δ σ_{v} = σ_{v} (0) - 2 σ_{0} \tanh^{- 1} [\tanh (\frac{σ_{v} (0)}{2 σ_{0}}) exp (- \frac{t}{τ_{r}})]

(7)

For tests conducted at constant temperature, activation volume V is expected to remain constant during the stress relaxation, and according to equation (5), so does $σ_{0}$ . As a result, the trend line change for $σ_{0}$ has been used to detect the change of microstructural phases involved in the deformation process of materials such as semi-crystalline polymers.¹³ For the study presented here, changes in both $σ_{0}$ and $σ_{v} (0)$ are considered for the possibility of using their trend line change to detect the onset of delamination in APL-FRP.

For simplicity, $τ_{r}$ value in equation (7) is assumed to be constant, independent of the deformation level applied for the stress relaxation, in spite of the well-known, complex dependence of $τ_{r}$ on the deformation behaviour at the molecular level.¹⁴ Based on equation (6), the assumption of constant $τ_{r}$ value requires constant reference strain rate ( ${\overset{\cdot}{ɛ}}_{0}$ ), but at this stage, relationship of ${\overset{\cdot}{ɛ}}_{0}$ and mechanical testing condition is not clear. Nevertheless, loading rate used in this study to reach all deformation levels for stress relaxation was kept constant to minimize its possible influence on ${\overset{\cdot}{ɛ}}_{0}$ value. With the above test conditions, experimentally measured variation of $Δ σ_{A}$ as a function of stress relaxation time t was fitted using equation (7) by adjusting $σ_{0}$ and $σ_{v} (0)$ values, in order to establish variation $σ_{0}$ and $σ_{v} (0)$ values as functions of stroke used for the stress relaxation. The corresponding $σ_{st}$ value was then determined using equation (3).

Material and test conditions

A tensile test program was designed based on the above concept to introduce multiple stress relaxation stages in a single specimen using stroke control to vary the deformation level. The test is thus named multi-relaxation (MR) test. Experimentally measured time functions of stress drop were then fitted using equation (7) by adjusting $σ_{0}$ and $σ_{v} (0)$ values, in order to explore the feasibility of using the trend line change of $σ_{0}$ and $σ_{v} (0)$ to detect the critical stroke for the on-set of delamination in APL-FRP.

Coupon specimens for the MR test were prepared from a carbon fibre-reinforced APL, with (±30°)_3S for the fibre layup, using unidirectional prepreg of carbon fibre/epoxy (T300/7901) of which the fiber volume fraction was in the range of 58–62%, provided by Guang Wei Carbon Fiber Co., Ltd. The APL was produced using a vacuum hot pressing technique for the consolidation, following the processing conditions recommended for the material, that is, at the curing temperature of 120℃ and with the maximum pressure of 0.65 MPa. Four coupon specimens with dimensions shown in Figure 2 were machined from the APL. Tabs were then attached to both ends of each specimen to ensure proper gripping of the specimens during the test.

The study was first to determine the suitable stroke increment that should be introduced before each stress relaxation stage. Ideally, the stroke increment should be as small as possible in order to provide a good stroke resolution for the on-set of delamination. However, decrease of the stroke increment increases test time required to reach the critical stroke for delamination, and also reduces the amount of viscous stress to be released at each stress relaxation stage. Furthermore, decrease of the stroke increment decreases the amount of delamination that can be generated at each loading stage, and thus may reduce detectability of the change in the stress relaxation behaviour. In view of these issues, three stroke increments of 0.1, 0.15, and 0.2 mm were first used to conduct the MR test in order to determine the most suitable stroke increment among the three for a clear indication of the change in the trend lines for $σ_{0}$ and $σ_{v} (0)$ . Data from MR test with the selected stroke increment were then used to investigate feasibility of using change in the trend lines for $σ_{0}$ and $σ_{v} (0)$ to detect the onset of delamination in APL-FRP.

The MR tests were conducted using a Qualitest universal test machine in a tensile mode, with the load cell capacity of 100 KN. The tests were carried out under stroke control, at the crosshead speed of 1 mm/min for the loading, and with at least nine intermittent crosshead halts for stress relaxation, each for a relaxation period of 10,000 s. As mentioned earlier, three specimens were first tested with stroke increments of 0.1, 0.15 or 0.2 mm. Once the appropriate stroke increment was identified, which as to be justified was 0.15 mm, the fourth specimen was then tested with this stroke increment to examine repeatability of the test results.

Curves of stress drop during stress relaxation were fitted using equation (7) by adjusting $σ_{0}$ and $σ_{v} (0)$ values, with $τ_{r}$ fixed at 16,000 s. Values for $σ_{0}$ and $σ_{v} (0)$ so-determined were then examined for their functions of stroke, to identify the critical stroke for the onset of delamination.

Test results, analysis and discussion

Results from the MR tests are summarized in Figure 3. Figure 3(a) and (b) present the same set of curves of engineering stress versus stroke from all MR tests conducted in the study. Figure 3(a) has all four curves overlapped to show the consistency of trend lines between engineering stress and stroke in spite of different stroke increments used for the testing. Figure 3(b) has the four curves separated using a stroke offset, which shows that except the curve with the stroke increment of 0.1 mm, all other curves contain very small load drops before the commencement of stress relaxation, starting at engineering stress levels above 300 and 400 MPa for curves with stroke increment of 0.15 and 0.2 mm, respectively. The first of such small load drops in each curve is indicated in Figure 3(b) using an arrow. As mentioned earlier, these load drops occurred during the loading phase, not after stress relaxation has started. Therefore, its occurrence is probably due to small matrix cracking introduced during the loading. Work in the literature^15–17 has suggested that matrix cracking does not always cause a load drop, which supports the phenomenon shown in Figure 3(b), for the curve with stroke increment 0.1 mm. Although this curve does not show any small load drop during the loading phase, post-test inspection of the specimen suggested that matrix cracking has occurred during the MR test.

Figure 3(c) presents an example of using equation (7) to fit the experimentally measured time function of Δσ_A. As shown in the figure, the curve generated from equation (7) could only fit part of the experimental curve, which was selected to be for the time period after 1000 s. Selection of this time period is because such curve fitting provides a better prediction of $σ_{v} (0)$ value which for the model in Figure 1 is equivalent to the long-term stress drop from the stress relaxation.

Figure 4 presents $σ_{0},$ $σ_{v} (0)$ , and $σ_{st}$ values determined from the curve fitting, plotted as functions of stroke for all four specimens used in the study. Figure 4(a) and (b) shows the same data sets of $σ_{0}$ . The former has all data sets overlapped, and the latter uses a stroke offset to separate the four data sets. The corresponding data sets of $σ_{v} (0)$ are presented in Figure 4(c) and (d) in a similar arrangement, and the data sets of $σ_{st}$ is presented in Figure 4(e). As shown in Figure 4(a) and (c), both $σ_{0} and$ $σ_{v} (0)$ values show some sensitivity to the change of stroke increment used for the testing, especially at stroke levels above 1 mm. Such dependence is expected, as change of the stroke increment affects test time needed to reach a given stroke level, which is expected to affect the viscous state of the deformation. Figure 4(c) also suggests that smaller the stroke increment is, smaller the $σ_{v} (0)$ value is at a given stroke, which is consistent with the expectation that longer is the time allowed for stress relaxation, smaller is the viscous stress component.

The trend of variation for $σ_{0}$ and $σ_{v} (0)$ with increase of stroke is better presented by separating the four data sets using a stroke offset, as shown in Figure 4(b) and (d). Figure 4(d) clearly shows a change of the trend line in the curves with stroke increment of 0.1 and 0.15 mm, in which arrows are used to point out the transition points. However, such transition is not shown in the curve with stroke increment of 0.2 mm in Figure 4(d). For $σ_{0}$ , as shown in Figure 4(b), none of the curves shows any clear change of the trend line. Rather, these curves show some scattering of $σ_{0}$ values determined using data from the MR tests. The corresponding curves for $σ_{st}$ , as indicated in Figure 4(e), are highly consistent with little dependence on the stroke increment used for the testing. This is expected, as $σ_{st}$ value should not be sensitive to the change of time involved for stress relaxation. Note that the three points circled in Figure 4(e) correspond to the $σ_{st}$ values at the strokes that show the change of trend lines for $σ_{v} (0)$ in Figure 4(c) and (d). These $σ_{st}$ values are 262, 267 and 278 MPa, with the average of 269 MPa.

Figure 4(c) suggests that the two sets of data for the stroke increment of 0.15 mm, presented using legends of diamond and square, provide the clearest indication for the change of trend line for $σ_{v} (0)$ . The two curves also suggested that their $σ_{v} (0)$ values have good reproducibility. However, as shown in Figure 4(a), reproducibility of the corresponding $σ_{0}$ values is poor, especially for strokes above 1 mm.

Figure 5 summarizes the stroke functions of $σ_{st}$ , $σ_{v} (0)$ and $σ_{A} (0)$ for one of the two specimens tested with stroke increment of 0.15 mm. The figure shows three changes of the data trend. The first change occurs at the stroke around 0.75 mm at which all three curves show a downward bend, but the change for $σ_{v} (0)$ is clearer than the changes for $σ_{A} (0)$ and $σ_{st}$ . The second change is at the stroke around 1.35 mm for $σ_{v} (0)$ and $σ_{st}$ . Change for $σ_{v} (0)$ at this stroke is an upward swing and change for $σ_{st}$ is a slightly further downward bend. But the trend line for $σ_{A} (0)$ shows little difference, continuing to increase with the increase of stroke. The third change is at the stroke around 1.65 mm, for $σ_{st}$ only, at which $σ_{st}$ reaches a peak value, but $σ_{v} (0)$ and $σ_{A} (0)$ values continue to increase with the increase of stroke. Note that increase of $σ_{A} (0)$ from this point to the next is 27 MPa, which is the result of a big increase of $σ_{v} (0)$ of 41 MPa and a relatively small decrease of $σ_{st}$ of 14 MPa. Due to the significant increase of $σ_{v} (0)$ , the decrease of $σ_{st}$ with further increase of stroke from 1.65 mm could not be detected using the experimentally measured $σ_{A} (0)$ value.

Failure in an APL-FRP specimen of $\pm 30 \deg$ is known to be associated with edge delamination.¹⁷ Figure 5 suggests that the edge delamination starts at the first change of the trend lines at the stroke around 0.75 mm. At the stroke around 1.35 mm when the second change of the trend line occurs, delamination growth starts in the gauge section of the specimen, which greatly increases the volume that is subjected to high stress, thus accelerating the $σ_{v} (0)$ increase with further increase of the stroke. At the stroke of 1.65 mm, the appearance of peak value for $σ_{st}$ is an indication that delamination-induced final fracture is imminent.

As mentioned earlier, the first change of the trend line occurs at an average $σ_{st}$ value of around 269 MPa, for the onset of edge delamination. This value is very similar to that reported in Lorriot et al.¹⁰ for delamination in a similar APL-FRP system, T800/914 of (±30°)_3S. In that work, critical stress for delamination was estimated based on the classical laminate theory (CLT) with the use of a correction factor to adjust the local stresses along the free edges.¹⁸ In contrast to the CLT approach which determines the critical condition for the delamination initiation based on the linear elasticity principle, the approach described here considers viscous behaviour of the polymer matrix and thus has the advantage of detecting the delamination initiation purely based on the experimental testing. This advantage allows determination of the critical condition for delamination initiation in a full-sized FRP structure. Such a capability is very difficult to achieve using any of the existing test methods due to size limitation or delicate instrumentation required to obtain ancillary signals to detect the on-set of delamination.

Conclusions

A new approach is proposed for determining the loading level for onset of delamination in FRP. The new approach was evaluated using an angle-ply laminate of $\pm 30 \deg$ . The measured stress level for delamination initiation is consistent with the stress level reported in the literature, for a similar type of FRP with the same fibre lay-up.

Since this new approach does not require specimens with any pre-notch or specific fibre layup, and can be used to determine critical stress for delamination purely based on the mechanical testing, it has the advantages of allowing the evaluation of delamination resistance for large FRP structures when subjected to the in-service loading mode. However, the current study only provides limited data for the evaluation. Further evaluation is needed to confirm its validity, which is being planned when this manuscript is prepared.

Footnotes

Acknowledgements

Idea for this new approach was generated during Jar’s visit to the Tongji University in Shanghai, China in the summer of 2018. Sincere appreciation is due to staff at the Tongji University for preparation of the angle ply laminate and staff at the University of Alberta for preparation of specimens used for the testing. Without these high-quality specimens, it is unlikely that evidence to support the idea could be obtained.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The subsequent work was sponsored by the National Natural Science Foundation of China (Grant No. 11832014) and the Natural Sciences and Engineering Research Council of Canada (NSERC).

ORCID iDs

Yi Xiao

P-Y Ben Jar

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