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A Split-Hopkinson Tension Bar study on the dynamic strength of basalt-fibre composites
Composites Part B 171 (2019) 310–319
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A Split-Hopkinson Tension Bar study on the dynamic strength of basalt-fibre composites G.C. Ganzenmüller a, b, *, D. Plappert b, A. Trippel b, S. Hiermaier a, b a b
Fraunhofer Ernst-Mach-Institute for High-Speed Dynamics, EMI, Freiburg i. Br, Germany Albert-Ludwigs Universit€ at Freiburg, Institute for Sustainable Systems Engineering, Freiburg i. Br, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Mechanical testing Strength Basalt fibre
This paper investigates the strain rate sensitivity of laminated composites made of plies of unidirectional basalt fibres and epoxy resin. We consider laminates with quasi-isotropic [0∘, 45∘,þ45∘,90∘]s and orthogonal [þ45∘, 45∘]4 layup. A Split-Hopkinson Tension Bar is used to generate accurate stress/strain data at elevated rates of strain of �3 � 102/s. Moderate strain rate effects are observed with strength increase of �3.5% per decade of increased loading rate for both laminate types.
1. Introduction The purpose of this work is to present reasonably accurate dynamic tensile strength data for composites made of plies of endless, unidirec tional basalt fibres and epoxy resin. Over the last two decades, basalt fibres have come into consideration as potential reinforcement for composite materials requiring high strength and temperature resistance. In contrast to the infamous asbestos fibre, which meets these application requirements from a mechanical point of view, basalt is not considered carcinogenic [1]. It is a mineral belonging to the group of silicates and as such similar to glass in its chemical composition but with more iron and less calcium content. While glass fibres are completely amorphous solids, basalt fibres can feature some degree of cristallinity [2]. Basalt’s mechanical properties strength and stiffness are lower than those of carbon fibres, and more akin to those of S-2 glass fibres. The quasi-static properties of basalt fibres and its composites are already well known [3, 4]. A promising field of application for basalt fibre composites are structural elements in the automotive sector [5], where it is important to predict the dynamic material behaviour at elevated strain rates relevant at crash, which are on the order of 10–100/s [6]. However, as of now, only a few studies are available on dynamic properties. This is particu larly true for composites composed of plies of endless, unidirectional (UD) fibres. To our knowledge, only one, very recent study is available [7], where the authors report that the dynamic tensile strength of a pure 0∘ basalt/epoxy composite doubles its strength when strain rate is increased from 10/s to 300/s. This amount of strain rate sensitivity appears exceptionally high, as UD composites of carbon or glass fibres
only exhibit strength increases of approximately 1%–10% per order of magnitude of increased strain rate [8–14]. In another study, the strain rate sensitivity of woven basalt/epoxy composites was studied, and a similarly exceptional high strain rate sensitivity was observed [15]. Is there something special about basalt fibres which makes them very strain rate sensitive? To shed some light on this issue, we investigate the strain rate dependency of the tensile failure strength and strain for quasi-isotropic [0∘, 45∘,þ45∘,90∘]s and orthogonal [þ45∘, 45∘]4 layups of unidirectional basalt fibres in a thermoset epoxy matrix within the strain rate regime 10 3 – 3 � 102/s. To obtain accurate data at high rates of strain, we employ a Split-Hopkinson Tension Bar, for reasons we feel important to briefly review in the following: Acquiring accurate stress/strain data at high rates of strain >100/s is not a trivial task, as has been discussed extensively in the literature [16, 17]. The difficulty may be attributed to three issues: (i) The testing apparatus must achieve a well defined, e.g., constant velocity at which the sample is loaded, typically on the order of a few m/s. This requires strong acceleration and good velocity control of the gripping devices used to interface the specimen with the testing apparatus. (ii) The stress state within the specimen needs to be inferred from a force sensor, which is usually not mounted on the specimen itself but rather located in-line with the loading axis at some distance from the specimen. During the short time span of a dynamic experiment, propagation effects of elastic waves must be considered. The force sensor will, in general, not sense the force experienced by the specimen at a given instant of time, but instead report another value which is influenced by its own acceleration. Furthermore, the dynamic bandwidth, i.e., its frequency response
* Corresponding author. Fraunhofer Ernst-Mach-Institute for High-Speed Dynamics, EMI, Freiburg i. Br, Germany. E-mail address: [email protected] (G.C. Ganzenmüller). https://doi.org/10.1016/j.compositesb.2019.04.031 Received 22 January 2019; Received in revised form 2 April 2019; Accepted 26 April 2019 Available online 4 May 2019 1359-8368/© 2019 Elsevier Ltd. All rights reserved.
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stress/strain data for composites, see e.g. Refs. [20,21], and a strong theoretical basis is available which rationalizes these beliefs [22]. In contrast, Split-Hopkinson Tension (SHTB) Bar experiments are more difficult to perform. The specimen needs to be attached to the ends of the input and output bars without introducing significant additional mass, e.g., specimen grips as used in conventional testing machines. The presence of such additional mass means additional changes in acoustic impedance, causing unwanted wave reflections. This violates the as sumptions made in the classic analysis based on simple wave propaga tion in media with constant acoustic impedance. The situation is fortunate for specimens with cylindrical shape, as these can be threaded at the ends and screwed into matching interior threads in the bar ends. Thus, no additional mass is introduced for affixing the specimen. Com posites, however, are typically tested in a planar strip form. Until now, the only solution has been has been to introduce slots either into the bar ends or into adapters, and bond the specimens into these slots with highstrength adhesives [9,12,23]. This process requires intensive labour and expensive machining. Recently, two of the current authors introduced a clamping device which allows reversible mounting of strip-type speci mens without violating the requirement of constant acoustic impedance [24]. Here, we utilize this clamping device to obtain SHTB data of reasonable accuracy and compare this data with results from quasi-static and moderate strain rates. The remainder of this article is organized as follows: The composite investigated here and the experimental methods employed are described in the next section. We then report our findings for strength and failure strains, analyse the strain rate dependency and conclude with a critical review of our data in the light of other researchers’ findings.
function, must be adequate to accurately describe the specimen’s force signal. Typically, dynamic bandwidths on the order of 1 MHz are required for experiments at strain rates between 102 – 103/s [18]. (iii) Determination of the specimen strain state requires fast and robust measurement techniques. Fast, because for sampling the strain evolution of a specimen with failure strain of 5% at a strain rate of 100/s with only 100 data points, already an acquisition rate of 200 kHz is required. Robust, because the sampling method must yield reliable measurements for a rapidly accelerating and deforming specimen. While strain gauges meet the requirement of fast acquisition rates, it can be difficult to maintain adhesion on the surface of a specimen until failure. Alterna tively, high-speed imaging and optical strain analysis is a universal yet expensive solution if high spatial resolution is required. For conducting experiments at high rates of strain, two popular experimental methods are available. The first method is given by the universal testing machine, powered by a servo-hydraulic actuator. While these machines can operate over large ranges of testing velocities and forces with typical upper limits of 10 m/s and 250 kN, difficulties arise for accurately measuring the force signal of a specimen under rapid loading. Taking reference to the points (i) and (ii) from above, this is due to the following reasons: (I) The apparatus needs to achieve a high testing velocity before the specimen is loaded. This is realized via a lost motion device, which lets the hydraulic piston travel freely under ac celeration. Once the piston is at speed, its movement is coupled instantaneously to the specimen’s grips. The instantaneous acceleration of the – typically heavy – specimen grips causes a strong inertial force response which overlays the true force signal of the specimen’s stress response. This effect may be theoretically visualized by considering the frequency response of a velocity signal with a step-like acceleration. Its Fourier transform shows amplitudes over an infinite frequency spec trum. (II) Force measurement is conventionally performed via a piezoelectric load cell with finite stiffness and its own limited frequency response. Such force sensors act as low-pass filters. The fact that the quasi-infinite frequency spectrum associated with the step loading character of the slack adapter is not present in the output signal is caused by this filtering effect. While servo-hydraulic universal testing machines have their warranted uses for specific applications, acquiring highquality stress-strain data, in particular for lightweight, non-metallic materials, is challenging. Naively applying such a machine at high rates of strain, one does not measure the true specimen response but instead a convolution of the specimen response with the acceleration response of the specimen grips and elastic wave reflection effects with visible oscillatory artifacts, see e.g. Ref. [19]. However, it needs to be pointed out that high-quality data may also be obtained using this approach if its limitations are considered carefully [17]. A more straightforward alternative for high rate testing is given by the Split-Hopkinson Bar method, also known as the Kolsky Bar method. It is out of the scope of the present work to detail this technique extensively, instead the reader is referred to Ref. [18]. Here, the spec imen is sandwiched between two long and straight bars of constant cross section, the input and output bars. An elastic wave of well defined amplitude is created in the input bar, either by impacting the bar with another long bar, the striker, or by pre-tensioning the bar and employing a release mechanism. The elastic wave then passes through the specimen into the output bar, creating secondary transmitted and reflected waves at the interfaces between specimen and input/output bars. These waves are measured using strain gauges mounted on the bars, and analysed using the theory of one-dimensional elastic wave propagation to yield both specimen strain and stress. Due to the simple geometry, all 1D wave effects are accounted for, and the resulting stress/strain curves can be of high quality. However, this fortunate situation applies only to compression experiments performed with the Split-Hopkinson Pressure Bar (SHPB). Here, a small amount of grease is sufficient to stick a cy lindrical sample between the faces of input- and output bar, and no explicit specimen grips are required. Due to this simplicity, the SHPB method is widely accepted for reliably producing accurate compression
2. Materials and methods 2.1. Material specimens Sheets of cured UD basalt fibre composites with layups [0∘, 45∘,þ45∘,90∘]s and orthogonal [þ45∘, 45∘]4 were provided tested. This composite is made from a prepreg of basalt fibres with filament diameter 17 μm and a low-viscosity epoxy resin system. Laminates were cured in an autoclave process resulting in fibre volume fraction of 60%. To illustrate the mechanical properties of this material, we quote the nominal strength and stiffness of pure 0∘ specimens according to ISO-527 as 1310 MPa and 44 GPa [25]. Testing specimens were cut from the sheets using a diamond coated saw blade to obtain good cut surface quality. The dimensions of the quasi-isotropic (QI) specimens are length � width � height ¼ 100 � 3 � 1.6 mm3 with a nominal gauge length of 20 mm. The di mensions of the �45∘ specimens are 100 � 5 � 2.0 mm3 with the same nominal gauge length. The same specimen geometry was used across all strain rates. Specimens were bonded to grooved 7075 aluminium tabs with a high-strength, thermally activated epoxy resin according to Fig. 1. We use split tabs as these can be easily manufactured in large quantities using a CNC router. Additionally, the split tabs provide good accessibility to the interior which makes it trivial to correctly apply the adhesive. The tabs feature holes for M4 through-bolts to affix the assembled specimen to the aluminium grips as shown in Fig. 2. The clamps are designed to fulfill three task: (i) they provide an ISO M12 � 1.5 thread to interface with either the Split-Hopkinson bars or the universal testing machine. (ii) The transition from the circular interfacing end with the testing apparatus to the rectangular shape used to attach the specimen maintains a constant cross section area. For dynamic testing, the SplitHopkinson bars are also made of aluminium, such that constant acous tic impedance is obtained. This avoids unwanted wave reflections due to the specimen gripping device, see Ref. [24] for details. (iii) The ø4.1 mm through holes allow for clamping the specimen inside the grip. To this end, lightweight M4 aluminium bolts with strength class ISO 5.6 may be used, yielding an axial force of 1.8 kN per bolt. With 4 bolts and a 311
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Fig. 1. CAD rendering of the specimen with its split aluminium tabs. Also shown is the fixture which is used to align all parts during assembly and curing of the epoxy adhesive. All dimensions in mm.
Fig. 2. Clamping fixture maintains cross section area and thus provides constant acoustic impedance which is required for reflection-free wave transmission. All dimensions in mm.
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conservatively assumed friction coefficient μ ¼ 1.0 between the clean and unlubed aluminium surfaces, this translates into a useful upper force limit of �14 kN for this clamp. This limit exceeds the expected forces in this work by a factor of 4. We note that those parts of the through-bolts and nuts protruding from the surfaces introduce additional mass, which causes unwanted wave reflections. However, this effect is so minor that its effects cannot be observed in our data.
a Wheatstone bridge circuit to eliminate bending information. The Wheatstone bridge circuit is driven in constant voltage mode and its output is increased by a factor of 100 using an amplifier with 1 MHz bandwidth. This signal is recorded by a data acquisition (DAQ) card operating at 10 MHz and 16 bit resolution. The conversion factor from strain to force is established via a calibration procedure, wherein a dedicated force sensor is placed between input- and output bars, while a static load is applied onto the bars. The signal of the force sensor is recorded using the same amplifier and data acquisition card as during a real experiment. The force sensor, in turn is calibrated against the already calibrated load cell in the universal testing machine described above. Thus, the entire force measuring system consisting of strain gauges, amplifier and DAQ card is calibrated, which compensates for any eventual misalignment of the strain gauges or similar constant systematic errors. In addition to the line-scan camera, we also use a high-speed area camera to obtain images of the deformation process. The highest sam pling rate of this camera is 38.65 kHz at a resolution of 336 � 96 pixels. This is not enough for accurate strain determination, but serves well to obtain a qualitative impression of the failure behaviour, see Appendix A. Both line-scan and area cameras are triggered by the DAQ card to start recording when the incident wave created by the striker reaches strain gauge no. 1, c.f. Fig. 3. In the case of the line-scan camera, a synchronization signal is fed back to the DAQ card, so precise timing information is available to correlate strain and force signals. It is important to realize that even though strain and force signals are perfectly well synchronized in this manner, these signals are measured in different locations: strain is measured on the specimen but force is inferred from the strain gages on the output bar, in our case 200 mm upstream in the transmitted elastic wave direction. The time taken by the elastic wave to travel this distance must be accounted for by shifting the force signal by Δt ¼ 200 mm/5090 mm/ms ¼ 0.039 ms, where 5090 mm/ms is the longitudinal wave speed in aluminium. In classic Split-Hopkinson Bar experiments, a check of dynamic equilibrium is possible by placing strain gages symmetrically around the specimen, i.e., also on the input bar near the specimen. This allows to check when the forces on both sides of the specimen are equal in magnitude, which marks the beginning of measured data validity and constant strain rate. Here, we cannot provide this information as no additional strain gauge was mounted on the input bar. However, the onset of dynamic equilibrium may also be calculated analytically, see e. g. Ref. [22]. Usually, the time taken for the elastic wave to perform 4 to 5 round-trips within the specimen is sufficient [28,29]. In case of the quasi-isotropic specimen, the longitudinal wave speed is � 3300 mm/ms, and the gauge length is 20 mm, resulting in an equilibration time of <60 μs. Our SHTB experiments were set up such that the incident wave at tains a force amplitude of 20 kN, corresponding to a particle velocity of 7.2 m/s. This marks the upper end of what is safely attainable with our aluminium setup due to its mechanical strength. Fig. 4 shows the recording of such an experiment on a quasi-isotropic specimen. The
2.2. Quasi-static and low strain rate testing For the quasi-static and low strain rates of 10 3/s and 10 1/s, we use a screw-driven universal testing machine with 100 kN capacity and a maximum testing velocity of 1500 mm/min. Specimens were attached using the gripping device described in section 2.1, which itself was mounted to the testing machine using cardanic joints to prevent off-axis loading. The testing velocity was adjusted in pilot experiments such that the target strain rates were obtained. Force was measured via the ma chine’s internal load sensor, which is calibrated to accuracy class 1/ISO 7500–1. Strain was measured optically using a black/white camera with resolution of 1280 � 1024 pixels and a maximum frame rate of 1500 fps at this resolution. To this end, white contrast marks were painted on the specimen at the edges on the gauge section with a paint marker. The displacement of the contrast marks was recorded and converted into a strain time series using a motion analysis software based on Digital Image Correlation (DIC). Both force and strain time series were com bined into nominal stress/strain graphs by eliminating the time infor mation. We note that for this universal testing machine in combination with the tested specimens, a strain rate of 10 1/s is the most dynamic experiment that can be achieved before acceleration effects cause ringing artifacts in the force signal can be observed. 2.3. Dynamic testing apparatus The Split-Hopkinson Tension Bar (SHTB) used here is sketched in Fig. 3 and described in detail in Ref. [26]. Compared to other SHTBs, this setup is optimized for low velocities, low forces and a long pulse dura tion of 1.2 ms. Specimens were attached to the bars using the grips detailed in section 2.1. Shortly before performing the experiment, white contrast marks were painted on the specimen at the edges of the gauge section using a paint marker. We have observed that it is crucial for the paint to be still compliant. If it is fully dried it will come off the specimen before actual failure occurs, thus voiding the strain measurement. For determining the strain, we employ a line-scan camera to track the displacement of the contrast marks. If only uniaxial displacement is of interest, line-scan cameras are advantageous over area cameras due to higher 1D resolution and increased light sensitivity due to larger pixel size. We employ a model with 1 � 4096 pixel resolution and a line scan frequency of 200 kHz. The strain is ultimately obtained by post-processing the line-scan data using a pattern matching algorithm with sub-pixel accuracy [27]. Force is measured via a pair of conven tional strain gauges mounted on the output bar, connected diagonally in
Figure 3. Sketch of the SHTB setup employed in this work. All dimensions in mm. Input and output bars are 16 mm diameter aluminium rods. The striker is a hollow aluminium tube of 40 mm outer diameter and 20 mm inner diameter. Two strain gauge stations on the input bar and output bar measure the incident wave, εinc, and transmitted wave, εtra. 313
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Figure 5. Force and strain time series for a single Split-Hopkinson Tension Bar experiment on a quasi-isotropic specimen at strain rate 300/s. Force data (black solid line) is sampled at 10 MHz, local specimen strain (red circular symbols) is sampled at 200 kHz using a line scan camera. The solid red line is a linear regression to the local strain data points with slope 0.295/ms, corresponding to the target strain rate of � 300/s. The classic, but erroneous, Split-Hopkinson estimate of the specimen strain as computed from the reflected wave is also shown.
Figure 4. Bottom plot: Recordings from a Split-Hopkinson Tension Bar experiment on a quasi-isotropic specimen. Both incident and transmitted waves are direct recordings from the strain gauges at the locations described in Fig. 3, scaled to units of force. The reflected wave is computed from the expression εref ¼ εinc εtra . Top plot: strain rate as obtained from the reflected wave and its time integral, which is the specimen strain.
incident wave has a pronounced rise time of �0.2 ms, which is due to the use of a pulse shaper between striker and transfer flange, a rubber disk of 1 mm thickness. This pulse shaper significantly reduces oscillations in the transmitted force signal, which would otherwise completely domi nate. However, the pulse shaper also implies that the strain rate is not constant at the start of loading but instead slowly ramps up. In this case, the pulse shape is chosen such that a nearly constant strain rate is ob tained before failure. The strain rate may be obtained in two different ways, either directly from the strain gauge signals or by measuring locally on the specimen using an independent method. In the following, we will compare these two approaches and argue why it is important for the SHTB to only use the independent local measurement. As a result of one-dimensional wave propagation theory, the classi cally used strain rate is proportional to the reflected wave originating at the interface between input bar and specimen [18],
ε_ classic ¼
2 c0 εref ; L0
Fig. 5 compares the local specimen strain measured using the line scan camera with the strain computed from the reflected wave. The locally measured strain is 4.9% and thus approximately only 65% of the classic estimate. Additionally the local strain rate as obtained from a linear fit to the specimen strain is 300 � 20/s, which is half of what is estimated by the classic approach. We would like to emphasize that this difference is not due to the two-wave approximation, but instead caused by the compliance of the specimen holding fixture. Our SHTB setup with its long pulse duration allows to reach rela tively low strain rates compared to what is common in the SHTB com munity. In principle, we could have realized even lower strain rates for same-sized specimens: in the quasi-isotropic case, only 0.25 ms of the available pulse duration of 1.2 ms is used, and for �45∘ specimens, failure is reached at approx 0.55 ms. However, such experiments would deliver little additional insight as strain rate effects typically depend logarithmically on the strain rate. It was therefore our aim to achieve the highest possible strain rate. The quality of our data compares well with other Split-Hopkinson studies [8,9,12,23], and we feel confident that quantitative conclusions can be drawn from these data.
(1)
where c0 is the wave speed in the bar and L0 is the initial separation of input and output bar, which is equal to the specimen length. This strain rate thus measures the relative movement of the specimen-bar in terfaces. With our setup, we cannot measure the reflected wave εref directly, as the geometry does not allow an isolated reading of the re flected pulse because the striker has a larger generalized impedance than the input bar which causes superpositions of waves. Instead, we approximate the reflected wave by assuming the condition of dynamic equilibrium, also referred to as the two-wave approximation:
εref ¼ εinc
εtra
3. Results and discussion 3.1. Data reduction Fig. 6 shows the stress/strain curves obtained at each combination of strain rate and laminate layup. Each testing series consists of N ¼ 5 valid experiments. An experiment is considered valid, if failure occurred within the gauge section, i.e., between the holding tabs. Within one series, the individual experiments exhibit scatter which necessitates an averaging procedure before multiple testing series can be compared to each other. The characteristic properties maximum stress, σ max and strain at maximum stress, εmax are represented using straightforward
(2)
The strain rate computed in this fashion, along with the resulting strain is depicted in the upper part of Fig. 4. This strain rate attains a value of ε_classic�600/s near failure and the failure strain is 7.6%. It is known that this approach works well in the case of a pressure bar, but we claim that is not apt in the case of the SHTB, where the specimen mounting introduces additional compliance. In the present case, the specimen is glued into a holder, and the deformation of the adhesive contributes erroneously to the nominal strain measured using the re flected wave. It is therefore necessary to employ a local measurement of specimen strain.
averages and uncertainty estimates, e.g. for the stress: σmax ¼ qffiP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN N σmax Þ2 =ðN 1Þ. j¼1 σ max;j =N and sðσ max Þ ¼ j¼1 ðσ max;j
In contrast, computing an average stress/strain curve for an entire testing series is not as straightforward. Within each testing series, the sequences of strain values εi are not the same for all experiments; they
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Figure 6. Stress/strain curves for quasi-isotropic and �45∘ specimens at different strain rates. Black lines show individual experiments and red lines represent average behaviour obtained with the histogram approach (see text). Error bars indicate the standard deviation of the average. An artificial steep drop in the average curves indicates our definition of failure strain, which is the strain at maximum stress.
Fig. 7. Comparison of average stress/strain data for quasi-isotropic laminates tested at different strain rates. Errorbars indicate the standard deviation and are only shown for the highest strain rate results to avoid clutter.
Fig. 8. Comparison of average stress/strain data for �45∘ laminates tested at different strain rates. Errorbars indicate the standard deviation and are only shown for the highest strain rate results to avoid clutter.
span different ranges and are sampled at different locations. Thus, a
naïve averaging process over j ¼ 1.N datasets, such as σðεi Þ ¼ PN j¼1 σ j ðεi Þ=N, cannot work. Instead, we employ a histogram-based
Table 1 Average stress and strain values for quasi-isotropic and �45∘ laminates, for the different strain rates considered in this work. The discrete standard deviation is used as uncertainty estimate following the � symbol. Note that for the �45∘ laminate, maximum strain is defined as the strain at maximum stress.
approach. First, a strain axis which extends from the smallest observed strain value (typically 0) to the average strain at maximum stress is defined and discretized with sufficiently high resolution – 1000 points in our case. Each individual stress/strain dataset is then interpolated using linear functions and sampled at every histogram bin which is covered by the dataset. Now, each histogram bin contains between 1 and N samples, which allows the definition of an average and an uncertainty estimate, for which we use the discrete standard deviation.
315
strain rate
10 3/s
10 1/s
3 � 102/s
QI max. strain [%] QI max. stress [MPa] �45∘ max. strain [%] �45∘ max. stress [MPa]
3.3 � 0.4 428 � 43 15.0 � 1.4 179 � 6
3.9 � 0.2 452 � 42 14.0 � 1.2 194 � 7
4.8 � 0.1 524 � 38 13.9 � 0.8 217 � 7
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3.2. Quasi-isotropic specimens Stress/strain curves for the quasi-isotropic specimens are shown in Fig. 6, along with their representative average curves. We note that the results obtained with the universal testing machine show some un dulations at strains <1%, which is due to initial movement of the car danic joints. Nevertheless, we are confident that the average curves are reliable. The material exhibits almost linear-elastic, brittle character. Maximum stress and strain at maximum stress coincide with the failure stress and strain. Comparison among the different strain rates shows that the failure stress and increases with strain rate from approximately 428 to 524 MPa, i.e. by 22 � 8%, see Table 1. To illustrate that the comparatively small specimen size used here allows representative re sults to be obtained, we note that the corresponding quasi-static strength according to ISO-527 using much larger, 25 mm wide specimens is 436 � 10 MPa [25]. Failure strain increases significantly from approxi mately 3.3%–4.8%. Classic visco-elastic behaviour, i.e., increased stiff ness and decreased failure strain with strain rate, behaviour is not observed. Instead, and according to Fig. 7, all curves are superimposed but extend to higher stress and strain as the strain rate is increased. Within the uncertainty estimate, no distinction in stiffness can be observed, see Fig. 10. The location at which failure occurred exhibited little systematic behaviour, independent of strain rate. Experiments
Fig. 9. Strain rate sensitivity analysis of failure stress for quasi-isotropic (solid circles) and �45∘ specimens (crosses). Error bars denote one standard deviation. Straight lines are fits using a sensitivity model which is proportional to the logarithm of the strain rate, see text.
Fig. 10. Photographs of representative specimens after testing. The upper part shows the �45∘ specimens with zoomed-in regions for the two different strain rates 10 3/s and 300/s. The lower part shows the same for quasi-isotropic specimens. The edge length of a grey square on the background paper is 10 mm. 316
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were only considered valid, if failure occurred in the gauge region. The failure mode could not be identified as the specimens disintegrated instantaneously at failure due to elastic energy release. This process was too fast to be captured even with the high-speed camera at a frame rate of 38.65 kHz, see Fig. A11.
arguments may be misleading, as the composite’s strength is also a function of the fibre-matrix interface strength. To the knowledge of the authors, no published information is available about the loading rate sensitivity of interface properties in the case of basalt fibres and epoxy resins.
3.3. �45∘ specimens
4. Discussion and conclusions
The �45∘ specimens exhibit classic strain rate dependent pseudoviscoplastic behaviour, see Fig. 8 Following an initial linear response, which is the same at all strain rates within the uncertainty estimate, a rounded transition to an inelastic regime with constant strain hardening is observed. According to Table 1 the stress level of the inelastic regime increases significantly with strain rate: the maximum stress increases from approximately 179 to 217 MPa, i.e. by 21 � 4%. The corresponding quasi-static results for conventionally-sized specimens of 25 mm width according to DIN ISO-527 is 203 � 8 MPa [25]. This differs slightly but finite-size effects are expected especially for this specimen type as the amount of shear deformation that can be sustained is proportional to the specimen width. Failure strain (strain at maximum stress) is approxi mately 14% and thus much higher than in the quasi-isotropic case. The failure strain appears to decrease slightly with increased strain rate, although this observation cannot be fully justified given the uncertainty estimate. No qualitative distinction could be made in failure behaviour at different strain rates. Failure occurred typically in the center of the gauge region between the holding tabs, and was less explosive in char acter compared to the quasi-isotropic specimens, see Fig. 10.
This work reports the strain rate dependency of failure stress and strain for composites made of basalt fibres and epoxy resin, produced as laminates with quasi-isotropic [0∘, 45∘,þ45∘,90∘]S and orthogonal [þ45∘, 45∘]4 layups. Three different strain rates in the range 10 3 – 3 � 102 are considered. Data at the highest strain rate is obtained using a Split-Hopkinson Tension Bar. Our data is of sufficient quality to conduct a meaningful strain rate sensitivity analysis of strength, i.e., our mea surements indicate significant changes in strength – relative to the ex periments’ standard deviation – as a function of strain rate. We quantify the strain rate sensitivity of strength as approximately 3.5% per order of magnitude of loading rate, both for quasi-isotropic and �45∘ specimens. This is in agreement with existing high-quality studies on comparable carbon and glass fibre epoxy composites in tension [8–13] and also similar to what is observed in compression [14,20,21]. Our results contradict the findings of a recent study on a similar basalt/epoxy composite [7], who report a much more pronounced strain rate sensi tivity. In their study, a doubling of strength for a strain rate increase from 10/s to 300/s was measured using a servo-hydraulic universal testing machine. We argue that the Split-Hopkinson Tension Bar should be used as the standard method for testing dynamic behaviour of com posites: Whilst servo-hydraulic machines are detrimentally affected by pronounced wave propagation effects on short time scales, the Hop kinson Bar method incorporates this very phenomenon as the basis for its measuring principle. To facilitate the use of the Split-Hopkinson Tension Bar for composites, we introduce a novel system for easily mounting specimens of strip shape to the Hopkinson bars. This new mounting system provides constant acoustic impedance and does not hinder wave propagation. Our results show that the strain rate sensi tivity is nearly identical for quasi-isotropic and �45∘ laminates. This is somewhat surprising, as �45∘ specimens load the thermoplastic and thus strain rate sensitive polymer matrix in shear, whereas much of the load in the quasi-isotropic specimens is carried by the 0∘ plies, which are not expected to exhibit pronounced strain rate sensitivity due to their mineral constitution. To further investigate this issue, future work should address accurate testing of 0∘ specimens with Split-Hopkinson methods. This, however, is challenging because i) such specimens require much stronger holding forces than the laminates considered here, and ii) the small failure strain reduces the time available for equilibration such that accurate measurements might prove difficult.
3.4. Analysis of strain-rate effects We analyse the strain rate dependency by plotting failure stress against the decadic logarithm of strain rate, see Fig. 9. Both quasiisotropic and �45∘ datasets are well described with a simple model, which is proportional to the logarithm of the strain rate, i.e., � � �� ε_ σ fail ¼ A � 1 þ B log : ε_ 0 Here, ε_0 ¼ 1/s is a reference strain rate which serves to render the argument of the log function dimensionless; A is the failure stress at the reference strain rate and B is the slope, i.e., the strain rate sensitivity. Fitting this function to the data points from Table 1 using the LevenbergMarquardt algorithm yields the parameters A and B including their uncertainty estimates. As is obvious from Fig. 9, the failure stress at ε_0 is higher for the quasi-isotropic specimens than for the �45∘ specimens, A ¼ 477.2 � 5.0 MPa vs. A ¼ 200.1 � 0.5 MPa. More interestingly, the strain rate sensitivity is identical within the uncertainty estimate: We observe B ¼ 0.038 � 0.005 in the quasi-isotropic case and B ¼ 0.035 � 0.001 for the �45∘ specimens. This is somewhat surprising, as most of the load within the quasi-isotropic specimen is carried by the fibres of the 0∘ ply, and the strain rate sensitivity of the fibres itself is believed to be weak [30]. In contrast, in the �45∘ specimens, the epoxy matrix is loaded in shear, and it is well known that epoxy resins exhibit pronounced strain rate sensitivity [31,32]. However, these simple
Funding This work was supported by the Gips-Schüle-Stiftung and the CarlZeiss-Stiftung [Sonderlinie 2017/2018 Grundlagenwissenschaften mit Anwendungsbezug].
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Appendix A. Example Split-Hopkinson High Speed Images
Fig. A11. Sequence of high-speed images showing the deformation behaviour of a quasi-isotropic basalt-fibre/epoxy composite test in our Split-Hopkinson Tension Bar at a loading rate of 300 /s.
References
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Composites Part B journal homepage: www.elsevier.com/locate/compositesb
A Split-Hopkinson Tension Bar study on the dynamic strength of basalt-fibre composites G.C. Ganzenmüller a, b, *, D. Plappert b, A. Trippel b, S. Hiermaier a, b a b
Fraunhofer Ernst-Mach-Institute for High-Speed Dynamics, EMI, Freiburg i. Br, Germany Albert-Ludwigs Universit€ at Freiburg, Institute for Sustainable Systems Engineering, Freiburg i. Br, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Mechanical testing Strength Basalt fibre
This paper investigates the strain rate sensitivity of laminated composites made of plies of unidirectional basalt fibres and epoxy resin. We consider laminates with quasi-isotropic [0∘, 45∘,þ45∘,90∘]s and orthogonal [þ45∘, 45∘]4 layup. A Split-Hopkinson Tension Bar is used to generate accurate stress/strain data at elevated rates of strain of �3 � 102/s. Moderate strain rate effects are observed with strength increase of �3.5% per decade of increased loading rate for both laminate types.
1. Introduction The purpose of this work is to present reasonably accurate dynamic tensile strength data for composites made of plies of endless, unidirec tional basalt fibres and epoxy resin. Over the last two decades, basalt fibres have come into consideration as potential reinforcement for composite materials requiring high strength and temperature resistance. In contrast to the infamous asbestos fibre, which meets these application requirements from a mechanical point of view, basalt is not considered carcinogenic [1]. It is a mineral belonging to the group of silicates and as such similar to glass in its chemical composition but with more iron and less calcium content. While glass fibres are completely amorphous solids, basalt fibres can feature some degree of cristallinity [2]. Basalt’s mechanical properties strength and stiffness are lower than those of carbon fibres, and more akin to those of S-2 glass fibres. The quasi-static properties of basalt fibres and its composites are already well known [3, 4]. A promising field of application for basalt fibre composites are structural elements in the automotive sector [5], where it is important to predict the dynamic material behaviour at elevated strain rates relevant at crash, which are on the order of 10–100/s [6]. However, as of now, only a few studies are available on dynamic properties. This is particu larly true for composites composed of plies of endless, unidirectional (UD) fibres. To our knowledge, only one, very recent study is available [7], where the authors report that the dynamic tensile strength of a pure 0∘ basalt/epoxy composite doubles its strength when strain rate is increased from 10/s to 300/s. This amount of strain rate sensitivity appears exceptionally high, as UD composites of carbon or glass fibres
only exhibit strength increases of approximately 1%–10% per order of magnitude of increased strain rate [8–14]. In another study, the strain rate sensitivity of woven basalt/epoxy composites was studied, and a similarly exceptional high strain rate sensitivity was observed [15]. Is there something special about basalt fibres which makes them very strain rate sensitive? To shed some light on this issue, we investigate the strain rate dependency of the tensile failure strength and strain for quasi-isotropic [0∘, 45∘,þ45∘,90∘]s and orthogonal [þ45∘, 45∘]4 layups of unidirectional basalt fibres in a thermoset epoxy matrix within the strain rate regime 10 3 – 3 � 102/s. To obtain accurate data at high rates of strain, we employ a Split-Hopkinson Tension Bar, for reasons we feel important to briefly review in the following: Acquiring accurate stress/strain data at high rates of strain >100/s is not a trivial task, as has been discussed extensively in the literature [16, 17]. The difficulty may be attributed to three issues: (i) The testing apparatus must achieve a well defined, e.g., constant velocity at which the sample is loaded, typically on the order of a few m/s. This requires strong acceleration and good velocity control of the gripping devices used to interface the specimen with the testing apparatus. (ii) The stress state within the specimen needs to be inferred from a force sensor, which is usually not mounted on the specimen itself but rather located in-line with the loading axis at some distance from the specimen. During the short time span of a dynamic experiment, propagation effects of elastic waves must be considered. The force sensor will, in general, not sense the force experienced by the specimen at a given instant of time, but instead report another value which is influenced by its own acceleration. Furthermore, the dynamic bandwidth, i.e., its frequency response
* Corresponding author. Fraunhofer Ernst-Mach-Institute for High-Speed Dynamics, EMI, Freiburg i. Br, Germany. E-mail address: [email protected] (G.C. Ganzenmüller). https://doi.org/10.1016/j.compositesb.2019.04.031 Received 22 January 2019; Received in revised form 2 April 2019; Accepted 26 April 2019 Available online 4 May 2019 1359-8368/© 2019 Elsevier Ltd. All rights reserved.
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stress/strain data for composites, see e.g. Refs. [20,21], and a strong theoretical basis is available which rationalizes these beliefs [22]. In contrast, Split-Hopkinson Tension (SHTB) Bar experiments are more difficult to perform. The specimen needs to be attached to the ends of the input and output bars without introducing significant additional mass, e.g., specimen grips as used in conventional testing machines. The presence of such additional mass means additional changes in acoustic impedance, causing unwanted wave reflections. This violates the as sumptions made in the classic analysis based on simple wave propaga tion in media with constant acoustic impedance. The situation is fortunate for specimens with cylindrical shape, as these can be threaded at the ends and screwed into matching interior threads in the bar ends. Thus, no additional mass is introduced for affixing the specimen. Com posites, however, are typically tested in a planar strip form. Until now, the only solution has been has been to introduce slots either into the bar ends or into adapters, and bond the specimens into these slots with highstrength adhesives [9,12,23]. This process requires intensive labour and expensive machining. Recently, two of the current authors introduced a clamping device which allows reversible mounting of strip-type speci mens without violating the requirement of constant acoustic impedance [24]. Here, we utilize this clamping device to obtain SHTB data of reasonable accuracy and compare this data with results from quasi-static and moderate strain rates. The remainder of this article is organized as follows: The composite investigated here and the experimental methods employed are described in the next section. We then report our findings for strength and failure strains, analyse the strain rate dependency and conclude with a critical review of our data in the light of other researchers’ findings.
function, must be adequate to accurately describe the specimen’s force signal. Typically, dynamic bandwidths on the order of 1 MHz are required for experiments at strain rates between 102 – 103/s [18]. (iii) Determination of the specimen strain state requires fast and robust measurement techniques. Fast, because for sampling the strain evolution of a specimen with failure strain of 5% at a strain rate of 100/s with only 100 data points, already an acquisition rate of 200 kHz is required. Robust, because the sampling method must yield reliable measurements for a rapidly accelerating and deforming specimen. While strain gauges meet the requirement of fast acquisition rates, it can be difficult to maintain adhesion on the surface of a specimen until failure. Alterna tively, high-speed imaging and optical strain analysis is a universal yet expensive solution if high spatial resolution is required. For conducting experiments at high rates of strain, two popular experimental methods are available. The first method is given by the universal testing machine, powered by a servo-hydraulic actuator. While these machines can operate over large ranges of testing velocities and forces with typical upper limits of 10 m/s and 250 kN, difficulties arise for accurately measuring the force signal of a specimen under rapid loading. Taking reference to the points (i) and (ii) from above, this is due to the following reasons: (I) The apparatus needs to achieve a high testing velocity before the specimen is loaded. This is realized via a lost motion device, which lets the hydraulic piston travel freely under ac celeration. Once the piston is at speed, its movement is coupled instantaneously to the specimen’s grips. The instantaneous acceleration of the – typically heavy – specimen grips causes a strong inertial force response which overlays the true force signal of the specimen’s stress response. This effect may be theoretically visualized by considering the frequency response of a velocity signal with a step-like acceleration. Its Fourier transform shows amplitudes over an infinite frequency spec trum. (II) Force measurement is conventionally performed via a piezoelectric load cell with finite stiffness and its own limited frequency response. Such force sensors act as low-pass filters. The fact that the quasi-infinite frequency spectrum associated with the step loading character of the slack adapter is not present in the output signal is caused by this filtering effect. While servo-hydraulic universal testing machines have their warranted uses for specific applications, acquiring highquality stress-strain data, in particular for lightweight, non-metallic materials, is challenging. Naively applying such a machine at high rates of strain, one does not measure the true specimen response but instead a convolution of the specimen response with the acceleration response of the specimen grips and elastic wave reflection effects with visible oscillatory artifacts, see e.g. Ref. [19]. However, it needs to be pointed out that high-quality data may also be obtained using this approach if its limitations are considered carefully [17]. A more straightforward alternative for high rate testing is given by the Split-Hopkinson Bar method, also known as the Kolsky Bar method. It is out of the scope of the present work to detail this technique extensively, instead the reader is referred to Ref. [18]. Here, the spec imen is sandwiched between two long and straight bars of constant cross section, the input and output bars. An elastic wave of well defined amplitude is created in the input bar, either by impacting the bar with another long bar, the striker, or by pre-tensioning the bar and employing a release mechanism. The elastic wave then passes through the specimen into the output bar, creating secondary transmitted and reflected waves at the interfaces between specimen and input/output bars. These waves are measured using strain gauges mounted on the bars, and analysed using the theory of one-dimensional elastic wave propagation to yield both specimen strain and stress. Due to the simple geometry, all 1D wave effects are accounted for, and the resulting stress/strain curves can be of high quality. However, this fortunate situation applies only to compression experiments performed with the Split-Hopkinson Pressure Bar (SHPB). Here, a small amount of grease is sufficient to stick a cy lindrical sample between the faces of input- and output bar, and no explicit specimen grips are required. Due to this simplicity, the SHPB method is widely accepted for reliably producing accurate compression
2. Materials and methods 2.1. Material specimens Sheets of cured UD basalt fibre composites with layups [0∘, 45∘,þ45∘,90∘]s and orthogonal [þ45∘, 45∘]4 were provided tested. This composite is made from a prepreg of basalt fibres with filament diameter 17 μm and a low-viscosity epoxy resin system. Laminates were cured in an autoclave process resulting in fibre volume fraction of 60%. To illustrate the mechanical properties of this material, we quote the nominal strength and stiffness of pure 0∘ specimens according to ISO-527 as 1310 MPa and 44 GPa [25]. Testing specimens were cut from the sheets using a diamond coated saw blade to obtain good cut surface quality. The dimensions of the quasi-isotropic (QI) specimens are length � width � height ¼ 100 � 3 � 1.6 mm3 with a nominal gauge length of 20 mm. The di mensions of the �45∘ specimens are 100 � 5 � 2.0 mm3 with the same nominal gauge length. The same specimen geometry was used across all strain rates. Specimens were bonded to grooved 7075 aluminium tabs with a high-strength, thermally activated epoxy resin according to Fig. 1. We use split tabs as these can be easily manufactured in large quantities using a CNC router. Additionally, the split tabs provide good accessibility to the interior which makes it trivial to correctly apply the adhesive. The tabs feature holes for M4 through-bolts to affix the assembled specimen to the aluminium grips as shown in Fig. 2. The clamps are designed to fulfill three task: (i) they provide an ISO M12 � 1.5 thread to interface with either the Split-Hopkinson bars or the universal testing machine. (ii) The transition from the circular interfacing end with the testing apparatus to the rectangular shape used to attach the specimen maintains a constant cross section area. For dynamic testing, the SplitHopkinson bars are also made of aluminium, such that constant acous tic impedance is obtained. This avoids unwanted wave reflections due to the specimen gripping device, see Ref. [24] for details. (iii) The ø4.1 mm through holes allow for clamping the specimen inside the grip. To this end, lightweight M4 aluminium bolts with strength class ISO 5.6 may be used, yielding an axial force of 1.8 kN per bolt. With 4 bolts and a 311
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Fig. 1. CAD rendering of the specimen with its split aluminium tabs. Also shown is the fixture which is used to align all parts during assembly and curing of the epoxy adhesive. All dimensions in mm.
Fig. 2. Clamping fixture maintains cross section area and thus provides constant acoustic impedance which is required for reflection-free wave transmission. All dimensions in mm.
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conservatively assumed friction coefficient μ ¼ 1.0 between the clean and unlubed aluminium surfaces, this translates into a useful upper force limit of �14 kN for this clamp. This limit exceeds the expected forces in this work by a factor of 4. We note that those parts of the through-bolts and nuts protruding from the surfaces introduce additional mass, which causes unwanted wave reflections. However, this effect is so minor that its effects cannot be observed in our data.
a Wheatstone bridge circuit to eliminate bending information. The Wheatstone bridge circuit is driven in constant voltage mode and its output is increased by a factor of 100 using an amplifier with 1 MHz bandwidth. This signal is recorded by a data acquisition (DAQ) card operating at 10 MHz and 16 bit resolution. The conversion factor from strain to force is established via a calibration procedure, wherein a dedicated force sensor is placed between input- and output bars, while a static load is applied onto the bars. The signal of the force sensor is recorded using the same amplifier and data acquisition card as during a real experiment. The force sensor, in turn is calibrated against the already calibrated load cell in the universal testing machine described above. Thus, the entire force measuring system consisting of strain gauges, amplifier and DAQ card is calibrated, which compensates for any eventual misalignment of the strain gauges or similar constant systematic errors. In addition to the line-scan camera, we also use a high-speed area camera to obtain images of the deformation process. The highest sam pling rate of this camera is 38.65 kHz at a resolution of 336 � 96 pixels. This is not enough for accurate strain determination, but serves well to obtain a qualitative impression of the failure behaviour, see Appendix A. Both line-scan and area cameras are triggered by the DAQ card to start recording when the incident wave created by the striker reaches strain gauge no. 1, c.f. Fig. 3. In the case of the line-scan camera, a synchronization signal is fed back to the DAQ card, so precise timing information is available to correlate strain and force signals. It is important to realize that even though strain and force signals are perfectly well synchronized in this manner, these signals are measured in different locations: strain is measured on the specimen but force is inferred from the strain gages on the output bar, in our case 200 mm upstream in the transmitted elastic wave direction. The time taken by the elastic wave to travel this distance must be accounted for by shifting the force signal by Δt ¼ 200 mm/5090 mm/ms ¼ 0.039 ms, where 5090 mm/ms is the longitudinal wave speed in aluminium. In classic Split-Hopkinson Bar experiments, a check of dynamic equilibrium is possible by placing strain gages symmetrically around the specimen, i.e., also on the input bar near the specimen. This allows to check when the forces on both sides of the specimen are equal in magnitude, which marks the beginning of measured data validity and constant strain rate. Here, we cannot provide this information as no additional strain gauge was mounted on the input bar. However, the onset of dynamic equilibrium may also be calculated analytically, see e. g. Ref. [22]. Usually, the time taken for the elastic wave to perform 4 to 5 round-trips within the specimen is sufficient [28,29]. In case of the quasi-isotropic specimen, the longitudinal wave speed is � 3300 mm/ms, and the gauge length is 20 mm, resulting in an equilibration time of <60 μs. Our SHTB experiments were set up such that the incident wave at tains a force amplitude of 20 kN, corresponding to a particle velocity of 7.2 m/s. This marks the upper end of what is safely attainable with our aluminium setup due to its mechanical strength. Fig. 4 shows the recording of such an experiment on a quasi-isotropic specimen. The
2.2. Quasi-static and low strain rate testing For the quasi-static and low strain rates of 10 3/s and 10 1/s, we use a screw-driven universal testing machine with 100 kN capacity and a maximum testing velocity of 1500 mm/min. Specimens were attached using the gripping device described in section 2.1, which itself was mounted to the testing machine using cardanic joints to prevent off-axis loading. The testing velocity was adjusted in pilot experiments such that the target strain rates were obtained. Force was measured via the ma chine’s internal load sensor, which is calibrated to accuracy class 1/ISO 7500–1. Strain was measured optically using a black/white camera with resolution of 1280 � 1024 pixels and a maximum frame rate of 1500 fps at this resolution. To this end, white contrast marks were painted on the specimen at the edges on the gauge section with a paint marker. The displacement of the contrast marks was recorded and converted into a strain time series using a motion analysis software based on Digital Image Correlation (DIC). Both force and strain time series were com bined into nominal stress/strain graphs by eliminating the time infor mation. We note that for this universal testing machine in combination with the tested specimens, a strain rate of 10 1/s is the most dynamic experiment that can be achieved before acceleration effects cause ringing artifacts in the force signal can be observed. 2.3. Dynamic testing apparatus The Split-Hopkinson Tension Bar (SHTB) used here is sketched in Fig. 3 and described in detail in Ref. [26]. Compared to other SHTBs, this setup is optimized for low velocities, low forces and a long pulse dura tion of 1.2 ms. Specimens were attached to the bars using the grips detailed in section 2.1. Shortly before performing the experiment, white contrast marks were painted on the specimen at the edges of the gauge section using a paint marker. We have observed that it is crucial for the paint to be still compliant. If it is fully dried it will come off the specimen before actual failure occurs, thus voiding the strain measurement. For determining the strain, we employ a line-scan camera to track the displacement of the contrast marks. If only uniaxial displacement is of interest, line-scan cameras are advantageous over area cameras due to higher 1D resolution and increased light sensitivity due to larger pixel size. We employ a model with 1 � 4096 pixel resolution and a line scan frequency of 200 kHz. The strain is ultimately obtained by post-processing the line-scan data using a pattern matching algorithm with sub-pixel accuracy [27]. Force is measured via a pair of conven tional strain gauges mounted on the output bar, connected diagonally in
Figure 3. Sketch of the SHTB setup employed in this work. All dimensions in mm. Input and output bars are 16 mm diameter aluminium rods. The striker is a hollow aluminium tube of 40 mm outer diameter and 20 mm inner diameter. Two strain gauge stations on the input bar and output bar measure the incident wave, εinc, and transmitted wave, εtra. 313
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Figure 5. Force and strain time series for a single Split-Hopkinson Tension Bar experiment on a quasi-isotropic specimen at strain rate 300/s. Force data (black solid line) is sampled at 10 MHz, local specimen strain (red circular symbols) is sampled at 200 kHz using a line scan camera. The solid red line is a linear regression to the local strain data points with slope 0.295/ms, corresponding to the target strain rate of � 300/s. The classic, but erroneous, Split-Hopkinson estimate of the specimen strain as computed from the reflected wave is also shown.
Figure 4. Bottom plot: Recordings from a Split-Hopkinson Tension Bar experiment on a quasi-isotropic specimen. Both incident and transmitted waves are direct recordings from the strain gauges at the locations described in Fig. 3, scaled to units of force. The reflected wave is computed from the expression εref ¼ εinc εtra . Top plot: strain rate as obtained from the reflected wave and its time integral, which is the specimen strain.
incident wave has a pronounced rise time of �0.2 ms, which is due to the use of a pulse shaper between striker and transfer flange, a rubber disk of 1 mm thickness. This pulse shaper significantly reduces oscillations in the transmitted force signal, which would otherwise completely domi nate. However, the pulse shaper also implies that the strain rate is not constant at the start of loading but instead slowly ramps up. In this case, the pulse shape is chosen such that a nearly constant strain rate is ob tained before failure. The strain rate may be obtained in two different ways, either directly from the strain gauge signals or by measuring locally on the specimen using an independent method. In the following, we will compare these two approaches and argue why it is important for the SHTB to only use the independent local measurement. As a result of one-dimensional wave propagation theory, the classi cally used strain rate is proportional to the reflected wave originating at the interface between input bar and specimen [18],
ε_ classic ¼
2 c0 εref ; L0
Fig. 5 compares the local specimen strain measured using the line scan camera with the strain computed from the reflected wave. The locally measured strain is 4.9% and thus approximately only 65% of the classic estimate. Additionally the local strain rate as obtained from a linear fit to the specimen strain is 300 � 20/s, which is half of what is estimated by the classic approach. We would like to emphasize that this difference is not due to the two-wave approximation, but instead caused by the compliance of the specimen holding fixture. Our SHTB setup with its long pulse duration allows to reach rela tively low strain rates compared to what is common in the SHTB com munity. In principle, we could have realized even lower strain rates for same-sized specimens: in the quasi-isotropic case, only 0.25 ms of the available pulse duration of 1.2 ms is used, and for �45∘ specimens, failure is reached at approx 0.55 ms. However, such experiments would deliver little additional insight as strain rate effects typically depend logarithmically on the strain rate. It was therefore our aim to achieve the highest possible strain rate. The quality of our data compares well with other Split-Hopkinson studies [8,9,12,23], and we feel confident that quantitative conclusions can be drawn from these data.
(1)
where c0 is the wave speed in the bar and L0 is the initial separation of input and output bar, which is equal to the specimen length. This strain rate thus measures the relative movement of the specimen-bar in terfaces. With our setup, we cannot measure the reflected wave εref directly, as the geometry does not allow an isolated reading of the re flected pulse because the striker has a larger generalized impedance than the input bar which causes superpositions of waves. Instead, we approximate the reflected wave by assuming the condition of dynamic equilibrium, also referred to as the two-wave approximation:
εref ¼ εinc
εtra
3. Results and discussion 3.1. Data reduction Fig. 6 shows the stress/strain curves obtained at each combination of strain rate and laminate layup. Each testing series consists of N ¼ 5 valid experiments. An experiment is considered valid, if failure occurred within the gauge section, i.e., between the holding tabs. Within one series, the individual experiments exhibit scatter which necessitates an averaging procedure before multiple testing series can be compared to each other. The characteristic properties maximum stress, σ max and strain at maximum stress, εmax are represented using straightforward
(2)
The strain rate computed in this fashion, along with the resulting strain is depicted in the upper part of Fig. 4. This strain rate attains a value of ε_classic�600/s near failure and the failure strain is 7.6%. It is known that this approach works well in the case of a pressure bar, but we claim that is not apt in the case of the SHTB, where the specimen mounting introduces additional compliance. In the present case, the specimen is glued into a holder, and the deformation of the adhesive contributes erroneously to the nominal strain measured using the re flected wave. It is therefore necessary to employ a local measurement of specimen strain.
averages and uncertainty estimates, e.g. for the stress: σmax ¼ qffiP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN N σmax Þ2 =ðN 1Þ. j¼1 σ max;j =N and sðσ max Þ ¼ j¼1 ðσ max;j
In contrast, computing an average stress/strain curve for an entire testing series is not as straightforward. Within each testing series, the sequences of strain values εi are not the same for all experiments; they
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Figure 6. Stress/strain curves for quasi-isotropic and �45∘ specimens at different strain rates. Black lines show individual experiments and red lines represent average behaviour obtained with the histogram approach (see text). Error bars indicate the standard deviation of the average. An artificial steep drop in the average curves indicates our definition of failure strain, which is the strain at maximum stress.
Fig. 7. Comparison of average stress/strain data for quasi-isotropic laminates tested at different strain rates. Errorbars indicate the standard deviation and are only shown for the highest strain rate results to avoid clutter.
Fig. 8. Comparison of average stress/strain data for �45∘ laminates tested at different strain rates. Errorbars indicate the standard deviation and are only shown for the highest strain rate results to avoid clutter.
span different ranges and are sampled at different locations. Thus, a
naïve averaging process over j ¼ 1.N datasets, such as σðεi Þ ¼ PN j¼1 σ j ðεi Þ=N, cannot work. Instead, we employ a histogram-based
Table 1 Average stress and strain values for quasi-isotropic and �45∘ laminates, for the different strain rates considered in this work. The discrete standard deviation is used as uncertainty estimate following the � symbol. Note that for the �45∘ laminate, maximum strain is defined as the strain at maximum stress.
approach. First, a strain axis which extends from the smallest observed strain value (typically 0) to the average strain at maximum stress is defined and discretized with sufficiently high resolution – 1000 points in our case. Each individual stress/strain dataset is then interpolated using linear functions and sampled at every histogram bin which is covered by the dataset. Now, each histogram bin contains between 1 and N samples, which allows the definition of an average and an uncertainty estimate, for which we use the discrete standard deviation.
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strain rate
10 3/s
10 1/s
3 � 102/s
QI max. strain [%] QI max. stress [MPa] �45∘ max. strain [%] �45∘ max. stress [MPa]
3.3 � 0.4 428 � 43 15.0 � 1.4 179 � 6
3.9 � 0.2 452 � 42 14.0 � 1.2 194 � 7
4.8 � 0.1 524 � 38 13.9 � 0.8 217 � 7
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3.2. Quasi-isotropic specimens Stress/strain curves for the quasi-isotropic specimens are shown in Fig. 6, along with their representative average curves. We note that the results obtained with the universal testing machine show some un dulations at strains <1%, which is due to initial movement of the car danic joints. Nevertheless, we are confident that the average curves are reliable. The material exhibits almost linear-elastic, brittle character. Maximum stress and strain at maximum stress coincide with the failure stress and strain. Comparison among the different strain rates shows that the failure stress and increases with strain rate from approximately 428 to 524 MPa, i.e. by 22 � 8%, see Table 1. To illustrate that the comparatively small specimen size used here allows representative re sults to be obtained, we note that the corresponding quasi-static strength according to ISO-527 using much larger, 25 mm wide specimens is 436 � 10 MPa [25]. Failure strain increases significantly from approxi mately 3.3%–4.8%. Classic visco-elastic behaviour, i.e., increased stiff ness and decreased failure strain with strain rate, behaviour is not observed. Instead, and according to Fig. 7, all curves are superimposed but extend to higher stress and strain as the strain rate is increased. Within the uncertainty estimate, no distinction in stiffness can be observed, see Fig. 10. The location at which failure occurred exhibited little systematic behaviour, independent of strain rate. Experiments
Fig. 9. Strain rate sensitivity analysis of failure stress for quasi-isotropic (solid circles) and �45∘ specimens (crosses). Error bars denote one standard deviation. Straight lines are fits using a sensitivity model which is proportional to the logarithm of the strain rate, see text.
Fig. 10. Photographs of representative specimens after testing. The upper part shows the �45∘ specimens with zoomed-in regions for the two different strain rates 10 3/s and 300/s. The lower part shows the same for quasi-isotropic specimens. The edge length of a grey square on the background paper is 10 mm. 316
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were only considered valid, if failure occurred in the gauge region. The failure mode could not be identified as the specimens disintegrated instantaneously at failure due to elastic energy release. This process was too fast to be captured even with the high-speed camera at a frame rate of 38.65 kHz, see Fig. A11.
arguments may be misleading, as the composite’s strength is also a function of the fibre-matrix interface strength. To the knowledge of the authors, no published information is available about the loading rate sensitivity of interface properties in the case of basalt fibres and epoxy resins.
3.3. �45∘ specimens
4. Discussion and conclusions
The �45∘ specimens exhibit classic strain rate dependent pseudoviscoplastic behaviour, see Fig. 8 Following an initial linear response, which is the same at all strain rates within the uncertainty estimate, a rounded transition to an inelastic regime with constant strain hardening is observed. According to Table 1 the stress level of the inelastic regime increases significantly with strain rate: the maximum stress increases from approximately 179 to 217 MPa, i.e. by 21 � 4%. The corresponding quasi-static results for conventionally-sized specimens of 25 mm width according to DIN ISO-527 is 203 � 8 MPa [25]. This differs slightly but finite-size effects are expected especially for this specimen type as the amount of shear deformation that can be sustained is proportional to the specimen width. Failure strain (strain at maximum stress) is approxi mately 14% and thus much higher than in the quasi-isotropic case. The failure strain appears to decrease slightly with increased strain rate, although this observation cannot be fully justified given the uncertainty estimate. No qualitative distinction could be made in failure behaviour at different strain rates. Failure occurred typically in the center of the gauge region between the holding tabs, and was less explosive in char acter compared to the quasi-isotropic specimens, see Fig. 10.
This work reports the strain rate dependency of failure stress and strain for composites made of basalt fibres and epoxy resin, produced as laminates with quasi-isotropic [0∘, 45∘,þ45∘,90∘]S and orthogonal [þ45∘, 45∘]4 layups. Three different strain rates in the range 10 3 – 3 � 102 are considered. Data at the highest strain rate is obtained using a Split-Hopkinson Tension Bar. Our data is of sufficient quality to conduct a meaningful strain rate sensitivity analysis of strength, i.e., our mea surements indicate significant changes in strength – relative to the ex periments’ standard deviation – as a function of strain rate. We quantify the strain rate sensitivity of strength as approximately 3.5% per order of magnitude of loading rate, both for quasi-isotropic and �45∘ specimens. This is in agreement with existing high-quality studies on comparable carbon and glass fibre epoxy composites in tension [8–13] and also similar to what is observed in compression [14,20,21]. Our results contradict the findings of a recent study on a similar basalt/epoxy composite [7], who report a much more pronounced strain rate sensi tivity. In their study, a doubling of strength for a strain rate increase from 10/s to 300/s was measured using a servo-hydraulic universal testing machine. We argue that the Split-Hopkinson Tension Bar should be used as the standard method for testing dynamic behaviour of com posites: Whilst servo-hydraulic machines are detrimentally affected by pronounced wave propagation effects on short time scales, the Hop kinson Bar method incorporates this very phenomenon as the basis for its measuring principle. To facilitate the use of the Split-Hopkinson Tension Bar for composites, we introduce a novel system for easily mounting specimens of strip shape to the Hopkinson bars. This new mounting system provides constant acoustic impedance and does not hinder wave propagation. Our results show that the strain rate sensi tivity is nearly identical for quasi-isotropic and �45∘ laminates. This is somewhat surprising, as �45∘ specimens load the thermoplastic and thus strain rate sensitive polymer matrix in shear, whereas much of the load in the quasi-isotropic specimens is carried by the 0∘ plies, which are not expected to exhibit pronounced strain rate sensitivity due to their mineral constitution. To further investigate this issue, future work should address accurate testing of 0∘ specimens with Split-Hopkinson methods. This, however, is challenging because i) such specimens require much stronger holding forces than the laminates considered here, and ii) the small failure strain reduces the time available for equilibration such that accurate measurements might prove difficult.
3.4. Analysis of strain-rate effects We analyse the strain rate dependency by plotting failure stress against the decadic logarithm of strain rate, see Fig. 9. Both quasiisotropic and �45∘ datasets are well described with a simple model, which is proportional to the logarithm of the strain rate, i.e., � � �� ε_ σ fail ¼ A � 1 þ B log : ε_ 0 Here, ε_0 ¼ 1/s is a reference strain rate which serves to render the argument of the log function dimensionless; A is the failure stress at the reference strain rate and B is the slope, i.e., the strain rate sensitivity. Fitting this function to the data points from Table 1 using the LevenbergMarquardt algorithm yields the parameters A and B including their uncertainty estimates. As is obvious from Fig. 9, the failure stress at ε_0 is higher for the quasi-isotropic specimens than for the �45∘ specimens, A ¼ 477.2 � 5.0 MPa vs. A ¼ 200.1 � 0.5 MPa. More interestingly, the strain rate sensitivity is identical within the uncertainty estimate: We observe B ¼ 0.038 � 0.005 in the quasi-isotropic case and B ¼ 0.035 � 0.001 for the �45∘ specimens. This is somewhat surprising, as most of the load within the quasi-isotropic specimen is carried by the fibres of the 0∘ ply, and the strain rate sensitivity of the fibres itself is believed to be weak [30]. In contrast, in the �45∘ specimens, the epoxy matrix is loaded in shear, and it is well known that epoxy resins exhibit pronounced strain rate sensitivity [31,32]. However, these simple
Funding This work was supported by the Gips-Schüle-Stiftung and the CarlZeiss-Stiftung [Sonderlinie 2017/2018 Grundlagenwissenschaften mit Anwendungsbezug].
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Appendix A. Example Split-Hopkinson High Speed Images
Fig. A11. Sequence of high-speed images showing the deformation behaviour of a quasi-isotropic basalt-fibre/epoxy composite test in our Split-Hopkinson Tension Bar at a loading rate of 300 /s.
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