1 Introduction

Bone, as an engineering material, has been studied, and its mechanical properties quantified, for over half a century [1]. However, there does not appear to be a consensus in the literature regarding definitive values of the various properties that have been studied [1,2,3,4,5,6,7]. Various authors have attributed these scattered results to the inherent complexity of bone as a material [8,9,10,11]. It is also possible that inconsistent storage and testing protocols have influenced the recorded results [12,13,14]. In fact, Lee et al. [15] have provided evidence that, even under ideal storage conditions, the properties of the bone can degrade.

Due to the material properties of bone not being consistent in all regions of a particular donor bone [3, 4], the testing of large or whole bone samples cannot offer definitive material characteristics. Therefore, bone testing is often conducted using small specimens, where the macroscopic material properties can be considered near constant throughout the test specimen. These small specimens would ideally be machined from a single donor bone in order to reduce the chance of different donor bones having different material properties, which has been demonstrated by Currey [16], even when bones are retrieved from the same donor animal. Therefore, a priority must be placed on being able to conduct a large number of tests in a small time period. This is due to bone being a biological material that is susceptible to drying out when removed from the donor, such that the mechanical properties would not be representative of in-vivo bone [17].

In addition to the unique challenges one faces when conducting mechanical tests on small cortical bone specimens, the testing machine compliance has an effect on the measured results. Due to the low failure strain of cortical bone when compared to other common engineering materials [3, 18], such as aluminum and polycarbonate, compliance issues associated with standard testing apparatuses become prominent. The existence of after-market measurement systems, such as a visual extensometer, highlights the need to eliminate compliance effects.

In this paper, a custom compression test set-up is presented, which is designed to operate in a Zwick universal testing machine. In keeping with the nomenclature defined in the ASTM [19] and ISO [20] specimen compression standards, the compression test set-up will hereafter be referred to as a subpress. This subpress aims to offer a simple and cost-effective means of measuring displacements in the order of a micron through the novel use of Hall effect sensors and multipole magnetic strips. The subpress was required to interface with the universal testing machine via a standard 10 kN load cell. An accuracy of 1 μm was specified and a focus placed on minimizing the effect of test machine compliance on the measured displacements. Compression tests were conducted on PMMA specimens whereby the subpress was compared to the integrated measurement system of the Zwick itself as well as a visual extensometer (VE). Additional tests on small cortical bone specimens were conducted to demonstrate the effectiveness of the custom subpress in its intended use, with novel data regarding the incipient fracture of cortical bone being obtained.

2 Design

2.1 Subpress

The subpress, shown in Fig. 1 consists of a base, a sleeve, a plunger and a pre-load nut, all of which, when assembled, fit into the Zwick. A specimen is compressed between two platen surfaces formed by the bottom of the plunger (upper platen) and the top of the base (lower platen). All the components were manufactured from a mild steel, with a Lathe Master GHK-1440-K conventional lathe used as the primary manufacturing machine. The pre-load nut interacts with the 10 kN load cell via a proprietary load transfer pin and threaded component supplied with the Zwick.

The function of the pre-load nut is to avoid backlash due to the load transfer pin. The plunger features a left-hand thread, while the load cell attachment features a right-hand thread. When the pre-load nut is tightened on these two threads, it pulls the plunger toward the load cell attachment, which pre-loads the load transfer pin in the same direction as it will experience during a compression test. This ensures that the plunger is firmly secured to the load cell and additionally minimizes the compliance between the various components.

The plunger features four recessed slots along its length. Bogen MS10-300 Magnetic Multipole Strips, used to measure the displacement of the plunger, are secured into three of these slots, while the fourth engages with a guide screw, which prevents any rotational motion between the plunger and the sleeve.

Prior to any test series, the plunger is secured to the crosshead of the universal testing machine and is displaced vertically, as it is in operation. At this point the base is lightly secured to the loading platform of the universal testing machine, which would allow the base to self-align to the motion of the plunger until there is a zero force reading, at which point the base is firmly secured to the loading platform. This would ensure that there is negligible vertical force transfer, firstly, between the guide screw and the recessed slot, and secondly between the plunger and the sleeve.

The magnetic strips are paired with AMS AS5311 Hall effect sensors housed in recessed slots machined into the sleeve, as seen in Fig. 1b. The use of three Hall effect sensors allows for any non-axial motion of the plunger to be accounted for. In addition, the potential for non-axial motion is minimized through the use of close machining tolerances, as discussed in Sect. 3. Nevertheless, non-axial motion can occur due to slight eccentricities in the load path. The arrangement of three Hall effect sensors is sensitive enough to detect and correct for these otherwise imperceptible deviations. In addition, a number of cut-outs are machined into the sleeve to allow for interior access to place the specimens for testing as well as exits for the wires from the Hall effect sensors to connect to a Pro Micro ATMEGA32U4 microcontroller.

Fig. 1
figure 1

a A schematic and cross-section of the subpress with a specimen in the testing location. b A side-on view of the assembled subpress. c A view of the sleeve and base assembly showing a Hall effect sensor in a recessed slot in the sleeve

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2.2 Hall effect sensors and magnetic strips

The resolution achieved with the Hall effect sensor is dependent on the specification of the magnetic strip secured to the plunger. The subpress is instrumented with three Hall effect sensors, each paired with a multipole magnetic strip which has repeating segments with a 2 mm pole-to-pole distance. The Hall effect sensor provides 12-bit digitization which, when paired with the magnetic strip, results in a displacement resolution of 0.488 μm.

The signal from the Hall effect sensor is captured using the microcontroller. The three data streams from each sensor are captured separately and averaged during signal processing to account for any slight differences in the readings as a result of non-axial motion.

3 Commissioning

The subpress was designed in such a way that the critical features could be machined, primarily, using a lathe, for its inherent accuracy. In particular, individual parts where designed such that the platen and mating surfaces could be machined in a single operation, i.e., without needing to reposition and re-secure a part. This ensured that the plunger and sleeve would have near-ideal concentricity, which, in turn, ensured that the upper and lower platens remained parallel within a close tolerance. Consequently, there is a high degree of confidence that a specimen will experience pure axial compression between two flat and parallel platen faces. Additionally, the bottom face of the sleeve is perpendicular to the axis of the plunger and sleeve, making it an ideal datum for validation testing.

The initial commissioning of the subpress was conducted using a Mitutoyo depth micrometer, which is rated to an accuracy of 10 μm. The base was removed, and the sleeve was secured to a mounting fixture such that all motion was restricted. The plunger was inserted to the bottom dead center of the sleeve such that the upper platen face was flush with the bottom face of the sleeve. A depth micrometer was secured against the bottom face of the sleeve and manually incremented in 0.01 mm steps across a total displacement of 0.1 mm. Between each step a period of time was allowed to pass in order to obtain distinct steps in the final displacement-time plots. The raw data from the subpress were analyzed and compared to the expected displacements. A typical displacement-time plot is presented in Fig. 2.

Fig. 2
figure 2

A displacement vs time plot from a Hall effect sensor during an incremented displacement test using a manually adjusted micrometer for the purpose of subpress commissioning. The inset shows the effect of ambient noise on the digital signal, with the data point divided into five bands. The moving average of this data provides an estimate of the true position

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The commissioning test described above was conducted ten times in order to ensure consistency and repeatability, and similar results were obtained. The subpress was sensitive enough to measure each step with clarity. Across all of the tests, the average difference between the subpress readings and the expected displacements is 0.087 μm with a standard deviation of 0.746 μm. As such, these validation tests, which include operator error when progressing the micrometer, demonstrate that the subpress is capable of measuring to an accuracy of at least 1 μm with potential to measure even smaller displacements.

Figure 2 shows that there is noise superimposed on the signal. This is particularly evident at the plateaus when the plunger in stationery. The envelope of the noise is ± 1 μm, which typically causes the data points to be distributed between five distinct bands of the digital output signal. The five bands were consistently present throughout the displacement histories of the commissioning tests. An example of this is given in the inset in Fig. 2 which spans 200 data points. The data presented in the inset have a skewness value of \(-0.04602\), which, based on the sensor discretising of the data into specifics bins, implies that the underlying noise signal is random with a normal distribution [21]. The central of the five bands contains the majority of the data (113 points), while the adjacent bands have less (49 and 33 points) and the outermost bands have the least (3 and 2 points). This distribution of data points in the outer bands is interpreted as low amplitude noise of both mechanical and electronic origin. Given that the sensors are only capable of providing digital signals at discrete position values, the majority of the data are expected to be in the band closest to the true position. Consequently, provided that the sampling rate is high enough, a normal analysis of this data should yield a low skewness value, as it does in this case. Hence, it is concluded that the noise observed in the form of the distributed data points in the outer bands of the inset in Fig. 2 are as a result of random noise and not an error due to the sensors. Furthermore, with the sufficiently high sampling rate mentioned previously, a moving average of the signal will provide greater accuracy than the resolution of the sensor. This process is similar to ‘dithering’ in image processing, where noise is used to achieve sub-pixel accuracy [22,23,24].

4 Specimen tests

A series of compression tests were conducted on polymethyl methacrylate (PMMA) specimens in order to compare the accuracy of the subpress to that of the integrated measurement system of the Zwick and an Allied Vision MG-505B visual extensometer (VE). Thereafter, tests were conducted on bone specimens in order to refine the data processing and analysis for future work. All tests were conducted at a nominal strain rate of 0.01s\(^{-1}\) and data acquisition rates for the crosshead displacement and the subpress displacement were set to 100 Hz. Furthermore, all tests were set to run until a maximum force of 6000 N was recorded by the load cell.

All three systems could not be compared simultaneously as the proprietary software of the Zwick does not allow for its integrated displacement measurement system to operate at the same time as the VE. The use of external softwares to accommodate this simultaneous testing was limited by the available hardware.

4.1 Specimen geometry

There are no standards for compression testing of bone specimens. The most applicable standard to use in this case, based on tests for materials with similar material properties, would be the standards used for rigid plastics. For example, the ASTM D695 [19] or ISO D604 [20] standards. However, these standards require that specimens intended to determine elastic modulus have a height of 50.8 mm and a diameter of 12.7 mm; a height to diameter ration of 4:1 [19]. Obtaining specimens that adhere to this ratio, with a reasonable diameter, is exceedingly difficult as a result of the inconsistent cross-section of cortical bone regions. Additionally, if one donor bone is to be used to produce all specimens, obtaining a large, statistically significant number of specimens, as outlined in the standards mentioned above, becomes impractical. As a result of these factors, it is not uncommon to find specimen geometries closer to a 1:1 ratio in the literature. Lee et al. [15], McElhaney [25] and Rampersadh et al. [26] tested cortical bone specimens with a maximum aspect ratio of 2:1, with dimensions in the order of millimetre magnitudes. Furthermore, the bone tests conducted using this custom subpress would be complimented by a series of dynamic tests to demonstrate the strain rate sensitivity of cortical bone. A requirement for these dynamics tests to be valid is that the specimens are in a stress equilibrium, and using such tall specimens would make achieving this stress state very difficult. Hence, for consistency between test configurations, these standards could not be adhered to.

For this study, a total 17 PMMA specimens were machined to have a diameter of 8 mm and a 5 mm height. The barrelling and friction effects of this unique specimen geometry were not of concern for these tests as the purpose was not to evaluate the properties of PMMA. Rather, these tests were strictly used to compare displacement data between the various measurement devices. Tests for anisotropy were conducted and it was found that the material directions from which the specimens were machined had no effect on the recorded material characteristics.

4.2 PMMA tests

Figure 3a shows results of 13 tests where the displacement measurements of the crosshead and the subpress are compared. The subpress recorded an average total displacement of 2436 μm with a standard deviation of 61 μm. The Zwick recorded an average displacement of 3120 μm with a standard deviation of 73 μm for the crosshead. This results in an average difference between the crosshead displacements and the subpress displacements of 684 μm. These results show that the crosshead is distinctly offset from the subpress. However, they both show consistent repeatability, which can be seen by the similar standard deviations.

Figure 3b shows the results from the four tests where the VE was compared to the subpress. The subpress measured an average total displacement of 2501 μm with a standard deviation of 51 μm. The VE recorded an average of 2582 μm with a standard deviation of 292 μm. An average difference of 81 μm was recorded between the subpress and the VE results. The VE records displacements much closer to those recorded by the subpress when compared to the results recorded by the Zwick, as seen by the similar averages. This comparatively higher accuracy is as a result of the VE measuring the displacements from lines marked onto the platens, as close to the specimen as possible, which avoids the compliance of the load train of the Zwick, as well as the majority of the compliance of the subpress. These line marks could not be placed on the specimens themselves as the specimens were too small, and the visual access was constrained.

Fig. 3
figure 3

a Comparison of the subpress and crosshead displacement measurements for the 13 PMMA specimen tests. b Comparison of the subpress and crosshead displacement measurements for the 4 PMMA specimen tests

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The reason for the discrepancy in the number of tests between the Zwick comparison and the VE comparison is that the purpose of this study is to demonstrate the accuracy of the subpress. When the VE demonstrated its inconsistency in the first four tests, no further investigation with this apparatus was deemed useful to the evaluation of the accuracy of the subpress. Furthermore, the Zwick comparison incorporated machine compliance, which is a primary hindrance in current testing that the subpress aims to address, thus, more specimens were dedicated to this comparison.

These tests yielded an average apparent elastic modulus for the PMMA specimens of 3.5 GPa, with a standard deviation of 0.27 GPa.

4.3 Bone tests

To assess the effectiveness of the subpress for its intended purpose, five tests were conducted on cortical bone specimens of 5 mm diameter and height, machined from a bovine femur. These tests highlighted a clear deviation of the magnetic sensor displacement data from the crosshead data (Zwick Data), as shown in Fig. 4. Furthermore, prior to the impact of the plunger on the specimen i.e., during a no-load condition where compliance is not a factor, the displacement-time data of the subpress is practically identical to that of the crosshead. During the no-load phase of all five bone tests these displacement-time values differed by less than two microns. The no-load phase allowed for a datum to be created whereby the subpress displacement data stream could be synchronized with the force data stream from the universal testing machine software. This also prevented any pre-loading of the specimen.

Fig. 4
figure 4

Typical comparison of the subpress and crosshead displacement vs time data for a bovine cortical bone test

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The dimensions of the specimens were measured using a Facom 1300E Digital Caliper, which is rated to an accuracy of 10 μm. The initial specimen cross-sectional area was calculated and used to calculate the engineering stress with the force being obtained from the load cell of the universal testing machine. The initial specimen height and the specimen displacement recorded by the subpress were used to calculate the engineering strain. These tests yielded an average apparent elastic modulus for the cortical bone specimens of 23.69 GPa, with a standard deviation of 1.55 GPa.

The stress–strain plot in Fig. 5 is used to determine where the elastic region started, representing the true specimen behavior if ‘settling’ was not evident, as well as the point of incipient failure. As shown by the strain rate data in Fig. 5, the crosshead does not maintain a constant speed. In fact, the true specimen strain rate is an order of magnitude lower than that specified in the Zwick software, which was 0.01s\(^{-1}\). Only once the specimen has failed does the crosshead move at the desired strain rate; as can be seen in Fig. 5 where the strain rate settles near the desired value after the peak of the stress–strain curve.

The strain rate curve showed that during the deformation of the specimen, prior to failure, the testing strain rate was an order of magnitude lower than that set by the Zwick. Only after failure did the strain rate settle to the desired value. Furthermore, as the specimen approached its maximum stress, there was a sharp increase in the strain rate curve. Before and after this, the regions were relatively linear and linear fits could be generated. It is proposed that the intersection of these two linear fits be interpreted as the point of incipient failure and will be discussed further in Sect. 5.

Fig. 5
figure 5

A typical stress and strain rate vs strain plot for a bone specimen. Linear fits are superimposed onto the strain rate plot in order to identify the transition between the elastic and plastic regions of the stress–strain plot, and hence the point of incipient failure

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5 Discussion

5.1 Subpress, crosshead and VE comparison

The micrometer stepping tests discussed in Sect. 3, demonstrate that the subpress is able to accurately measure displacements at the desired resolution of one micron. This conclusion is supported by Fig. 4, where, under a no-load condition, it is evident that the subpress records the same displacement-time data as the crosshead to within two micron. Given the scale of the load frame, this level of agreement is satisfactory. However, when the plunger makes contact with the specimen, the Zwick continues to record linear displacement-time data, while the subpress is able to record the take-up in compliance of the load frame.

Considering the results of the crosshead PMMA test series, it is evident that the subpress is at least as consistent as the integrated measurement system of the Zwick as shown in Fig. 3a. This is shown by the relatively low and similar standard deviations in the comparison of these two systems. Furthermore, as seen in Fig. 3b, the VE PMMA test series, the subpress appears to be at least as accurate as a VE system, as seen by the comparatively similar overall displacement measurements of these two systems. Thus, the subpress combines accuracy that is, at worst, equivalent to a VE, while having the repeatability equivalent to the Zwick, which uses a rotary encoder. Furthermore, the subpress is a more cost-effective means of achieving the desired resolution of displacement data when compared to the VE as it does not require specialized cameras, nor proprietary digital image correlation (DIC) software.

When comparing the four averages of the overall displacements from the two sets of tests, it was noted that the average displacement of the subpress differed by 65 μm between the crosshead PMMA test series and the VE PMMA test series. This amounts to a difference of 2.6 %, which is equivalent to one standard deviation of the displacement readings from the subpress in both test sets. Given the small sample sizes, this small amount of variation is reasonable for samples cut from the same stock.

The effect of compliance is once again evident when the stress vs strain and strain rate vs strain results are considered, as seen in Fig. 5. The offset displacements recorded under load by the integrated displacement measurement system of the Zwick, shown in Fig. 4, are attributed to this inherent compliance of the machine, due to its large multi-component load train. The load train between those encoders and the specimen includes the load screws themselves, the crosshead, the load cell, the plunger and all of their connections, which contribute to the overall compliance of the load frame. Even a small percentage compliance as a result of this load train has a distinct influence on the resulting displacement readings. The subpress removes the load train from the displacement measurements entirely, hence eliminating the impact that machine compliance has on the final displacement measurements.

In principle, the desired resolution may be achieved using DIC. Although some studies testing cortical bone have used dried specimens with speckled patterns for DIC [27,28,29,30], the initial drying and time outside of the donor can have an effect on the measured properties of the bone specimens [17, 31]. An alternative would be to apply the speckle pattern to the platens. Assuming that DIC would be more accurate than a VE system, this would, at best, provide results similar to that of the subpress, but with a significant increase in costs due to additional equipment, software, calibration time and processing time. Hence, DIC was not considered to be a viable option for bone material characterization.

Finally, despite these tests primarily being used as validation of the displacement results of the subpress, the measured apparent elastic moduli obtained through these tests was well within the expected range of PMMA material properties reported in the literature, at this strain rate and temperature [32,33,34]. This further reinforces the operation of the subpress for its intended use case and confirms that it is capable of recording the apparent elastic moduli of small specimens for material characterization tests.

5.2 Bone tests

The strain rate data plotted in Fig. 5 offers valuable information on the parameters of the test that would have otherwise been overlooked. The strain rate data obtained through the subpress demonstrates that the Zwick was not operating at the intended strain rate. Rather it was operating at a tenth of the desired strain rate during the elastic compression of the specimen. This observation is critical when conducting tests of viscoelastic materials, such as cortical bone, where the strain rate of the test has an influence on the measured properties of the material.

Figure 5 clearly depicts that once the specimen has failed, the velocity of the crosshead rapidly increases, and after a period of settling behavior, it equilibrates to near the desired strain rate. This inconsistent crosshead velocity, and hence strain rate, stems from the compliance of the load train of the Zwick. Only when the specimen cannot offer any resistance to the crosshead, as a result of the specimen failure, can it move freely at the intended velocity.

The point of incipient failure of the specimen was identified in Fig. 5 using two linear fits. The first linear fit is generated over the region of the strain rate plot where the specimen is deforming at a near constant strain rate. This can be seen by the initial flat portion of the strain rate data. The second linear fit is generated over the region of rapid increase in strain rate. This is interpreted as the region wherein the elastic energy built up during the take-up in machine compliance is being released as the specimen begins to lose structural integrity, i.e., incipient failure. It is proposed that the intersection of these fits denotes the end of the elastic response, i.e., the point of incipient failure of the specimen. It is interesting to note that there is no perceptible deviation in the stress vs strain curve at this point, i.e., without a detailed strain rate history the point of incipient failure is undetectable.

With the point of incipient failure and the ultimate compressive strength of cortical bone occurring at an approximate strain of 0.02, the machine compliance is an order of magnitude larger than the specimen displacements at that point. Based on the results, the difference between the Zwick displacements and the subpress displacements, which is attributed to machine compliance, is seven times the expected specimen deformations at these aforementioned critical strains. This demonstrates the need for such a measurement device that eliminates machine compliance when testing small cortical bone specimens.

Finally, the average apparent elastic modulus of the bone specimens tested in this study was approximately 23 GPa. This is slightly above the upper range of literature data at this strain rate, as reported by Johnson et al. [1]. Furthermore, a study investigating the degradation behaviour of cortical bone specimens demonstrates that this slightly higher apparent elastic modulus could be as a result of fresh specimens being tested, as opposed to the potentially degraded specimens results quoted in the literature [35].

5.3 Design limitations

The subpress has been designed with the intention of characterizing materials with relatively low elastic moduli, while reducing machine compliance to near-negligible levels. However, a combination of the small specimen geometry and a material much stiffer than the PMMA and bone tested in this study could result in a specimen stiffness in the same order of the local deformation of the plunger and lower platen of the subpress. In this case, more significant local deformation would be expected on the surface of the plunger and lower platen, as well as global deformation of the plunger, between the specimen interface and the point on the plunger where the magnetic sensor is reading displacement.

Modifications could be made to the subpress to better account for these anticipated compliance effects. For example, the subpress could be manufactured from a material stiffer than mild steel, such as tungsten, or inserts of stiffer materials can be integrated into the design of the subpress at the specimen interfaces. However, even with these modifications, the deformations as a result of the compliance could be reduced, but not removed entirely. Nevertheless, these compliance deformations are minimal, typically in the order of less than 10 microns, and can be accounted for theoretically should a stiffer material need to be tested using this subpress. For the cases stated in this study, the compliance as a result of local and global deformations did not emerge as a primary point of consideration.

6 Conclusions and recommendations

In this study a custom compression subpress for quasi-static tests on small specimens is commissioned and compared to alternative measurement methods. The results show that the subpress has an accuracy in the order of at least one micron. Furthermore, the results suggest that the subpress is at least as accurate as the VE and as consistent as the universal testing machine, but shows features that offer improvements on these current displacement measurement techniques for this given task. The subpress offers a cost-effective means of measuring displacements in the order of a micron in a repeatable, reliable, relatively simple and robust manner. Many tests can be conducted in a short time frame, thus reducing the effects of degradation on biological specimens, which is the primary intent of the subpress.

The results presented in Sect. 4 show that the compliance of a large testing machine can lead to large errors when assessing specimen strain and strain rate, particularly when testing small specimens of quasi-brittle materials. Additionally, these machines can generate non-axial motion, which can have an effect, particularly when considering small deformations, as in the case of small cortical bone specimens. The use of three magnetic sensors in the subpress allows both the compliance and non-axial motion to be accounted for.

Finally, further development of the subpress can improve the resolution of the system. The use of higher bit sensors and/or employing techniques similar to dithering could allow for finer resolution, potentially to the sub-micron level.