Detailed knowledge of the dynamic viscoelastic properties of bone is required to understand the mechanisms of macroscopic bone fracture in humans, and other terrestrial mammals, during impact loading events (e.g. falls, vehicle accidents, etc.). While the dynamic response of bone has been studied for several decades, high-quality data remain limited, and it is only within the last decade that techniques for conducting dynamic compression tests on bone at near-constant strain rates have been developed. Furthermore, there appears to be a lack of published bone data in the intermediate strain rate (ISR) range (i.e. 1–100 s−1), which represents a regime in which many dynamic bone fractures occur. In this paper, preliminary results for the dynamic compression of bovine cortical bone in the ISR regime are presented. The results are obtained using two Hopkinson-bar-related techniques, namely the conventional split Hopkinson bar arrangement incorporating a novel cone-in-tube striker design, and the recently developed wedge bar apparatus. The experimental results show a rapid transition in the strain rate sensitive behaviour of bovine cortical bone in the ISR range. Finally, a new viscoelastic model is proposed that captures the observed transition behaviour.
The macroscopic fracture of bone in humans, and other terrestrial mammal bones (e.g. bovine, equine, porcine, etc.), typically occurs during impact loading events (e.g. accidental falls, sporting collisions, vehicle accidents, etc.). Therefore, a detailed knowledge of the dynamic viscoelastic properties of bone is required to model the macroscopic deformation and fracture of bone during such events.
Bone is a complex material with a hierarchical microstructure that provides a combination of high strength, stiffness and toughness that exceeds the performance of the constituent parts. The structure of bone and the corresponding influence this has on its macroscopic mechanical properties is discussed in several excellent reviews [1–4]. At the macroscopic level, bone is classified into cortical bone, the hard and compact outer shell, and cancellous (trabecular) bone, the relatively soft and spongy inner material. Cortical bone is the primary load-bearing component of the human body and is the principal bone type found in the central portions of long bones. In more complex bone geometry (e.g. joints, flat bones, etc.), cancellous bone acts as a supportive core material surrounded by a thin shell of cortical bone.
At the microscopic level, bone is a composite material composed of an organic matrix (mainly collagen fibres) and mineral crystals (hydroxyapatite). The mineral component provides mechanical stiffness and strength while the collagen content increases the toughness . Furthermore, the pressure developed by the viscous fluid constituent in the bone cavities may increase the load-carrying capacity of bone during dynamic loading events .
The mineralized collagen fibrils form the basic building block of all bone types. The most abundant bone type in large mammals is lamellar bone, which has a microstructure composed of thin planar sheets of parallel fibrils (3–7 μm), referred to as lamellae . A microstructural element composed of lamellae arranged in small concentric layers (three to eight lamellae) surrounding a central canal is referred to as an osteon (Haversian system) and is the predominant bone type in adult human long bones [2,4]. In bovine bone, by contrast, the lamellar layers are relatively thick (150–300 μm) and arranged in an alternating pattern with layers of parallel-fibred bone that extends around the complete circumference of long bones to form a structure referred to as plexiform (fibrolamellar) bone .
The preceding discussion indicates that bone is a multi-phase, non-homogeneous and anisotropic material  that can exhibit behaviour that is viscoelastic–brittle or viscoplastic. This necessitates the use of dynamic mechanical testing to obtain the relevant mechanical properties of bone. Johnson et al.  reported an extensive survey of the rate dependence of the apparent Young's modulus of cortical bone. They note that most of the work has focused on strain rates in the range of 0.001–0.1 s−1, which are relevant to activities such as walking and running . By contrast, high strain rate bone characterization tests are conducted using drop testing  or split Hopkinson bar (SHB) [6,10,11] techniques, which are typically restricted to strain rates above 300 s−1 and correspond to impact loading conditions. Consequently, the data compiled by Johnson et al. show that the intermediate strain rate (ISR) range (i.e. 1–100 s−1) is sparsely populated. Furthermore, high-quality data remain limited , and it is only within the past decade that techniques for conducting dynamic compression tests on bone at near-constant strain rates have been developed [6,11].
The mechanical testing of bone in the ISR range is challenging because of the unique combination of bone properties. The reported viscoelastic nature of bone dictates that tests must be conducted at near-constant strain rates to prevent false results where strain rate effects appear to be spread over a wider strain rate range . Furthermore, cortical bone tends to exhibit brittle failure at small strains [6,9], which implies that the testing machine must be able to rapidly attain the required strain rate without significant transient strain rate fluctuations.
Adharapurapu et al.  noted that the viscoelastic nature of bone causes a specimen to harden significantly during a dynamic compression test. They achieved near-constant dynamic strain rate SHB tests by placing a sacrificial pulse shaper, i.e. a deformable metal disc, on the impact face of the incident bar, so as to produce a pulse with an extended rise time that closely matched the hardening rate of the bone specimen. However, the rise time was a relatively small portion of the incidence pulse, which restricted the maximum test duration.
Cloete et al.  achieved similar pulse shaping results using a conical striker technique, which allows for good control and repeatability of the pulse shape, as it is reusable. However, the conical striker technique has the undesirable feature of a stress wave ‘tail’ that restricts the maximum test duration and the minimum strain rate that can be achieved.
Material testing in the ISR range has no universally accepted testing method, even though a variety of techniques have been developed [12–17]. The unique challenge of the ISR regime is that it lies above the upper limit of standard tensile test machines and below the lower limit of typical SHB arrangements and is therefore also referred to as the medium strain rate regime  or the sub-Hopkinson bar regime . Furthermore, while ISR test speeds are modest, they must be attained quickly, causing inertia effects in the load frame and load sensors. Therefore, several researchers have focused on small custom-built machines with lightweight components and integral load sensors [12,15,17].
Modern electro-servo-hydraulic (ESH) machines can achieve the speeds required for ISR testing provided that the actuators are first brought up to speed prior to engaging with the specimen. In tension tests, this is accomplished by using ‘run-up’ fixtures, such as ‘fast jaw’ grips  or ‘slack adaptors’ ; while, in compression tests, an initial gap is left between the specimen and the upper anvil .
All the above-mentioned techniques tend to induce stress wave oscillations in the load frame and transducers that obscure the specimen response. Several approaches have been pursued to address this problem. The relatively short load paths of custom-built machines [12,15,17] are intended to minimize the effect of stress wave oscillations. Similarly, short load cells [16,18] or the strain gauging of specimens [14,19] offer increased frequency response, and signal processing techniques that incorporate vibration models can compensate for some dynamic effects.
An alternative approach to dealing with stress wave oscillations is to use Hopkinson bar techniques, where the incident bar is replaced by a flywheel  or hydraulic actuator , and the extended test durations are accommodated by using atypically long bars  or by accounting for stress wave reflections using wave separation techniques [13,21,22].
In general, the above-mentioned techniques are focused on capturing the stress histories in the ISR regime, and less attention has been given to the strain rate histories. Typically, only the nominal or average strain rates are reported, or strain rate histories are reported that vary significantly. Custom-built lightweight machines are sensitive to changes in the specimen stiffness, which results in nonlinear displacement histories [12,15,17], while the inherent compliance of ESH machines causes a similar effect [13,16]. Therefore, specimen-mounted strain gauges [14,16] or optical techniques [13,21] are often required to capture the strain history.
An alternative ISR testing concept, referred to as the ‘wedge bar’ technique, has recently been developed by Cloete & co-workers [23,24]. The wedge bar technique uses a wedge mechanism to compress small cylindrical specimens and has produced ISR data for various metals and polymers at near-constant strain rates. The concept is closely related to the Hopkinson bar technique and is discussed in further detail in §2d.
In this paper, preliminary results for the dynamic compression of bovine cortical bone in the ISR regime are presented. The results were obtained using two Hopkinson-bar-related techniques. The first technique is the conventional split Hopkinson pressure bar arrangement incorporating a novel cone-in-tube striker design, which extends the practical testing range down to 100 s−1. The second technique is the recently developed wedge bar apparatus. In addition, the results are compared to published data obtained under quasi-static and high strain rate conditions. Finally, a new viscoelastic model is proposed that captures the observed transition behaviour.
2. Experimental methods
Although the ultimate aim of this work is to develop ISR techniques to test human bone, this study is limited to bovine bone for several practical reasons. Fresh material is readily available from animals of a consistent age, physical state and dietary history. Furthermore, the cortical cortex of bovine bones is thick, which simplifies the specimen manufacturing process, and bovine material requires less ethical clearance while conforming to strict health and safety standards .
From a material modelling perspective, bovine bone is a suitable material because, though its microstructure and composition vary from those of human bone, its mechanical behaviour is qualitatively similar. In particular, the literature indicates that the macroscopic elastic behaviour of bone is mostly due to the properties of individual lamellae and not dependent on the particular structures that they form (e.g. osteonal versus plexiform), although the fracture behaviour is distinct . Furthermore, data for nominally similar materials are available in the literature [6,7,9].
(a) Specimen preparation
Whole fresh femurs of slaughter-age cattle were sourced from a butcher and were stored in a frozen condition. During specimen preparation, the bones were thawed and cortical specimens were extracted from the mid diaphyses of two femurs. Sixty cylindrical specimens with nominal dimensions of 4 mm diameter and 5 mm length were machined and individually inspected, weighed, measured and refrozen. Several randomly selected specimens were examined in greater detail using optical microscopy to confirm that the specimens contained no flaws or damage. The results reported in this paper are for specimens originally orientated in the longitudinal direction relative to the bone axis. In all cases, the specimens were tested in a random order to prevent any systematic deviation in the test results.
(b) Quasi-static testing
Quasi-static compression tests were conducted on a Zwick universal testing machine. In total, 34 specimens were tested at three different strain rates (14 at 0.003 s−1, 10 at 0.01 s−1 and 10 at 0.08 s−1).
(c) Split Hopkinson bar testing using a cone-in-tube striker
Dynamic bone compression tests at the upper range of the ISR regime (i.e. 100 s−1) were performed using the classic SHB technique  in combination with a new cone-in-tube (CiT) striker configuration. As the name suggests, a CiT striker consists of a slender cone made from 6063-T6 aluminium coaxially mounted inside a tube of the same material, as shown in figures 1 and 2. The CiT striker had a length of 870 mm, with inner cone end diameters of 9 mm and 25 mm. The outer tube has a standard section and was chosen to have approximately the same cross-sectional area as the large end of the inner cone.
The Hopkinson bars were 3 m long and made of 19.05 mm diameter maraging steel with a density of 7950 kg m−3 and a longitudinal wave propagation speed of 4575 m s−1. Diametrically opposed foil strain gauges (2 mm, 120 Ω) were bonded to the midpoint of the Hopkinson bars. The signals were amplified using custom-built amplifiers (gain of 1000, bandwidth of 100 kHz) and digitized using an ADLINK PCI-9812 card (10 MHz, 12 bit).
The CiT striker design is intended to produce an incidence pulse that allows ISR compression tests to be conducted at near-constant strain rates on material with steep hardening characteristics, for example cortical bone. Therefore, a shaped incidence pulse that has a steeply rising slope with the same gradient as that of the specimen response is required. Furthermore, the end of the pulse should ideally be sharply truncated to allow rapid load removal. This additional requirement has two purposes. Firstly, it allows the test duration to be maximized for a given SHB length, and hence allows the strain rate to be minimized for a given final strain. Secondly, it allows for the possibility of conducting dynamic viscoelastic recovery tests. In addition, it would allow for the recovery of an intact specimen for microstructural analysis. Previously published pulse shaping concepts have only been able to meet the steep slope requirement [6,11].
The CiT concept is a modification of the conical striker technique . The CiT striker is launched in a conventional manner using a gas gun. However, only the inner cone impacts upon the incident bar, whereas the outer tube impacts upon a large reaction mass mounted coaxially around the incident bar. The resulting stress waves in the cone and tube simultaneously reach the free end of the striker where the cone and tube are connected. In isolation, the stress wave produced in a slender cone with a large ratio between its end diameters would not be sufficient to reverse the velocity of the large end and is the reason that conical strikers produce pulses with residual ‘tail’ features. However, in the CiT striker, the reflected velocities of the cone and tube are summed and, if the tube is large enough, results in the velocity of the large end of the conical striker being reversed. This velocity reversal induces a tensile stress wave in the cone that, upon arrival at the impact face, causes the conical striker to separate from the incident bar, thus abruptly terminating the incidence pulse. In this way, a steeply rising incidence pulse, with a large ratio between the initial and final stresses, can be created without any residual ‘tail’ features, as shown in figure 3. Additional practical benefits of the CiT striker configuration are that buckling is avoided and mounting of the striker in the gas gun barrel is simplified in comparison to the conical striker.
Also shown in figure 3 are typical reflected and transmitted signals. The reflected signal has a distinct plateau, which indicates that a constant strain rate test has been achieved. The data can be processed using the standard SHB formulae  and typical results are shown in figure 4. Figure 4a shows the combined stress history obtained by shifting the pulses to the specimen-bar interfaces, and figure 4b shows the resulting stress and strain rate curves as a function of strain. Figure 4 shows that specimen equilibrium has been achieved and that a near-constant strain rate of 115 s−1 was achieved throughout the test duration. In total, five specimens were tested at an average strain rate of 120 s−1.
A minor limitation of the CiT technique is that the required cone angle is not known a priori, and, in general, a set of inner cones with a range of cone angles would be required to cover possible variation in specimen responses. The cone angle used in this work was selected based on previous experimental results [6,11], while the diameter ratios were obtained using a simple spreadsheet-based one-dimensional mass–spring code  and did not require any iterations.
(d) Wedge bar intermediate strain rate testing
Dynamic bone compression tests in the lower range of the ISR regime (i.e. 1–10 s−1) were performed using the wedge bar technique [23,24]. The general layout of a wedge bar system consists of three axially aligned high-strength steel bars (i.e. the striker, wedge and stopper bars) and a miniaturized load frame assembly, as shown schematically in figure 5. The bars are nominally identical except for a shallow wedge machined into the wedge bar (from which the technique derives its name), either in the central third of a long wedge bar [23,24] or in the distal end of a short wedge bar .
The components of the miniaturized load frame assembly are shown in figure 6. The elements of the load path are the monolithic load frame, wedge bar, sliding anvil, load cell and specimen. The load frame assembly is designed to maximize the load path stiffness and minimize inertia effects and vibrations. To accomplish this, the load path is short (50 mm) and, other than the specimen, contains no compliant fluid or viscoelastic components. The load frame is monolithic in the sense that it was machined from a single block of grey cast iron, to maximize its stiffness and reduce the number of surfaces from which stress waves could reflect back towards the load path. Grey cast iron was chosen for its inherent structural vibration damping properties due to the presence of graphite flakes in the microstructure , which also tend to reduce the friction between the load frame and the wedge bar.
The components of the actuation mechanism are the wedge bar (made of 20 mm diameter silver steel) and the sliding anvil (made of mild steel) and are the only moving parts of the load frame assembly. The sliding anvil is designed to be as small and lightweight as possible. A bearing surface, referred to as a ‘slipper’, was mounted on the bottom of the sliding anvil. The slipper was made of brass, i.e. a dissimilar material to steel to avoid friction welding , and its lower surface machined to the same angle as the shallow wedge of the wedge bar such that the specimen platform of the sliding anvil would be horizontal. The load cell is made of stainless steel and is adjustable to allow the specimen to be preloaded. This design feature was included to allow the specimen to be reliably positioned using a slight preload, which also reduced the initial transient ‘take up’ phase of the test. Furthermore, bones typically function in a preloaded state and this feature will allow future investigations of the effect of preloading on bone behaviour.
To perform a test, a small cylindrical specimen is positioned between the sliding anvil and the load cell, and lightly preloaded to secure it. The striker bar is fired from a gas gun at the desired test velocity and impacts upon the wedge bar. As the striker and wedge bar are virtually identical and experience a near-perfect elastic impact, the principles of the conservation of momentum and energy dictate that the striker comes to rest while the wedge bar proceeds at the desired test velocity. Thereafter, the wedge bar moves through the load frame assembly, forcing the sliding anvil and the lower portion of the load frame apart, thus compressing the specimen between the anvil and the load cell. Finally, the motion of the wedge bar is fully arrested through elastic impact with the stopper bar.
The wedge bar concept has several features that make it suited to testing rate sensitive quasi-brittle materials at small strains in the ISR regime. Firstly, the acceleration (and deceleration) of the actuation mechanism (i.e. the wedge bar) through the collinear elastic impact of closely matched bars results in the full test speed being attained in a small fraction of the overall test duration, without any significant speed fluctuations. This feature of the wedge bar concept, which is based on the same principles as the Hopkinson bar technique, enables the small strain specimen response characteristics to be captured prior to incipient failure, and at the desired strain rate. Furthermore, this feature allows for the possibility of specimen recovery with a well-defined strain rate history through accurate positioning of the stopper bar, which determines the stroke of the wedge bar and, consequently, the final strain of the specimen.
Secondly, the shallow wedge angle enables large forces to be generated, while also providing a dramatic speed reduction between the wedge bar and the sliding anvil. The speed reduction feature is crucial in that it allows the sliding anvil to move at the low speeds (≈10 mm s−1) required for ISR testing, while the wedge bar moves at the higher speeds (≈10 m s−1) required to store a large amount of kinetic energy, which is much greater than that dissipated through sliding friction and specimen deformation, and therefore ensures a near-constant rate of deformation.
Thirdly, the use of a wedge mechanism allows for a short load path that can rapidly attain equilibrium. This feature is enhanced by the damping properties of the cast iron load frame and the absence of load path components with large masses (e.g. pistons) or low stiffness (e.g. air or hydraulic fluid).
An additional practical benefit of the wedge bar concept is that it has several components in common with a typical SHB installation. The wedge bar configuration reported in this work used the same gas gun, bar supports and electronic data capture system as that used for the SHB tests reported in the previous section. Hence, the velocity of the striker bar just prior to impact is recorded using a light trap, while the specimen load signal is captured using diametrically opposed strain gauges bonded to the load cell. The only additional data capture device is a reflective object sensor system , shown in figure 7, which captures the velocity history of the wedge bar.
The wedge bar lengths (wedge ratios and corresponding strain rates) used in this work were 1.5 m (1:500, 3–4 s−1), 1 m (1:300, 6–8 s−1) and 0.5 m (1:200, 10–12 s−1). In total, 17 specimens were tested at four different nominal strain rates (four at 3.6 s−1, four at 6.5 s−1, six at 7.9 s−1 and seven at 11 s−1). Typical raw signals obtained from a wedge bar test and processed results are shown in figure 8.
3. Experimental results
The results from the three test programmes described in §2 are summarized in figure 9 along with data due to Adharapurapu et al.  and Cloete et al. . The stress values at a given strain are plotted as a function of strain rate and represent the mean values of all the tests conducted at a given strain rate. The apparent Young's moduli have a similar distribution, with values from 9.4 GPa at 0.003 s−1 to 15.8 GPa at 120 s−1. Also shown is the mean value of the maximum stress obtained at each strain rate, although the maximum stress did not occur at a consistent strain. Each testing technique gave repeatable results with consistent standard deviations, in the order of 14% for the quasi-static, 5% for the wedge bar and 3% for the SHB tests. For more detailed data and analysis, the reader may refer to the work of Paul .
4. Viscoelastic model
Cortical bone is typically modelled as a viscoelastic material [7,28] owing to its rate-dependent behaviour and its dissipative qualities. These models are generally based on classical elastic spring and dissipative viscous damper elements. These elements can be arranged in two fundamental configurations: the Kelvin–Voigt solid, consisting of a spring and damper in parallel; and the Maxwell fluid, consisting of a spring and damper in series. A combination of a Kelvin–Voigt solid and a Maxwell fluid in parallel is referred to as the standard linear viscoelastic model, although various combinations have been proposed to model complex viscoelastic behaviour.
Shim et al.  described the behaviour of cancellous bone using a model similar to the standard linear viscoelastic model but containing a nonlinear damper in the Kelvin–Voigt element in order to capture the rapidly increasing stiffness of the material at high strain rates. Johnson et al. , by contrast, proposed a generalized Maxwell model, consisting of two Maxwell elements in parallel with a single spring, in order to capture the increasing apparent modulus of cortical bone over a wide range of strain rates.
The inclusion of a Maxwell element in the model suggests that a transition will occur between two strain rate regimes with distinct apparent moduli. The experimental data in figure 9 show that this transition behaviour occurs in the ISR regime and that three distinct behaviours are evident. At low strain rates, the material displays a relatively small apparent Young's modulus with mild rate sensitivity. By contrast, high rate dependence is evident in the ISR regime, where a rapid transition in the apparent Young's modulus occurs. Finally, at high strain rates the material displays a relatively large apparent Young's modulus, and published data indicate  the onset of rapid strain rate hardening.
To capture these effects, a model is proposed that consists of one nonlinear Kelvin–Voigt and two nonlinear Maxwell elements, as shown in figure 10. The Kelvin–Voigt element provides the static strength, given by the modulus E0, while the nonlinear damper ηp is used to capture the high strain rate sensitivity. Similarly, the two Maxwell elements also have dampers that are slightly nonlinear. The damper of the first Maxwell element has a strain rate term that is raised to a power less than 1, which provides the mild strain rate sensitivity at low rates, while in the second Maxwell element the strain rate term is raised to a power greater than or equal to 1, which allows the transition behaviour to be sharpened to match the experimental data.
The time-dependent stress can thus be expressed as 4.1where m<1 and n≥1.
For constant strain rates, equation (4.1) can be analytically evaluated and simplified to give 4.2
Note that the standard linear viscoelastic model is recovered by setting the values of ηp and ηm to 0 and the value of n to 1. Constant strain curves obtained from equation (4.2) with the parameters E0=9 GPa, ηp=0.2 MPa sp, p=0.7, Em=1 GPa, ηm=50 MPa sm, m=0.5, En=5.5 GPa, ηn=0.15 MPa sn and n=3 are shown in figure 9. The parameter values were determined using a heuristic procedure and provide a fit with an average error of 3.97 MPa between the mean of each test series and the model, which is well within the standard deviation of the experimental data.
5. Discussion of results
Johnson et al.  provide an extensive survey of the rate dependence of the apparent Young's modulus of cortical bone, and several of those studies [6,9,11] report data that show distinct behaviour at quasi-static and high strain rates, similar to that shown in figure 9. However, only a small number of those studies report ISR data of the order of 10 s−1 and only for a single strain rate within the ISR regime. Consequently, while some published material models [7,11,28] suggest a sudden transition in strain rate sensitivity in the ISR regime, there appears to be no published experimental verification of this behaviour. On the contrary, several studies [6,9] depict a logarithmic relationship between the apparent Young's modulus and strain rate, even though this incorrectly predicts a negative elastic modulus at extremely low strain rates .
The wedge bar and CiT techniques presented in §2, while applicable to a wide variety of materials , were specifically developed to investigate the response of bone in the ISR regime. The results depicted in figures 4 and 8 show that the techniques provide near-constant strain rate compression tests in the ISR regime.
The data summarized in figure 9 show that, in the region of 10 s−1, the rate sensitivity of bovine cortical bone changes significantly over a relatively small strain rate range. Furthermore, adequate resolution of the shape of the transition required repeated testing at several narrowly spaced strain rates. These data emphasize the requirement for a near-constant strain rate testing technique, because, as argued by Adharapurapu et al. , any significant strain rate variation during a test would cause the observed behaviour to be averaged over a range of strain rates, potentially leading to a false result, for example a gradual transition in strain rate sensitivity.
The substantial difference in the apparent Young's moduli between the quasi-static and 100 s−1 data reported in figure 9 is consistent with previously published results for bovine cortical bone under similar conditions [6,11]. Furthermore, the low rate sensitivity observed in the quasi-static regime is consistent with published data for cortical bone [7,11] in general, although Adharapurapu et al. report conflicting trends for dry and wet bones in this regime .
Figure 9 shows that equations (4.1) and (4.2) provide a good fit to the experimental data. While, at present, this model is strictly phenomenological, some physically based models that show similar trends are available in the literature. For example, Liebschner & Keller  present a rate-dependent strength model for cortical bone based on the hydraulic strengthening (HS) hypothesis. The predicted range of behaviour is similar to that which can be obtained from a Maxwell element incorporating a nonlinear damper. In other words, the nonlinear Maxwell elements in the proposed model are intended to represent an HS-type mechanism, whereby, as the strain rate and pore fluid velocity increase, the additional flow resistance leads to greater pressure in the fluid component such that a certain portion of the load is transferred via structural load paths that would not be active at low strain rates. These secondary load paths provide additional stiffness, as represented by the spring in the Maxwell element. Furthermore, owing to the hierarchical structure of bone, there may be several fluid–pore systems that become active at distinct strain rates, which implies that more than one Maxwell component may be required.
In the proposed model, the spring in the Kelvin–Voigt element provides the bulk of the quasi-static stiffness, while the first Maxwell element (m<1) provides the mild strain rate sensitivity at low strain rates. This latter effect must scale with increasing strain and, therefore, cannot be accomplished by the nonlinear damper in the Kelvin–Voigt element, which is strain independent. The second Maxwell element (n≥1) provides the sharp transition in rate sensitivity in the ISR regime, the degree of which is also strain dependent. Finally, the nonlinear damper in the Kelvin–Voigt element provides for the dramatic increase in the apparent strength of cortical bone from strain rates above 100 s−1 , which, in contrast to the previous phenomena, appears not to be strain dependent.
In this paper, novel results for the dynamic compression of bovine cortical bone in the ISR regime are presented. The ISR data were obtained using an SHB with the new CiT striker design and the recently developed wedge bar technique, which is closely related to Hopkinson bar techniques. The CiT striker generated a steeply rising yet well-truncated incidence pulse that allowed near-constant strain rate tests to be performed in the upper range of the ISR regime (≈100 s−1). To complement these data, wedge bar tests were performed at near-constant strain rates in the lower range of the ISR regime (≈10 s−1). In addition, quasi-static compression results were obtained, which, along with the 100 s−1 data, were consistent with previously published results. The experimental results show a rapid transition in the strain rate sensitive behaviour of bovine cortical bone in the ISR range. This novel result emphasizes the requirement for constant strain rate testing in the ISR range, and hence the development of the techniques reported in this paper. Finally, a new viscoelastic model comprising one nonlinear Kelvin–Voigt and two nonlinear Maxwell elements is presented that successfully captures the various features of the macroscopic bone response.
The financial assistance of the National Research Foundation and the University of Cape Town towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the National Research Foundation.
The authors thank Mr C. Harris, of the University of Cape Town (UCT) Medical School, for the machining of the bone specimens, Mr G. Newins and Mr P. Smith, of the Department of Mechanical Engineering UCT, for machining the wedge bar components, CiT striker and reaction mass, and Ms P. Park-Ross, of the Centre for Materials Engineering UCT, for assistance with the Zwick machine.
One contribution of 12 to a Theme Issue ‘Shock and blast: celebrating the centenary of Bertram Hopkinson's seminal paper of 1914 (Part 1)’.
- © 2014 The Author(s) Published by the Royal Society. All rights reserved.