Materials and Design 154 (2018) 153–169

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Materials and Design

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Adaptive head impact protection via a rate-activated suspension

Devon J. Spinelli a,b, Thomas A. Plaisted a,EricD.Wetzela,⁎ a U.S. Army Research Laboratory, Materials and Manufacturing Science Division, Aberdeen Proving Ground, MD 21005, United States b Drexel University, Department of Materials Science and Engineering, Philadelphia 19104, PA, United States

HIGHLIGHTS GRAPHICAL ABSTRACT

• A helmet suspension containing shear thickening fluid is designed and tested under conditions representative of head impacts. • The helmet suspension exhibits rate- sensitive behavior, increasing in resis- tance to extension as impact velocity increases. • Impact accelerations for the rate- sensitive suspension are ~50% lower than observed for a conventional suspension. • Model calculations reveal ideal suspen- sion characteristics: a rate-sensitive force that is steady over large displacements.

article info abstract

Article history: The design of an adaptive helmet suspension system that provides optimized head protection under variable im- Received 9 January 2018 pact conditions is reported. The adaptive response is achieved through the use of rate activated tethers (RATs), a Received in revised form 6 April 2018 flexible strap-like material that uses shear thickening fluids to generate speed-sensitive extensional resistance. Accepted 30 April 2018 The RATs are integrated into a helmet by replacing the webbing system of a traditional construction hardhat Available online 04 May 2018 with a network of RATs, and performing impact attenuation testing over a range of velocities. Impacts to the region of the helmet demonstrate a 50% reduction in peak acceleration experienced by the headform Keywords: Impact for impact velocities between 1.5 and 3.5 m/s compared to the conventional webbing system, and comparable Brain injury response to the conventional system at 4.5 m/s. Complementary RAT extensional testing and low velocity helmet Helmet compression tests confirm that the rate-sensitive response of the RATs contributes significantly to improved sys- Head injury criterion tem performance. Additionally, calculations for suspension model systems show that the steady yield force ex- Shear thickening fluid hibited by RATs over long strokes is a critical feature for minimizing head acceleration, and that the RAT Rate-activated tether suspension systems are achieving responses remarkably close to the ideal suspension response. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction preventing severe injuries such as intracranial bleeding and skull frac- ture [1,2]. Helmet test standards, such as NOCSAE performance stan- Head protection is a vital requirement for military, sports, and indus- dards [3], FMVSS 2218 [4], ASTM F1446-13 [5], and ANSI/ISEA Z89.1- trial safety. Most are primarily designed to avert fatality by 2014 [6], were created to determine a helmet's ability to attenuate im- pact forces, where the acceptance criteria were derived from the obser- ⁎ Corresponding author. vation of cranial fracture in cadaveric studies [2]. Generally, these E-mail address: [email protected] (E.D. Wetzel). standards require mounting the helmet to an instrumented headform,

https://doi.org/10.1016/j.matdes.2018.04.083 0264-1275/© 2018 Elsevier Ltd. All rights reserved. 154 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 subjecting the head and helmet to an impact, measuring translational In the present study, we evaluate a new helmet webbing suspension head response, and then comparing head response to injury limits. design, in which the web elements consist of “rate activated tethers” Most helmet test standards report translational acceleration metrics, (RATs) [39,40]. RATs are flexible straps that exhibit a low force, elastic both peak value and time-integrated, which correlate well with likeli- response at low velocities, but exhibit increasing resistance to extension hood of skull fracture [1,7]. Due to the emphasis on averting fatal inju- as extension rates increase. RATs consist of an outer elastic tube, with ries, these tests evaluate helmets under relatively high impact two enclosed ribbons, immersed in a shear thickening fluid (STF) energies where skull fracture is likely to occur for an unprotected (Fig. 1). The STF imbues speed sensitivity to the device, and consists of head and seldom provide additional requirements for reducing head colloidal particles stabilized at high volume fraction in a carrier fluid. loads at lower impact energies. As a result, many helmets are designed At low shear rates, the STF exhibits a flowable low viscosity state, but with relatively stiff webbed or foam suspension systems that are opti- at high shear rates the STF becomes solid-like [41–43]. These unique mized for high energy impact, but provide little compliance during properties have been exploited for a range of energy absorbing applica- lower energy events. tions, including protective textiles [44–48], shock absorbers [49,50], and More recently, the medical community has recognized that concus- foam pads [51–53]. For the RATs, the detailed mechanism of interaction sion and repetitive brain trauma are also a serious health concern between the STF and ribbons is uncertain, but it is likely that the [8–10], leading to interest in designing helmets that also reduce the transitioned STF transfers load between ribbons through a combination likelihood and severity of brain injuries [11]. Translational acceleration of viscous forces and particle-particle force chains [54–56]. Increasing has been shown to correlate with [12–15], although newer RAT extension rates lead to high shear rates between the ribbons, hypotheses propose that rotational loads more directly generate brain resulting in the aforementioned STF transition which drives an in- tissue strains that are believed to induce axonal damage associated creased resistance to RAT extension. For typical RATs, a 10–100× in- with long term diffuse brain injury [16–19]. In the present study we crease in extension forces are possible with an increase in extension focus on translational dynamics, consistent with most existing head rate of 10–100×. Factors such as RAT diameter, length, ribbon material, protection performance requirements. Studies have also suggested and STF composition can be selected to tune mechanical response to suit that multiple impacts, even if each impact is at low energy, can cause cu- specific application needs. mulative effects that lead to serious injury [20–24]. Therefore, an The objective of this study is to compare the impact response of a emerging for head protection is to minimize head loads at all veloc- helmet with a RAT-based suspension system to a conventional web- ities, rather than designing helmets to meet only maximum thresholds based suspension. First, details on the construction of the RATs, and at high energies. their assembly into a helmet suspension, are provided. Then, mechani- To minimize translational head accelerations at all impact velocities, cal and impact testing of RATs and RAT-based helmet suspensions are an energy absorber with uniquely tuned properties is required. First, the conducted and compared with a conventional helmet suspension sys- energy absorber must yield at a near constant force over the full stroke tem. A construction hard is used as a simple and low cost test plat- of the energy absorber [25–28]. A constant resistance force induces con- form, and testing follows the ANSI/ISEA Z89.1-2014 test standard. stant head deceleration, which minimizes peak and time-averaged ac- Impact results are then compared with analytical impact models to celeration metrics when the translational body is brought to rest at identify key features of system response, and quantify the behavior of the point of maximum stroke. The stroke of the energy absorber is typ- the helmet suspensions relative to ideal performance. ically limited by a fundamental geometric constraint such as the shell- to-skull gap in a helmet, and there is no penalty associated with using 2. Experimental methods that full stroke during the impact event. Second, the energy absorber should generate a resistance force that is proportional to impact energy, 2.1. Materials thereby minimizing acceleration by using the full stroke of the energy absorber under all impact conditions. This requirement is met by de- The RAT assemblies utilized two commercially available STFs, STF- signing an energy absorber that is compliant under lower velocity im- PO-52 and STF-PO-50 (STF Technologies, New Castle, DE). These fluids pacts, but becomes increasingly resistive to displacement as impact consist of precipitated calcium carbonate (PCC) particles with a mean velocity increases. This combination of properties – a velocity- size of 600 nm, suspended at 52 vol% and 50 vol%, in a paraffinic oil car- sensitive yield force, which, upon yielding, displaces at a constant rier fluid. Fig. 2 shows viscosity (resistance to flow) as a function of force – is rarely found in energy absorbers, including those used in cur- shear rate and shear stress, measured by an AR2000 (TA Instruments, rent helmet technologies. New Castle, DE) stress-controlled rheometer with 40 mm diameter par- The three main components of a helmet are the shell, the suspension allel plates and a 1 mm gap. Both fluids exhibit steady shear thinning, a system, and the retention system. The suspension system maintains a reduction in viscosity as shear stress increases, at low shear rates. For standoff distance between the shell and the skull, and provides the pri- both fluids, the viscosity reaches a minimum value at a “critical shear mary means of absorbing energy during impact. The two most common rate”, γc, as marked on Fig. 2a. The 0.52 volume fraction STF shows a suspension designs are a webbed suspension or compression pads. The sharp rise in viscosity, or shear thickening, at higher shear rates. Addi- main advantage of a webbed design is improved thermal comfort, made tionally, a similar viscosity rise would be expected at higher shear possible by the large air gap between the suspension and the helmet rates for the 0.50 volume fraction STF, but the rheometer is unable to shell. Compression pads have proven to be more efficient for energy ab- operate at shear rates higher than 100 s−1. The critical shear rate for sorption and can provide a better helmet fit[29]. For these reasons, pad- each fluid occurs at a common critical shear stress value, τc,of100Pa ded suspensions are currently used in most military and sports helmets. as indicated in Fig. 2b. Two ascending and two descending shear rate Compression pads include open cell and closed cell foams [30–32], sweeps were performed for each STF to confirm repeatability of the pneumatic pads [33,34], and elastomeric trusses [35–38]. Closed cell STF formulations. Critical shear stress values over these four sweeps ex- polystyrene foams are commonly used in bicycle and motorcycle hel- hibited a coefficient of variation of 10% for the 0.50 volume fraction STF, mets, but are only designed to withstand a single impact event and and 11% for the 0.52 volume fraction STF. are then discarded. For most sports and military applications, suspen- RATs were constructed using elastomeric tubing, a polymer spacer, sion energy absorbers (both webbed and pad systems) must retain steel ribbons, heat shrink tubing, and epoxy (Fig. 3a). Viton tubes their protective attributes over multiple impacts. Other suspension ap- (10 mm inner diameter and 12 mm outer diameter, cut to 127-mm proaches include slip layers to reduce rotational loads, and hybrid sys- lengths; McMaster Carr, Robbinsville, NJ, product number 5102K15) tems that combine both webbing and compression pads into a single were pre-conditioned in an oven at 100 °C for 24 h to remove any mois- system. ture, and placed in a bag with desiccant for storage prior to RAT D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 155

Fig. 1. Construction and operation of a rate-activated tether (RAT). assembly. Steel ribbons (93 × 7.3 mm active area) were cut out of spring STF. To fabricate a RAT, one Viton tube was placed over the first ribbon steel sheets (0.64-mm-thick, type 1095, “blue state”; McMaster Carr (with adhered spacer). The tubing was stretched open with reverse- product number 9072K14) using a waterjet. Once cut, they were action pliers to pull the tubing past the barbed section of the ribbon. Re- prepped with a Scotch-Brite surface conditioning disc (3M, St. Paul, moving the pliers then allowed the tubing to contract and mechanically MN), and then grit blasted using 60 grit aluminum oxide blast media. lock around the ribbon barb. The assembly was then positioned verti- Immediately after grit blasting, the steel ribbons were placed in mineral cally, with the ribbon head at the top, and a measured volume of STF oil for storage to ensure strong oil wetting to the ribbon surface, and to was injected through the bottom (open) end of the assembly, allowing prevent ribbon oxidation. A spacer was designed to the same dimen- air to vent from the top of the assembly as the device was filled. After sions of the ribbons, with three 3-mm-high projections to maintain a filling, to seal the first ribbon in place, epoxy was injected between the constant gap distance between ribbons (Fig. 3b). The spacer was 3D ribbon barb and the tubing. Heat-shrink tubing was placed over the printed from poly(lactic acid) (PLA) thermoplastic, placed in an oven barb and heat was applied using a heat gun to activate the heat-shrink at 212 °F for 24 h, then placed in a bag with desiccant for storage. The tubing and accelerate the epoxy cure, locking and sealing the first rib- spacer was adhered to one ribbon using epoxy. Heat shrink tubing bon in place. The assembly was then flipped and degassed on a vortex (19 mm inner diameter, McMaster Carr product number 8195K35) mixer to remove trapped air. The second ribbon was then carefully and five-minute cure, two-part Gorilla epoxy (Sharonville, OH) was inserted past the barb, and the adhesive and heat shrink process was re- used as received. peated to bond and seal the second ribbon in place, creating the final A construction hardhat (Peak View-PV50, Portwest, Carrowbeg, RAT device (Fig. 3c). Ireland) shell was used as a low cost test platform for helmet suspen- sions. The shell is transparent , which allowed for better 2.3. Suspension system fabrication visualization during testing. Four types of suspension systems were tested, and are denoted as 2.2. Tether fabrication “conventional”, “T52”, “T50”,and“empty”. The conventional suspension is the as-received hardhat suspension, consisting of nylon webbing con- Two types of RATs were constructed and tested. “T52” RATs use the figured in a six-spoke design (Fig. 4a, b). This design will serve as the 0.52 volume fraction STF, while “T50” RATs use the 0.50 volume fraction benchmark when evaluating RAT suspension performance.

Fig. 2. Rheological behavior of PCC-oil STFs at particle volume fractions of 0.52 and 0.50. (a) Viscosity versus shear rate, and (b) viscosity versus shear stress. 156 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169

steel ribbons 20 mm

(b)

spacer Viton tubing

heat-shrink tubing (a) (c)

Fig. 3. (a) RAT components, (b) ribbon and spacer assembly, and (c) assembled RAT.

Preliminary impact testing using a 6-spoke array of RATs resulted in shaped assembly (Fig. 4c). A steel coupling piece was cut and attached displacements significantly less than the acceptable displacement to the free end of each tether, allowing it to hook into the existing limits, indicating that the suspension was resisting with forces that shell mounting points (Fig. 4d), like the conventional system. No mod- were greater than optimal. The RAT suspension design was then ifications were made to the shell. changed to a 4-spoke array, which resulted in lower peak acceleration The empty suspension was made using the same steps as the RAT values within acceptable displacement limits for the test conditions. suspensions, but without filling the RATs with STF. The performance Based on these results, for all testing the T52 and T50 suspensions of an empty suspension provides a control condition that clarifies the consisted of four T52 or T50 RATs assembled into a 4-spoke, cross- contribution of the STF to the performance of the RAT suspensions.

Fig. 4. Conventional suspension system (a) isolated and (b) integrated with helmet shell. 4-spoke RAT suspension system (c) isolated and (d) integrated with helmet shell. Front and side views of the conventional and RAT systems, respectively, mounted on a headform can be seen in Figs. 5band6b. D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 157

Each RAT suspension design was constructed such that there was a 10 mm due to the high stiffness of the webbing. Because the response 30–40 mm gap between the crown of the headform and the interior is not speed sensitive, only one force history (the third repetition) per of the helmet shell, consistent with the gap for the conventional helmet extension rate will be reported. design. High rate tensile testing was performed on T52 and T50 RATs using an MTS 810 servo-hydraulic load frame with MTS 458.20 MicroConsole 2.4. Extensional testing (Eden Prairie, MN) and a 2 kN load cell. The tethers were mounted on custom pin grips inserted through closely fitting holes of each RAT Tensile testing was performed on individual T52, T50, and empty end effector. An output signal was processed with an oscilloscope RATs using an Instron 8871 servo-hydraulic load frame (Norwood, using Win600e software (Hi-Techniques, Madison, WI). Each tether MA) with a 5 kN load cell and WaveMatrix control software. This load was subject to a randomized matrix of 24 extensional tests, consisting frame reduces ringing in the force data by means of “inertial dampen- of four repetitions of six different extensional rates: 160, 250, 500, ing”, using an integrated accelerometer on the load cell to measure 750, 1000, and 1250 mm/s. Data acquisition rates at each speed were and subtract inertial effects from the reported sample loads. Mechanical set to record force histories for at least every 0.1 mm extension. Using screw-action grips were used to clamp the ends of the specimens. A se- Matlab software, average force data was binned into displacement ries of tensile experiments were conducted in random order, consisting steps in a similar manner as the low rate extensional test data. Zero dis- of five repetitions at six rates: 0.833, 8.33, 16.7, 50.0, 83.3, and placement was defined as the last point where force was less than 1.1 N. 167 mm/s. Data acquisition rates were set for each extension rate so The global average of the force CV values during high rate tensile testing there were force values recorded for at least every 0.1 mm of extension, was 10.2% for the T52 RAT and 8.6% for the T50 RAT. Force- and experiments were conducted to extensions of 40 mm. Data is re- displacement, force-time, and energy-rate curves were generated and ported as force-displacement, rather than stress-strain, because the merged with those of the low rate extensional test. concept of an “average stress” or “average strain” acting over these multi-material devices is not rigorously meaningful; additionally, the 2.5. Helmet low velocity testing force-displacement data is more directly applicable for designing a RAT-based suspension that results in a desired headform acceleration- Suspension designs were tested at low speeds in an Instron 8871 displacement behavior during helmet impact. The data was analyzed load frame using a 5 kN load cell (Fig. 5). A Department of Transporta- using Matlab software (Mathworks, Natick, MA) where average force tion (DOT) headform size C (CADEX, QC, Canada) was mounted to the response histories at each extension rate were created by binning base of the load frame, using mechanical screw-action grips and a each set of five repetitions onto discrete 0.1 mm displacement steps, short length of aluminum extrusion fastened to the headform. A helmet up to 30 mm. To account for inconsistent specimen pre-tension or containing one of the suspension assemblies was placed on the slack, zero displacement was defined as the last point where force was headform, and the crosshead moved downward at constant velocity to less than 3.3 N. The consistency of the RAT response was characterized compress the helmet system to 20 mm of displacement. Experiments by calculating a coefficient of variation (CV), over the five experimental were run at six different rates in random order: 0.833, 8.33, 16.7, 50, repetitions at each rate, for the binned force values at each displacement 83.3, and 167 mm/s. Force versus displacement histories were recorded for each rate. The global average of these CV values during low rate ten- at each rate and reported up to 15 mm. Experiments were performed on sile testing was 7.1% for the T52 RAT and 7.5% for the T50 RAT, indicating the conventional, T52, T50, and empty RAT suspensions. that the force-displacement response of the RATs is very repeatable at each extension rate. Tether energies at each extension rate were calcu- 2.6. Helmet impact testing lated by integrating the force-displacement data from 0 to 30 mm. An additional tensile test sequence was performed on a sample of Impact testing experiments were conducted based on the “Impact nylon webbing with a gauge length of 145 mm, extracted from the con- Energy Attenuation” requirements and test descriptions provided in ventional suspension and mounted to the load frame using capstan fab- the American National Standards (ANSI) and International Safety Equip- ric grips. The webbing was tested at the same five repetitions and six ment Association (ISEA) standard Z89.1-2014 [6]. Helmets were rates as the RAT testing protocol, but only to a peak displacement of mounted on a DOT headform size C on a guided uniaxial monorail

Fig. 5. Helmet system low velocity test platform. (a) Schematic and (b) photograph, also showing front view of conventional suspension system mounted on headform. 158 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169

(CADEX, QC, Canada) (Fig. 6), constrained to allow motion in the verti- cal direction. The total drop assembly, with mass 4.955 kg, was raised to a prescribed drop height, then released to impact a hemi-spherical anvil with a radius of 48 mm and a chord length of 76 mm. The helmet and headform were oriented to achieve a crown hit on the helmet, and the anvil was positioned directly in line with an accelerometer (model 353B18 with ±500 g peak from PCB Piezotronics, Depew, NY) inte- grated into the drop assembly. Impact velocity was measured using an optical sensor offset 2 mm above the horizontal plane of impact. A high speed video camera (Photron FASTCAM SA1.1, San Diego, CA, USA) operating at 2000 frames per second (1 frame per 500 μs) was po- sitioned to view the back of the helmet upon impact. To achieve a passing test, the standard requires that the peak accel- eration of the headform must not exceed 150 g for an impact velocity of 3.5 m/s. For the present study, experiments at additional impact veloc- ities were conducted to determine a more comprehensive evaluation of suspension response. RAT suspensions were tested at 1.5, 2.5, and 3.5 m/s with three replicates at each velocity, in a random order, and one replicate at 4.5 m/s. These target velocities correspond to heights of 11.9, 35, 65 cm, and 107 cm respectively. After each impact, helmets were removed and inspected for damage. Approximately 10–15 min elapsed between successive impacts on the same suspension system, allowing for a cursory data review, test sample inspection, and resetting the test system. An empty tether suspension was tested with three rep- etitions of 1.5 m/s, but only one test at each higher speed to reduce the Fig. 7. Conventional suspension strap tensile behavior at different extension rates. likelihood of damaging the shell. The conventional helmet was also tested with three repetitions at 1.5 and 2.5 m/s, but only one test at the higher speeds, as damage to the shell and suspension (most typi- headform displacement values are likely to be 2–3 mm smaller than the cally at the connection point between shell and suspension) was ob- total vertical suspension travel relative to its initial position for all served for drops at 3.5 and 4.5 m/s. For any experiment that resulted experiments. in helmet damage, the helmet was not tested further and was replaced Helmet impact data was also characterized via head injury criterion with an untested helmet. Therefore the conventional helmet behaviors (HIC) values, calculated as at 3.5 and 4.5 m/s could include some energy absorption from irrevers- ible damage mechanisms, unlike the RAT systems which were fully re- 8 2 3 9 2:5 settable at all velocities. <> Zt2 => 6 1 7 fi ¼ ðÞ− 4 ðÞ 5 ð Þ Raw acceleration data was sampled at 33 kHz, and ltered using a HIC Max> t2 t1 ðÞ− atdt > 1 fi : t2 t1 ; CFC 1000 low-pass lter with a corner frequency of 1650 Hz [57]. To re- t1 port data, the acceleration histories for the three replicates at each drop velocity were averaged at each time increment, and then this averaged response was integrated to create representative displacement histo- where a is acceleration, t is time, and integration occurs over the time ries. Comparison of integrated accelerometer data with displacement interval from t1 to t2 where t1 is selected to provide the maximum HIC histories measured from analysis of high speed video confirmed the ac- value. The HIC captures the combined effects of intensity and duration curacy of the integrated displacement data to within video resolution, of acceleration, and provides a complementary metric to peak accelera- approximately 1 mm. Video analysis revealed that the headform had tion values. A t2 − t1 value of 15 ms (HIC-15) was used for all calcula- typically displaced 2–3 mm towards the crown of the helmet before tions, consistent with prior studies showing a correlation between this the headform accelerometer registered over 1 g. Therefore, our reported metric and concussion risk [12,58].

Fig. 6. Helmet impact test platform. (a) Schematic and (b) photograph, also showing side view of RAT suspension system mounted on headform. D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 159

Fig. 8. Tether T52 response as a function of extension rate. (a) Force versus displacement and (b) force versus time.

3. Results displacement for an extension rate of 1250 mm/s, while at the same dis- placement the force is only 11 N at 0.83 mm/s. At extension rates of 3.1. Extensional response of suspension components 17 mm/s and below, the loads are small and increase gradually with dis- placement. At higher extension rates, the force shows an initial peak Fig. 7 shows the force response as a function of displacement for the value followed by a long, relatively steady plateau force. The value of nylon webbing from the conventional suspension. The force is linear this plateau force increases gradually with extension rate up to with displacement, and shows negligible rate dependence. A spring 750 mm/s. At extensional velocities of 750, 1000, and 1250 mm/s, the constant was calculated by taking an average of the spring constants initial peak forces continue to increase, but the force values at higher at each extension rate between 2 mm to 5 mm, giving a value of 133 displacement are similar. The peak force value at 1250 mm/s occurs at ± 6 N/mm. a time of 7 ms, demonstrating the extremely fast response time for The extensional behavior of RAT T52 is shown in Fig. 8.Theresponse RATs, and suggesting their relevance for impact applications. The is highly rate-sensitive, with a peak force of 319 N at 4 mm of cause of the fluctuations in the plateau force with displacement is not

Fig. 9. Tether T50 response as a function of extension rate. (a) Force versus displacement and (b) force versus time. 160 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169

Fig. 10. Empty tether response as a function of extension rate. (a) Force versus displacement and (b) force versus time. known, but could be associated with stick-slip behavior at the STF- displacement. In contrast, the T52 assembly shows a strong speed sen- ribbon interface. sitivity (Fig. 12b). A transition in the response is evident between The extensional behavior of RAT T50 is shown in Fig. 9.Comparedto 50 mm/s and 83.3 mm/s for this assembly, where force values jump the T52 RAT, the T50 RATs shows lower forces at similar extension rates, from 200 N to 360 N, respectively, at 15 mm displacement. The T50 as- while not exhibiting a strong initial force peak. The highest force value sembly also shows clear speed sensitivity, but with lower forces and a at 1250 mm/s for the T50 RAT is 190 N at a displacement of 18 mm more gradual dependence on velocity (Fig. 12c). The maximum forces and time of 17 ms. While the T50 force values are lower than the T52 reported for the T52 and T50 suspensions at 167 mm/s and 15 mm of force values, at velocities of 500 mm/s and above a prominent, steady displacement are 413 and 186 N, respectively. The empty assembly force plateau over long displacements is apparent. At very low speeds shows a small amount of compression rate dependence. At 15 mm dis- of 50 mm/s and below, the T50 RAT exhibits a low force, linear elastic placement, this system reaches a peak force of 179 N at 167 mm/s. response, similar to that of the T52 RAT at 0.83 mm/s. In contrast, the tensile response of the empty RAT was linear elastic at all extension rates (Fig. 10). At a displacement of 30 mm, the empty tether demonstrated an average force of 30 N at each rate. This response is nearly identical to the low rate responses for T52 and T50 RATs, confirming that at low rates, RAT response is equivalent to the response of the elastic tubing. Fig. 11 shows the total extensional energies at each extension rate for each RAT. Extensional energy is compared, rather than for example plateau force, since the extensional energy better captures the average system response over long strokes. Comparing the STF filled RATs, T52 consistently absorbed greater energy than T50 at the same speed. At 1250 mm/s, the T52 and T50 RATs absorbed 8.0 and 4.4 J of energy, re- spectively. However, the energy plots suggest that the T52 response could be plateauing, while the T50 energy is continuing to increase with extension rate. It is possible that at higher velocities (such as 3.5 m/s) the T50 and T52 exhibit similar energy absorption. In contrast, the energy absorbed by the empty tether at 167 mm/s is 0.72 J. The per- cent increase of absorbed energy between extension rates of at 0.83 mm/s to 1250 mm/s is 1200% for T52, and 800% for T50, while for the empty tether it is only 23%.

3.2. Helmet system compression response

Fig. 12 shows the force-displacement curves of the different helmet systems tested in low velocity compression. The conventional suspen- sion is consistent with the extensional behavior of the nylon webbing, fi demonstrating high forces with linearly elastic response, and insigni - Fig. 11. Comparison of energy versus extension rate for T52, T50, and empty tether cant rate dependence (Fig. 12a). Forces reach 2000 N at 15 mm of systems. D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 161

Fig. 12. Force versus displacement at different helmet displacement rates during low velocity testing, for (a) conventional suspension system, (b) T52 system, (c) T50 system, and (d) empty system.

3.3. Helmet system impact response average peak acceleration of the T52 assembly was 38 g, or 25% lower than the conventional suspension, at a peak displacement of 28 mm. Figs. 13 and 14,andTable 1, summarize the impact test results at the The T50 provided an even more compliant response, with a very long four target velocities of 1.5, 2.5, 3.5, and 4.5 m/s. In Table 1, standard de- force plateau and peak displacement of 37 mm. The peak acceleration viation values are reported for cases where three drop repetitions were for the T50 system was only 25 g, or 51% lower than the conventional performed. suspension. All systems show a rapid response, achieving significant We first consider the response of the conventional, T52, and T50 sus- forces within 5 ms of impact (Fig. 13b). pensions at 3.5 m/s (Fig. 13). All suspensions meet the ANSI/ISEA Fig. 14a shows the conventional suspension response for all impact hardhat requirement of that the headform experience less than 150 g velocities. A small amount of hysteresis is present in the acceleration- peak acceleration. The conventional helmet suspension displays a displacement curves up to 3.5 m/s. At 4.5 m/s, more hysteresis and a peak acceleration of 51 g and a peak displacement of 21 mm longer force plateau is noted. The sudden, short peak in force at (Fig. 13a). Only one drop was performed at 3.5 m/s, leading to evident 30 mm is most likely due to contact between the headform and the in- deformation of the components that connect the nylon webbing side of the helmet shell. At 4.5 m/s failure of components was apparent. to the helmet shell (this shell was not tested further). The T52 system, in For the conventional helmet, the plastic buckles that connect the straps contrast, shows a more prominent force plateau and hysteresis. The to the shell failed while, for the RAT systems, the slots molded into the 162 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169

Fig. 13. Comparison of different helmet designs subject to impact at 3.5 m/s. (a) Acceleration versus displacement, and (b) acceleration versus time. helmet shell that hosted the steel RAT-to-shell couplers broke. Com- and 170 mm/s, in agreement with the rheological predictions (Figs. 8 bined with the observation of component deformation for the impact and 9). Furthermore, the rise in forces and energy for the T52 RAT at 3.5 m/s, it is likely that the conventional helmet is designed to absorb with respect to extension rate is more rapid than the T50 RAT, which ex- energy via deformation and failure of components, and is meant to be hibits a more gradual rise in extensional resistance (Fig. 11). This behav- discarded after a serious impact. ior correlates with the steeper rise in viscosity typical for higher particle A different response is apparent for the T52 suspension system loaded STFs, and suggested by the rheology data in Fig. 2 (although, (Fig. 14b). A force plateau and significant hysteresis is apparent at however, further rheology data for the 0.50 system above 100 s−1 each impact velocity, with the plateau force value increasing with veloc- would be necessary to confirm this relationship). We also note that ity. Compared to the conventional system, the accelerations are lower, the empty tether system shows low forces and negligible rate depen- and displacements are higher. Similarly, the T50 system (Fig. 14c) dence (Fig. 10), confirming that the STFs are responsible for the unusual shows long plateaus with lower force and higher displacement than behaviors of the RATs. the conventional or T52 system. At 4.5 m/s, the T50 system extends to Of less certainty is the relationship between STF rheology and RAT the point that the headform contacts the helmet shell, while the T52 plateau force. The T52 RAT generates approximately 250 N at system does not. Finally, the empty RAT system shows very little force 1250 mm/s, while the T50 RAT resists with around 150 N (Figs. 8 and below 30 mm of displacement, and high peak displacements. The 9). It is tempting to correlate the higher forces in the T52 system to its headform contacts the shell at velocities above 2.5 m/s. higher viscosities measured in the rheology data. However, we do not Table 1 also tabulates head injury criterion (HIC) values for each ve- believe this correlation to be rigorously meaningful. At an extensional locity and design. The results show that HIC values for the RAT designs rate of 1.25 m/s, the STF is at a shear rate of around 400 s−1 (assuming are significantly lower than for the conventional suspension at all veloc- a ribbon gap of 3 mm), well beyond the critical shear rates and mea- ities. At 3.5 m/s, for example, the HIC values for the T52 and T50 designs sured rheology data. Under these conditions, the STF is likely to be in a are 69% and 82% lower than the conventional suspension HIC, solid-like state. Prior studies have shown that dilatancy during shear respectively. thickening causes particles to push outward from the bulk STF and pro- trude from free fluid surfaces, creating a rough surface that appears 4. Discussion “dry” due to scattering [43,55,60]. Therefore, during rapid RAT exten- sion, RAT forces are generated by particle-particle force chains bridging 4.1. Relationship between STF rheology, RAT behavior, and RAT suspension between the ribbons, and mediated by particle-ribbon friction and response fluid-ribbon wetting. The quantitative origin of these forces is elusive, and indeed the physics and mechanics of the interactions between The rheology of the STFs (Fig. 2) shows that the 0.52 STF has a lower transitioned, confined STFs and solid surfaces is not well understood. critical shear rate compared to the 0.50 STF, higher viscosity at all shear The low velocity helmet suspension response data also correlates rates, and a steeper rise in viscosity above the critical shear rate. These well with the extensional RAT data (Fig. 12). For the T52 system, a features are consistent with prior studies on volume fraction effects on clear jump in force is observed between helmet compression rates of STF rheology [59], and appear to directly determine RAT behavior. As- 50 and 83 mm/s. If we assume that the RATs are at a 45° angle relative suming that the ribbon-to-ribbon gap is determined by the spacer to the compression direction, then these compression rates correspond height of 3 mm, then the rheological critical shear rates of 13 s−1 and to tether extension rates of 30–60 mm/s, similar to the critical rates ob- 33 s−1 correspond to critical extension rates of 39 mm/s and 99 mm/s served in extension experiments. For the T50 system, similar arguments for the T52 and T50 RATs, respectively. For the T52 system, extension would suggest that the compression force should increase at compres- forces begin to increase significantly between 17 and 50 mm/s of exten- sion rates of around 140 mm/s, which is reasonably consistent with ob- sion rate, and for the T50 system significant forces appear between 83 servations. We also note that the steeper force and viscosity responses D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 163

Fig. 14. Acceleration versus displacement for helmets subject to impact testing at different impact velocities. (a) Conventional helmet, (b) T52 system (c) T50 system, and (d) empty system.

measured for the 0.52 STF and T52 RATs also agree well with the more We also note that the impact velocities of 1.5–4.5 m/s correspond to ex- distinct jump in force observed for the T52 system during compression. tensional rates of approximately 1–3 m/s, above the limits of our exten- The empty RAT system, consistent with prior data, show no rate depen- sional RAT data. The extensional energy data (Fig. 11)suggeststhat dence, although the compression tests results show surprisingly high resistance to extension rises steadily within this range, and this increas- resistance forces of 200 N at 15 mm of displacement. This behavior sug- ing resistance is likely responsible for the increasing head acceleration gests that friction between the ribbons is significant for the empty sys- plateaus with increasing drop velocity. tem when subject to transverse compression from the headform. Based on these supporting experimental results, the behaviors ob- 4.2. Comparison of RAT and conventional webbed suspensions served during helmet impact are not surprising. The T52 system, com- pared to the T50 system, demonstrates higher acceleration values and The webbing used in the conventional suspension system exhibits lower displacements, consistent with the higher forces observed in ex- rate-insensitive, linear elastic behavior, with loads that extrapolate to tensional and compression experiments. The empty tether system approximately 3500 N at 30 mm of displacement (Fig. 7). In contrast, shows poor performance in impact, demonstrating that ribbon-to- RATs exhibit highly rate-sensitive behavior, requiring only 30 N to ex- ribbon friction is not sufficient to resist impact in these experiments. tend to 30 mm at low rates (below 10 mm/s), and up to 300 N at higher 164 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169

Table 1 the force required to bring the body to rest at a distance H is (see Summary of impact results, and comparison with model systems. Standard deviation Appendix A). values are calculated from sets of three drops; table entries without standard deviation values indicate that a single experiment was performed, or that results are exact analytical ! solutions. A spring constant of K = 133 N/mm is used for the invariant spring model. V 2 F ¼ Mgþ o ð2Þ Nominal impact velocity c 2H 1.5 m/s 2.5 m/s 3.5 m/s 4.5 m/s

Conventional Impact velocity (m/s) 1.56 ± 0.09 2.50 ± 0.01 3.52 4.62 where g is gravitational acceleration. This body is decelerated at a con- Peak accel. (g) 27.5 ± 1.0 39.1 ± 1.0 51.0 61.0 stant value ac,givenby Peak disp. (mm) 8.1 ± 0.45 13.7 ± 0.08 20.5 40.3 HIC 27 67 137 141 2 V o T52 axðÞ¼ ac ¼ − ð3Þ Impact velocity (m/s) 1.68 ± 0.04 2.61 ± 0.11 3.55 ± 0.02 4.50 2H Peak accel. (g) 21.8 ± 1.5 31.0 ± 0.5 37.7 ± 2.5 48.6 Peak disp. (mm) 11.4 ± 0.3 18.1 ± 1.0 28.0 ± 1.1 38.4 HIC 14 29 43 81 We will refer to this behavior as a “constant force” system. Relating this model to our helmet problem, we assume that the fixed distance T50 Impact velocity (m/s) 1.49 ± 0.01 2.54 ± 0.11 3.55 ± 0.01 4.54 H is equal to the standoff distance between the helmet shell and the sus- Peak accel. (g) 11.8 ± 0.3 18.8 ± 1.0 25.3 ± 1.3 65.8 pension, and there exists an Fc value at each impact velocity that will de- Peak disp. (mm) 13.9 ± 0.3 25.6 ± 1.5 37.1 ± 1.5 41.6 celerate the headform to zero velocity utilizing this full standoff HIC 4 11 25 110 distance. This constant force model represents the best possible suspen- Empty sion system response, by minimizing peak acceleration at every impact Impact velocity (m/s) 1.69 ± 0.01 2.49 3.52 4.62 velocity. Peak accel. (g) 9.06 ± 0.48 31.5 92 147.8 Now consider a mass M with initial velocity Vo, decelerating due to Peak disp. (mm) 26.7 ± 0.6 39.3 40.5 46.7 HIC 3 27 177 543 the action of a spring force that increases linearly with displacement ac- cording to F = K · x,whereK is a spring constant. The spring constant Constant force model necessary to bring the mass to rest at a distance H is (see Appendix A). Peak accel. (g) 3.8 10.6 20.8 34.4 Peak disp. (mm) 30 30 30 30 HIC 0.4 5.5 29.7 104 ! 2Mg V 2 K ¼ 1 þ o ð4Þ Optimal spring model H 2gH Peak accel. (g) 8.7 22.3 42.7 69.9 Peak disp. (mm) 30 30 30 30 HIC 3.1 29.3 126 347 The acceleration of this body as a function of distance is Invariant spring model Peak accel. (g) 25.0 41.6 58.3 74.9 ! Peak disp. (mm) 9.6 15.7 21.8 28.0 V 2 x HIC 24.2 86.8 201 377 axðÞ¼g−2g 1 þ o ð5Þ 2gH H rates (above 1 m/s). Importantly, RATs also exhibit force plateaus over long distances, rather than a linearly proportional force displacement trend. These effects lead directly to 10× lower forces during low velocity helmet compression, and 2× lower acceleration values during helmet impact, compared to the conventional suspension. Given the even greater differences between webbing and RATs dur- ing tensile testing at low speeds (i.e. 100× lower forces for RATs at 10 mm/s), we would expect even more dramatic differences in peak ac- celeration values at impact velocities below 1.5 m/s. Unfortunately, we were unable to collect data at impact velocities lower than 1.5 m/s for our test platform due to limitations in the control software. Although the head accelerations would be relatively minor (less than 20 g for the conventional helmet) at these low speeds, these impacts are far more common than the 3.5 m/s impacts required in the test standard. Data collected using instrumented headforms during indicates that typical players receive roughly 1000 significant head impacts per season, and nearly half of those impacts result in head accelerations of less than 20 g [15]. As concern grows about the role of repetitive low en- ergy head impacts in long term brain health, technologies that can re- duce head accelerations at low velocities could prove critically important.

Fig. 15. Acceleration versus displacement for ideal “constant force”, “optimal spring”,and 4.3. Behavior of suspension systems relative to ideal behaviors “invariant spring” systems for a mass of 5 kg at an initial velocity of 1.5, 2.5, 3.5, and 4.5 m/s. For the constant force and optimal spring models, the mass is brought to rest at a distance of 30 mm. For the invariant spring model, a spring constant of K = Consider a mass M with an initial velocity Vo, decelerating due to the 133 N/mm is used and each labeled data point indicates the final displacement and action of a constant force Fc. Standard dynamics calculations predict that acceleration when the mass is brought to rest. D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 165 and the peak acceleration, which occurs at x = H,is a different deceleration distance H for each initial velocity. In this ap- proach, which we refer to as the “invariant spring” system, the drop ! 2 height is given by Vo ac ¼ − g þ ð6Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ! gM V 2K H ¼ 1 þ 1 þ o ð7Þ K Mg2 In this treatment, which we refer to as the “optimal spring” system, the spring constant varies as a function of initial velocity (but remains and the acceleration history and peak acceleration are given by constant during an impact event) to bring the body to rest at a selected Kx distance of H. This model represents the best possible performance for a axðÞ¼g− ð8Þ suspension system consisting of a linear elastic strap or linear elastic M pad suspension system, in which the spring constant is optimally sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tuned for each impact velocity. 2 The same mass-spring solution can be re-expressed for the case of a ¼ − þ VoK ð Þ ac g 1 2 9 spring constant K that does not change with initial velocity, resulting in Mg

Fig. 16. Comparison of model acceleration versus displacement responses with experimental data for conventional, T52, and T50 designs at (a) 1.5 m/s, (b) 2.5 m/s, (c) 3.5 m/s, and (d) 4.5 m/s. Experimental data is plotted up to the peak displacement value for each case. 166 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169

The invariant spring model best represents conventional elastic and similar to the ideal constant force response, in some cases achieving HIC linear elastic pad suspension systems, which exhibit a linear force- values below the ideal value due to displacements beyond the 30 mm displacement behavior that is relatively invariant with deformation model limit. Previous studies associate HIC values of 250 [12] to 700 rate. [58] with a significant probability of concussion, so that only the Fig. 15 shows acceleration versus displacement for the constant empty RAT suspension at 4.5 m/s impact velocity introduces a signifi- force, optimal spring, and invariant spring systems. Peak acceleration cant risk of concussion (HIC of 543). However, the significant reduction and displacement values, as well as HIC values (see Appendix A for der- in HIC for RAT suspensions relative to conventional webbing, and their ivation), are tabulated in Table 1.AvalueofM = 5 kg is used here and similarity to ideal behavior, suggests that these designs could signifi- for all subsequent calculations, and for the constant force and optimal cantly improve outcomes from low velocity head impact. spring systems the mass is brought to rest at a distance of H = 30 mm. For the invariant spring case, we use a spring constant of K = 4.4. Comparison of RAT suspension relative to foam pads 133 N/mm, consistent with the slope of the force-displacement data for the conventional helmet subject to quasistatic loading (Fig. 7). The Although the current tests use a webbed suspension as a control, invariant spring model shows that stopping distance increases with RAT suspensions may also offer superior energy attenuating character- drop velocity. For all velocities up to 4.5 m/s, the final displacement is istics compared to energy attenuating foams. Helmets designed for less than 30 mm, leading to peak acceleration values significantly higher multi-impact will use padding materials, such as expanded polypropyl- than the optimal spring model. Comparing the optimal spring and con- ene or vinyl nitrile foam, that retain their compression characteristics stant force models, the constant force system brings the drop mass to rest at approximately half the peak acceleration value of the optimal spring system. These model comparisons reveal the two key features of an ideal suspension system, with the goal of minimizing peak acceler- ation at every impact speed: (i) a resistance force that is constant throughout the deceleration distance, and (ii) a resistance force that in- creases proportionally with impact velocity. The invariant spring system performs the worst, because it does not meet either of these ideal char- acteristics; the constant spring system meets the second criterion (re- sponse proportional to speed), but does not provide a resistance force that is constant with displacement. Fig. 16 superimposes experimental acceleration versus displacement behavior for the conventional, T52, and T50 suspension systems relative to the model systems. The conventional suspension system exhibits a behavior that is most similar to the invariant spring model, with a roughly linear acceleration-displacement response and a consistent slope for impacts at 1.5–3.5 m/s. At 4.5 m/s, interestingly the conven- tional suspension system exhibits a more ideal force plateau response. This force plateau could be due to one-time mechanical damage of the suspension components, as the second impact at this velocity caused catastrophic failure at the strap-to-shell mounting locations. The T52 and T50 RAT suspension systems, in comparison, show behavior more similar to the ideal, constant force response. Both of these systems show force plateaus, with plateau values that increase with increasing impact velocity. The T50 most closely matches the ideal peak accelera- tion values for the constant force system, although the significant accel- eration rise period (5–10 mm) leads to peak displacements that exceed the 30 mm limit at higher velocities. The fact that the RAT responses so closely follow ideal response is primarily the result of empirical tuning. A number of other RAT suspen- sion configurations (e.g. smaller diameter tubing, a six-spoke design, long straps parallel to the coronal plane of the head) and RAT compo- nent details (RAT length, ribbon material, ribbon spacing) were attempted before arriving at the presently reported designs. Additional design modifications could be performed to continue to tune perfor- mance empirically and approach ideal response even more closely. A computational design and optimization approach would be possible by collecting RAT extensional data through 5 m/s, and coupling these constitutive responses with a more detailed model of the impact event. A dynamic model would need to explicitly treat the RATs as rate-dependent elements, capture the details of headform engagement with the suspension system, and model compliance and deformation of the helmet shell. This model would provide a more detailed under- standing of the suspension response, as the analytical treatment of RATs as constant force elements during the entire impact event is an oversimplification that does not consider the continuous decrease in RAT extensional velocity as the headform decelerates. Fig. 17. (a) RAT suspension design for protection in multiple impact directions. Table 1 also compares theoretical HIC values (see Appendix A for (b) Suspension between inner and outer helmet shells, to provide energy absorption derivation) with measured values. The RAT suspensions are remarkably during rotational loading. (c) Suspension design with RAT outside of the helmet shell. D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 167

Fig. 17 (continued).

over repetitive impacts. These materials exhibit a compression response that allow us to relate RAT suspension behavior back to details of the characterized by an initial linear region followed by yielding that ex- enclosed STF and RAT design features. Therefore, great opportunity ex- tends over a “plateau region” as the cellular structure collapses [30]. ists to further improve RAT suspension performance relative to the This plateau is rarely a true constant force response, as closed cell presently reported designs. foams usually exhibit some degree of strain hardening over the plateau The present testing uses plastic industrial hard , but we expect region as the entrapped gas is progressively pressurized [31]. Instead, the results to carry over to other helmet systems including military unless the pads have been carefully designed and optimized [35–37], and sports helmets. While the present experiments focus on the ANSI their behavior tends to be intermediate between a “constant force” “impact energy attenuation” test requirement, we would also expect and “invariant spring” response. Therefore, a well-tuned RAT system good performance for the “force transmission” requirement, in which has a high potential to provide a more ideal response, and consequently a falling mass strikes a stationary, helmeted headform. The current lower head accelerations, than current foam pad systems. strap configuration appears best suited for crown helmet strikes, but ad- In addition, although some pads have viscoelastic responses that ditional strap configurations could be designed to resist loading from provide a moderate increase resistance with impact velocity [31], multiple directions (Fig. 17a). We also hypothesize that RATs could be these resistance changes are difficult to tune and do not appear to used to control rotational accelerations, for example by creating inner have been highly optimized for head protection. Generally foams that and outer helmet shells that are coupled with an array of RATs, oriented are highly rate sensitive are also highly temperature sensitive which parallel to the shell walls (Fig. 17b). can limit the conditions over which the helmet protects the head. The use of extensional energy absorbers such as RATs also opens up These factors suggest that our rate-dependent suspension system has completely new suspension concepts, such as coupling conventional strong potential to exceed the impact performance of foam pad suspen- webbing inside the helmet to RATs positioned outside of the helmet sion systems over a wide range of impact energies. We also expect that shell (Fig. 17c). This ability to transmit head loads to a remote energy creating a hybrid protection system with webbed suspension, RAT sus- absorber has broad design implication and, importantly, can potentially pension, and foam pads in parallel or series could provide advantages lead to dramatically increased head displacement allowables within for tuning impact response and comfort. Practically, we note that RATs current helmet shell envelopes. Current foam pad suspensions begin are relatively straightforward to manufacture and do not require exotic to significantly densify at compressions of 50–75%, so 25–50% of the materials, but likely would be somewhat more expensive to produce head-to-shell standoff is not available for deceleration distance. With compared to existing resettable foam pads. Therefore we would expect thin webbing and a remote RAT, the full head-to-shell standoff gap cost-performance tradeoffs to be one consideration when designing a would be available for managing head acceleration, leading to even commercial helmet using a RAT-based suspension. lower injury risk within existing helmet profiles.

5. Conclusions Acknowledgements

This study has identified two key features for optimal helmet energy The authors would like to acknowledge John Lim and Matt absorbers: constant resistance force over large displacements, and in- Langenstein for their contributions to preliminary helmet suspension creasing resistance with increasing impact velocity. The experimental experiments; Mike Neblett, Larry Long, and Ryan Dunn for fabrication data shows that RATs provide both of these characteristics, leading to assistance; Mike Leadore for high rate testing support; and Rich significantly lower head accelerations compared to a conventional Dombrowski of STF Technologies for the rheology data. Mr. Spinelli webbed suspension system. Rheology, extensional testing, and low ve- was supported in part by an appointment to the Undergraduate Re- locity helmet compression testing demonstrate behavior correlations search Participation Program at the U.S. Army Research Laboratory 168 D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169

2 administered by the Oak Ridge Institute for Science and Education where ω = K/M. If the mass comes to rest at time tc then, according to through an interagency agreement between the U.S. Department of En- Eq. (16), ergy and USARL. ¼ g ðÞþω ðÞω ðÞ 0 ω sin tc V o cos tc 18 Appendix A or. A.1. Solution for constant force deceleration of a falling mass V ω tanðÞ¼ωt − o ð19Þ For the “constant force” system shown in Fig. A1a, the acceleration, c g velocity, and displacement history are given by Eq. (19) can be re-expressed as. F ðÞ¼ − c ð Þ ! at g 10 −1=2 M ω 2ω2 V o V o  sinðÞ¼ωtc − 1 þ ð20Þ g g2 Fc vtðÞ¼ g− t þ Vo ð11Þ M ! −1=2 2ω2  ðÞ¼ω þ V o ð Þ 1 F cos tc 1 21 xtðÞ¼ g− c t2 þ V t ð12Þ g2 2 M o At time t the mass is at position +H. From Eq. (11),wefind that the velocity reduces to a value of zero at c the critical time, tc,of g V H ¼ ðÞþ1− cosðÞωt o sinðÞωt ð22Þ ω2 c ω c − F 1 t ¼ V c −g ð13Þ c o M Substituting Eqs. (20) and (21) into this equation yields. 0 !1 ! −1=2 −1=2 g V 2ω2 V V ω V 2ω2 H ¼ @1− 1 þ o A− o o 1 þ o ð23Þ ω2 g2 ω g g2 For the displacement to be H at a time of tc, then combining Eqs. (10) and (12) reveals that Fc must be equal to the value given in Eq. (2). Substituting this value back into Eq. (10) gives the constant acceleration This equation can be solved for ω,giving. value of the mass given in Eq. (3). V 2 2g g g ω2 ¼ o þ ð24Þ H2 H Vo M Vo M or ! Fc F=Kx H x H x 2Mg V 2 K ¼ 1 þ o ð25Þ H 2gH

This equation provides the spring constant value necessary for the

optimal spring system to bring a body of mass M and initial velocity Vo to rest at a distance H. Substituting this value back into Eq. (15) gives (a) (b) the acceleration as a function of displacement as given in Eq. (5).

Fig. A1. Schematic of (a) constant force model and (b) optimal spring and invariant spring models A.3. Solution for invariant spring deceleration of a falling mass

A.2. Solution for optimal spring deceleration of a falling mass For the case of an invariant spring, the solution to the equations of motion are identical to the optimal spring case, but we now invert For the “optimal spring” system shown in Fig. A1b, the acceleration Eq. (25) to calculate the final drop distance H as a function of spring con- is dependent on the position x of the mass according to stant K as given in Eq. (7). The solution has two roots, with the larger root being equal to the downward deflection H (the second, negative Kx root with slightly smaller magnitude, is the value of the upward re- atðÞ¼g− ð14Þ fl M bound de ection). Acceleration as a function of distance is simply given by Eq. (8) using the fixed K value.

Solving this differential equation with the initial conditions that a = A.4. Head injury criterion (HIC) values g, v = Vo and x = 0 yields. For the constant force scenario, acceleration is constant and the HIC from Eq. (1) reduces to. atðÞ¼ g cosðÞωt −ωVo sinðÞωt ð15Þ 5=2 HIC ¼ ðÞΔt ac ð26Þ g vtðÞ¼ sinðÞþωt V o cosðÞωt ð16Þ ω where ac is given by Eq. (3). For the constant and invariant spring models, the acceleration is si- ðÞ¼ g ðÞþ− ðÞω V o ðÞω ðÞ nusoidal with respect to time, and reaches peak acceleration at a time xt 2 1 cos t sin t 17 ω ω of t = tc,wheretc is given by Eq. (19). We can therefore calculate a D.J. Spinelli et al. / Materials and Design 154 (2018) 153–169 169

HIC by substitution a(t) from Eq. (15) into Eq. (1), and integrating from [26] T.A. Schaedler, et al., Designing metallic microlattices for energy absorber applica- – − Δ Δ tions, Adv. Eng. Mater. 16 (3) (2014) 276 283. (tc t/2) to (tc + t/2), giving. [27] M.R. Begley, F.W. Zok, Optimal material properties for mitigating brain injury during head impact, J. Appl. Mech. 81 (3) (2014) 031014. = 1 g 5 2 [28] O. Nazarian, M. Begley, F. Zok, A concept for mitigating head injury under transla- ¼ ðÞΔ ðÞω ðÞωΔ = − ðÞω ðÞωΔ = – HIC t Δ 2 ω cos tc sin t 2 2V o sin tc sin t 2 tional blunt impact, Int. J. Crashworthiness 20 (5) (2015) 483 494. t [29] B.J. McEntire, P. Whitley, Blunt Impact Performance Characteristics of the Advanced ð27Þ and the Paratrooper and Infantry Personnel Armor System for Ground Troops Helmet, U.S. Army Aeromedical Research Laboratory, Fort Rucker, AL, 2005. which reduces to. [30] E.T. 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