proc imeche part p: j sports engineering and technology ......j sports engineering and technology...

Special Issue Article

Proc IMechE Part P:J Sports Engineering and Technology226(2) 77–85� IMechE 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1754337112441112pip.sagepub.com

Dynamic characterization of rigid foamused in finite element sports ballsimulations

Scott D Burbank and Lloyd V Smith

AbstractNumerical simulation of sport ball impacts is challenging due to the varied contact conditions involved and the difficultyin characterizing nonlinear materials at high strain rates. The following considers rigid polyurethane foam used in soft-balls. Past works have shown that load displacement curves from viscoelastic material models do not completely agreewith experiment, suggesting incorrect mechanisms of compressive deformation. Additionally, dynamic testing using apressure bar apparatus was unable to achieve strain rates low enough, and dynamic mechanical analysis was unable toachieve strain magnitudes large enough to represent play conditions.

A method was developed to impact polyurethane foam samples over a range of displacement rates and magnitudesrepresentative of play conditions. A characteristic stress–strain loading curve of polyurethane foam was produced andincorporated into a finite element model. Comparison of instrumented and numerical impact properties producedagreeable results. The foam material model was then applied to a simulated softball which impacted surfaces of varyinggeometry. Results showed that the foam material model better predicted softball deformation mechanisms than previouslinear–viscoelastic models both as a function of speed and surface geometry. Persistent discrepancies in rate dependenceindicate a lack of complete characterization, however.

KeywordsSoftball, polyurethane foam, finite element analysis, ball characterization

Date received: 17 August 2011; accepted: 8 February 2012

Introduction

Finite element analysis (FEA) has been widely imple-mented in sport ball simulation including golf, cricket,baseball, tennis, and softball. As technology and com-petition in the field of sports equipment increases,numerical modeling and simulation becomes a moreviable and economical option in research and develop-ment. An accurate ball impact model may be used, forinstance, to simulate sport equipment performance,protective equipment or personal injury. Compared tothe nearly elastic behavior of bats and clubs however,1

sports balls are difficult to model due to their time-dependent material response.2

Sports ball modeling in FEA extends to a range ofdisciplines and applications. Fuss simulated the impactforce of golf balls by implementing a logarithmicviscoelastic model.3 Tanaka et al. constructed a multi-layered golf ball as part of a simulated ball-club colli-sion.4 Using a combination of hyperelastic andviscoelastic materials, ball rebound velocities and spin

angles correlated closely with experiment. Allen et al.investigated the coefficient of friction of tennis ballswhen subjected to oblique impacts with both racketsand court surfaces.5 Finite element models were used todetermine the effects of sliding, rolling and the effectson horizontal rebound velocities. A linear viscoelasticmaterial was used to model a cricket ball by Singh andSmith as part of a bat–ball simulation resulting in goodexperimental agreement.6

Numerical modeling of softballs has received limitedattention compared to the baseball, which has been thefocus of previous works by Mustone and Sherwood,7

Shenoy,8 Axtell9 and others. Early softball models by

School of Mechanical and Materials Engineering, Washington State

University, USA

Corresponding author:

Lloyd V Smith, School of Mechanical and Materials Engineering,

Washington State University, Pullman, WA, USA.

Email: [email protected]

Sandmeyer10 and Duris11 used a three parameter powerlaw viscoelastic material model with a time dependentshear modulus defined by

G tð Þ=G‘ +(G0 � G‘)e�bt ð1Þ

where GN and G0 were the long term and instantaneousshear moduli, and b was the decay constant. Duris11

also investigated a general viscoelastic material modelbased on a six-term Prony series with a relaxation curvegenerated from dynamic mechanical analysis (DMA).The power law characterization lacked agreement withexperiment at speeds above 26.8m/s and when theimpact surface was altered. The Prony series model wasbased on small deformation DMA results, resulting inpoor correlation with experiment.

Bryson used a split Hopkinson pressure bar (SHPB)device to describe the behavior of polyurethane (PU)softball foam at high strain rates.12 Experimentalresults were compared with a corresponding linear vis-coelastic numerical model (equation (1)). The operatingprinciple of a pressure bar test requires the transmis-sion of an elastic wave through the sample. The highenergy dissipation inherent in the PU foam used insoftballs required relatively thin samples which resultedin minimum strain rates above 2700 s21 (which washigher than the 1000–2500 s21range typical of play).12

Bryson attributed lack of experimental and numericalagreement to deviant numerical and experimentalstrain rates.

Finite element models of a bat and softball were cre-ated by Faber as part of a simulated bat–ball colli-sion.13 Parameters for the power law viscoelasticmaterial model, though not derived or verified experi-mentally, were systematically adjusted to achieve vary-ing amounts of dissipated energy and stiffness. Thepeak impact force and dissipated energy were in goodagreement with experiment at the simulated speed(42.5m/s), but the stiffness of the ball model was persis-tently higher than experiment.

The following will discuss a new test method toobtain dynamic impact properties of PU foam used insoftballs at strain rates representative of play condi-tions. The development of a numerical material modeldescribing the impact response of softball foam is alsodiscussed. Experimental and numerical softball impactsare compared as a function of impact speed and surfacegeometry.

Ball tests

Softballs usually consist of a core made from closed-cell, rigid PU foam (98mm diameter) wrapped in aleather or synthetic cover. Because the cover is only1.5mm thick, its effect on impact behavior is relativelysmall. Softballs have an impact response dependent ontemperature, humidity, composition and rate.11,14 Toreduce the effect of environmental conditions, the tem-perature and humidity of the following tests were

regulated by an environmentally controlled chamberset to conditioning parameters defined in ASTM F2845(22.2 6 2�C, 50 6 10% RH).15 Balls were acclimatizedin the chamber for three weeks prior to testing. Toeliminate variation in ball weight, size, and materialproperties, one ball model was studied (DeMariniA9044).

Instrumented ball impacts were performed on adynamic stiffness test apparatus as shown in Figure 1.11

The test consisted of firing a ball from an air cannonagainst a fixed solid-steel half cylinder (57mm dia-meter) and a flat plate. Load cells (PCB, model208C03) were placed between the impact surface and amassive support where measurements were taken at asample rate of 100 kHz. While inside the cannon barrel,the ball was cradled by a sabot which controlled speedand launched the ball with no spin. An arrester platecaptured the sabot at the end of the cannon, while theball continued forward. Light screens, placed betweenthe cannon and impact surface, measured the inbound,vi, and rebounding, vr, ball speeds. Cannon alignmentwith respect to the impact surface was adjusted tomaintain a rebound path within 610� of the inboundpath.

The coefficient of restitution (COR) from flat plateimpacts was calculated as the ratio of inbound andrebounding ball speed as

COR=vrvi

ð2Þ

Equation (2) was also used for impacts with a cylindri-cal surface, where the coefficient of restitution wasdenoted as CCOR. Impulse is the change in ballmomentum and was calculated from

I=

ðt

0

f tð Þdt ð3Þ

where f(t) was the measured load at time t.Displacement of the ball center of mass (COM) wascalculated using

d tð Þ=ðt

0

vi �ðt

0

f(t)

mdt

0@

1Adt ð4Þ

where m was the ball mass.

Figure 1. Instrumentation diagram of the ball dynamic stiffnesstest apparatus.

78 Proc IMechE Part P: J Sports Engineering and Technology 226(2)

Duris previously found that softballs impacted atthe same location repeatedly at 40.2m/s and allowed torecover between impacts maintained their liveliness upto 100 hits.11 This suggests that damage to the softballfoam structure caused by impact at the speeds used inthis study is negligible.

To determine an average ball response, 16 new soft-balls were impacted on a cylindrical surface at 26.8m/s(60mile/h). A single ball was selected which had a load-ing curve, weight and CCOR that was representative ofthe group of balls. The ball was impacted six times at26.8, 42.5 and 53.6m/s (60, 95 and 120mile/h) on bothflat and cylindrical surfaces.

The experimental results are summarized in Table 1,and agree with trends and values of previous works.11,13

The COR was observed to linearly decrease withincreasing ball speed, while impulse, peak force andpeak displacement increased with speed. For impactson the cylindrical surface the peak displacement wasgreater than the flat surface while the COR and peakforce were lower. A flat surface has a larger contactarea, which lowers both the ball deformation anddissipated energy and results in a higher peak force.Figure 2 illustrates the effect of speed and surface geo-metry on ball impact response.

Foam sample tests

Most of the previous works involving numerical soft-ball models have used a viscoelastic material model.This study relies on a structural foam material modelwhich, among other things, is better able to describe thesmall Poisson effect, characteristic of softballs and base-balls, than a viscoelastic material model. In response tostress, linear–viscoelastic materials exhibit a timedependent response that is linearly proportional to theapplied stress.16 Foams exhibit a nonlinear responsethat consists of three distinct regions of behavior17. Theinitial elastic region resists compression with the inher-ent stiffness of the foam structure. At a critical strain,the foam structure begins to collapse, causing a plateauregion in the stress–strain response. As the cells becomefully compressed, a densification region begins and thematerial stiffens, behaving much like the matrix mate-rial. A single curve representing the compressive load-ing response of the PU foam used in softballs wasneeded to numerically model the foam material. Thus,a method was developed to obtain a master compres-sive loading curve.

Apparatus

The apparatus shown in Figure 3 was constructed toimpact foam samples, and can be used to test otherpolymers or similarly compliant materials. The deviceused a small, horizontally-mounted air cannon thatachieved a larger range in impact speed than is possiblewith standard drop towers.18 When the cannon wasfired, a pneumatic valve released compressed air froman accumulator tank to a 610mm long barrel. An

Figure 2. Load vs time measurements of softball impacts atspeeds of: (I) 26.8 m/s, (II) 42.5 m/s, and (III) 53.6 m/s (60, 95 and120 mile/h). Comparison of flat and cylindrical impact surfaceshighlight the differences in impact response caused by thesurface geometry.

Table 1. Impact properties of DeMarini A9044 softball. Values are the averaged result of six impacts 6 SD.

Impact speed (m/s) Surface COR Impulse (N s) Peak force (kN) Peak disp. (mm)

26.9 6 0.11 Cylinder 0.419 6 0.004 7.75 6 0.09 11.5 6 0.10 10.8 6 0.3442.3 6 0.26 Cylinder 0.376 6 0.003 11.9 6 0.08 18.4 6 0.79 16.0 6 0.1953.6 6 0.24 Cylinder 0.350 6 0.002 14.6 6 0.12 23.6 6 1.08 19.8 6 0.3226.8 6 0.16 Flat 0.442 6 0.003 8.00 6 0.02 13.3 6 0.23 9.60 6 0.1942.3 6 0.12 Flat 0.391 6 0.003 12.1 6 0.03 21.4 6 0.27 14.4 6 0.3353.7 6 0.31 Flat 0.357 6 0.003 14.9 6 0.07 28.3 6 0.40 17.5 6 0.83

Figure 3. Instrumentation diagram of the apparatus used tomeasure dynamic impact response of softball foam samples.

Burbank and Smith 79

18.3 g aluminum striker bar, 13mm in diameter and51mm long, was projected out of the barrel and speedwas measured as it passed in front of infrared sensorsspaced 38mm apart. The striker bar then impacted a10.7mm diameter and 7.6mm long cylindrical PU sam-ple machined from a softball core. The foam samplewas mounted to a stationary piezoelectric load cell(PCB, model 208C03) by a thin coating of syntheticlubricating grease. Force was measured at a sample rateof 1.0MHz.

Foam sample displacement, x(t), was assumed to beequal to the striker bar COM displacement found usingequation (4), where m and vi were the striker bar massand speed, respectively. The average strain in the foamsample, e(t), was found from the sample thickness, l0,and displacement as

e tð Þ= x(t)

l0ð5Þ

Stress, s(t), was found from the load signal, f(t), andsample cross sectional area, A, as

s(t)=f(t)

Að6Þ

To verify velocity and displacement measurements, aPhantom V711 high speed camera was used to recordtwenty impacts ranging in speed between 10 and20.3m/s at 60,000 frames per second. Motion trackingsoftware was used to measure both initial velocity ofthe striker bar and foam sample displacement. Barvelocity and displacement from the video and speedmonitors agreed on average within 1.7% (6 1.7%) and1.5% (6 1.7%), respectively (values in parenthesis rep-resent standard deviation).

Ball impacts have local regions of high stress nearthe contact region, while the foam test samples have anearly uniform stress state. Thus, structural damagedue to repeated impacts would affect the response of afoam sample more than a softball. To minimize theinfluence of potential damage from cell wall collapse,each foam sample was tested once.

Results

After being environmentally acclimatized, 30 foam sam-ples were impacted at speeds ranging from 9 to 20.1m/swhich resulted in peak strain rates between 1000 and2700 s21. The peak strain rate occurred as contact wasinitiated and exponentially decreased at higher speeds(. 17m/s) while decreasing nearly linearly to zero asthe bar slowed down (\ 14m/s). The stress–straincurves from each impact, along with the master com-pressive loading curve are shown in Figure 4. The threephases of polymeric foam compression were evident.Foam collapse occurred at approximately 0.1e, andcontinued until the densification region at approxi-mately 0.5e. This is consistent with the compressivebehavior of PU foam published by Chen et al.14

An oscillation in the load signal, consistent withelastic wave propagation, may be observed in Figure 4.The oscillation in the stress–strain response is a resultof measuring load at the fixed end of the sample andderiving strain from the displacement at the free end ofthe sample. Fortunately, the magnitude of the stressoscillations is relatively small, so that the average mate-rial stress–strain response could be readily discerned.Elastic wave propagation was also observed in thenumerical simulations as will be discussed in the nextsection.

Many of the stress–strain curves depart from themaster curve as their peak strain is approached and thestrain rate approaches zero. This is consistent with vis-coelastic effects where creep tends to increase as strainrate decreases.

Finite element models

Many sport ball models (including baseball, tennis andgolf balls) are developed phenomenologically, by tailor-ing models to describe measured ball response. The fol-lowing describes an FEA foam model with anexperimentally derived material loading response and aphenomenologically developed unloading response.Models developed in this way have the potential ofdescribing and comparing structural response prior toproduct fabrication.

Foam sample impact

A collision of the aluminum striker bar with a PU foamsample was modeled using the dynamic finite elementcode LS-DYNA (Version 971, LSTC, Livermore, CA).Because the foam cell size (;0.2mm) was significantlysmaller than the specimen diameter, the foam wasassumed to be isotropic and homogeneous. The foamsample, shaded black in Figure 5, was modeled with2772 linear, solid elements with two symmetry planes.Large strain magnitudes in the model necessitated theuse of fully integrated elements to minimize hourglas-sing. The elastic aluminum striker rod, shaded white in

Figure 4. Stress–strain compressive response of softball PUfoam including the master loading curve used to controlcompression in the FEA material model.


Figure 5, was similarly modeled with 2709 elements.The rod length was shortened to 2.5mm to reduce thenumber of elements, while density was correspondinglyincreased to achieve the correct mass. The striker barvelocity was directed normal to the impact surface andthe specimen face opposite of the impact was con-strained in-plane. The contact type was ‘‘surface to sur-face’’ where friction was neglected, as the impact hadno obliquity. The simulation duration was 2ms.

A standard foam material model was selected tocharacterize the PU foam (Mat #57 low density foam)as used by Sambamoorthy and Halder.19 The mastercurve, developed from impacting foam samples,became the input parameter controlling the compres-sive loading response of the material. The primarydeformation mode in the foam sample was compres-sion which, due to the Poisson effect, caused trans-verse tensile strain. As the Poisson ratio of softballs isrelatively low (0.1),11 the tensile strains were small.Young’s modulus (E) was set as the elastic slope ofthe master curve (Figure 4) and governed tensioncaused by the Poisson effect.20

The shape of the unloading curve was controlled bytwo non-dimensional parameters: a hysteretic unload-ing (HU) factor and a ‘‘shape’’ factor. HU can range invalue between zero and one. Low values of HU shiftedthe unloading path downward. The lowest value of HU(maximum hysteresis or energy loss) did not sufficientlyaccount for energy loss, so the shape factor wasadjusted to increase energy loss by further shifting theunloading curve down. The shape factor can range

from zero (inactive) to 500 (above which hysteresis didnot significantly change).

Viscous effects were controlled by a ‘‘damp’’ factor.Increasing damp values tended to lower displacementanalogous to adding a material damper. High dampvalues, above the suggested range of 0.05–0.5, pro-duced unstable deformations.20 The non-dimensionalvalues of HU, damp and shape factors were iterativelyadjusted to align the load displacement curve withexperiment. Table 2 summarizes the foam materialmodel parameters.

The model was run with initial striker bar speeds of8.9, 13.4, and 18.5m/s to capture the low, mid-rangeand upper strain rates typical of play. The load–displacement response of the numerical model is com-pared with experiment in Figure 6 and shows generallygood agreement. A notable departure between themodel and experiment occurs near the peak strain ofeach test. The model shows no evidence of strain soft-ening in this region, which suggests an insensitivity totime dependence. Oscillations in the FEA results atsmall strain magnitudes indicate the presence of elasticwaves propagating through the sample.

The impulse, peak displacement and peak force fromthe model agreed with experiment on average within5%, 6% and 4%, respectively. The COR from thesesimulations was found from the ratio of the outgoingto incoming striker bar speed. The COR from the FEAmodel is compared with experiment in Figure 7 andshows the predicted COR changing less as a function ofspeed than was observed experimentally. Additionalmaterial parameters not listed in Table 2, but includedin the MAT #57 material model and designed to con-trol rate dependence (Young’s relaxation modulus anddecay constants), were unable to improve the depen-dence of COR on impact speed.

Softball model

In the following, the softball and cover were modeledas a single isotropic homogeneous sphere as done else-where.11–13 The softball was modeled using 7168 linear,fully integrated, solid elements and the foam materialmodel described above. The softball, impacting acylindrical surface of 4864 nodes, is shown in Figure 8.Impacts were also simulated against a flat impact sur-face of 500 nodes. Both surfaces were elastically charac-terized. Quarter symmetry was applied as well ascontact and boundary conditions as was done with thefoam sample model described above. Mesh density was

Figure 5. FEA model of an aluminum striker bar (white)colliding into a foam sample (black) using quarter symmetry.

Table 2. FEA material model parameters for LS-DYNA’s Mat#57 low density foam: Density (r), Young’s modulus (E), hystereticunloading factor (HU), shape factor, and damp factor.

Model r (kg/m3)/(lb s2/in4) E (MPa)/(kpsi) HU Shape Damp

Foam sample 398/3.90E-05 138/20 0.01 30 0.75Softball 398/3.90E-05 138/20 0.01 9 1.6


determined to be sufficient, as doubling the number ofelements altered impulse, displacement and peakimpact force by only 2%.

Mesh alignment between the ball and impact surfaceas well as contact surface penetration were investigated.It was found that as the ball and contact surface meshaligned, contact penetration was minimized. These vari-ables had a minimal effect on impact properties com-pared to that caused by material type and surfacegeometry.

Applying the material parameters from the foamsample model to the softball model resulted in pooragreement with experiment. Cylindrical surface impactsproduced an impulse, loading stiffness and peak displa-cement within 13%, 18% and 15% of experiment,respectively, while the CCOR was 36% lower thanexperiment. The damp factor was increased and theshape factor was decreased to improve correlation withthe loading stiffness and CCOR respectively. Theimproved impulse, loading stiffness, peak displacement,

and CCOR of the modified ball model were, on aver-age, within 3%, 8%, 9% and 4% of experiment, respec-tively. The parameters of both the foam sample andsoftball models are summarized in Table 2.

Figures 9a and 9b compare experimental and numer-ical softball response on a cylindrical and flat surface at26.8, 42.5, and 53.6m/s (60, 95, and 120mile/h). Forthe flat surface results, the peak force, COR, andimpulse maintained excellent agreement with experi-ments across all speeds (2%, 4% and 5% average,respectively), while peak displacement was slightlyhigher (9% average) with increasing deviation at lowerspeeds.

Model comparison

Previous work by Duris, Faber and Bryson producedfinite element models of softballs in LS-DYNA usingthe standard viscoelastic material model.11–13 Theauthors each published material parameters for theirmodels as summarized in Table 3. Each material modelwas applied to the conditions described previously, andthe results were compared to the current foam model.

The cylindrical surface load–displacement respon-ses of the foam model and viscoelastic models arecompared in Figure 10(a). Both peak displacementand peak force of the foam model more closely agreewith experiment than was found from the viscoelasticmodels. The foam ball model more accuratelydescribed the load–displacement curves than the vis-coelastic models during both the loading and unload-ing phases. The generally oval-shaped curve,characteristic of the viscoelastic models, indicates adifferent mechanism of deformation than occurs withfoam. A similar comparison, but of a flat surfaceimpact is shown in Figure 10(b). The foam materialmodel again shows the best agreement with experi-ment. The comparison is less favorable for the viscoe-lastic models, showing a greater departure fromexperiment than was observed in the cylindrical sur-face comparisons. The COR and impulse of the foammaterial model also tended to agree better withexperiment than the viscoelastic models.

The COR from the foam and viscoelastic FEA mod-els on flat and cylindrical surfaces is compared withexperiment in Figure 11(a). Since impacts on a flat sur-face have less ball deformation and less energy loss, theCOR should be greater than the CCOR. The foammaterial model was the only model that correctly cap-tured the effect of surface geometry on COR (viscoelas-tic models show CCOR . COR). The magnitude ofCOR and CCOR from the foam model was also closerto experiment than with the viscoelastic models.

The dependence of COR and CCOR on impactspeed was observed to be linear in Table 1 and in previ-ous work.7 The slope of numerical COR and CCOR (asa function impact speed) is compared with experimentin Figure 11(b). The viscoelastic models tended tomatch the experimental COR and CCOR rate

Figure 6. Comparison of experimental and numerical load–displacement curves of foam sample impacts at the low,medium, and upper strain rates typical of play conditions: (I)8.9 m/s, (II) 13.4 m/s, and (III) 18.5 m/s.

Figure 7. Comparison of experimental and numerical COR ratedependence for the foam sample model.


dependence more closely than the foam material model.While the foam softball model showed less rate depen-dence than the viscoelastic models, the rate dependenceof the foam softball model was higher than the rate

dependence of the foam sample model (Figure 7). Thus,while the foam model improves the stress–strain com-parison with experiment, its insensitivity to time depen-dence inhibits its ability to describe rate dependence.

Figure 8. Strain distribution at maximum deflection for FEA foam softball impacting a cylindrical surface at: (I) 26.8 m/s, (II) 42.5 m/s, and (III) 53.6 m/s (60, 95, and 120 mile/h). Shades corresponding to strains of \ 0.1e (white), \ 0.5e (grey), and . 0.5e (black)represent the three regions of compressive foam response as shown in Figure 5.

b

Figure 9. Comparison of experimental and FEA foam softball force–displacement response during impacts against (a) cylindricalsurface and (b) flat surface: (I) 26.8 m/s, (II) 42.5 m/s, and (III) 53.6 m/s (60, 95, and 120 mile/h).

Table 3. FEA material parameters from previous viscoelastic softball models: density (r), bulk modulus (K), instantaneous shearmodulus (G0), long term shear modulus (GN) and decay constant (b).

Model r (kg/m3)/(lb s2/in4) K (Pa)/(lb/in2) G0 (Pa)/(lb/in2) GN (Pa)/(lb/in2) b (Hz)

Duris 398/3.90E-05 6.89E10/1E7 6.89E7/1E4 4.82E6/700 68,000Faber 398/3.90E-05 6.89E8/1E5 1.38E8/20,000 6.89E6/1000 68,000Bryson 398/3.90E-05 6.89E9/1E6 2.41E7/3500 2.75E6/400 40,000


Interestingly, both the foam and viscoelastic materialmodels correctly described the higher rate dependenceof the flat surface impacts over the cylindrical impactsobserved experimentally.

While the foam material model considered in thiswork did not sufficiently characterize rate dependence,other material models hold promise for future work.LS-DYNA’s Mat #83 allows input of multiple com-pressive stress–strain curves as a function of strain rate.Another low density viscous foam model uses onestress–strain loading curve and incorporates a relaxa-tion curve for rate effects (LS-DYNA Mat #73). Animproved description of time dependence will be key tofuture improvements of solid ball impacts using foammaterial models.

Conclusion

The foregoing has been concerned with improvingnumerical predictions of solid sport ball impacts. A testmethod has been proposed to characterize ball material

response at strain rates and magnitudes that are repre-sentative of play conditions. Results of this methodshowed a material response that is consistent with struc-tural foam. Incorporating a foam material model into anumerical softball model significantly improved corre-lation with measured ball impact response compared toprevious viscoelastic models. Optimal agreement wasonly achieved, however, when the foam material para-meters were tailored to the measured ball response.While it is unfortunate that coupon testing alone wasnot sufficient to create the ball material properties, theresulting ball model was able to describe the load–displacement response and the effect of a changingimpact surface better than any existing numericalmodel. A persistent shortcoming of foam and viscoelas-tic material models concerns the effect of changingimpact speeds on COR. Both types of numerical modelsare less sensitive to changing speed than is observedexperimentally. Further material model improvements,such as a viscoelastic foam material model with the abil-ity to account for rate effects in loading and unloading,are needed.

(a) (b)

Figure 10. Comparison of load–displacement response of linear-viscoelastic and foam material models on (a) cylindrical surfaceand (b) flat surface at 42.5 m/s (95 mile/h). The viscoelastic models were from (1) Bryson, (2) Faber, and (3) Duris.

(a) (b)

Figure 11. A comparison of (a) COR on flat and cylindrical surfaces at 42.5 m/s (95 mile/h) and (b) the dependence of the COR andCCOR on impact speed from experiment and finite element models. The viscoelastic models were from (1) Bryson, (2) Faber, and (3)Duris.


Obtaining an appropriate description of the ballremains the primary impediment to modeling sportsball impacts with high energy dissipation. While prog-ress in this area continues to be made, the models havenot evolved to the point where they can be used effec-tively in product development or design. The currentresults suggest that future model improvements willdepend on extensive material characterization which, inturn, will limit the application of these methods.

Funding

This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.

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