Dynamic interaction between a fingerpad and a flat surface experiments and analysis.pdf
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Medical Engineering & Physics 25 (2003) 397406/locate/medengphyShort communicationDynamic interaction between a fingerpad and a flat surface:experiments and analysisJ.Z. Wua, R.G. Donga, W.P. Smutza, S. RakhejabaNational Institute for Occupational Safety & Health (NIOSH), 1095 Willowdale Road, Morgantown, WV 26505, USAbConcordia University, Montreal, CanadaReceived 28 June 2002; received in revised form 24 January 2003; accepted 7 February 2003AbstractMany neural and vascular diseases in hands and fingers have been related to the degenerative responses of local neural andvascular systems in fingers to excessive dynamic loading. Since fingerpads serve as a coupling element between the hand and theobjects, the investigation of the dynamic coupling between fingertip and subjects could provide important information for theunderstanding of the pathomechanics of these neural and vascular diseases. In the present study, the nonlinear and time-dependentforce responses of fingertips during dynamic contact have been investigated experimentally and theoretically. Four subjects (2 maleand 2 female) with an average age of 24 years participated in the study. The index fingers of right and left hands of each subjectwere compressed using a flat platen via a micro testing machine. A physical model was proposed to simulate the nonlinear andtime-dependent force responses of fingertips during dynamic contact. Using a force relaxation test and a fast loading test at constantloading speed, the material/structural parameters underlying the proposed physical model could be identified. The predicted rate-dependent force/displacement curves and time-histories of force responses of fingertips were compared with those measured in thecorresponding experiments. Our results suggest that the force responses of fingertips during the dynamic contacts are nonlinear andtime-dependent. The physical model was verified to characterize the nonlinear, rate-dependent force-displacement behaviors, forcerelaxations, and time-histories of force responses of fingertips during dynamic contact.Published by Elsevier Science Ltd on behalf of IPEM.Keywords: Nonlinear; Viscoelastic; Force/deflection responses; Fingertip; Compression tests1. IntroductionExtended exposure of the human fingertips to repeatedloading has been associated with many vascular, sensori-neural, and musculoskeletal disorders, such as carpaltunnel syndrome, handarm vibration syndrome, andflexor tenosynovitis 1,2. Since fingerpads serve as acoupling element between the hand and objects, investi-gation of the dynamic coupling between fingertip andobjects could provide important information for theunderstanding of the pathomechanics of these diseases.From a biomechanical point-of-view, a fingertip hasa complex anatomical structure, composed of skin layers(epidermis and dermis), subcutaneous tissue, bone, andCorresponding author. Tel.: +1-304-285-5832; fax: +1-304-285-6265.E-mail address: jwu (J.Z. Wu).1350-4533/03/$30.00. Published by Elsevier Science Ltd on behalf of IPEM.doi:10.1016/S1350-4533(03)00035-3nail 3,4. The material properties of the subcutaneousand skin tissues are known to be nonlinear and time-dependent 57; consequently, the response of fingertipsto mechanical loading are expected to be nonlinear andtime-dependent. Rempel et al. 8 measured the forceresponse of fingertip during keyboard strokes, and foundthat the peak forces on the fingertip ranged from 1.65.3 N. Serina et al. 9 studied the force-deformationbehavior of the fingerpad during key tapping at fre-quencies of 0.5, 1, 2, and 3 Hz and at different incli-nations; they found that force/deformation responses ofthe fingerpad are highly nonlinear and time-dependent.Pawluk and Howe 10,11 further investigated thedynamic contact pressure distributions between a fing-erpad and a flat surface. The time-dependent forceresponse and force relaxation behavior of the fingertipsunder physiological loading conditions, however, havenot been explored systematically in a quantitative man-ner in these previous studies.398J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406From the modelling point of view, two types of mod-els of fingertips have been proposed in the literature:structural models and physical models. The structuralmodels consider simplified anatomical structures of thefingertip and can predict the stress/strain in the tissues.These models have been applied to predict the staticforce-deflection characteristics 12 and fingertip surfacedeflection under a static, line load 13. These structuralfingertip models are static in nature and cannot beapplied to cases involving dynamic loading. Pawluk andHowe 10,11 used the physical (lumped element) modelto simulate dynamic contact of fingerpads with a flat sur-face. However, the force response and force relaxationbehavior of fingertips under physiologic loading con-ditions (e.g. low loading rates) were not investigated.Although the physical model did not consider anatomicalstructures of the fingertip and could not simulate thestress/strain properties within the soft tissues of finger-tips, it is mathematically simple and can be readily usedto estimate time-dependent force responses for manyindustrial and ergonomic applications.The studies of the dynamic force response and forcerelaxation of finger-tips during grasping may provideimportant information to the understanding of the patho-mechanics of work-related musculoskeletal disorders,and may help industrial ergonomic designers in theirefforts to prevent musculoskeletal injuries. The purposesof the present study are: (a) to analyze, experimentallyand theoretically, the time-dependent force responsesand viscous relaxation of human fingertips duringdynamic contacts with a flat surface, and (b) to developa simple physical model which describes the nonlinearand time-dependent force response of fingertips.2. Physical modelIn classical viscoelastic theory, the constitutive equa-tions are typically expressed in terms of stress, s(t), andstrain,?(t), using Bolzmanns principle of superposition,as shown by Fung 14. The dependence of the instan-taneous stress on the deformation history of a linearlyviscoelastic material is expressed by a hereditary inte-gral:s(t) ?t?E(t?t)e (t)dt(1)where E(t) is a time-dependent relaxation modulus thatcharacterizes the materials time-dependent response.For many practical problems, it is convenient to writethe hereditary integral (1) in terms of stress, s:s(t) ?t?E(t?t)E0E0e (t)dt ?t?x(t?t)s 0(t)dt(2)where x(t) = E(t)/E0, with E0being the instantaneousmodulus E0= E(0), is the normalized relaxation modu-lus and s 0( = e E0) is the time derivative of the instan-taneous stress.Considering the force (F) and displacement (?) asdirectly measurable quantities, Bolzmanns superpo-sition principle can be reformulated in terms of F(?) and?(t). Assuming negligible contributions of pre-historyloading for t ? 0, Eq. (2) is rewritten asF(t) ?t0g(t?t)dF0(?)d?dt(3)which can be reformulated using a mathematical conver-sion into:F(t) ? F0?(t) ?t0F0?(t?t)g (t)dt(4)where g(t) is the dimensionless relaxation modulus orthe relaxation function, and F0(?) represents the instan-taneous force/displacement relationship. Eq. (4) impliesthat the force response of the fingertip can be decom-posed into two components. The first term describes theinstantaneous force response, while the latter termcharacterizes the delayed force response that takes intoaccount the effects of loading histories on the currentdeformation state.For many practical problems (e.g. the force responsesand viscous relaxations of a fingertip subjected to a sud-den step displacement), the time-delivery of the pre-scribed displacement, ?, is noncontinuous. Conse-quently, the time-delivery of the instantaneous forceresponse, dF0?(t)/dt =dF0(?)d?also becomes noncon-tinuous. The numerical singularities associated with dif-ferentiations of the instantaneous force responses in Eq.(3) can be overcome by computing the differentiationsof the relaxation modulus, g (t) in Eq. (4), which is con-tinuous in the time domain. Therefore, in the presentstudy, Eq. (4) is used to compute the force response ofthe fingertips.The application of the proposed model requires thedetermination of two material/structural functionals, g(t)and F0(?). A series of experiments were performed onhuman fingertips to identify these parameters, which arefurther described in the following section. In order todetermine g(t), a step displacement, ?0, was applied tothe fingerpad at t = 0, and the displacement was thenkept constant for t ? 0. The time-history of the responseforce of fingerpad, F(t), was measured. Using Eq. (3),the relaxation function, g(t), is then derived from themeasured force response, in the following manner:g(t) ?F(t)F0(?0), F0(?0) ? F(0)(5)In the present study, a Prony series expansion for thedimensionless relaxation modulus 15 was employed to399J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406characterize the normalized relaxation modulus, suchthat:g(t) ? 1?Ni ? 1gi(1?e?t/ti)(6)where giand ti(i = 1,2,3,.) are the material/structuralconstants of the fingertip; and N is a sufficiently largenumber to achieve a good fit of Eq. (6) to the experi-mental data.The mechanical stability restrictions require that theinstantaneous force response function satisfiesF0(0) ? 0 anddF0(?)d?d? ? 0(7)which implies that F0(?) is an increasing function of ?.The instantaneous force response can be consideredto follow a power law of the form:F0(?) ? A?b(8)where A (N) and b (?) are positive material/structuralconstants; ?(=1.00 mm) is the reference displacement,a characteristic displacement of fingerpad; and the dis-placement, ?, is in mm. This formulation satisfies themechanical stability restrictions, as described in Eq. (7).3. Experimental methods3.1. Experimental setupAn experimental setup was designed to study themechanical response of a fingertip to dynamic loading.The setup comprises a 25 mm 25 mm flat steel platenand a finger hold, as schematically shown in Fig. 1. Auniversalmicromechanicaltestingmachine(Type:Mach-I, Biosyntech, Montreal, Canada) was used to gen-Fig. 1.Experimental setup for the finger compression tests. (a): Side view. The subjects finger was rested on a plastic finger rest, which keptthe angle between the dorsum of the distal part of the index finger and the table top to be 20 degrees for all tests. (b): Cross section view. Thenail was fixed onto the finger rest using a thin double-sided adhesive tape. The tests were conducted using a displacement-controlled protocol.erate the platen motion using a displacement-controlledprotocol. The testing machine was equipped with a dis-placement sensor with a resolution of 0.5 m and a 9.8N (1 kg) load cell with a resolution of 0.49 mN (500mg). During the experiments, each subject was seatedassuming a relaxed posture with forearm supported onan arm rest. Subjects were advised to place their indexfingers in the finger hold that was designed to produceapproximately a 20 angle of the dorsum of the distalpart with respect to the horizontal table top (or the com-pression platen).The steel compression platen was covered by asmooth plexiglass sheet (3 mm thick) in order to minim-ize the effects of temperature difference between theplaten and the fingertip on the mechanical response ofthe fingerpad. In order to keep contact between the fingerand the support surface during the test, a double-sidedadhesive tape was applied on the nail of the subjectsindex finger before placing it on the finger rest. Sincethe thickness of the tape (0.10 mm) is small comparedto the dimensions of the finger, the error associated withthe deformation of the double-sided tape (i.e. the vari-ation in tape thickness under compression) was negli-gible. We have calibrated the system to evaluate theerror that may be introduced by the adhesive tape. Arubber block was compressed on the testing machine,once with the adhesive tape between the rubber blockand compression platen and once without the adhesivetape. No measurable difference was found between theforce responses obtained from these two measurements.3.2. Test proceduresThe experiments were performed by displacing theplaten against the subjects fingertip by a specified mag-nitude using the displacement-controlled protocol. Theresulting force response and platen displacement wererecorded at a sampling frequency of 33 Hz. Four adult400J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406subjects (2 males and 2 females) participated in thestudy. The subjects had an average age of 24 years (2130 years). The experiments were conducted on the indexfingers of the right and left hand of each subject. Eachsubject gave written consent to the tests that had beenapproved by the NIOSH Human Subject Review Board.Each subject was permitted a few practice runs beforedata collection. The average width and height of the dis-tal phalanx of the subject fingers were 16.5 1.5 mmand 12.0 2.0 mm, respectively.Two series of experiments involving different magni-tudes of fingertip compression were conducted. In testseries A, the fingerpad was compressed to a displace-ment of 2.00 mm using six different rates of loading(0.1, 0.2, 0.4, 0.9, 1.5, and 5.7 mm/s), as demonstratedin Fig. 2a. The maximum displacement of the fingertip(2.00 mm) was held constant for approximately 30 s,allowing the response forces of the fingertip to stabilize.The platen displacement was then reduced to 1.00 mmat a speed of 1.00 mm/s, and the displacement (1.00 mm)was held constant for another 30 s.Fig. 2.The prescribed displacement histories of the compressionplaten. (a) Test series A. (b) Test series B.Test series B involved a controlled displacement to3.00 mm (Fig. 2b). The fingertip was loaded to the peakdeformation (3.00 mm) using five different loadingspeeds (0.3, 0.5, 0.9, 2.3, and 4.0 mm/s), and the dis-placement was held constant at 3.00 mm for approxi-mately 30 s. The platen displacement was then reducedfrom 3.00 mm to 2.00 mm at a speed of 1.00 mm/s, andheld at 2.00 mm for another 30 s.Force relaxation measurements were conducted priorto test series A and B in order to calibrate the proposeddimensionless relaxation modulus, g(t), described in Eqs.(5) and (6). Fingertips were subjected to step displace-ments of 2.00 and 3.00 mm (at a loading speed of 5.00mm/s), corresponding to test series A and B, respect-ively, that were kept constant for 30 s to permit the forceresponse of the fingertip to approach steady-state values.All subjects underwent the two force relaxation testsand two series of tests (A and B), with a break of 20 sbetween two successive runs within each series of tests,allowing for a recovery of the viscous deformation ofthe soft tissues. Each subject was permitted a 15 minrest period between the two series of tests to enable thesubjects to recover from any musculoskeletal fatigue infingers and hands. The experimental study involved eighttests, i.e. the index fingers of right and left hands of foursubjects. Two of the tests were considered unsuccessfulbecause the subjects were unable to keep the index fingerstable during compression; the corresponding data wereexcluded from subsequent analysis. In the test resultsreported below, the mean values are calculated using theremaining six tests. The force relaxation tests were con-ducted only on the right index finger of each subject.4. ResultsFig. 3a and b shows the time histories of the nor-malized force responses obtained from the force relax-ation tests corresponding to test series A and B, respect-ively, together with the fitting curves, g(t), as describedin Eq. (6). Four sets of test data acquired from four sub-jects for each series (A and B) were used to obtain thedimensionless relaxation modulus functions. The resultsshow relatively small variations among the data acquiredfrom the four subjects. The averaged values of the testdata for all subjects were used for the curve fitting. Thestandard lease square method was used and a criteria of2% error tolerance was applied for the curve fitting. Theresults further revealed that satisfactory fitting can beachieved using two terms (N = 2) for the Prony seriesexpansion Eq. (6). The parameters defining g(t) weredetermined from the measured data, as: gi= (i = 1,2)0.2359 and 0.1541, and ti= (i = 1,2) 0.1182 and 5.45s, for series A (Fig. 3a); gi= (i = 1,2) 0.3866 and 0.1560,and ti= (i = 1,2) 0.1802 and 5.45 s, for series B (Fig.3b).401J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406Fig. 3.The dimensionless relaxation modulus of the proposed modelg(t) compared to the time histories of the normalized contact forceresponses of fingertips obtained under a step displacement during fourtrials. (a) Test series A. (b) Test series B.Fig. 4 depicts comparisons between the rate-depen-dent force/displacement data obtained from experimentsand those predicted using the proposed physical modelfor test series A and B. These results were obtained forthe loading parts of test series A and B, and were theaveraged values for all six available test data. The rate-dependent force/displacement curves were predictedusing Eq. (4), which was based on two previouslydetermined material/structural parameters underlyingfunctions, g(t) and F0(?).The instantaneous force/displacement relations, F0(?),should, theoretically, be determined at an extremely highloading speed at which the viscous deformation is negli-gible. In the present study, the compression tests wereperformed using a conventional micro testing machinewith limited loading and data acquisition speeds. There-fore, F0(?) was obtained using an iterative scheme: (a)An initial instantaneous force/ displacement relationship,F0(?) is assumed by multiplying the force/displacementcurves corresponding to the tests at the highest loadingspeed in each test series (i.e. v = 1.5 and 4.0 mm/s fortest series A and B, respectively) by a factor of p (p?1.2 for the first trial). It is to be noted that the highestloading speed in series A (5.7 mm/s) could not be usedbecause of insufficient data points; the test data acquiredat the next highest speed (1.5 mm/s) were used. (b) Theforce/displacement curves corresponding to the highestloading speed were then derived using the force relax-ation parameters (giand ti) obtained previously (Fig. 3)together with the initial estimate of F0(?). (c) The mag-nification factor p was modified until an optimal fit ofthe model predictions with the experimental data wereachieved. The optimized magnification factors, p, were1.29 and 1.20, respectively, for test series A and B. (d)Finally, the instantaneous force/displacement relationEq. (8) was fitted to the optimized force/displacementrelationships, and the constants A and b were found tobe 0.2368 (N) and 2.0696 for series A; and 0.1727 (N)and 2.8694 for series B, respectively.Thetime-dependentforce/displacementcurvesobtained from the experimental data, together with theoptimizedinstantaneousforce/displacementcurves(labelled vmax, in the figures), are depicted in Figs. 4aand c, corresponding to test series A and B, respectively.The predicted rate-dependent force/displacement curvesfor test series A and B are shown in Figs. 4b and d,respectively. The results suggest that fingertips stiffnessand force magnitudes increase with increasing loadingspeeds. The results show good agreements between theexperimental data and the model predictions in the entirerange of loading speeds and deflections considered inthis study.The time-histories of the averaged values of the meas-ured force responses for all subjects in test series A andB are depicted in Figs. 5a and c, respectively. The corre-sponding model predictions for the test series A and Bare shown in Figs. 5b and d, respectively. Our resultsshow that the predicted time-histories of the forceresponses are consistent with those obtained experimen-tally.Fig. 6 illustrates the mean, minimum, and maximumvalues of the force responses as a function of time, corre-sponding to three different loading speeds in test seriesA and B. The mean values of the test data were calcu-lated from six independent measurements; these test dataare scattered within the region enclosed by the upper andlower bound curves. The predicted time-histories of theforce responses corresponding to these test conditionsare also shown in the figures. It is seen that the predictedforce responses agree well with the mean experimentaldata over the entire range of loading speeds and time his-tories.The instantaneous stiffness of the fingerpad used inthe present study were compared with those measured402J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406Fig. 4.The rate-dependent force-displacement curves of fingertips: experimental data vs. model predictions. (a) Series Ameasurements; (b):Series Amodel predictions; (c) Series Bmeasurements; (d) Series Bmodel predictions. The experimental data were the averaged values forall six available tests data.Fig. 5.The force responses as a function of time for fingertips subjected to the prescribed displacement histories as shown in Fig. 2. (a) SeriesAmeasurements; (b): Series Amodel predictions; (c) Series Bmeasurements; (d) Series Bmodel predictions. The experimental data werethe averaged values for all six available tests data.403J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406Fig. 6.The comparison of the force responses predicted using the proposed model to those obtained from the experiments. (a),(c), and (e): SeriesA. (b),(d), and (f): Series B. The mean values of the experimental data were calculated from six independent measurements which are scatteredwithin the upper and lower bounds.by Pawluk and Howe 10, as shown in Fig. 7. Pawlukand Howes data were obtained using a similar test pro-cedure as proposed in this study; however, the angle ofthe dorsum of the distal part with respect to the contactplaten was not fixed in their tests, and the subject wasallowed to take a comfortable posture during themeasurements. Tests #1#4 from Pawluk and Howes10 data were obtained using four different subjectsunder the same test procedure. The comparison showsgood agreement in view of the trends, while the magni-tudes differ, especially for high forces. The differencesin magnitudes are attributed to the differences in subjectsand test conditions (e.g. ambient temperature and contactangle between fingerpad and platen).No significant difference between the force responsebehaviors between right and left index fingers wasobserved in the test. However, due to the small samplingnumber we cannot get a conclusive result from thisobservation.The repeatability of the experiments has been tested404J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406Fig. 7.The instantaneous stiffness of fingertips obtained in thepresent study compared to those reported by Pawluk and Howe 11.Tests #1#4 from Pawluk and Howes data were obtained using fourdifferent subjects under the same test procedure.on one subject by repeating the same loading procedurefor three times. The subject was allowed to have a breakof ten minutes after each run. The measured force valuesvaried within a range of 15% (results not shown).5. Discussion and conclusionHandarm vibration syndrome (HAVS) has beenassociated with the degenerations of the local neural andvascularsystemsinthefingersduringextensiveexposure to vibration 1618. These hypotheses havebeen supported by numerous diagnostic tests of HAVSperformed on the fingers or fingertips 19,20. Severalinvestigators 21,22 suggested that vibration energyabsorption (VEA) may be a significant factor leading tothe vibration injuries. These hypotheses obtained supportfrom the epidemiological investigations 23 which indi-cated that HAVS might be correlated to VEA. Somestudies have been conducted to characterize the totalVEA in the handarm system 24,25. A fundamentaldeficiency of these previous methods is that they couldnot determine the location of the energy dissipation inhand and arm. The local energy dissipation in fingertipsmight be an essential factor leading to the developmentof the vibration-induced disorders, since the most severeHAVS symptoms have been observed to be localized onfingers or fingertips. The study of the local VEA in thefingertips may, therefore, provide essential informationto elucidate the mechanisms of HAVS. It is technicallyextremely difficult to experimentally separate the localVEA in the fingertips from the total energy absorptionin the hand-arm system at low vibration frequency(?100 Hz). The most critical step to determine the localVEA is to obtain the dynamic force response of finger-tips. In the present study, a physical model was proposedto simulate the nonlinear and time-dependent forceresponse of fingertips during dynamic contact with a flatobject. The proposed approach can be used to calculateforce response and estimate the energy dissipation infingertips during vibration and dynamic loading. Ourapproach could, therefore, be used to provide insight intothe pathomechanics of occupational-related musculoske-letal disorders, and help in the design of hand tools andoffice equipment to prevent occupational injuries.Thepredictedrate-dependentforce/displacementcurves and time-histories of force responses under differ-ent loading speeds agreed well with the correspondingexperimental measurements. The results thus suggestthatcouplingofthenonlinearforce/displacementrelations and time-dependent force responses of finger-tips under dynamic contact conditions can be adequatelycharacterized using the proposed model. Both experi-mental results and the model predictions reveal generalfeatures of the force response of fingertips when con-tacting a flat surface: (a) the force/displacement relation-ship of fingertips is nonlinearthe stiffness of thefingertips under compression increases dramatically withincreasing deformation and follows a power law; (b) theforce/displacement curves are rate- or time-dependenta higher loading speed yields higher fingerpad stiffness;(c) the loading or deformation histories influence thetransient force responses of the fingertips; and (d) theresponse forces of the fingertips, which are compressedand then held constant for a relatively long period, tendto reach a steady-state value that depends on theimposed displacement.The present study represents an extension of the inves-tigations conducted by Pawluk and Howe 10,11. In ourproposed model, the relaxation modulus is expandedusing a general Prony series, which describes the forcerelaxation curves accurately with two terms. The analy-sis of the force response using Bolzmanns superpositionprinciple directly, however, would pose difficultiesbecause of the numerical singularities associated withthe discontinuous nature of the force time-history (Eq.(3). Alternatively, we propose another form to derivethe force responses (Eq. (4). The solution of Eq. (4)requires knowledge of the relaxation modulus, g(t), andthe instantaneous force-displacement relationship, F0(?),which can be determined using two sets of experiments:(a) a force relaxation test, and (b) a fast loading test ata constant loading speed (v ? 1.0 mm/s). These testscan be conducted using commercially available testingmachines.Theoretically, the instantaneous force/displacementrelationship should be obtained by loading the fingertipat a very high speed, so that the viscous effects duringthe loading process can be neglected. However, in realtests, it is technically difficult to avoid the measurementerrors produced by the inertial effects of the loadingequipment and soft tissues during high speed loading.Consequently, it would be a technical challenge to obtainrepeatable data of the instantaneous force/displacementrelationships using “direct” measurements. Using the405J.Z. Wu et al. / Medical Engineering & Physics 25 (2003) 397406proposed “indirect”, iterative approach, reliable instan-taneous force/displacement relationships can be obtainedby performing tests at “slow” loading speeds.In order to restrict the movement of fingers during thetests, the subjects fingers had to be kept in place.Because the lateral restriction on deformation of softtissues of the fingerpads will affect the force responsecharacteristics, previous researchers have usually gluedthe nails on the testing table using commercial super glue10,26. In the present experiments, a thin double-sidedtape was used to fix the nail on the testing table. Whilethe fixation strength of the doubled-sided tape is rela-tively poor compared to that using super glue, thismethod was considered to be more convenient, as itallowed subjects to relax between successive tests.The time histories of the measured force responseshows force fluctuations when the displacement was keptconstant, and the posture of the subjects was not changed(Figs. 3, 5a and c and 6). Although the subjects main-tained a fixed posture during the tests, involuntary mus-cle activation and pulsing blood flow in the fingers arebelieved to cause these force fluctuations. The use of alow-pass filter would eliminate these “high” frequencyfluctuations. In our physical model, the anatomicalmicro-structures were not considered, therefore, theseforce fluctuations would not be simulated.The proposed model simulates the force response ofthe soft tissues for dynamic loading, while the inertialeffects of the tissues were neglected. The proposedmodel is quasi-static in nature, and the force responsewas considered to depend on the deformation historiesof the soft tissues exclusively. Therefore, the proposedmodel is not suitable for predicting impact forces, whereinertial effects would become important.The model predictions agree well with the experi-mental results. They indicate that the force-displacementcurves at high loading speeds (e.g. ?1.0 mm/s, as in Fig.4) differ distinctively, while those for low loading speeds(e.g. ?0.5mm/s) are similar and tend to converge to asteady-state. Experimental and theoretical results suggestthat the peak resultant contact force at the fingertipincreases by 50% to 75% at high (?1.0 mm/s) comparedto slow (?0.2 mm/s) speeds of loading.The measured time-histories of the force responseexhibit high transient forces near the end of the displace-ment ramps (Fig. 5). These “singularities” in the time-histories of the force response may be caused by thesudden transition of the loading speed to zero, therebyinducing large decelerates. Such transient effects weresmaller for the low compared to the high loading speeds.Consequently, the measured rate-dependent force-dis-placement curves are presented in the deformationranges of 0.01.9 mm (Fig. 4a,b) and 0.02.9 mm (Fig.4c,d) for test series A and B, respectively.Many factors may affect the stiffness and forceresponse of the soft tissues in the fingerpads. Preliminaryevidence indicates that the ambient temperature andfinger posture during the test may have great effects onthe force response. In order to reduce temperatureeffects, we conducted all tests at a strictly controlledroom temperature (approximately 22 C), and all sub-jects were climatized while staying in the room for atleast 30 min prior to testing. All components of the testequipment that contact with the hand and fingers of thesubjects were covered with plexiglass. All compressiontests were performed using a contact angle of 20between the fingertip and the flat contact surface, so thatthe results can be compared across tests and subjects.The fingertip is complex in anatomical structure andcontains biological materials of totally different charac-teristics. One way to model such complicated problemsis using finite element, such as 27. However, in manycases bioengineers are mainly interested in the globalforce responses of fingertips, not the detailed stress/straindistributions in tissue level. The proposed model wasintended for these practical applications. One-dimen-sional quasi-linear viscoelastic models have been pre-viously used for other soft tissues, such as cartilage 28,soft tissues 29, and skeletal muscles 30. Since themajor interest for such phenomenological models is thetotal force responses of the system, the biological sub-structures of the fingertip are not considered, while theeffects of the different material properties in fingerpadare included in the global force/deformation responsemeasures. The limitation of such phenomenologicalmodels, in general, is that the local stress/strain distri-butions in the soft tissues cannot be predicted.In summary, the force response of fingertips duringdynamic contact with a flat surface was investigatedexperimentally and theoretically in the present study.Our results suggest that the force response of fingertipsduring dynamic contacts is nonlinear and time-depen-dent. The physical model is able to characterize the non-linear, rate-dependent force-displacement, force relax-ation, and force time-histories of fingertips duringdynamic contacts.References1 Beck-Foehn M. Occupationally-induced carpal tunnel syndrome.Nervenarzt 1992;63(8):46772.2 Kulick R. Carpal tunnel syndrome. Orthop Clin North Am1996;27(2):345.3 Gray H. Anatomy of the human body., 29th ed Philadelphia: Leaand Febider, 1973.4 Fawcett D. Text book of histology., 11th ed Philadelphia: W.B.Saunders Co, 1986.5 Wan AW. Biaxial tension test of human skin in vivo.
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