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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: (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.398 J.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, H9280(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) H11005H20885tH11002H11009E(tH11002t)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) H11005H20885tH11002H11009E(tH11002t)E0E0e(t)dtH11005H20885tH11002H11009x(tH11002t)s0(t)dt (2)where x(t) = E(t)/E0, with E0being the instantaneousmodulus E0= E(0), is the normalized relaxation modu-lus and s0( = eE0) is the time derivative of the instan-taneous stress.Considering the force (F) and displacement (H9004)asdirectly measurable quantities, Bolzmanns superpo-sition principle can be reformulated in terms of F(H9004) andH9004(t). Assuming negligible contributions of pre-historyloading for t H11021 0, Eq. (2) is rewritten asF(t) H11005H20885t0g(tH11002t)dF0(H9004)dH9004H9004dt (3)which can be reformulated using a mathematical conver-sion into:F(t) H11005 F0H9004(t) H11001H20885t0F0H9004(tH11002t)g(t)dt (4)where g(t) is the dimensionless relaxation modulus orthe relaxation function, and F0(H9004) 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, H9004, is noncontinuous. Conse-quently, the time-delivery of the instantaneous forceresponse, dF0H9004(t)/dt =dF0(H9004)dH9004H9004also 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(H9004). 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, H90040, was applied tothe fingerpad at t = 0, and the displacement was thenkept constant for t H11022 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) H11005F(t)F0(H90040), F0(H90040) H11005 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) H11005 1H11002H20888Ni H11005 1gi(1H11002eH11002t/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) H11005 0 anddF0(H9004)dH9004dH9004H110220 (7)which implies that F0(H9004) is an increasing function of H9004.The instantaneous force response can be consideredto follow a power law of the form:F0(H9004) H11005 AH20873H9004H9004H20874b(8)where A (N) and b (H11002) are positive material/structuralconstants; H9004(=1.00 mm) is the reference displacement,a characteristic displacement of fingerpad; and the dis-placement, H9004, 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.Auniversal micromechanical testing machine (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 adult400 J.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(H9004).The instantaneous force/displacement relations, F0(H9004),should, theoretically, be determined at an extremely highloading speed at which the viscous deformation is negli-gible. In the present study, the compression tests wereperform
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