Evaluation of contact stiffness between solid-solid interfaces using dual-frequency ultrasound

Transcription

1 Proceedngs of 0 th Internatonal Congress on Acoustcs, ICA August 00, Sydney, Australa Evaluaton of contact stffness between sold-sold nterfaces usng dual-frequency ultrasound Xasheng Guo (), Dong Zhang () and Xufen Gong () () Insttute of Acoustcs, Key Laboratory of Modern Acoustcs, Mnstry of Educaton, Nanng Unversty, Nanng, P.R.Chna PACS: d, y, 8.70.Cv ABSRAC Acoustcal methods are playng an mportant role for nondestructve evaluaton (NDE) of adhesve-bonded compostes and components n ndustral applcatons. A dual-frequency ultrasonc technque s proposed for the quanttatve evaluaton of contact strength between pressed sold surfaces. An ultrasonc exctaton conssts of two prmary frequency components s appled perpendcular to the nterface, and the transmtted wave s examned. heoretcal study s based on a perturbaton analyss of contact acoustc nonlnearty (CAN) model, predctng the generaton of dfference and sum frequency waves, together wth the second harmoncs. Nonlnear parameters are defned to descrbe the nonlnearty generaton effcences. Experments are performed for three types of nterfaces,. e. the nterfaces of two alumnum alloy blocks wth and wthout couplant and two glass blocks. he dfference frequency wave component has bgger generaton effcency than other nonlnear components, whch offers an advantage of hgh SNR and good detecton capablty of contact stffness (nterfacal stffness). For each nterface, the frst and second-order nterfacal stffness are measured wth contact pressure ncreasng from near zero to about 0.8 MPa wth the ad of a laser nterferometer. Fnally, numercal smulatons are also carred out, and a consstency s found between measurements and calculatons. he dual-frequency ultrasound sent to the nterface generates at least four second-order nonlnear components, whch enrches the CAN technque for nterface qualty examnatons. Both measured and smulated results ndcate an ncrease of nterfacal stffness and decrease of nonlnear parameters wth growng contact pressure. Moreover, measured results show that couplant between nterfaces nfluences the contact stffness evaluatons n an enhanced manner, whle the contact pressure determned by measured nterfacal stffness values are underestmated due to the couplant. he man problem comes from the contact between the transducersample nterfaces, whch brngs extra nonlnearty to the detected sgnals and affect the accuracy of measurng. INRODUCION In the nondestructve evaluaton (NDE) of adhesve-bonded compostes and components, ultrasonc methods have attracted much nterest. In ndustral applcatons such as electroncs, aerospace, and shpbuldng etc., ultrasonc evaluatons allow the detecton of potental fracture rsks, such as delamnatons and debondngs. he state of the art of contact evaluatons ncludes pulse-echo, resonant ultrasound spectroscopy and acoustc magng technques []. However, quanttatve evaluatons have not been acheved yet. For cracks or defects, a hgh level nonlnearty mght be nduced by ther contact behavours, whch s known as the contact acoustc nonlnearty (CAN) and has promsng potental applcatons for dscrmnatng flaws and nhomogenetes n samples. heoretcal [] and expermental [3] studes conducted by Rchardson et al revealed that CAN s caused by passage of a longtudnal acoustc wave across the nterface, and the harmonc ampltude s a functon of the pressure appled normal to the nterface. However, the hypothetcal perfect smooth nterface, whch s the base of ther works, does not exst n real applcatons. In realty, contact surfaces have certan roughness, and are accounted for by several models. Representtve works carred out by Pecorar [4] and Gusev et al [5-7] ndcate that nonlnearty s generated at the nterface. However, the pressure-dependent characterstcs of generated nonlnearty requre further nvestgatons. o descrbe the topology and ultrasonc response of rough nterfaces, varous statstcal models are developed. he frst concern s the pressure dependent contact stffness (nterfacal stffness) between sold nterfaces, whch s lnked to the roughness topology by Rudenko et al [8], Drnkwater et al [9] and then Km et al [0, ]. he nteracton of nterfaces could be descrbed usng a sprng boundary or other models such as a relaxator, and the nterfacal stffness changes senstvely wth the contact pressure []. In the theoretcal [] and expermental [3] studes by Bwa et al, the nterfacal stffness was obtaned from measured ultrasonc reflecton/transmsson coeffcents together wth nonlnearty coeffcents. However, t s mpossble to drectly compare the theoretcal and expermental results n a quanttatve manner, as there s a lack of absolute calbraton of the measured wave ampltude [3]. hs study ams at quanttatve evaluaton of the nterfacal stffness of contactng nterfaces by usng a dual-frequency ultrasonc technque. A theoretcal analyss usng a perturbaton method ndcates that dfference- and sumfrequency waves, together wth second harmoncs, are generated from contactng nterfaces rradated by dualfrequency ultrasound. heoretcal analyss s testfed by ICA 00

2 3-7 August 00, Sydney, Australa Proceedngs of 0th Internatonal Congress on Acoustcs, ICA 00 numercal smulatons and experments performed on dfferent samples. he frst- and second-order nterfacal stffness values are obtaned from measured spectral ampltudes calbrated wth a laser nterferometer. he pressure dependences of defned nonlnearty parameters are also examned. A A% A% (8a) ( 0) ( + ) A () ( 3) KA% %, ( A ) 3 KA % % (8b) A 4κ AAK % % K, ( 4) A 4κ AAK % % K (8c) ( 5) κ 4 A A %, ( 6) A κ A% (8d) n whch the followng symbols are ntroduced, K % K c K % K %, (0a) ρω, κ +,,,3,4 A % A K %, (0b) + 4,, Fgure. Geometrcal sketch of the consdered model. MEHODS heoretcal Analyss A geometrcal sketch of the consdered model s llustrated n Fgure. As has been consdered by other researchers [,, ], two elastc solds of dentcal materal are pressed together by an external pressure p 0. he contactng surfaces are consdered as flat wth certan roughness, whle ther average heght n x-drecton are defned as the reference planes at X- and X+, whch leads to a gap dstance of h X+ - X-. Actually, the gap dstance s a functon of the dynamc pressure p and has an ntal equlbrum dstance h 0 at the contact pressure p 0. Assumng that u I, u R and u are the dsplacement felds of the ncdent, reflected and transmtted waves, X(t)[u(X -, t)+u(x +, t)]/ s ntroduced as the center of gravty of the contactng nterfaces, whle Y(t) u(x +, t)- u(x -, t) s the dynamc gap dstance. he vbraton of the gap could then be descrbed by, I X& cu ct (4a) ( ) Y& cu ct + KY + KY (4b) I he ncdent dual-frequency ultrasound s composed of two monochromatc snusodal components wth zero ntal phase angles, whle ω and ω are the two prmary angular frequences (assumng that ω >ω ), A and A are the correspondng dsplacement ampltudes. A perturbaton method s therefore appled for analytcal nonlnear solutons of equaton (4b). he gap dstance Y s composed of lnear component Y and second-order perturbaton Y, determned by, he phase angles of each wave component are not concerned, and are omtted here. he analytcal results ndcate the second harmoncs, sum- and dfference-frequency waves and DC components are contaned n the transmtted waves. he second harmonc ampltudes are proportonal to the square of correspondng prmary wave ampltudes; whle the ampltudes of dfference- and sum-frequency waves are lnear functons of both prmary wave ampltudes. o quanttatvely measure the generaton effcences of sumand dfference-frequency waves, followng nonlnear parameters are defnded, ( sum) ( 3) () A A A 4κ (a) ( dff ) ( 4) () 3 A A A 4κ (b) 4 whch are correlated to the contact pressure p 0 and ndependent on the ampltudes of the ncdent ultrasonc waves A and A. In the prevous studes by other researchers [, ], nonlnearty parameters were defned as the rato of the second harmonc ampltude to the squared prmary wave ampltude, () ( 5) () A A 4κ (a) ( 6) A A 4κ (b) whch reflect the generaton effcences of second harmoncs and are also ndependent of the ampltude of exctaton. K Y& + Y Aωsnωt+ Aωsnωt (6a) Y& + Y Y (6b) he transmtted waves are hence obtaned, ( 0) () () ( τ) + cos ωτ +Φ u c A A 6 ( ) ( ) A sn ωτ 3 + +Φ n whch τ t-x/c, ω 3 ω +ω, ω 4 ω -ω, ω 5 ω and ω 6 ω. he ampltudes A () are expressed as, (7) Expermental setup Fgure.. Expermental setup. In experments, two peces of sample blocks were pressed together by a mechancal system as s shown n Fgure. he emttng and recevng transducers were pressed towards the samples usng Vaselne as couplant medum, whle rubber ICA 00

3 3-7 August 00, Sydney, Australa Proceedngs of 0th Internatonal Congress on Acoustcs, ICA 00 pads were placed at ther backs to mantan constant pressure at the transducer-sample nterfaces. wo knds of sample materals were used. One was alumnum alloy, wth ts densty and sound velocty measured to be kg/m 3 and m/s; the other was glass, whose densty and sound velocty were measured to be 57.kg/m 3 and 596.3m/s, respectvely. For the alumnum alloy samples, the contactng nterface wth and wthout Vaselne as couplant were examned. All samples used had a rectangular contact surface of dmensons 30 mm 30 mm; the length along the wave propagaton drecton was 5 mm. Before the measurements, the contactng surfaces of alumnum alloy samples were polshed wth No. 000 sand-paper, all samples were subected to three loadng/unloadng cycles to flatten the contact aspertes, whch ensured that pressure-dependent hysteress was elmnated [9, ]. A functon generator (Aglent 3350A, USA) was used to produce repeatng dual-frequency electrc pulses (500 and 800 khz) at a repetton frequency of 0. khz and the pulse length was 80 µs. he sgnal was wndowed wth a Hannng functon, and amplfed by a broadband 55 db RF power amplfer (ENI 50A, USA), then used to drve the emtter. he emtter was a broadband pezoelectrc transducer wth a nomnal frequency of 500 khz (Panametrcs V43-SB). wo pezoelectrc transducers were chosen as recevers: recever # (Panametrcs V43-SB) was used to detect wave components that have frequences less than MHz and recever # (Panametrcs V40-SB, nomnal frequency MHz) for hgher frequency wave packets. he detected sgnals were sent to a dgtal osclloscope (Aglent 54830B, USA) at a samplng frequency of 50 MHz. A personal computer was used to control the waveform generator and the osclloscope. Pror to all other measurements, the vbraton ampltudes produced by the emtter at the angular frequences of ω (,, 4, 6) were measured wth a laser nterferometer (Polytec OFV-505/5000, Germany), when the exctaton voltage (peak-to-peak value) was 48V. Consequently, measured voltage values produced by the recevers n experments were calbrated to dsplacement ampltudes. For each par of samples, total of measurements were performed at dfferent pressure levels as contact pressure ncreases from 0.03 MPa to 0.8 Mpa. In each measurement, the exctaton voltage appled to the emtter (peak-to-peak value) was kept to be 48V. he detected transmtted sgnals were averaged over 3 traces to enhance the sgnal nose rato, and then analyzed n the frequency doman usng a Fast- Fourer-ransform program, fnally converted to absolute dsplacement ampltudes. Fgure. 3. Frequency spectrum of transmtted waves for uncoupled alumnum nterfaces. Numercal smulatons he ordnary dfferental equaton 4(b) was drectly solved wth fourth-order Runge-Kutta algorthm to provd a comparson wth the analytcal solutons and measured results. In the smulatons, the pressure dependent nterfacal stffness values were determned by the power-law model [], K Cp (3a) m 0 m K mc p (3b) 0 n whch C and m are postve constants correlated to the materal and topology of the nterfaces, such as the radus of aspertes and ther heght dstrbutons [9, ]. In ths study, C and m used n numercal smulatons were determned from the measured pressure dependence of lnear nterfacal stffness K ftted by equaton (3a); and K was therefore obtaned usng equaton (3b). Fgure. 4 Measured lnear nterfacal stffness K. RESULS AND DISCOSSIONS Generaton of nonlnearty As s descrbed n equaton (7), nonlneartes generate from the nteracton between the dual-frequency ultrasound and the contactng nterface. Fgure 3 shows the spectrum of the measured transmtted ultrasonc sgnal through the alumnum alloy nterface under the contact pressure of 660 kpa. Result of numercal smulaton s also provded n ths fgure. Obvously, both smulaton and experment ndcate the generaton of dfference- and sum-frequency waves, second harmoncs and the zero-frequency DC component. he dfference-frequency wave s found to have bgger generaton effcency than other nonlnear components n both cases, whch offers an advantage of hgh SNR and good detecton capablty of contact stffness. Noted that the extra spectral components at other frequences, such as 00 khz,. MHz and.5 MHz, etc, are not predcted n the theoretcal analyss but observed n both smulatons and experments. he reason s that the used governng equatons only contan the frst- and second-order perturbatons of the dynamc gap dstance Y. Consderaton of hgher order perturbaton equatons s unnecessary, as the ampltudes of the extra waves are relatvely low. he nterfacal stffness he rato of ether two nonlnearty parameters defned n equatons (0) and () s a functon of lnear nterfacal stffness K and ndependent on K, thus could be an effectve ICA 00 3

4 3-7 August 00, Sydney, Australa Proceedngs of 0th Internatonal Congress on Acoustcs, ICA 00 evaluaton of the lnear nterfacal stffness. In ths study, (dff) / () s adopted for the calculatons. For all three tested nterfaces, the pressure dependences of K were presented n Fgure 4. In all the cases, the frst-order nterfacal stffness ncreases monotonously wth the contact pressure, whch ndcates a decrease of nonlnearty n the transmtted ultrasonc waves. Furthermore, the alumnum alloy nterfaces coupled by Vaselne show the strongest contact and the uncoupled alumnum alloys s the weakest. he measured pressure dependences of K were ftted by equaton (3a), and the nterface constants C and m were thus determned as: a) C and m0.54 for the glass nterface, b) C and m0.6 for the uncoupled alumnum alloy nterface and c) C and m0.68 for alumnum alloy nterface wth couplant. he calculated K values were then used for the determnaton of second-order nterfacal stffness K va equaton (b) and (b), whch was selected as the average of that determned by both equatons. he obtaned contact pressure dependences of K are llustrated n Fgure 4. he case of a coupled alumnum alloy nterface has the largest second-order stffness whle the same uncoupled nterface has the smallest. Besdes, K shows no obvous ncreasng or decreasng wth the growng contact pressure. become possble, whch makes the dual-frequency technque more valuable n the NDE of apparently closed cracks or mperfect bonds. In measurements, errors come from the contactng at transducer-sample nterfaces and mult-reflectons nsde the sample blocks, whch generates extra nonlneartes. Snce the calculaton regards all the nonlneartes n detected sgnals to be generated from the sample-sample nterface, the contact between nvestgated surfaces are mstaken as softer than they really are. Accurate evaluaton of the contact strength requres an approach of all three nvolved nterfaces. CONCLUSION Inspecton of the qualty of nterfaces by examnaton of the contact strength s of great sgnfcance n modern NDE technology. A dual-frequency ultrasonc technque has been studed n ths work for the quanttatve predcton of lnear and nonlnear nterfacal stffness. he nteracton between rradated dual-frequency ultrasound and the nterfaces generates not only harmonc waves, but also dfference- and sum-frequency components, whch has been testfed by both theoretcal and expermental studes. Accordng to the measurements from three types of nterfaces, t s found that generated nonlneartes decrease wth the growng contact strength, whle couplant between nterfaces enhances the strength of contact. he man problem comes from the contact between the transducer-sample nterfaces, whch brngs extra nonlnearty to the detected sgnals and affect the accuracy of measurng. For a better nterpretaton and evaluaton of the contactng behavor of sold nterfaces, the nonlnearty caused by the contact-type measurement should be taken nto account. More accurate measurements requre a combned study of the reflecton/transmsson technque and quanttatve methods, such as the dual-frequency evaluaton. AKNOWLEDGEMENS Fgure. 5 Measured second-order nterfacal stffness. Other dscussons In ths study, the nterfacal stffness values are derved from measured nonlnearty parameters, whch are obtaned after a calbraton usng the optcal nterferometer. In tradtonal studes usng monochromatc exctatons, lnear/nonlnear reflecton/transmsson coeffcents were utlzed to acqure K and K. In one of the expermental studes by Bwa et al [3], the measured pressure dependences of K were ftted by a polynomal expresson and then used to determne K. o some extent, the use of reflecton/transmsson coeffcents s more reasonable n the determnaton of nterfacal stffness, as t covers the nfluence of measurng system more or less. However, t demands the measurements of both reflected and transmtted waves, as n the representatve works by Pecorar et al [4] and Bwa et al [3]. For the choce of ultrasonc nonlnearty parameters, there are two optons: a) second harmonc ampltude to fundamental ampltude and b) second harmonc ampltude to squared fundamental ampltude. he latter s preferred n our vew, because t s ndependent on the ampltude of exctaton, as s predcted n equatons (), () and the work of other researchers [, ]. he latter choce also keeps formal consstence wth the two addtonal nonlnearty parameters defned for dfference- and sum-frequency waves. Wth the adopted nonlnearty parameters, quanttatve comparson between theoretcal and correspondng expermental results hs work s supported by the Chna Postdoctoral Scence Foundaton (Contract No ), Natonal Natural Scence Foundaton of Chna (Grant No ), and Natonal Basc Research Program 973 (Grant No. 00CB73600) from Mnstry of Scence and echnology of Chna. REFERENCES E. Maeva, I. Severna, and S. Bondarenko et al, Acoustcal methods for the nvestgaton of adhesvely bonded structures: A revew, Can. J. Phys., (004) J. M. Rchardson, Harmonc Generaton at an Unbonded Interface I. Planar Interface between Sem-Infnte Elastc Meda, Int. J. Engng. Sc., 7, (979) 3 O. Buck, W. L. Morrs, and J. M. Rchardson, Acoustc harmonc generaton at unbounded nterfaces and fatgue cracks, Appl. Phys. Lett (978) 4 C. Pecorar, Adheson and nonlnear scatterng by rough surfaces n contact: Beyond the phenomenology of the Presach-Mayergoyz framework, J. Acoust. Soc. Am., (004) 5 V. Gusev, B. Castagnede, and A. Moussatov, Hysteress n response of nonlnear bstable nterface to contnuously varyng acoustc loadng, Ultrasoncs 4, (003). 6 Gusev, Frequency dependence of dynamc hysteress n the nteracton of acoustc wave wth nterface, J. Acoust. Soc. Am. 5, (004). 7 A. Moussatov, V. Gusev, and Bernard Castagnède, Selfnduced hysteress for nonlnear acoustc waves n cracked materal. Phys. Rev. Lett (003) 4 ICA 00

5 3-7 August 00, Sydney, Australa Proceedngs of 0th Internatonal Congress on Acoustcs, ICA 00 8 O. V. Rudenko and C. A. Vu, Nonlnear acoustc propertes of a rough surface contact and acoustodagnostcs of a rough heght dstrbuton, Acoust. Phys (994). 9 B. Drnkwater, R. Dwyer-Joyce, and P. Cawley, A study of the nteracton between ultrasound and a partally contactng sold-sold nterface, Proc. R. Soc. London, Ser. A 45, (996). 0 J. Y. Km, A. Baltazar, J. W. Hu, and S. I. Rokhln, Hysteretc lnear and nonlnear acoustc responses from pressed nterfaces, Int. J. Solds. Struct. 43, (006). J. Y. Km, and J. S. Lee, A mcromechancal model for nonlnear acoustc propertes of nterfaces between solds, J. Appl. Phys., 0, (007) S. Bwa, S. Nakama, and N. Ohno, On the acoustc nonlnearty of sold-sold contact wth pressuredependent nterface stffness, J. Appl. Mech-. ASME 7, (004). 3 S. Bwa, S. Hrawa, and E. Matsumoto, Expermental and theoretcal study of harmonc generaton at contactng nterface, Ultrasoncs 44, e39-e3 (006). 4 C. Pecorar, Nonlnear nteracton of plane ultrasonc waves wth an nterface between rough surfaces n contact, J. Acoust. Soc. Am. 3, (003) ICA 00 5