Finite-size effects on the quasistatic displacement pulse in a solid specimen with quadratic nonlinearity

Transcription

1 Finite-size effets on the quasistati displaement pulse in a solid speimen with quadrati nonlinearity Peter B. Nagy a) Shool of Aerospae Systems, University of Cininnati, Cininnati, Ohio 451 Jianmin Qu Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois 68 Laurene J. Jaobs College of Engineering, Georgia Institute of Tehnology, Atlanta, Georgia 333 (Reeived 6 Marh 13; revised 16 June 13; aepted 18 July 13) There is an unresolved debate in the sientifi ommunity about the shape of the quasistati displaement pulse produed by nonlinear aousti wave propagation in an elasti solid with quadrati nonlinearity. Early analytial and experimental studies suggested that the quasistati pulse exhibits a right-triangular shape with the peak displaement of the leading edge being proportional to the length of the tone burst. In ontrast, more reent theoretial, analytial, numerial, and experimental studies suggested that the quasistati displaement pulse has a flat-top shape where the peak displaement is proportional to the propagation distane. This study presents rigorous mathematial analyses and numerial simulations of the quasistati displaement pulse. In the ase of semi-infinite solids, it is onfirmed that the time-domain shape of the quasistati pulse generated by a longitudinal plane wave is not a right-angle triangle. In the ase of finite-size solids, the finite axial dimension of the speimen annot simply be modeled with a linear refletion oeffiient that neglets the nonlinear interation between the ombined inident and refleted fields. More profoundly, the quasistati pulse generated by a transduer of finite aperture suffers more severe divergene than both the fundamental and seond order harmoni pulses generated by the same transduer. VC 13 Aoustial Soiety of Ameria. [ PACS number(s): 43.5.D, 43.5.Ed, 43.5.Qp, 43.5.Ba [ANN] Pages: I. INTRODUCTION A harmoni aousti tone burst propagating through an elasti solid with quadrati nonlinearity produes both a ollinear burst of the well-known seond harmoni, and an often negleted quasistati displaement that is assoiated with the aousti radiation-indued eigenstrain. There is an unresolved debate in the sientifi ommunity about the time-domain shape of this quasistati displaement pulse. Early analytial and experimental studies suggested that the pulse has a right-triangular shape with the peak displaement of the leading edge being proportional to the length of the tone burst. Based on Cantrell s theoretial analysis of the aousti-radiation stress in solids, 1 Yost and Cantrell in a follow-up paper predited that the quasistati displaement pulse u (t) produed by a longitudinal plane wave propagating through a semi-infinite elasti solid with quadrati nonlinearity must be of right-triangular shape as illustrated in Fig. 1. The leading edge of the quasistati displaement pulse deteted at arrives with an L/ propagation delay after it was generated at t ¼ by an infinitely large harmoni displaement radiation soure loated at the surfae of the half-spae (x ¼ ). The length of the quasistati displaement pulse is equal to the temporal length s of the exitation pulse. Cruially, Yost and Cantrell also predited that the slope of a) Author to whom orrespondene should be addressed. Eletroni mail: peter.nagy@u.edu the right-triangular shape is a measure of the nonlinearity ¼ bx U ; (1) 8 where U is the displaement amplitude of the harmoni aousti tone burst, x is the angular frequeny, and is the longitudinal sound veloity in the solid. Sine they also predited that the trailing edge of the quasistati displaement pulse vanishes, this means that the leading edge of the quasistati displaement pulse must be independent of the propagation distane, and proportional to the duration of the pulse u 1 ¼ bx U s : () 8 As pointed out in Ref. 3, this is diffiult to reonile, sine it suggests that information an be arried by an elasti disturbane faster than the speed of sound; an observer stationed at annot instantaneously determine that the tone burst was turned off at time t ¼ s at a transmitter loated at x ¼, sine this information is not available at the point of observation () before t ¼ L/ þ s. Yost and Cantrell onduted arefully designed and exeuted measurements to validate their analytial preditions in the [11] rystallographi diretion of single rystal silion and isotropi vitreous silia. Later, Cantrell et al. suessfully used the slope of the quasistati displaement pulse 176 J. Aoust. So. Am. 134 (3), September /13/134(3)/176/15/$3. VC 13 Aoustial Soiety of Ameria

2 FIG. 1. (Color online) Shemati illustration of the right-triangular shape predited by Yost and Cantrell for the quasistati displaement pulse (Ref. ). to measure the nonlinearity parameter of rystalline silion in all three rystallographi symmetry diretions. 4 Although these studies did not diretly present the experimentally reorded quasistati pulse that exhibits the right-triangular shape with a sharp leading edge and a uniformly dereasing slope until the end of the pulse, their measurements of the aousti nonlinearity parameter using the slope of the quasistati displaement pulse did show good agreement with known values. Later on, additional experimental, omputational, and analytial evidene emerged whih shows that Yost and Cantrell s predition of a right-triangular shape for the quasistati displaement pulse annot be independently verified. Jaob et al. 5 onduted displaement measurements with an optial interferometer in fused silia and aluminum alloy samples and found that the quasistati displaement pulse produed by a longitudinal aousti wave exhibited a flat-top shape with amplitude independent of the duration of the tone burst, but proportional to the propagation distane. Renier et al. 6 also found that the amplitude of the quasistati displaement generated by an ultrasoni tone burst propagating through water was linearly proportional to the propagation distane. Narasimha et al. 7 onduted similar experiments in Al7175-T7351 alloy using a piezoeletri reeiver and onfirmed that the deteted quasistati pulse exhibited a flat-top shape and its amplitude was independent of the duration of the tone burst. In response, Cantrell 8 argued that the experimental results of Jaob et al. 5 were the spurious onsequene of unorreted diffration and attenuation effets in their measurements and suggested that the results of Narasimha et al. 7 were a onsequene of the harateristis of their reeiving transduer. In their rebuttal, Narasimha et al. 9 onduted onedimensional numerial simulations using a finite-differene method to determine the harateristis of the quasistati pulse and onfirmed the pulse exhibited a flat-top shape with the pulse amplitude independent of the duration of the tone burst and proportional to the propagation distane. Subsequently, Cantrell and Yost 1 further argued that the ever inreasing amplitude of the quasistati displaement pulse presented in Ref. 9 would mean that the energy density must inrease uniformly with propagation distane, thus violating the law of energy onservation. Reently, Qu et al. 3,11 studied this ontroversial issue of quasistati pulse shape by obtaining an analytial solution for the propagation of tone burst in elasti solids with quadrati nonlinearity. They showed that as the eigenstrain pulse produed by the nonlinearity moves through the medium at the speed of sound, it ontinuously generates a quasistati elasti wave, like an airplane flying at the speed of sound, whih results in a umulative effet and produes a quasistati displaement pulse that is proportional to the propagation length, but independent of the duration of the tone burst. They also analyzed the effets of displaement-presribed versus tration-presribed boundary onditions at the radiating plane. They showed that the quasistati displaement pulse depends on the boundary onditions, so are must be taken when using the quasistati displaement to measure the aousti nonlinearity parameter of a solid. In their response, Cantrell and Yost 1 suggested that the analytial solutions derived in Ref. 11 violate the Law of Energy Conservation. The publiation of Ref. 1 prompted us to arry out a areful re-examination of our early work. 3,11 Our objetives are twofold. First, we want to hek the orretness of our analytial solutions, speifially heking if they violate the law of energy onservation. Seond, we want to understand the mathematial and physial underlining of the right-triangular shape of the quasistati pulses reported in Refs. 1,, 4, 1, and1. We will aomplish the first objetive by offering a rigorous analytial proof that the earlier results presented in Ref. 11 are indeed in strit agreement with the Law of Energy Conservation, and by onduting a omprehensive numerial study using the finite element (FE) method. In pursuing the seond objetive, we unovered several interesting phenomena related to the effets of the finite size of the speimen and the ultrasoni beam on the propagation of a tone burst in elasti solids with quadrati nonlinearity, although none of these effets fully explains the triangular quasistati displaement pulse measured experimentally by Yost and Cantrell. The outline of this paper is as follows. In Se. II we review the analytial preditions of Qu et al. for the quasistati displaement pulse generated by a plane wave propagating in an elasti half-spae of quadrati nonlinearity. 11 We also provide a lear physial interpretation of these analytial solutions, and rigorously prove that they do not violate the law of energy onservation. In addition, we also present new FE simulation results that unequivoally prove that these analytial results are orret. In Se. III we present both analytial and FE results to show that the finite length of the speimen annot simply be modeled with a linear refletion oeffiient that neglets the nonlinear interation between the ombined inident and refleted fields. In Se. IV we present FE results to show that the divergene of the quasistati wave is not negligible, even when the divergenes of the fundamental and seond harmoni are both negligible and illustrate how the lateral dimension of the aousti beam affets both the shape and amplitude of the quasistati displaement pulse. Finally, we onlude in Se. V. J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets 1761

3 II. PLANE WAVE PROPAGATION IN AN ELASTIC HALF-SPACE Assume that a harmoni aousti tone burst of displaement amplitude U and angular frequeny x propagates through an elasti solid with weak quadrati nonlinearity. The solid is haraterized by its longitudinal sound veloity and nonlinearity parameter b. The governing equation is given in Eq. (A3). Perturbation solutions to Eq. (A3) that are valid up to the terms proportional to b were derived in Ref. 11 for both the displaement-presribed and trationpresribed boundary onditions at the surfae of the half-spae. In Appendix A, we show that, up to the terms proportional to b, the energy flux omputed based on the solutions in Ref. 11 is the same regardless of the loation where the energy flux is omputed. This indiates that energy arried by the tone burst remains the same as it propagates. Therefore, the law of energy onservation is not violated, ontrary to the laims made in Ref. 1. The aousti radiation-indued quasistati eigenstrain that aompanies the harmoni tone burst as it propagates through the elasti medium is 3 e ¼ ð1 þ bþ x U 4 : (3) The nonlinearity parameter b aounts for both the nonlinearity of the displaement-strain relationship for finite amplitude waves, and the nonlinearity of the stress-strain relationship that gives rise to ubi terms in the strain energy density. The material effet an be haraterized with the seond- and third-order elasti oeffiients of the material. In an isotropi solid, it is ustomary to use two of the three Murnaghan oeffiients k and m to obtain b as follows: b ¼ 3 þ þ 4m k þ l ; (4) where k and l are Lame onstants. Alternatively, b ¼ (3 þ C 111 /C 11 ), where C 11 and C 111 are seond- and third-order elasti oeffiients, respetively. While the eigenstrain is intrinsi to the material undergoing harmoni deformation of given amplitude and frequeny, the total quasistati strain and stress also depend on the boundary onditions. 3 Let us assume that the aousti tone burst was produed by a harmoni displaement presribed on an elasti solid half-spae at x ¼, uð; tþ ¼ UPðtÞ sinðxtþ: (5) Here, the pulse shape is given by PðtÞ ¼HðtÞ Hðt sþ; (6) where H(t) is the Heaviside step funtion. Later we will assume that s ¼ pt, where p ¼ 1,, is an integer, T ¼ 1/f is the period of the fundamental harmoni, and f denotes the yli frequeny, i.e., x ¼ pf. The quasistati displaement is then defined somewhat arbitrarily as u ðx; tþ ¼ ð t þ T= t T= uðx; tþ dt: (7) In the partiular ase of presribed displaement at the transmitting plane, the quasistati displaement pulse grows proportionally to the propagation distane x, and exhibits a flattop shape of length s as given by 11 u D ðx; tþ ¼ bx U x 8 P t x : (8) In the ase of presribed trations at the transmitting plane, we an hoose the tration vetor so that stress produed in front of the transmitter is 11 rð; tþ ¼ q x UPðtÞ osðxtþ: (9) Although the polarity of the exitation pulse does not affet the nonlinear terms, the negative sign in Eq. (9) was hosen so that the polarity of the generated fundamental harmoni is the same as in the displaement-presribed ase before. The quasistati displaement pulse also grows proportionally to the propagation distane x, but it exhibits a trapezoidal shape u T ðx; tþ ¼ bx U 8 ð x tþ P t x s H t s x : (1) We should point out that Eq. (1) was misprinted in Ref. 11. The leading edge of the quasistati displaement pulse is the same as in the previous displaement-presribed ase u T ðx; x=þ ¼ bx U x 8 ; (11) but the shape of the pulse is not flat-top in this ase. Rather, the quasistati displaement dereases with the same slope as the one predited by Yost and Cantrell and previously given in Eq. ¼ bx U : (1) 8 It should also be mentioned that a true stati displaement term is left when the tone burst passes the observer u T ðx; t > x= þ sþ ¼ b U x s ; (13) 8 and the material extension aused by the eigenstrain is in the region behind the point of observation, therefore it auses a negative displaement behind the propagating pulse. It does not matter that the eigenstrain pulse is moving away from the observer at the speed of sound; there is a remnant of the negative displaement due to the bakwards push of the eigenstrain pulse. In this way, the drop at the trailing edge of the quasistati displaement pulse at t ¼ x/ þ s is exatly the same as the rise at the leading edge given in Eq. (11). Formally, Eq. (13) is of the same magnitude and onveys the 176 J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets

4 same information about the duration of the tone burst, as Eq. (), but of ourse at t ¼ x/ þ s this information is legitimately available at the point of observation. We should point out that, in a pratial situation, the remnant stati displaement will inevitably deay and ultimately vanish due to the effets of absorption, sattering, divergene, and refletion. The differene Du ðx; tþ ¼u T ðx; tþ u D ðx; tþ between the quasistati displaement pulses produed by tration- and displaement-presribed boundary onditions is a tapered step funtion Du ðx; tþ ¼ bx U t x H t x 8 t s x H t s x : (14) In the displaement-presribed ase, a restraining fore is ating on the transmitter that assures that the average displaement during exitation and the remnant displaement after the end of it remain zero. The differene between the two quasistati displaement pulses is the additional quasistati displaement pulse produed in the material by this restraining fore ating on the transmitter. This pulse is generated at the transmitting plane at x ¼ and it propagates through the medium as a purely linear elasti wave Du ðx; tþ ¼Du ð; t x=þ, where Du ð; tþ ¼ bx U ½tHðtÞ ðt sþhðt sþš: (15) 8 After ramping up during the duration of the exitation, this displaement assumes a truly stati nature with a remnant stati value of Du ð; t > sþ ¼ bx U s=ð8 Þ that is proportional to the duration of the exiting tone burst. It is well known that when suh a downward-step waveform passes through an alternating urrent (a)-oupled (i.e., highpass filtering) detetion system it will produe an initial negative slope, whih is the behavior observed by Yost and Cantrell. It should be pointed out that the results in Se. II are limited to the one-dimensional (1D) plane-wave ase. Compared to the fundamental and seond harmonis, the low-frequeny disturbane propagates through the medium with a very large divergene and quikly deays away from a finite-diameter transmitting soure, as will be shown later. of the omputational prowess of today s omputers and the omputational effiieny of state-of-the-art simulation software like COMSOL, one an easily eliminate any disturbing artifats, suh as numerial dispersion aused by disretization, and obtain irrefutable evidene as to whether an analytial predition based on a given model is valid or not. Speifially, we used the properties of generi aluminum available from the material library of COMSOL. The relevant properties of the material are listed in Table I. We ran simulations with the temporal period of the signal disretized into 1 time steps and the spatial period (wavelength of the fundamental harmoni) disretized into 1 elements. Earlier studies 13 have shown that suh disretization is more than fine enough to eliminate spurious numerial dispersion. We avoided using any post proessing or filtering that ould possibly distort the reeived displaement signals. As an example, Fig. shows FE simulation results of a p ¼ 8 yle, f ¼ 1 MHz frequeny, U ¼ 1 nm amplitude tone burst after L ¼ 5 mm propagation in aluminum for the ase of a displaement-presribed plane radiator. Although suh simulations ould be readily onduted on a 1D model, we used an axisymmetri two-dimensional model with rolling boundary (no normal displaement, no tangential tration) ondition on the outside surfae of the ylindrial model so that finite-beam effets ould be also simulated with the same model later. In order to obtain the signals produed by the nonlinear interation between the material and the propagating aousti tone burst without resorting to filtering that might affet these pulse shapes, we reorded the reeived signals with both positive and negative polarities of the transmitted burst and averaged the two signals [Fig. (a)]. The average signal suppresses all even harmonis, and most importantly the fundamental one. 14,15 Therefore, the average signal is entirely due to nonlinear interation and in our ase it is a superposition of the seond-harmoni and the quasistati displaement pulses [Fig. (b)]. The transient spikes seen at the beginning and the end of the burst are indiations of the high frequeny omponents of the signal aused by using an untapered retangular window funtion P(t) to modulate the tone burst. The effet is magnified in the nonlinear signal partially beause the nonlinear effet is proportional to the square of frequeny, and partially beause higher frequenies are less perfetly resolved by disretization in COMSOL. Finally, we used running time averaging (smoothing) aording to Eq. (7) to reover the A. FE simulation The analytial solutions derived in Ref. 11 are asymptoti solutions to Eq. (A4) for weak nonlinearity. To further onfirm the validity of these asymptoti solutions, FE analyses were onduted to solve the nonlinear wave equation Eq. (A4). To this end, we exploited the unique apabilities of the COMSOL Multiphysis FE simulation software that has a built-in option alled Murnaghan material to simulate the nonlinear interation between an aousti wave and an isotropi elasti solid of quadrati nonlinearity. The material is haraterized by two Lame onstants, k and l, three Murnaghan onstants, k, m, and n, and its density q. Beause TABLE I. Physial properties of aluminum used in the FE simulations. Property Symbol Value Unit Density q 7 kg/m 3 Lame onstant k 51 GPa Lame onstant l 6 GPa Murnaghan onstant k 5 GPa Murnaghan onstant m 33 GPa Murnaghan onstant n 35 GPa Veloity a 6176 m/s Nonlinearity a b a Calulated from listed properties. J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets 1763

5 FIG. 3. (Color online) Analytial preditions and FE simulation results for the amplitude of the quasistati displaement pulse produed by an f ¼ 1 MHz frequeny, U ¼ 1 nm amplitude tone burst in aluminum (displaement-presribed plane radiator). FIG.. (Color online) FE simulation results of tone burst propagation in aluminum (displaement-presribed plane radiator, p ¼ 8, f ¼ 1 MHz, U ¼ 1 nm, L ¼ 5 mm). envelope of the seond-harmoni and the quasistati displaement pulses [Fig. ()]. The gradual rather than abrupt transitions at the leading and trailing edges of the pulses are aused by the averaging sheme used to separate the seondharmoni and quasistati displaement omponents without the use of filtering. These results are in good agreement with our analytial predition previously given in Eq. (8), as both pulses exhibit flat-top shapes and the amplitude of both pulses are found to be idential to the predited u ¼.9488 nm value within.1% numerial trunation error. The auray of this agreement itself leaves no doubt about the validity of our analytial predition of a flat-top quasistati displaement pulse in the ase of a displaementpresribed plane radiator. Still, for illustration purposes, Fig. 3 shows the analytial preditions and FE simulation results for the amplitude of the quasistati displaement pulse produed by an f ¼ 1 MHz frequeny, U ¼ 1 nm amplitude tone burst in aluminum in the ase of a displaementpresribed plane radiator. For simpliity, the essentially idential amplitude of the seond harmoni is not shown. The amplitude of the quasistati displaement pulse is plotted (a) as a funtion of the duration of the tone burst at a onstant observation distane of L ¼ 5 mm and (b) as a funtion of the observation distane for a onstant pulse duration of s ¼ 8 ls (p ¼ 8). As expeted for the ase of a displaement-presribed plane radiator, the amplitude of the quasistati displaement pulse is independent of the pulse duration and linearly proportional to the propagation distane. Results of Qu et al. suggested that the shape of the quasistati displaement pulse depends on the boundary onditions that prevail in the transmitting plane. Displaementpresribed boundary onditions orrespond to infinitely high transmitter aousti impedane, whih is totally unrealisti in an experimental test. The initial goal of Qu et al. was only to point out that, although the eigenstrain is independent of the boundary onditions, the total strain is the sum of the eigenstrain and the elasti strain, and that the latter is strongly influened by the boundary onditions. 3 Subsequently, Qu et al. illustrated the importane of the boundary onditions, and therefore the aousti impedane of the transmitter, by analyzing the ase of a radiator of negligible aousti impedane using tration-presribed boundary onditions. 11 They found that in this ase, although the negative slope of the quasistati displaement pulse as given by Eq. (1) is idential to the slope of the right-triangular pulse shape predited by Yost and Cantrell, the amplitude of its leading edge is independent of the pulse duration, and proportional to the observation distane. Yost and Cantrell used a narrow-band lithium niobate transduer bonded to the surfae of their speimen. Suh a transduer is likely to present a low aousti impedane to the elasti solid, espeially below its resonane frequeny. We also ran COMSOL simulations similar to the previously desribed ones for the ase of a trationpresribed plane radiator. In order to produe the same U ¼ 1 nm amplitude for the fundamental harmoni, we applied a tone burst of normal tration with qxu ¼ N/m amplitude alulated from Eq. (9) without the negative sign as the tration ating on the half-spae with a negative surfae normal. The simulated signals were proessed exatly as before. Figure 4 shows the FE simulation 1764 J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets

6 earlier analytial preditions based on a infinite plane radiator ating on the surfae of a semi-infinite elasti solid of quadrati nonlinearity in both the displaement- and tration-presribed boundary onditions. In Ses. III and IV, we investigate several fators that may affet the shape of the quasistati displaement pulse reorded by the reeiving transduer, suh as the finite size of the speimen used in the measurements, and the finite beam width generated by the transmitting transduer. FIG. 4. (Color online) FE simulation results of tone burst propagation in aluminum (tration-presribed plane radiator, p ¼ 8, f ¼ 1 MHz, U ¼ 1 nm, L ¼ 5 mm). results for the ase of p ¼ 8 yles, f ¼ 1 MHz frequeny, U ¼ 1 nm amplitude tone burst after L ¼ 5 mm propagation in aluminum. First, the reeived signals are reorded with flipping the polarity of the transmitted signal [Fig. 4(a)]. As before, the average signal is a superposition of the seond-harmoni and quasistati displaement pulses [Fig. 4(b)]. Finally, time-averaging of the average pulses yields the envelopes of the seond-harmoni and the quasistati displaement pulses [Fig. 4()]. These results are again in exellent agreement with our analytial predition previously given in Eq. (1), as the quasistati displaement pulse exhibits the expeted trapezoidal shape. The heights of the leading and trailing edges of the quasistati displaement pulse are found to be idential to the u ¼.9488 nm value alulated from Eq. (11) within.1% numerial unertainty. Furthermore, the slope of the pulse agrees with the.117 nm/ls value alulated from Eq. (1) within.5% numerial unertainty. In summary, COMSOL simulations left no doubt about the validity of our III. PLANE WAVE PROPAGATION IN A FINITE-LENGTH SOLID SPECIMEN Finite-size speimens are used in most ultrasoni tests. Typially, a tone burst is generated by a transduer at one end of the speimen, and is reeived by a reeiving transduer at the other end of the speimen. If the reeiving transduer s mass is muh smaller than that of the speimen, one may assume that the reeiving end of the speimen is under tration-free ondition. Thus, the signal reeived is different from that propagating inside the speimen. This is partiularly the ase when the signal is reeived by a apaitanebased reeiver as used in Ref.. To interpret suh reeived signals, solutions to a tone burst refleted at a tration-free surfae are needed. First, assume that the same harmoni tone burst displaement previously given in Eq. (5) is presribed at the transmitting plane at x ¼, while the reeiver is plaed on the tration-free surfae at. An analytial solution to this problem is derived in Appendix B. It follows from Eq. (B) that the quasistati displaement pulse at the tration-free surfae is given by u D ðl;tþ¼ bx U th t L ðt sþh t s L : 4 (16) When the spatial extent of the pulse is muh less than the thikness of the speimen (s L), the quasistati displaement pulse amplitude essentially doubles due to the refletion at the tration-free surfae. The slope of the top of the quasistati displaement pulse ¼ bx U : (17) 4 The stati displaement left at the reeiving plane after the pulse has been fully refleted is u D ðl; t > L= þ sþ ¼ bx U s : (18) 4 This is understandable beause refletion traps the eigenstrain pulse between the transmitter and the reeiver. In the ase of the presribed-displaement boundary ondition at the transmitting plane, the trapped part of the eigenstrain pulse produes an additional displaement that inreases with time as more and more of the eigenstrain pulse reflets from the free surfae bak to the spae between the transmitter, whih annot move, and the reeiver, whih an move J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets 1765

7 freely. Therefore, the top of the quasistati pulse will exhibit a positive slope and a residual stati strain remains when the tone burst fully reflets from the reeiving plane and all of it is trapped. In the ase of a tration-presribed plane radiator, the situation reverses. The quasistati displaement pulse an be obtained by integration from Ref. 11 after aounting for the presene of the tration-free surfae as follows (see Appendix C for details): u T ðl; tþ ¼ bx U L 4 P t L ; (19) whih is the flat-top pulse shape obtained for the displaement-presribed transmitter on a semi-infinite halfspae exept that the amplitude is doubled due to refletion at the free surfae. In addition to the analytial solutions, we also onduted COMSOL simulations similar to those shown in Se. II. First, assume that the same harmoni tone burst displaement previously given in Eq. (5) is presribed at the transmitting plane at x ¼, while the reeiver is plaed on the trationfree surfae at. Figure 5 illustrates the nonlinear signal for a tone burst propagating through an L ¼ 5 mm thik aluminum plate for (a) displaement-presribed and (b) tration-presribed plane radiators (p ¼ 8 yles, f ¼ 1MHz frequeny, U ¼ 1 nm amplitude). Here, and in all that follows, the nonlinear signal means the two even harmonis (quasistati and seond harmoni) obtained by averaging the FE simulation results for opposite polarities of the exitation tone burst. Only the averaged signals are shown that inlude both the seond harmoni and quasistati pulses, but the main features are very lear. As a result of refletion at the free surfae, the amplitude of the seond harmoni doubles, but its envelope retains its retangular shape. 16 However, in agreement with our analytial preditions above, the shape of the quasistati displaement pulse is dramatially hanged by the refletion at the free surfae and annot be desribed simplistially by a linear refletion oeffiient. IV. FINITE-DIAMETER BEAMS IN AN ELASTIC HALF-SPACE Up to this point we have only onsidered plane waves propagating through an elasti medium of quadrati nonlinearity. Therefore, the above results do not aount for the divergene of the primary aousti beam, or that of the seondary nonlinear omponents produed by the nonlinear interation between the harmoni tone burst and the material. This is a serious limitation of the plane wave model that signifiantly influenes both the magnitude and shape of the quasistati displaement pulse. In pratie, the finite lateral dimension of the aousti beam annot be negleted beause the divergene of the quasistati wave is not negligible, even when the divergenes of the fundamental and seond harmonis are negligible. Assume that the aousti pulse is generated by a rigid piston transduer of radius a and reeived by a similar piston reeiver of the same radius after propagation over a path of length L. No apodization is onsidered over the transduer area, i.e., every element of the transmitter moves at the same displaement u t and the reeived eletri signal is proportional to an un-weighted average of the vibration displaement u r over the whole aperture of the reeiver. Then, the D ¼ u r /u t ratio is a measure of the diffration loss aused by beam divergene. For example, attenuation measurements must be orreted for this diffration loss to aurately assess the true attenuation oeffiient of the material from the measured total loss. In linear ultrasonis, the standard way of obtaining an analytial orretion for diffration losses is to use the Lommel integral that is exat for fluids. The Lommel diffration orretion D L an be most onveniently written as a funtion of the normalized separation distane s ¼ L/N between the transmitter and the reeiver, where N ¼ a /k is the near field/far field transition distane of a transduer for an aousti wavelength k. Rogers and van Buren derived the following exat analytial solution for the Lommel diffration orretion 17 D L ðsþ ¼1 e i p=s ½J ð p=sþþij 1 ð p=sþš; () FIG. 5. (Color online) Nonlinear signals generated by a harmoni tone burst propagating through an aluminum plate for (a) displaement-presribed and (b) tration-presribed plane radiators (p ¼ 8, f ¼ 1 MHz, U ¼ 1 nm, L ¼ 5 mm). where J and J 1 are zeroth- and first-order Bessel funtions of the first kind. Although, stritly speaking, the Lommel orretion is limited to fluids, it is also an exellent approximation for 1766 J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets

8 most pratial ases involving solids and it an be used to illustrate the underlying physial problem behind the above desribed plane wave solutions for the quasistati displaement pulse produed by a harmoni tone burst. Figure 6 illustrates the magnitude of the Lommel diffration orretion as a funtion of transduer radius. The propagation distane, sound veloity, and inspetion frequeny are all hosen to orrespond to the harmoni tone burst onsidered in the above examples of nonlinear wave propagation in aluminum (L ¼ 5 mm, ¼ 6176 m/s, f ¼ 1 MHz). Similar diffration orretions are not available for the nonlinear seond harmoni or quasistati displaement pulses. Therefore, COMSOL simulations were used to ondut fast and aurate numerial experiments to examine the divergene behavior of the tone burst in elasti solids with quadrati nonlinearity. In partiular, we are interested in whether the quasistati pulse is more divergent than the umulative seond harmoni that propagates. In pratie, the main reason why the seond harmoni exhibits lower magnitude than expeted based on the previously desribed plane wave model is not divergene, but signifiant losses due to material attenuation at higher frequenies. In extreme ases, the attenuation indued loss is so high that the aumulation of the seond harmoni is limited by the harateristi attenuation length, i.e., the inverse of the attenuation oeffiient, rather than the total propagation length. 18 In our study the frequeny-dependent attenuation of the material is ompletely negleted, therefore the seond harmoni is proportional to the propagation distane and its diffration loss is expeted to be fairly well desribed by the Lommel diffration orretion shown in Fig. 6. In the plane wave approximation, the quasistati displaement pulse was also found to exhibit a umulative effet in the sense that its magnitude was proportional to the propagation distane. Although the eigenstrain remains onstant as the harmoni tone burst propagates through the medium, the elasti part of the total strain exhibits a umulative effet sine the elasti strain pulse propagates at the same aousti veloity as the onstant eigenstrain that generated it in the first plae. Sine the quasistati strain pulse might exhibit very strong divergene in the ase of finite-diameter transduers, its ontribution to the total strain at the distant point of observation might beome all but negligible. Therefore, for a slender ollimated aousti beam (k < a < L < a /k) the diffration loss of the quasistati pulse reahing the observer will be muh stronger than that of the fundamental and seond harmonis. In this setion we will present COMSOL simulation results to illustrate the signifiant diffration loss of the quasistati displaement pulse even in ases when the seond harmoni is barely affeted by the divergene of the aousti beam that generated both. Figure 7 shows an example of two aousti tone bursts of opposite polarity generated by a displaement-presribed finite aperture radiator in an aluminum half-spae (p ¼ 6, f ¼ 1MHz, U ¼ 1 nm, a ¼ 15 mm). The snapshots were taken at t ¼ 8 ls just before the leading edge of the tone burst reahed the reeiver loated at L ¼ 5 mm (only halves of the axisymmetri distributions are shown). In this example, the wavelength is k 6. mm and the near field/far field transition distane is N 36 mm, slightly less than the distane between the transmitter and the reeiver, therefore some signs of divergene and interferene in the near field are visible. Figure 8 shows examples of the reeived nonlinear signals for three different transduer radii. Due to the high-pass filtering effet of the frequeny-dependent diffration orretion, the small-diameter ase (a) appears to be a oupled with the weak quasistati omponent while the largediameter ase () appears to be diret urrent oupled with the quasistati pulse having the same amplitude as the seond harmoni. Quantitative assessment of the diffration loss is made diffiult by the fairly omplex waveforms of the reeived nonlinear signals. We further separated the quasistati pulse from the seond harmoni tone burst by averaging the nonlinear signal over half a period of the fundamental signal, i.e., over a full period of the seond harmoni. Sine the fundamental harmoni is already suffiiently suppressed FIG. 6. (Color online) Lommel diffration orretion as a funtion of transduer radius (L ¼ 5 mm, ¼ 6176 m/s, f ¼ 1 MHz). FIG. 7. (Color online) Example of two aousti tone bursts of opposite polarity generated by a displaement-presribed finite aperture radiator in an aluminum half-spae (p ¼ 6, f ¼ 1 MHz, U ¼ 1 nm, a ¼ 15 mm, L ¼ 5 mm, t ¼ 8 ls). J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets 1767

9 FIG. 9. (Color online) Examples of the (a) envelope of the seond harmoni tone burst and (b) quasistati displaement pulse for four different transduer radii and the infinite plane wave predition. FIG. 8. (Color online) Examples of the reeived nonlinear signals for three different transduer radii. by averaging the two reeived signals for opposite exitation polarities, this hoie of integration length represents a good ompromise between the rejetion of unwanted harmonis and retaining subtle features of the quasistati pulse shape. Figure 9 shows examples of the (a) envelope of the seond harmoni tone burst and (b) quasistati displaement pulse for four different transduer radii and the infinite plane wave predition. In the infinite ase, for the parameters hosen in this example (b ¼ 14.67, ¼ 6176 m/s, f ¼ 1 MHz, U ¼ 1 nm, L ¼ 5 mm), the amplitude of the seond harmoni and the peak of the quasistati displaement pulse are both equal to u ¼.9488 nm. As the transduer radius dereases, the envelope of the seond harmoni essentially retains its retangular shape, though there is a pereivable drop in its amplitude. In ontrast, the drop in the amplitude of the quasistati displaement pulse is muh more signifiant and the shape of the pulse also gets distorted with a mostly negative slope for small transduer radii. The highly omplex shape of the quasistati displaement pulse is due to the frequeny-dependene of the relevant diffration loss, and the downward slope of the top of the quasistati displaement pulse indiates inreasing diffration loss at very low frequenies. It should be mentioned that the numerially omputed waveforms shown in Fig. 9(b) are similar to the experimental waveforms observed by Renier et al. in water. 6 Until similar measurements are onduted in solids, this good qualitative agreement an be onsidered as initial experimental proof of the desribed diffration effets on the quasistati displaement pulse. Beause of the omplex shape of the quasistati displaement pulses deteted for small transduer radii, it is diffiult to quantitatively haraterize the diffration loss as a funtion of transduer radius using a single parameter. Still, in order to indiate the main trends, we hose the average of the envelopes between 1 and 1 ls, i.e., in a -lslong window around the enter of the reeived pulse. Using 1768 J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets

10 FIG. 1. (Color online) The mean displaement amplitudes of the first and seond harmonis and the quasistati pulse as funtions of the transduer radius for L ¼ 5 mm transduer separation. this definition, Fig. 1 shows the mean displaement amplitudes of the first (fundamental) and seond harmonis and the quasistati pulse as funtions of the transduer radius for L ¼ 5 mm transduer separation. The fundamental harmoni was normalized to the amplitude of the presribed displaement at the transmitter (u t ¼ 1 nm) while the amplitudes of the two nonlinear omponents were normalized to their ommon plane wave asymptoti limit (u ¼.9488 nm). The first harmoni behaves exatly the way one would expet based on the Lommel diffration orretion previously shown in Fig. 6. The seond harmoni behaves similarly, but exhibits slightly larger osillations, whih is not surprising onsidering that its level is proportional to the square of the amplitude of the fundamental wave. At the a ¼ 1 mm transduer radius, the normalized amplitudes of the first and seond harmonis are both 8% based on the nonlinear COMSOL simulation while the linear predition based on the Lommel orretion of the fundamental harmoni is about 76%. In omparison, the normalized amplitude of the quasistati displaement pulse is only %, whih learly indiates the inreased role of beam divergene in the magnitude (and shape) of the quasistati displaement pulse. It should be mentioned that the higher diffration loss of the quasistati displaement pulse relative to the first and seond harmonis also means that the differene between the displaement- and tration-presribed ases will be suppressed for finite transduer diameters. V. CONCLUSIONS This paper re-examined the ontroversial issue related to the shape of the quasistati displaement pulse produed by nonlinear aousti wave propagation in an elasti solid of quadrati nonlinearity. Early results suggested that the quasistati displaement pulse has a right-triangular shape with a peak displaement of the leading edge proportional to the length of the tone burst; this is in ontrast to a flat-top shape with a peak displaement that is proportional to the propagation distane suggested by reent researhers. This study uses a numerial simulation to settle this debate, and then analyzes the finite-size effets in this problem. These numerial simulation results unequivoally show that a quasistati displaement pulse has a flat-top shape with a peak displaement that is proportional to the propagation distane, onfirming the analytial results of Qu et al. 3,11 We also present a new analytial proof that, in ontrast to the statement in Ref. 1, these analytial results do obey the Law of Energy Conservation. In an effort to understand the experimentally observed right-triangular shape of the quasistati displaement pulse reported by Yost and Cantrell, we explored several other aspets of the problem that might potentially be responsible to suh observations. Speifially, we investigated the effets of finite speimen length and finite beam diameter, two pratial aspets of the experiment of Ref.. First, we found that the finite axial dimension of the speimen annot be simply modeled with a linear refletion oeffiient that neglets the nonlinear interation between the ombined inident and refleted fields. These omputational results are in agreement with new analytial results presented for the nonlinear refletion phenomenon at a tration free surfae of a finite-length speimen. Seond, we determine that the finite lateral dimension of the aousti beam annot be negleted sine the divergene of the quasistati wave is not negligible, even when the divergenes of the fundamental and seond harmoni are both negligible. Under ertain onditions, both of these finite-size effets an influene the shape of a quasistati displaement pulse. However, neither effet an lead to a right-triangular shape of the quasistati displaement pulse. ACKNOWLEDGMENTS J.Q. and L.J.J. aknowledge the finanial support by the Department of Energy through NEUP and APPENDIX A: ENERGY CONSERVATION This appendix proves that the perturbation solution derived for the propagation of an aousti pulse in an elasti medium with weak quadrati nonlinearity satisfies the ondition of energy onservation. To begin, onsider a half-spae defined by x, where x is the Lagrangian (or material) oordinate desribing the loation of the material partile in the initial (t ¼ ) state. At any given time t, the displaement of the partile x from its initial position is denoted by uðx; tþ. Deformation of the elasti body an then be desribed by the Lagrangian strain e þ : (A1) We assume that the half-spae is made of an elasti solid with quadrati nonlinearity, i.e., the normal (first Piola- Kirhhoff) stress is related to the Lagrangian strain/displaement gradient in the x-diretion through r ¼ q e b þ 1 " e ¼ b ; (A) where q is the mass density, is the longitudinal phase veloity, and b is the aousti nonlinearity parameter, all for the elasti solid in the undeformed (initial) state. J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets 1769

11 The displaement equation of motion governing the wave propagation in the x-diretion is u ¼ u : (A3) By a standard perturbation proedure, one may write the solution to Eq. (A3) as uðx; tþ ¼u 1 ðx; tþþu ðx; tþ; where ju 1 ðx; tþj ju ðx; tþj,oru ¼ Oðu 1Þ, and (A4) u 1 ¼ ; u u ¼ u 1 : (A5) Making use of the onstitutive law Eq. (A), one may expand the stress into rðx; tþ ¼r 1 ðx; tþþr ðx; tþ; where jr 1 ðx; tþj jr ðx; tþj and (A6) r 1 ðx; tþ ¼q ; " r ðx; tþ b # : (A7) The solution to the first of Eq. (A5) that represents a forward propagating wave an be written as u 1 ðx; tþ ¼f ðt x=þ: (A8) It then follows that the seond part of Eq. (A5) an be written as where u u ¼ gðt x=þ; (A9) gðsþ ¼ b 3 f ðsþ f ðsþ; (A1) and the prime denotes the derivative with respet to the argument of the funtion. By a diret substitution, one an show that the solution to Eq. (A9) is given by u ðx; tþ¼ b x ð t x= f ðsþ f ðsþ ds þ DxþBðt x=þ; þ (A11) where BðyÞ is an arbitrary funtion of y and D is an integration onstant, both need to be determined by the boundary onditions and/or the onsisteny ¼ 1 # 3= 1 : (A1) 3 b If f ðsþ is a smooth funtion for s ð; t x=þ, the integral in Eq. (A11) an be arried out u ðx;tþ¼ bx 4 ð½f ðt x=þš ½f ð þ ÞŠ ÞþDxþBðt x=þ: (A13) This is the general solution to the seond order governing equation Eq. (A5), whih was previously derived as Eq. (11) of Ref Energy flux To demonstrate that the solution we have obtained does not violate energy onservation, we onsider the energy flux Fðx; tþ. If the time integral of Fðx; tþ over the duration of the pulse is independent of x, then the total energy is onserved, beause it means that energy is not added nor subtrated from the wave paket as it propagates along the positive x-diretion. The energy flux in the positive x-diretion is defined by Fðx; tþ ¼ rðx; tþ _uðx; tþ: (A14) Thus, the total energy that passes through loation x over the duration of the pulse s is given by E ¼ Fðx; tþdt ¼ rðx; tþ _uðx; tþ dt : (A15) In what follows, we show that E is independent of the loation x for the solution derived in Qu et al. 11 To this end, we make use of Eqs. (A4), (A6), and (A7) in Eq. (A14) to arrive at the asymptoti expression where Fðx; tþ ¼F 1 ðx; tþþb F ðx; tþþ; (A16) F 1 ðx; tþ¼ q _u 1 ðx; tþ ðx; tþ " ¼ q _u _u # þ _u 1 : (A17) Sine the wave is non-dispersive, one an easily show that for the u 1 given by Eq. (A8), E 1 ¼ F 1 ðx; tþ dt ¼ q ½f ðsþš ds is independent of x. Furthermore, one an write E ¼ þ q 4 F ðx;tþdt¼ qx 3 ð½f ðsþš 3 ½f ðþš 3 Þ ð½f ðþš f ðsþþ½f ðsþš 3 þ8b ðsþf ðsþ 4D f ðsþþds: (A18) (A19) 177 J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets

12 Clearly, E is also independent of x if f ðsþ ¼f ðþ, i.e., the slope of the pulse is the same at the front and trailing edges. This is obviously the ase when the exitation is timeharmoni. To verify the above derivation, onsider the displaementpresribed boundary ondition, uð; tþ ¼ f ðtþ ¼ U PðtÞ sinðx tþ: (A) It follows from Ref. 11 that BðtÞ ¼ and D ¼ b ½f ðþš = ð4 Þ. Thus, E 1 ¼ 1 qx U s; E ¼ q 4 ½f ðsþš 3 dsþ qx 3 ð½f ðsþš 3 ½f ðþš 3 Þ¼: (A1) Thus, the total energy that passes through any loation is the same. Next, onsider the ase when a tration is presribed on the boundary rð; tþ ¼r ðtþ ¼ q x UPðtÞ osðxtþ: In this ase, aording to Qu et al., 11 and Thus, and u 1 ðx; tþ ¼ 1 q D ¼ E 1 ¼ ð t x= r ðsþ ds; (A) (A3) b 4 q 4 ½r ð þ ÞŠ ; BðtÞ ¼ b ð t 4 q 3 ½r ðsþš ds: þ (A4) F 1 ðx; tþdt ¼ 1 q r ðsþds ¼ 1 qx U s (A5) E ¼ 1 q x 6 q 3 ½r3 ðsþ r3 ðþš 3 q 4 ½r3 ðsþ r3 ðþš 1 4 q 3 ½r ðsþš3 ds ¼ : (A6) Again, the total energy passing through any loation x is the same. Further, we notie that in both ases, the energy assoiated with F is zero, indiating that the net energy assoiated with u ðx; tþ that passes through any given loation over the duration of the pulse is zero. This means that E 1 must be equal to the work done by the tration on the surfae at x ¼ over the pulse s duration, i.e., E 1 ¼ Ð s r _uj x¼dt. This an be easily verified by arrying out the integral. APPENDIX B: REFLECTION AT A FREE SURFACE, DISPLACEMENT-PRESCRIBED TRANSMITTER To begin, onsider a slab defined by x L, where x is the Lagrangian (or material) oordinate desribing the loation of the material partile in the initial (t ¼ ) state. First, let us now assume that a harmoni displaement pulse is presribed at the slab s left surfae x ¼, and the tration remains zero at the slab s right surfae. Thus, the boundary onditions an be written as uð; tþ ¼ UPðtÞ sinðxtþ; rðl; tþ ¼ ; (B1) where PðtÞ ¼HðtÞ Hðt sþ with HðtÞ being the Heaviside step funtion and s represents the duration of the pulse. For pratial interest, we assume that s L. In terms of the expansions, it follows from Eqs. (A4) and (A6) that u 1 ð; tþ¼upðtþsinðxtþ; u ð; tþ¼; r ðl; tþ ¼: (B) By making use of Eq. (A7), the last two equations in Eq. (B) an be written ¼ ; ¼ b ¼ : (B3) Summarizing the above, we have the following boundary value problems for u 1 ðx; tþ and u ðx; tþ, respetively, u 1 ¼ ; u 1ð; tþ¼upðtþsinðxtþ; ¼ ; u u ¼ ¼ u 1 ; u ð; tþ¼; (B4) (B5) The solution to Eq. (B4) an be found in many standard textbooks, u 1 ðx; tþ¼usin x t x P t x þusin x tþ x L P tþ x L : (B6) The solution to Eq. (B5) is a little more ompliated. To proeed, we substitute Eq. (B6) into the first part of Eq. (B5), u u ¼ g f ðx; tþþg b ðx; tþþg m ðx; tþ; (B7) where g f ðx; tþ¼ bx3 U 3 sin x t x P t x ; (B8) g b ðx; tþ¼ bx3 U 3 sin x t þ x L P tþ x L ; (B9) g m ðx; tþ ¼ bx3 U 3 sin x ðl xþ P t x P tþ x L : (B1) J. Aoust. So. Am., Vol. 134, No. 3, September 13 Nagy et al.: Finite-size nonlinear effets 1771