Focusing Flexural Waves in Beams for Precisely Controlled Dynamic Fracture

We show that flexural waves can be focused in a beam to an extent that induces precisely controlled dynamic fracture of the beam. Flexural waves are excited at one end of a finite beam and focus at another location along the beam to form a shorter but much larger bending-moment pulse. The strong focusing is achieved by use of the frequency dependence of the phase and group velocity of flexural waves, as well as the superposition of multiple reflections. Amplification of the actuator input by a factor of 20 is achieved solely by wave focusing, proving the potential of such a technique.


I. INTRODUCTION
Some mechanical waves, such as flexural waves and Lamb waves, show dispersion, meaning that their phase and group velocities are frequency dependent [1]. This causes the distortion of propagating broadband pulses of such waves, and spreads out the energy that is carried by the pulse in space and time. While the distortion of a propagating wave package due to dispersion may be considered detrimental in some applications [2], it also opens opportunities in others. For instance, the amount of distortion a wavelet has undergone reveals information about the distance it has covered. This feature has been used in guided-wave-testing applications to locate cracks in beams [3,4] and plates [5] with use of only a single-point measurement of the acoustic emission. Similarly, cracks were identified in tubes with use of only circumferential measurements at one location along the tube [6].
The aforementioned methods rely on the time-reversal symmetry of acoustic waves to retrace the propagation path of the measured waves. The time-reversal symmetry has been studied extensively by Fink with both acoustic waves [7][8][9][10] and electromagnetic waves [11] and has been used to focus waves via randomly distributed scatterers [12] and metamaterials [13][14][15], with applications in medicine, undersea communications, and hydrodynamics [7].
Another application of mechanical waves can be found in material testing at high strain-rates with split Hopkinson (Kolsky) pressure bars [16,17]. Here, a propagating pressure-wave pulse is used to induce fracture of the * vangemmeren@imes.mavt.ethz.ch material sample under dynamic conditions. The pressurewave pulse is created by hitting the bar with a striker, and it can be shaped, for example, by use of a specially designed striker [18] or by insertion of tip material between the striker and the bar [19]. While the classic Kolsky bar setup is used for compression experiments, modifications for tensile [20,21] and torsional [22] experiments exist. Different experimental implementations for bending have been reported as well, but are rare [23][24][25][26]. They offer an alternative to bending tests using drop-weight towers, which struggle with a moving specimen [27]. Especially for brittle materials, such as glass, ceramics, or concrete, questions regarding the fracture mechanisms at high strain rates remain open, and appropriate experimental setups are being developed [28].
Here, we propose a method that uses time-reversal techniques and dispersion to create a precisely controlled, localized, high-power wave pulse in a glass beam to induce dynamic fracture. Exploiting the dispersive nature of flexural waves, which causes the separation of different wavelengths, allows an actuation time that is much longer than the duration of the focused pulse. A somewhat similar method is used in optics for pulse shaping [29]. The time-reversal invariance is used to compute the actuation corresponding to the desired pulse shape and focusing location.
A schematic of the energy-focusing procedure is shown in Fig. 1: An electromechanical transducer is glued to one end of a glass beam and is excited in such a way that the resulting bending waves focus at an arbitrarily chosen location along the beam. At this location, which we call the "focal point," the time history of the bending moment is that of a Ricker wavelet with a certain peak width (Fig. 2). To obtain the corresponding excitation signal, the defocusing of the bending-moment pulse is simulated by its being set as a boundary condition at the focal point (time t S running to the left in the time-space map). Taking the Wavelets are excited at one end of a glass beam with an electromechanical transducer. The propagation paths of the three wavelets W 0 , W 1 , and W 2 are shown exemplarily in a time-space map. After being excited, W 0 focuses to a narrow peak at the focal point, whereas W 1 and W 2 are reflected once and twice, respectively, before focusing at the same point in space and time. The excitation signal is obtained by simulation of the defocusing of the narrow peak (time t S running to the left) and subsequent reversal of the time (t E running to the right). simulated time history of the bending moment at the end of the beam and reversing the time to t E (running to the right) yields the appropriate excitation signal. This is done not only for one wavelet W 0 taking the direct path from the transducer to the focal point but also for wavelets W r undergoing r = 0, 1, . . . , n reflections at the ends of the beam before focusing at the focal point. Thus, the time in which the transducer is active and energy can be pumped into the beam is increased. Finally, the excitation signal for the transducer is composed of the weighted sum of all computed wavelets W 0 to W n .
The excitation of a new wavelet W r coincides with the reflection of the previously created wavelets W r+m , where m is an even number, as illustrated in Fig. 1 for wavelets W 0 and W 2 . The reflection of a previous wavelet, however, does not affect the excitation of a new one, as the deformations stay small and the system behaves linearly. The propagating part of the reflected wavelet and the propagating part of the newly created wavelet are identical in shape and phase, as they are designed to focus at the same point in space and time.
In Sec. II, we discuss the computation of the excitation signal for wave focusing. This comprises the modeling of the beam and the magnetic sphere using spectral elements, the simulation of the defocusing, some considerations regarding the compensation of the evanescent mode, and finally the optimal composition of the excitation signal from multiple wavelets. In Sec. III, the experimental setup is detailed, the characterization of the beam and transducer is explained, and the results are presented. Finally, the results and limitations of the method are discussed, and a conclusion is given.

A. Modeling of the beam
To simulate the propagation of flexural waves, the glass beam is modeled according to Timoshenko beam theory [30]: The deformation is described by the lateral displacement w(x, t) of the centerline and the angle of rotation φ(x, t) of the cross section. The governing equations are where G is the shear modulus, A the cross-sectional area, ρ the density, E the Young's modulus, and I the second moment of inertia. K 1 and K 2 incorporate the effect of shear and are set to π 2 /12 and 1, respectively [31]. Two wave modes with wave numbers k 1 and k 2 satisfy these equations and are referred to as the "flexural-wave mode" and "thickness-shear mode." In the relevant frequency domain, the first wave number k 1 is real (i.e., a propagating wave), whereas the second wave number k 2 is purely imaginary (i.e., an evanescent mode). In general, when the beam is actuated, both wave modes are excited. So, while only the propagating mode is useful for wave focusing, the evanescent mode has to be considered in the boundary conditions too.
The equations of motion are solved in the frequency domain with use of the spectral-element approach proposed by Doyle [31]. We follow Doyle's notation and use semi-infinite spectral elements to model the glass beam. For a semi-infinite beam extending from x = 0 into the positive x-direction, the deformation is given by whereÂ andB are the frequency-dependent coefficients defining the amplitude and phase of the two wave modes. The two amplitude ratios R j are given by The dynamic-stiffness matrixK R , relating the nodal displacementŵ N =ŵ(0) and rotationφ N =φ(0) to the nodal forceF and momentT, is wherek 11 Similarly, the dynamic-stiffness matrixK L for a semiinfinite beam extending from

B. Modeling of the transducer
The transducer, a spherical magnet that is fixed to one end of the beam, is modeled as a rigid sphere with mass m M and moment of inertia I M . The equations of motion of the spherical magnet are set up in accordance with the freebody diagram shown in Fig. 3 and small angles of rotation φ N are assumed: where a is the distance between the end of the tube and the center of mass of the sphere. Assuming all functions to be harmonic in time and suppressing the time dependence e iωt yields the dynamic-stiffness matrix:

Without reflections
When computing the excitation moment for the transducer, we first consider the simplest case, where the wavelet undergoes no reflections between the focal point and the transducer. Here, the glass tube, which is of finite length in reality [ Fig. 4(a)], is modeled as two semi-infinite spectral elements connected by a single node [ Fig. 4 The bending moment in the left beam iŝ As we want the shape of the Ricker wavelet to be contained only in the first mode, we setM Ricker = ik 1 EIÂ and obtain a value forÂ. The value forB can be chosen arbitrarily. Thus, we can compute the shape of the dispersed Ricker waveletˆ at the transducer by evaluating the dynamic shape functionˆ =φ L (0).
To obtain the excitation signal from the dispersed Ricker waveletˆ , we first have to time reverse the wavelet by taking the complex conjugate [7] in the frequency domain * . Again, we want the wavelet to be contained in the first mode only and setÂ =ˆ * . It remains to determine the momentT on the spherical magnet that would produce exactly this wavelet. To this end, we set up another spectral-element model consisting of one semiinfinite beam element connected to the spherical magnet [ Fig. 4(c)]. Equating the loads at the (imaginary) joint yields the dynamic stiffness: where the external forceF is set to zero because the transducer cannot produce an external force. Thus,B is determined such that the shear force of the second mode cancels 044021-3 out the shear force of the first mode, and subsequently the excitation momentT is computed.

With reflections
For wavelets going through reflections, the simulation of the defocusing is extended. For every reflection, a separate semi-infinite beam element is used to simulate the propagation of the reflected wave away from the reflecting end of the beam. Separate elements are used to clearly distinguish between incident and reflected waves. To determine the shape of the reflected wave, we consider the free-body diagram [ Fig. 5(a)] illustrating a reflection at the spherical magnet. Equating the loads at the tip of the beam yields The loadingF(0) consists of a part produced by the incident waveF in and a part produced by the reflected waveF out . Furthermore, the dynamic-stiffness matrixK M of the spherical magnet is used to expressF M in terms of u(0) =û in +û out : where I is the identity matrix. Thus, the boundary condi-tionF out for the simulation of the reflected wave can be computed in terms of the incident waveF in [Fig. 5(b)]. For reflections at the free end, the dynamic-stiffness matrix K M is set to zero, which corresponds to the zero-shearforce and zero-bending-moment condition. On the basis of the deflection of the reflected waveû out , the shape of the dispersed and reflected Ricker wavelet at the transducerˆ is determined and the excitation momentT is computed as explained above.

D. Optimal signal composition
Subsequently, the n + 1 excitation signals for the individual wavelets, which focus at the focal point after zero to n reflections, have to be summed up to one excitation signal that can be fed to the transducer. To do this in an optimal way, we introduce the amplification factor A r = m r /T r . This is the ratio of the maximum bending moment m r produced by a wavelet W r at the focal point to the maximum moment T r required at the transducer [ Fig. 6(a)], which is also proportional to the current flowing through the coil. In other words, the amplification factor A r quantifies the efficiency of the wavelet W r .
A typical measurement of the amplification factor is shown in Fig. 6(b). It is obtained by our measuring the maximum bending moment at the focal point produced by 31 distinct wavelets having undergone up to 30 reflections. For small numbers of reflections r, the amplification is close to 1, because the covered distance between the transducer and the focal point is still small and the effect 044021-4 of dispersion is not strong. The amplification increases for larger r, as the distance covered by the wave increases and the dispersion spreads the frequency content more and more. For large distances, however, the energy loss through damping becomes dominant and the amplification factor decreases and goes to zero.
To summarize, we obtain n + 1 excitation signals for wavelets, which will each produce a maximum bending moment m r at the focal point. The final excitation signal is the weighted sum of all n + 1 signals and is delimited by the maximum current provided by the current source. Likewise, the resulting maximum bending moment at the focal point is the weighted sum of all n + 1 bending moments m r and shall be maximized. This corresponds to the problem statement of linear programming, for which an optimal solution can be found with use of the simplex algorithm, for example.

III. EXPERIMENTAL RESULTS
The wave-focusing method is demonstrated in experiments, where the strong bending-moment pulse at the focal point leads to dynamic fracture of the glass beam. After presentation of the experimental setup, the mensuration of the glass beam and the transducer is discussed. Finally, the results of the wave-focusing experiments are illustrated.

A. Experimental setup
In the experiments, a 1.5-m-long glass tube with circular cross section is used as a waveguide (for more details see Table I). The surface of the glass tube is slightly roughened at multiple locations with use of grade-60 sandpaper to decrease its bending strength [ Fig. 7(a)]. The transducer consists of a spherical magnet (diameter 6 mm), which is glued to one end of the glass tube, with the polarity of the magnet perpendicular to the axis of the tube. The spherical magnet is then inserted into an electromagnetic coil, which is aligned with the axis of the tube [ Fig. 7(b)]. This constellation of the magnetic fields of the magnet and the electromagnetic coil results in a moment acting on the magnet that is perpendicular to the polarity of the magnet and the axis of the beam [32]. Thus, bending waves are excited at the end of the beam with the displacement lying in the plane perpendicular to the moment. The electromagnetic coil is driven with a voltage-controlled current source because the electromagnetic field produced by the coil and the resulting moment on the magnet are proportional to the current flowing through the coil. The glass beam is placed horizontally on several foam supports to reduce disturbances resulting from the mounting. The lateral displacement of the beam is measured with a laser Doppler vibrometer (Polytec OFV 303 sensor head, VD-02 velocity decoder), which can be moved in the axial direction to observe the wave propagation. For this purpose, the beam is fitted with reflective tape at several locations.

B. Pulse shape
For the shape of the focused pulse, the Ricker wavelet (also known as a "Mexican hat wavelet") is chosen because it has zero mean and a well-defined frequency content. The pulse width, and thereby the frequency content, is varied in multiple experiments. For the results presented here, a pulse width of σ = 0.1 ms is used, which corresponds to a center frequency of 2.5 kHz (see Fig. 2).

C. Phase velocity
For the energy focusing to work, the frequencydependent phase velocity of the propagating wave mode has to be characterized with high precision. Therefore, the phase velocity is measured for every beam by excitation of the transducer with a chirp and measurement of the resulting wavelet at different positions along the beam. The 044021-5 comparison of the phase of the measured wavelets yields the phase velocity. Subsequently, the phase velocity curve obtained with Timoshenko beam theory is fitted to the experimental data by adjustment of the Young's modulus E and the shear modulus G (Fig. 8). These two parameters, together with the geometry of the beam, suffice to describe the phase velocity of the first bending wave mode of the glass beam.

D. Measurement of the bending moment
To assess the wave-focusing procedure, the shape and amplitude of the bending-moment pulse at the focal point have to be determined. As the bending moment cannot be measured directly, it is derived from the measurement of the lateral surface velocity obtained with the laser Doppler vibrometer as follows.
As long as we are far enough from the ends of the beam, the deformation of the beam is governed only by the propagating wave mode, because the evanescent wave mode has decayed substantially. Thus, the deflection of the beam can be written as the superposition of a left-traveling wave and a right-traveling wave: Accordingly, the velocity normal to the surface and the bending moment arê which can be rewritten aŝ

E. Measurement of the transducer strength
To measure the amplification of the bending moment experimentally, both the bending moment at the focal point and the moment produced by the transducer have to be determined. The bending moment at the focal point can be derived from the lateral surface velocity as shown earlier [Eq. (16)]. However, this cannot be done for the 044021-6 moment produced by the transducer, as the electromagnetic coil obstructs the laser. Moreover, the evanescent mode is present close to the transducer, so the derivations presented before are not valid.
Instead, the strength of the transducer is determined by measuring the lateral surface velocity close to the transducer and deducing the moment acting on the magnet with a simulation. The same excitation signal is applied as the current in the experiment and as the external moment in the simulation. Subsequently, the measured and simulated lateral surface velocities at a certain point along the beam are compared. They differ only in a scaling factor (Fig. 9), because the moment on the spherical magnet is proportional to the magnetic field inside the coil, which is again proportional to the current through the coil. This scaling factor corresponds to the strength of the transducer in newton meters per ampere.

F. Measurement of the beam length
Aside from the phase velocity, the beam length is an important quantity that has to be measured to achieve precise wave focusing. If the length of the beam in the simulation does not match the exact length of the physical beam, the individual wavelets will not focus at the same location, which may lead to destructive interference.
To determine the length of the beam, a procedure similar to that for the measurement of the transducer strength is chosen. Again, the measured and simulated lateral surface velocities at a certain point along the beam are compared. Subsequently, the length of the beam is adapted iteratively in the simulation such that the arrival times of the direct wavelet, the first reflection, and the second reflection coincide in the simulation and the experiment. A typical comparison between simulated and measured lateral surface velocities is presented in Fig. 9, showing that very close agreement between the two can be achieved.

G. Results
In a set of experiments, a time-space map of the wave field inside the beam is constructed to observe the wave focusing. For these experiments, the entire length of the beam is fitted with reflective tape necessary for the laser Doppler vibrometer measurements and the beam is not sanded, so it would not break. The mensuration of the beam and the computation of the excitation signal are done as explained earlier. Subsequently, the beam is excited with the resulting excitation signal repetitively, and the lateral VIDEO 1. The fracturing of the glass beam (recording 2000 frames/s, replay 24 frames/s). The recording starts with the excitation of the beam (t = 0 ms), whereas the replay starts at t = 276 ms. The beam breaks at t = 349 ms. After breaking, the righthand side of the beam starts to move because it is supported by only one foam cushion and loses balance. The surface of the beam was sanded only in the vicinity of the focal point. In later experiments, larger areas of the beam were sanded. 044021-7 FIG. 11. Measurement of the lateral surface velocity at a distance x = 0.5 m from the transducer, while the waves focus at x = 0.65 m. For the blue curve, only 40% of the available power is used so as not to break the beam. Subsequently, the blue curve is scaled up by 2.5 to match the red curve, for which full power is applied to break the beam. Both curves match very well before the beam breaks at approximately t = 351 ms and differ considerably afterwards. Furthermore, waves that have much higher frequency content and result from the energy released during the fracture (acoustic emission) can be seen in the red curve. surface velocity is measured at 1-mm intervals along the beam. The resulting time-space map of the bending moment is depicted in Fig. 10. It shows how the bending waves, which are excited at x = 0 m between t = 0 ms and t = 357 ms, focus at x = 0.65 m and produce a strong bending-moment pulse.
In experiments with beams that had not been sanded, a maximum bending moment of 184 × 10 −3 N m is measured at the focal point when 30 reflections are taken into account. This is more than 20 times larger than the maximum bending moment the transducer is capable of producing (around 8.3 × 10 −3 N m). After the beams are sanded and thereby their bending strength reduced to about 117 × 10 −3 N m (measured in quasistatic four-point bending tests), we are able to induce dynamic fracture at the focal point as can be seen in Video 1 and Fig. 11.

IV. DISCUSSION AND CONCLUSION
In summary, we present a method to create a short, highpower bending-wave pulse in a beam at a position away from the transducer. The excitation signal is computed in a time reversed spectral-element simulation and consists of 31 wavelets, for which up to 30 reflections are taken into account. The 31 wavelets are composed with linearprogramming algorithms to obtain an optimal excitation signal. Thus, a bending-moment pulse with a maximum amplitude of 184 × 10 −3 N m is achieved in experiments, where a transducer capable of producing only 8.3 × 10 −3 N m is used. This corresponds to an amplification of the bending moment by a factor of more than 20 by wave focusing. Moreover, the amplitude of the bending-moment pulse proves to be large enough to induce dynamic fracture of the glass beam.
The position of the focal point can be set with very high accuracy as both the phase velocity of the flexural waves and the length of the beam are determined precisely. Larger inaccuracies in the measurement of these two quantities would lead not to imprecision in the position of the focal point but rather to a complete breakdown of the focusing method. This is because the individual wave packages would focus at different locations and at different instances of time, possibly leading to destructive interference.
The location of fracture, on the other hand, is dictated by the critical bending moment required to break the beam, and the amplitude and width of the bending-moment pulse. Moreover, the critical bending moment is not constant along the beam, due to the sanding. Hence, we can only guarantee the location of fracture to be inside a certain perimeter of the focal point, which is governed by the wavelength chosen. In the experiments, the beams broke within a distance of ±5 mm from the focal point, where the bending moment is still 95% of its peak value.
For all experiments presented here, the same pulse width with a center frequency of 2.5 kHz is used. However, it is worth mentioning that the wave-focusing method works in the complete frequency domain below the cutoff frequency of the second wave mode. The accuracy of the location of fracture could be increased by going to higher frequencies, thereby reducing the wavelengths and the size of the focal spot. Yet, dispersion is less pronounced at higher frequencies (see Fig. 8), and the transducer bandwidth and material damping may add further difficulties. By increase of the number of reflections, the amplification could be increased even further, but the gain due to additional reflections decreases, as more energy is lost through material damping. On the other hand, the number of reflections required to achieve dynamic fracture could be reduced by use of a stronger electromechanical transducer (e.g., by use of a more powerful current source).
By precisely modeling the propagation of flexural waves in a glass beam, we design and create a bending-moment pulse at an arbitrary location away from the actuator. With the same method, other quantities, such as shear force, deflection, or surface velocity, could also be tailored. What is more, the wave-focusing technique could be transferred to other dispersive waves (e.g., water waves, or light in dispersive media) to focus energy.
Applying the wave-focusing technique to induce dynamic fracture at an arbitrary location might be interesting for high-strain-rate bending tests in fracture mechanics. By setting the shape of the pulse, one is able to directly control the strain rate at the location of fracture. Moreover, no external loads are applied during the fracturing process. This eliminates the stress concentrations that typically occur in the vicinity of the indenter and the support in 044021-8 classic mechanical tests, such as three-and four-point bending [33].