The thermal and elastic properties of materials are dependent on temperature, with an increase in temperature generally resulting in a decrease in both material density and stiffness. A uniformly heated sample of a titanium alloy (Ti-6Al-4V) that is assumed to be homogeneous and isotropic is considered first. The temperature-dependent density, elastic modulus, and Poisson’s ratio are obtained from the software JMatPro32 and are used to determine the longitudinal, \({c}_{L}\), and shear, \({c}_{T}\), wave speeds as a function of temperature. These values are, in turn, used to calculate the Rayleigh wave speed using the characteristic equation33. The result is shown in Fig. 1 where the Rayleigh wave speed is seen to decrease from 3000 m/s at room temperature (293 K) to 1880 m/s at the melting temperature of 1943 K. The decrease is relatively monotonic outside of a small region between 1100 and 1275 K, the temperature range at which Ti-6Al-4V undergoes an hcp (α) → bcc (β) phase transformation34. In the case of uniform heating, the thermal field and elastic properties are not depth-dependent and the Rayleigh waves propagate without dispersion.
Rayleigh wave propagation becomes more complex in the case of transient heating, such as that produced by a high power laser source, as the thermal field and temperature-dependent elastic properties are functions of both time and space. Furthermore, the thermal properties are also temperature-dependent, and, at sufficiently high heating powers, the material will undergo a phase transformation resulting in surface melt. Here, the elastic displacement response generated by a nanosecond pulsed laser source incident upon a surface that is being heated by a spatially uniform continuous wave (CW) laser with a step-function time dependence is calculated. First, the temperature field produced by a CW laser is calculated using the implicit finite difference method presented by Singh and Narayan35. Temperature-dependent thermal properties and density are included for both the solid and liquid Ti-6Al-4V phases36–39. These properties are summarized in Appendix A. The model is used to determine the temperature as a function of time (t) at each depth (d) below the surface. Figure 2(a) shows the surface temperature (at d = 0) as a function of time where the heating laser is turned on at t = 0 and a heating laser power density of 6 kW/cm2. The surface temperature rises until it reaches the melting temperature of 1943 K at a time of 465 ms, where it briefly remains until the net heat absorbed exceeds the latent heat of the phase change35,40. The melt front then begins to propagate into the material and, as shown on the right axis of Fig. 2(a), proceeds rapidly to a depth of over 35 µm. Figure 2(b) shows the full extent of the calculated thermal data, with the color bar indicating the temperature rise at each depth and heating time.
Surface acoustic waves are confined to propagate in the near surface region with a penetration depth on the order of the wavelength. For example, for a frequency of 30 MHz, surface waves in Ti-6Al-4V will be sensitive to mechanical property changes that occur over a depth of approximately 100 µm. In Fig. 2(b), the temperature rise is somewhat uniform over the near surface region, and thus it is expected that the mechanical property changes will also be relatively constant over the penetration depth, resulting in a surface acoustic wave delay that is frequency independent. The surface temperature for a significantly higher heating power density of 250 kW/cm2 is shown in Fig. 2(c). Here, surface melting occurs at about 300 µs and there are marked thermal gradients in the near surface region within the 500 µs time window as shown in Fig. 2(d). These thermal gradients can cause dispersion of surface waves since the higher frequency waves, with a shorter wavelength, will be more influenced by the near surface region while lower frequency waves will penetrate further into the cooler substrate.
This model allows for the calculation of temperature as a function of depth and melt front position at any time after the heating laser is turned on. Next, the pulsed laser excitation and interferometric detection of the CW laser-heated surface at a given time is modeled. The material near the surface is discretized into 400 layers, with a layer thickness of 0.6 µm. The elastic properties of each layer are calculated from the mean temperature of the layer, and the elastic wave propagation problem is then reduced to an analogous problem of wave propagation in a homogeneous, isotropic layered media41. When surface melting occurs, the thickness of the surface layer is set as the thickness of the melt pool and the density38 (3920 kg/m3) and longitudinal wave velocity39 (4407 m/s) of liquid Ti-6Al-4V are used. Laser generation of ultrasound in plates42–44 and in multi-layer plates on a semi-infinite substrate45 have been previously addressed by others. The approach presented by Cheng et al. is followed in which the excitation laser source is represented as an equivalent elastic boundary source (Gaussian in space with a 10 ns pulse width), and the transfer matrix technique is used to enforce the continuity of stress and displacement across all homogeneous and isotropic layer boundaries41. The problem is solved in cylindrical coordinates using the integral transform technique where a Hankel transform of the elastic wave equation is taken with respect to the radial coordinate (r) and a Laplace transform is taken with respect to time. The normal surface displacement as a function of time at a given r is found through numerical inversion of the Hankel-Laplace transforms.
The excitation laser spot size was set to 100 µm full width at half maximum (FWHM) with a detection location at r = 1.0 mm. The normal surface displacement as a function of time is given in Fig. 3(a) for a heating power of 6 kW/cm2. The top curve shows the room temperature response in the absence of CW laser surface heating. A small wave amplitude arrival corresponding to the surface skimming longitudinal wave is seen at 0.16 µs followed by the larger surface acoustic wave (SAW) amplitude arrival at about 0.30 µs. The other curves show the displacement response at various times after the CW heating laser is turned on. For the signals between t = 0 and t = 450 ms, the shape of the surface acoustic wave remains relatively uniform, but the arrival is delayed as heating proceeds. During the last three time steps: t = 508.5, 517.0, and 525.6 ms, melting has occurred with melt depths of 6.0, 16.8 and 27.0 µm, respectively. More prominent dispersion is seen when the surface waves traverse the molten layer; the higher frequency, short wavelength, components are delayed due to the strong interaction with the melt layer. Note that in this case, the longitudinal wave velocity in the molten liquid is higher than the shear wave velocity in the substrate so the surface waves are not leaky46. In general, the velocity of waves propagating on a liquid-covered half space transition from the Rayleigh wave velocity at zero thickness to the Scholte wave velocity when the thickness of the liquid is large with respect to the wavelength47 − 48.
Figure 3(b) shows the evolution of the displacement field calculated throughout the heating time. Here the abscissa gives the time after the heating laser is turned on while the ordinate gives the time after the excitation laser pulse. The color bar represents the normal displacement of the surface. In this image, the SAW arrival has the largest negative amplitude and is shown in red. The initial pronounced change in the SAW arrival time is associated with the rapid rise in the near surface temperature as seen in Figs. 2(a) and 2(b). The SAW arrival time is then relatively constant between 75 and 100 ms during which the α → β phase transformation in Ti-6Al-4V occurs. After this transition region, the arrival of the SAW continues to be delayed with heating time in a monotonic fashion until approximately 508 ms when a sharp break in the curve associated with surface melting is observed. The SAW signal sensitivity to the presence of melt makes it an attractive option for sensing melt depth.
Figure 3(c) shows the normal displacement of the surface for a higher heating power density, 250 kW/cm2. The excitation source characteristics are the same as those given above. In this case, however, the heating takes place much more rapidly and surface melting starts at about 305 µs. Such rapid heating leads to strong near-surface thermal gradients (see Figs. 2(c) and 2(d)) which, in turn, lead to sharp changes in the mechanical properties within the wavelength range of the broadband surface acoustic wave. At heating times between t = 0 and t = 300 µs, a significant degree of surface acoustic wave dispersion is evident, with the higher frequency components that probe the near-surface temperature delayed with respect to the lower frequencies that penetrate further into the cooler bulk of the material. This effect is more pronounced at later times (t > 300 µs) where the higher frequency SAW components are also delayed by the presence of surface melt. Figure 3(d) shows the temporal evolution of the displacement field with surface heating. While the dispersion is certainly more pronounced than in Fig. 3(b), the onset of surface melting is not as evident. Note that the dispersion of the SAWs in a multilayer system can be used to back out the depth-dependent mechanical properties using a model-based inversion approach49. For the heating case, depth-dependent mechanical properties could ultimately be related to the subsurface temperature profile.