*3.1. Plasma Pressure Model*

For more realistic simulation of LIW, it is hypothesized that incorporating the transient plasma pressure generated due to laser ablation will provide for improved modeling. Therefore, an axisymmetric Gaussian spatial laser pulse profile (fitted to the measured spatial profile), as well as the measured temporal profile of the laser beam pulse (Figures 4 and 5), were applied in conjunction with the 1D hydrodynamic plasma pressure model by Fabbro et al., which was developed for use in laser shock peening simulation [41]. In this model, during the heating phase, the transient plasma pressure, *<sup>P</sup>*(*t*), is a function of shock impedance *Z*, plasma thickness *<sup>L</sup>*(*t*), and laser intensity *<sup>I</sup>*(*t*), as shown in Equations (1)–(3).

$$\frac{2}{Z} = \frac{1}{Z\_1} + \frac{1}{Z\_2},\tag{1}$$

$$\frac{dL(t)}{dt} = \frac{2P(t)}{Z},\tag{2}$$

$$I(t) = P(t)\frac{dL(t)}{dt} + \frac{3}{2\alpha} \frac{d[P(t)L(t)]}{dt},\tag{3}$$

where *Z*1 and *Z*2 are shock impedances of glass and aluminum, respectively, and α is the fraction of internal energy that dissipates into heat. Coupling Equations (2) and (3) gives Equation (4) [37].

$$\frac{d^2L(t)}{dt^2} = \frac{I(t)}{c\_1L(t)} - \frac{c\_1 + c\_2}{c\_1} \left(\frac{dL(t)}{d(t)}\right)^2 \frac{1}{L(t)},\tag{4}$$

where *c*1 = 3*Z* 4α and *c*2 = *Z* 2 .

As soon as the laser is switched o ff (at time t = τ = full width, half maximum (FWHM) of the laser pulse), an adiabatic cooling phase is initiated during which plasma pressure decreases with time as shown in Equation (5).

$$P(t) = P(\pi) \left( \frac{L(\pi)}{L(t)} \right)^{\circ},\tag{5}$$

where γ is the adiabatic constant. Parametric constants for modeling of the plasma pressure are listed in Table 2.

**Table 2.** Parameter values used in plasma pressure modeling [37,42], with permission from ASME, 2019.


Using a substitution, the second-order ordinary di fferential Equation (4) is rewritten as a system of first-order di fferential equations and solved numerically using the ode45 solver function in MATLAB ® software. Using this solution (in terms of plasma thickness) in Equation (5) and incorporating the measured laser pulse temporal distribution (see Figure 5), the variation of peak plasma pressure over time for an axisymmetric Gaussian spatial pressure was obtained, as shown in Figure 8.

**Figure 8.** Variation of peak plasma pressure in time.
