*2.4. Pressure Model*

In this work, the laser-matter interaction and subsequent plasma formation were not modeled directly; rather, the resulting pressure pulse was applied to the FE model over the area corresponding to the laser pulse as a spatially-uniform, time-varying surface traction. While small transition regions likely exist along the spot perimeters, measurements by the LP vendor indicate the generated pressures are essentially uniform across the spot area.

The confined ablation model developed by Fabbro et al. [42], modified to account for the absorption fraction of the laser energy in the treated plasma, was used to determine the applied pressure *P*(*<sup>x</sup>*,*<sup>t</sup>*) = *P*(*t*). We can express the energy density at the plasma, *E*(*t*), to the nominal energy density of the laser, *I*(*t*), using the expression *E*(*t*) = *A*(*t*)*I*(*t*), where *A*(*t*) = 1 − *R*(*t*) is the absorbed energy and *R*(*t*) represents the change in energy density. However, owing to the very short time between the laser pulse and the plasma formation, it is generally assumed that the loss *R*(*t*) is negligible and hence *A*(*t*) is approximately 1.0 [43–46].

During the heating phase (while the laser is on), this relationship is then described by:

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

while during the cooling phase (after the laser is switched o ff):

$$P(t) = P(\tau) \left( \frac{L(\tau)}{L(t)} \right)^{\vee} \tag{3}$$

Here, *L*(*t*) is the thickness at time *t* of the interface between the ablative coating and the transparent overlay, given by:

$$\frac{dL(t)}{dt} = \frac{2}{Z}P(t) \tag{4}$$

and α is the fraction of the internal energy transferred to the workpiece, γ is the adiabatic cooling rate, and *Z* is the e ffective acoustic impedance of the interface, defined by:

$$\frac{2}{Z} = \frac{1}{Z\_{\text{coating}}} + \frac{1}{Z\_{\text{overlap}}} \tag{5}$$

*<sup>Z</sup>*coating and *<sup>Z</sup>*overlay are the acoustic impedances of the ablative coating and the transparent overlay, respectively. With water as the overlay ( *<sup>Z</sup>*overlay ≈ 0.15 × 10<sup>6</sup> g cm<sup>−</sup><sup>2</sup> s<sup>−</sup>1) and aluminum tape as the coating ( *<sup>Z</sup>*coating ≈ 1.7 × 10<sup>6</sup> g cm<sup>−</sup><sup>2</sup> s<sup>−</sup>1), *Z* has a value of about 0.3 × 10<sup>6</sup> g cm<sup>−</sup><sup>2</sup> s<sup>−</sup>1.

Assuming a Gaussian laser energy density *E*(*t*), a typical pressure profile resulting from these equations is shown in Figure 2 for a maximum energy density of 3 GW/cm2, laser pulse width of 18 ns, α = 0.09, and γ = 1.3. For these parameters, the maximum applied pressure, *P*max is about 1.95 GPa and the width (FWHM) of the pressure pulse is about 47 ns.

**Figure 2.** Typical temporal profile of the applied peening pressure.
