*2.5. Numerical Model*

The residual stresses induced by laser peening were computed by finite element analyses in the commercial software Abaqus/Explicit v6.12 (Dassault Systemes, Paris, France). Several models can be found in the literature to characterize the material response in the context of laser shock peening. The Johnson–Cook constitutive model considers the effect of large strains and high strain rates on material behavior, and was applied in an explicit/implicit finite element model by the authors of [18]. Ivetic et al. [8] optimized the non-linear response through full-explicit numerical analyses, taking into account the actual overlapping of the peening shots. Nevertheless, the material response to the shock waves propagation can be thought of as a condition of high-strain rate cyclic loading. The effect of the cyclic deformation induced by laser peening was already studied by Angulo et al. [19]. In this regard, a nonlinear isotropic/kinematic hardening plastic model, which also includes the change of properties due to the Bauschinger effect, can be used to simulate the material behavior. This model consists of isotropic hardening (Equation (1)) plus a nonlinear kinematic hardening component. The back stresses in kinematic hardening can be computed by Equation (2), according to Chaboche [20].

$$
\sigma\_Y = \sigma\_0 + Q\_\infty \left[ 1 - \exp\left(-b\varepsilon^{pl}\right) \right] \tag{1}
$$

$$d\alpha = \mathbb{C}d\varepsilon^{pl} - \gamma \alpha |d\varepsilon^{pl}| \tag{2}$$

The material properties of Al 6082-T6 were selected according to the works by Chen et al. [21] and Chen et al. [22], which present thorough measurements of the hardening parameters, including testing at high strain rates via a split-Hopkinson tension bar. The maximum strain rate measured in the tests was around 3400 s<sup>−</sup>1, which is lower than that occurring during laser peening (i.e., in the order of 10<sup>6</sup> s<sup>−</sup>1), according to the authors of [15]. However, Al 6082-T6 is reported to have a low sensitivity on the strain rate. Moreover, a study by Langer et al. [23] investigated the effects of using conventional test data to model the laser peening process; they concluded that consistent results could be obtained with strain rates in the order of 10<sup>3</sup> s<sup>−</sup><sup>1</sup> (i.e., similar to those encountered in the literature of [21,22]). The hardening parameters input into the material model are summarized in Table 1.


**Table 1.** Kinematic hardening parameters for Al 6082-T6.

Instead of direct modelling of the laser and the plasma layer, the equivalent pressure of the plasma was considered, following the approach described by the authors [18]. As a result of the low energy of the laser pulses and the short pulse duration, the thermal effect on the material is negligible compared to the effect of shock waves, as reported by the authors of [24]. The simulation is split into the following two steps: first, pressure is applied in the peened area and the waves propagate elastically in the material; second, the load and constraints are released and residual stresses develop, following material relaxation. An excessive computational time would have been required to model multiple laser shots. As a result, only one single shot was modelled, with a spot size equivalent to the treated area of the specimens.

The pressure exerted by the plasma depends on the laser parameters, namely, wavelength, power density, and pulse duration. The laser spot was simulated by applying a pressure load with a triangular temporal profile and uniform spatial distribution. The triangular temporal profile approximates the results reported in the literature [25] for a water confined plasma, with a maximum pressure of 4 GPa attained after 25 ns, which decays to zero at 50 ns.

The geometry of the finite element model reproduces that of the real coupons, cut on the two sides so that the total width is 48 mm. The size of the peened region is 40 × 4 mm2. The mesh size is not homogeneous, as shown in Figure 8, as thin elements are needed in the thickness direction to capture the residual stress gradient at the notch; the minimum element is 0.13 × 0.20 × 0.03 mm3. Clamped boundary conditions are applied at the edges of the specimen, and the bottom surface is constrained against displacements.

**Figure 8.** Finite element model of the specimen. Built-in constraints at the bottom edges are shown. The midline path is highlighted in red and its origin is denoted by the letter O.
