*2.3. Material Model*

During the laser peening process, the induced shock waves can produce very high strain rates, on the order of 10<sup>6</sup> s<sup>−</sup>1, resulting in a material response that is significantly di fferent from that under static or quasi-static loading conditions [35]. To capture these strain rate e ffects, a Johnson–Cook [36] material model was used. Following earlier work [11], thermal e ffects resulting from the LP process are considered to be negligible and thus temperature dependence is eliminated from the material model:

$$\overline{\sigma} = \left[ A + B \left( \overline{\varepsilon}^p \right)^n \right] \left[ 1 + C \ln \left( \overline{\varepsilon}^p / \dot{\varepsilon}\_0 \right) \right] \tag{1}$$

Here, σ is the flow stress; ε*p* is the e ffective plastic strain; . ε *p* is the e ffective plastic strain rate; . ε0 is a reference plastic strain rate (typically taken to be 1.0 s<sup>−</sup>1); *n* is the work hardening exponent; and *A*, *B*, and *C* are empirically-derived constants.

Lesuer [37] used split-Hopkinson-bar testing to measure the deformation response of Al 2024-T3 at moderately high strain rates (103–104 s<sup>−</sup>1), and used the data to define the material constants in Equation (1) (Table 1). Although the strain rates typically experienced during LP are an order of magnitude higher, recent flyer-plate testing has confirmed that Lesuer's constants reasonably model the LP material response, with a maximum deviation of less than 10% as compared to the measured values [38]. Comparisons between predicted residual stresses based on a Johnson–Cook material model and measured values have also validated the use this mode for LP simulations [39,40]. Thus, the Johnson–Cook material constants for Al 2024-T3 as shown in Table 1 were used for all simulations in the current work.

**Table 1.** Johnson–Cook material constants [37].


It should be noted that the constitutive model was not updated in the multi-shot simulations to account for LP-induced cyclic deformation e ffects [41].
