2.5.2. Constitutive Material Model

For shock produced by laser-induced high strain rate (i.e., up to 10<sup>6</sup> s<sup>−</sup>1), the appropriate behavior law of the treated target must consider the effect of the high strain rate. In this context, the Johnson–Cook constitutive model [20] was used. In this model, the Von Mises yield criterion is defined as:

$$
\sigma = (\sigma\_{\mathcal{Y}} + K \varepsilon\_p^n) \left( 1 + \mathbb{C} \ln \left( \frac{\dot{\varepsilon}}{\varepsilon\_0'} \right) \right) \left( 1 - \left( \frac{T - T\_0}{T\_{mclt} - T\_0} \right) \right). \tag{1}
$$

The first part describes the strain-hardening effect. The second part characterizes the strain rate effect. The last part of the Johnson–Cook constitutive law associates the stress with material

temperature evolution during the plastic deformation. *<sup>σ</sup>y* is the yield stress, *B* is the hardening modulus. It encompasses the second member of the above mentioned equation, *p* is the equivalent plastic deformation, *n* is the hardening coefficient, *C* is the strain rate sensitivity parameter, ˙ is the strain rate during the process, ˙ 0 is the reference strain rate, *Tmelt* is the fusion temperature, *T*0 is the room temperature, *E* is Young's modulus, and *ν* is Poisson's ratio.

In the present study, a preliminary simulation using CAX4RT elements (Continuum, A 4-node thermally coupled axisymmetric quadrilateral, bilinear displacement and temperature, reduced integration, hourglass control) confirmed that the thermal softening effect could be neglected (for the case of laser intensity *I* = 2.7 GW/cm2, plasma pressure FWHM equal to 18 ns, and spot size of 3 mm). In fact, the local temperature increases due to the plastic deformation and the wave propagation did not have a significant influence on the rear free velocity. For this reason, the thermal part in Johnson–Cook's constitutive law was neglected [21]. For pure aluminum, Table 1 [22] gives the Johnson–Cook parameters used for the simulation.

**Table 1.** Parameters used for the Johnson–Cook material model with an aluminum target (99.00%) [22].


### 2.5.3. Spatial and Temporal Pressure Profiles

To generate spatial and temporal pressure profiles *P* = *f*(*<sup>x</sup>*, *y*, *t*), a VDOLAD subroutine was used. The *P*(*t*) profiles given in Figure 5 [23] were adjusted to provide coherence between the experimental and the numerical results. The *<sup>P</sup>*(*<sup>x</sup>*, *y*) distribution was obtained (Figure 6) from beam analysis. The intensity profile from Figure 2b, obtained through a camera, was used to generate the *<sup>P</sup>*(*<sup>x</sup>*, *y*) distribution. Previous work on the subject showed the equivalence between maximum pressure and intensity [18,24]. Preliminary simulation showed that modulation had no effect on the rear free-velocity profile.

**Figure 5.** *P*(*t*) used for the simulation.

**Figure 6.** *<sup>P</sup>*(*<sup>x</sup>*, *y*) distribution simulation obtained from the intensity profile from Figure 2b.

### **3. Results and Discussion**
