## 3 Experimental Setup and Testing
### Dynamic Setup for High Strain Rate Tests
#### 3.1.1 Optimization of Samples for Dynamic Tests  
The geometries of the samples used in dynamic tests basically consist of modifications of those used in quasi-static tests, with the intention of adapting the sample to the equipment available for the dynamic testing described in the previous section. In fact, the dynamic test samples must meet the geometrical requirements for being mounted on the experimental facility (by means of M12 threads), and they must have the proper lengths and section sizes in order to be successfully tested up to failure with the displacements and forces allowed by the adopted SHB. An example of the adapted sample geometries is reported in Figure 2.  
An inverse optimization procedure was numerically implemented in order to define the geometric parameters of the samples reported in Figure 2, which would allow us to obtain the desired stress and strain evolutions in the sample. The targets of such optimization are:  
* average strain rate of about 2000 s\({}^{-1}\) for all three geometries;
* equivalent plastic strain rate at the specimen core that is as constant as possible;  
Figure 1: (**a**) Scheme of the direct Split Hopkinson bar, (**b**) direct Split Hopkinson bar facility [33].  
Figure 2: Shapes of test samples used in dynamic tests and FEM models used for their optimization.  
* triaxiality at the specimen core that is as constant as possible;
* much higher triaxiality in the round notch specimen (RN) than in the round bar (RB) and plane strain specimens (PS).  
The input parameters to be optimized are gauge length \(L_{0}\), fillet radii \(R\), section diameter \(D\) for the RB and RN specimens, thickness \(t\) for the PS specimen, and impact speed or, equivalently, incident wave amplitude \(\sigma_{i}\).  
The optimization is conducted by launching several finite element simulations with the ANSYS(r) commercial FE code. The mesh and boundary conditions are also illustrated in Figure 2. The RB and RN sample models are 2D axisymmetric, while the PS sample model is 2D plane strain. The bottom edges of the models representing the symmetry plane are constrained in the axial direction, while a positive axial displacement is applied to the top edge. Since the behavior of the material at high strain rates is not, the plastic flow curve of the material is assumed to be represented by the quasi-static curve given in the previous section, increased by a tentative dynamic correction factor of 1.08, accounting for the possible strain rate sensitivity. More details on the numerical procedure, especially about the method used for generating a reasonable displacement history, are given in a previous work by some of the authors [43].  
A full factorial Design of Experiments plane has been simulated for each of the three geometries by varying the input parameters on several levels, for a total of approximately 450 simulations. The output parameters of each simulation are the time averages of the plastic strain rate and of the triaxiality, as well as their variance over time, and the maximum equivalent plastic strain that can be reached with the given incident wave amplitude. For instance, Figure 3 illustrates the trends of the average triaxiality, the average strain rate and the maximum equivalent plastic strain for the RB, RN and PS geometries, reconstructed as continuous response surfaces (second-order polynomial) that best fit the results of the 450 simulations.  
Figure 3: Response surfaces (the black markers refer to the selected geometries).  
In order to select the best combination of input parameters, a cost function has been defined as follows:  
\[cost=\sum_{g}\Bigg{\{}\frac{\sqrt{\sum_{t}\big{[}T_{g}(t)-\overline{T_{g}}\big{]}^ {2}}}{\overline{T_{g}}}+\frac{\sqrt{\sum_{t}\big{[}\tilde{\varepsilon}_{g}(t)- \overline{\tilde{\varepsilon}_{g}}\big{]}^{2}}}{\overline{(\tilde{\epsilon})} }\Bigg{\}}\cdot\Bigg{\{}\frac{2\overline{T}}{\big{|}\overline{T_{RB}}+ \overline{T}_{PS}-2\overline{T}_{RN}\