## 3 Numerical Simulation of UDRC Spallation
### Stress State Analysis in UDRC and Comparison with Plain Concrete  
The internal stress state within UDRC and PC specimens was meticulously investigated to assess the validity of the one-dimensional stress assumption. The analysis focused on radial distributions of stress components and derived stress state parameters at a specific cross-section located \(535.5\,\mathrm{mm}\) from the impact end. These distributions were captured at the time instant when the axial compressive stress at this section reached its maximum magnitude, assuming an elastic material response throughout this pre-spall phase. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Density (kg/m\({}^{3}\))** & **Poisson’s Ratio** & **Tensile Strength/MPa** & \(\mathbf{B_{1}}\) & \(\mathbf{B_{2}}\) & \(\mathbf{B_{3}}\) & \(\mathbf{\Omega}\) & \\ \hline
2379 & 0.2 & 4.6 & 1.6 & 5.5 & 1.15 & 0.5 & \\ \hline \(A_{0}\) & \(A_{1}\) & \(A_{2  
Two key parameters are used to characterize the stress state:  
* Stress Triaxiality (\(\mathbf{\sigma_{m}}\)/\(\mathbf{\sigma_{eq}}\)): The ratio of mean stress to von Mises equivalent stress. For ideal uniaxial compression, this ratio is \(-1/3\). * Lode Parameter (\(\mathbf{L}\)): Defined as \(L=\cos(3\theta)\), where \(\theta\) is the Lode angle. For ideal uniaxial compression, \(L=-1\). Comparative Radial Distribution of Stress Components: Figure 4 presents the comparison of radial distribution profiles for axial stress (\(\sigma_{zz}\)) and secondary stress components (radial \(\sigma_{rr}\), hoop \(\sigma_{\theta\theta}\), and shear \(\tau_{rz}\)) between UDRC and PC specimens. As observed in Figure 4a, the axial compressive stress (\(\sigma_{zz}\)) profiles in the concrete matrix region of the UDRC specimen essentially coincide with those in the plain concrete (PC) specimen, maintaining a stable magnitude of approximately \(-42\) MPa throughout the bulk of the concrete. The secondary stress components in the UDRC specimen (series ending in '_UDRC' in Figure 4b-d) are most pronounced near the rebar-concrete interface (\(r\approx 3\) mm). Even at their peak magnitudes near the interface--approximately \(-3.1\) MPa for radial stress, \(-2.9\) MPa for hoop stress--these are less than \(8\%\) of the \(-42\) MPa axial stress. These secondary stresses exhibit a rapid, approximately exponential decay with increasing radial distance, diminishing to about \(10\%\) of their respective interface peak values by a radial position approximately \(5\) mm from the rebar surface. In contrast, for the plain concrete specimen, the corresponding secondary stresses are generally negligible across the radius, typically remaining below \(0.1\) MPa. Consequently, in the bulk concrete region of UDRC (\(r>8\) mm), its secondary stresses are very small and comparable to the already low levels in plain concrete. Comparative Radial Distribution of Lode Parameter and Stress Triaxiality: Figure 5 illustrates the radial profiles of the Lode parameter (\(L\)) and stress triaxiality (\(\sigma_{m}\)/\(\sigma_{\alpha q}\)) for both UDRC and PC specimens. The LODE parameter profiles (Figure 5a) show that for plain concrete, \(LODE\) is consistently very close to -1 across the radius. For UDRC, \(LODE\) is also predominantly around \(-1\) in most of the concrete matrix. A slight deviation occurs near the rebar-concrete interface (\(r\approx 3\) mm to \(8\) mm from axis), where \(LODE_{UDRC}\) reaches approximately \(-0.94\), still indicating a state very close to uniaxial compression. Similarly, stress triaxiality profiles (Figure 5b) for plain concrete remain very close to \(-1/3\). For UDRC, the stress triaxiality essentially coincides with the PC profile in the bulk concrete. Figure 3: Prescribed velocity history applied as the input boundary condition at the left end of the FEM specimen model. The internal stress state within the UDRC and plain concrete (PC) specimens was meticulously investigated to assess the validity of the one-dimensional stress assumption. The analysis first considered the peak compressive phase of the wave.