## 2 Theoretical Framework of the Pull-Back Method  
The dynamic tensile strength of materials at high strain rates is often investigated using spalling techniques, where tensile fracture is induced by the reflection of a compressive stress wave from a free surface. The pull-back method, based on one-dimensional stress wave theory, is commonly employed to determine the spall strength from the free surface velocity history.  
For a homogeneous elastic bar of density \(\rho\) and [PERSON]'s modulus \(E\), the propagation of a longitudinal stress wave can be described by the one-dimensional wave equation:  
\[\frac{\partial^{2}u}{\partial t^{2}}=C_{0}^{2}\frac{\partial^{2}u}{\partial x^{ 2}} \tag{1}\]  
where \(u(x,t)\) is the axial displacement at position \(x\) and time \(t\), and \(C_{0}=\sqrt{E/\rho}\) is the longitudinal wave speed. The general solution to this equation, according to [PERSON], consists of forward and backward traveling waves:  
\[u(x,t)=f(x-C_{0}t)+g(x+C_{0}t) \tag{2}\]  
From this, particle velocity \(v=\partial u/\partial t\) and axial stress \(\sigma=E(\partial u/\partial x)\) can be derived.  
The interaction of these waves governs the stress state within the specimen. When an incident compressive wave (C) travels through the specimen and reflects from the free end, it becomes a tensile wave (T). The superposition of the incident compressive wave and the reflected tensile wave leads to a region of high tensile stress within the material, as illustrated in the x-t diagram (Figure 1). If this tensile stress exceeds the material's dynamic tensile strength, spall fracture occurs.  
Spall fracture creates a new free surface within the specimen, generating unloading waves that propagate towards both the impact end and the original free end. The unloading wave reaching the original free surface causes a characteristic deceleration or \"pull-back\" in the free surface velocity. This pull-back velocity, \(\Delta v_{pb}\), is defined as the difference between the peak free surface velocity (\(v_{peak}\)) and the first minimum (rebound velocity, \(v_{rebound}\)) after the peak, as shown in Figure 2.  
\[\Delta v_{pb}=v_{peak}-v_{rebound} \tag{3}\]  
Figure 1: Schematic x–t diagram illustrating stress wave propagation and spall phenomenon: (C) incident compressive wave, (T) reflected tensile wave, and the spall event. Unloading waves (\(U_{L},U_{R}\)) are generated from the spall plane.  
Based on an acoustic approximation and assuming elastic behavior between the spall plane and the free surface, [PERSON] et al. [19] proposed a relationship to calculate the spall strength \(\sigma_{spall}\) directly from \(\Delta v_{pb}\):  
\[\sigma_{spall}=\frac{1}{2}\rho\text{C}_{0}\Delta v_{pb} \tag{4}\]  
This formula, often referred to as the pull-back method, is widely used due to its simplicity and direct reliance on the easily measurable free surface velocity. The application of Equation (4) relies on certain ideal conditions, such as one-dimensional wave propagation and material homogeneity. The validity of these conditions for unidirectional reinforced concrete (UDRC) systems is a key focus of this study and will be assessed using numerical and experimental methods.