# Development of Energy-Based Impact Formula--Part I: Penetration Depth
## 3 Development of New Energy-Based Penetration Formula
### Concept of New Formula
#### 3.1.4 Properties of Materials Affected by Strain Rate  
In this impact study, the material properties are related to the strain rate under the huge short-term load (Figure 5). The material under dynamic loading exhibits properties different from those exhibited under quasi-static loading, and the yielding stress and elastic modulus of the material increase with the strain rate [38; 39]. As the material strength increases rapidly at the high strain rate, it is essential to study and consider the strain-rate effect in impact load studies. In particular, the strain rate is closely related to the velocity of a projectile, and is very sensitive and difficult to predict. As it is difficult to study these matters separately, the strain rate value is referenced from [PERSON] [40], and the properties of materials affected by the strain rate are referenced from the _fib_ Model Code 2010 [32]. Strain rate is an important factor for evaluating the impact resistance and is dependent on the material properties of the projectile and the target. For a general estimate of the dynamic properties of materials, the Split-Hopkinson Pressure Bar (SHPB) and Kolsky Bar are used, which exhibit a difference. The SHPB indicates the performance of compression experiments, whereas the Kolsky Bar covers various properties such as compression, tension, and torsion [40]. Thus, this study refers to the Kolsky Bar. When imposing impact loading on the input bar, an incident pulse (\(\varepsilon_{l}\)) is generated,after which that bar transfers the incident pulse at its one end to the specimen connected to the other end. The incident pulse first reaches the test specimen, after which it is separated into a reflected pulse (\(\varepsilon_{R}\)) and a transmitted pulse (\(\varepsilon_{T}\)), as shown in Figure 6. The particle velocity (\(v_{1}\)) at interface 1 is defined as Equation (67) and that at interface 2 (\(v_{2}\)) is defined as Equation (68). \[v_{1}(t)=c_{b}(\varepsilon_{I}-\varepsilon_{R})=\sqrt{\frac{E_{b}}{\rho_{b}}}( \varepsilon_{I}-\varepsilon_{R}) \tag{67}\]  
\[v_{2}(t)=c_{b}\varepsilon_{T}=\sqrt{\frac{E_{b}}{\rho_{b}}}\varepsilon_{T} \tag{68}\]  
where \(v_{1}\) and \(v_{2}\) are the particle velocity at interfaces 1 and 2 (m/s), respectively; \(\varepsilon_{I}\) is the incident pulse; \(\varepsilon_{R}\) is the reflected pulse; \(\varepsilon_{T}\) is the transmitted pulse; \(c_{b}\) is the bar wave speed; \(E_{b}\) is the elastic modulus of a bar (MPa); and \(\rho_{b}\) is the density of an input bar (kg/m\({}^{3}\)). The mean axial strain rate (\(\dot{\varepsilon}_{s}\)) in the specimen can be obtained from these pulses, the specimen length, and the elastic modulus and density of the input bar (Equation (69)). \[\dot{\varepsilon}_{s}=\frac{1}{l_{0}}\sqrt{\frac{E_{b}}{\rho_{b}}}(\varepsilon _{I}-\varepsilon_{R}-\varepsilon_{T}) \tag{69}\]  
where \(l_{o}\) is the length of a specimen (mm); \(E_{b}\) is the elastic modulus of the bar (MPa); and \(\rho_{b}\) is the density of a bar (kg/m\({}^{3}\)). Figure 5: Strength of brittle materials affected by strain-rate (redrawn from [PERSON] et al. [38]). Figure 6: Wave propagation diagram (redrawn from [PERSON] [40]). Given that the initial length (\(l_{o}\)) of the specimen is known and the input and output bars are maintained under elastic conditions, the normal forces at interfaces 1 and 2 can be derived as Equations (70) and (71), respectively. In addition, the mean nominal axial stress (\(\sigma_{s}\)) in the specimen is derived using \(P_{1}\) and \(P_{2}\) (Equation (72)).