# Non-Contact and Real-Time Measurement of Kolsky Bar with Temporal Speckle Interferometry
## Principles  
### Principle of the Kolsky Bar Based on the Velocity Method  
As shown in Figure 1, a typical Kolsky bar includes a striker bar, incident bar, transmitter bar, specimen, and strain gauge 1 and strain gauge 2 which are pasted at \(X_{A}\) of the incident bar and \(X_{B}\) of the transmitter bar, respectively. Figure 1: A typical Kolsky bar. Before the measurement, the specimen is placed between the incident bar and transmitter bar. Using a hammer or other tools, the strike bar is driven so that it hits the incident bar at a certain speed, thereby generating an incident wave. The specimen is deformed at high speed under the loading of the incident wave, propagates the reflected wave into the incident bar, and propagates the transmitted wave into the transmitter bar. The incident wave and the reflected wave are recorded by strain gauge 1, and the transmitted wave is recorded by strain gauge 2. The dynamic mechanical properties of the specimen can be solved from the records of these waves by theoretical calculation. If the length of the strike bar is \(L_{0}\), the impact velocity is \(\
u\), then its duration \(T_{M}\) is:  
\[T_{M}=\frac{2L_{0}}{C_{0}} \tag{1}\]  
The magnitude of the incident pulse is:  
\[A_{M}=\frac{\rho_{0}C_{0}\
u}{2} \tag{2}\]  
where \(\rho_{0}\) is the density of the bars. \(C_{0}\) is the longitudinal speed of wave in the bar, and can be expressed as:  
\[C_{0}=\left(\frac{E_{0}}{\rho_{0}}\right)^{1/2} \tag{3}\]  
where \(E_{0}\) is the elastic modulus of the bars. If the four parameters include stress \(\sigma(X_{1},t)\) and \(\sigma(X_{2},t)\), particle velocity \(\
u(X_{1},t)\) and \(\
u(X_{2},t)\) can be measured, then the dynamic mechanical properties of the specimen can be calculated according to Formula (4)-(6). \[\sigma(t)=\frac{A_{0}}{2A_{S}}[\sigma(X_{1},t)+\sigma(X_{2},t)]=\frac{A_{0}}{ 2A_{S}}[\sigma_{I}(X_{1},t)+\sigma_{R}(X_{1},t)+\sigma_{T}(X_{2},t)] \tag{4}\]  
\[\dot{\epsilon}\left(t\right)=\frac{\
u(X_{2},t)-\
u(X_{1},t)}{L_{S}}=\frac{ \
u_{T}(X_{2},t)-\
u_{I}(X_{1},t)-\
u_{R}(X_{1},t)}{L_{S}} \tag{5}\]  
\[\varepsilon(t)=\int_{0}^{t}\dot{\epsilon}\left(t\right)dt=\frac{1}{L_{S}}\int _{0}^{t}[\
u_{T}(X_{2},t)-\
u_{I}(X_{1},t)-\
u_{R}(X_{1},t)]dt \tag{6}\]  
where \(\sigma(t)\) is stress, \(\dot{\epsilon}\left(t\right)\) is strain rate, \(\varepsilon(t)\) is strain. \(L_{S}\) is the length of the specimen. \(A_{0}\) and \(A_{S}\) are the cross-sectional area of the bar and specimen, respectively. \(\sigma_{I}(X_{1},t)\) is incident wave, \(\sigma_{R}(X_{1},t)\) is reflected wave, and \(\sigma_{T}(X_{2},t)\) is transmitted wave. \(\
u_{I}(X_{1},t)\), \(\
u_{R}(X_{1},t)\) and \(\
u_{T}(X_{2},t)\) are incident velocities, reflected velocities, and transmitted velocities, respectively. According to the assumption, both the one-dimensional stress waves of the bar and the short specimen strain are evenly distributed over its length, and the three waves have the following relationship:  
\[\varepsilon_{I}+\varepsilon_{R}=\varepsilon_{T} \tag{7}\]  
where \(\varepsilon_{I}\), \(\varepsilon_{R}\) and \(\varepsilon_{T}\) is strain corresponding with incident pulse, reflected pulse, and transmitted pulse, respectively. Moreover, there is a linearly proportional relationship between strain, stress, and particle velocity [10] in the elastic region of the Kolsky bar:  
\[\sigma(X_{1},t)=\sigma_{I}(X_{1},t)+\sigma_{R}(X_{1},t)=E_{0}[\varepsilon_{I} (X_{1},t)+\varepsilon_{R}(X_{1},t)] \tag{8}\]  
\[\sigma(X_{2},t)=\sigma_{T}(X_{2},t)=E_{0}\varepsilon_{T}(X_{2},t) \tag{9}\]  
\[T_{M}=\frac{2L_{0}}{C_{0}} \tag{1}\]0  
\[T_{M}=\frac{2L_{0}}{C_{0}} \tag{1}\]1In summary, the stress, strain, and strain rate of the specimen can be expressed [11] as follows:  
\[T_{M}=\frac{2L_{0}}{C_{0}} \tag{1}\]2  
\[T_{M}=\frac{2L_{0}}{C_{0}} \tag{1}\]3  
\[T_{M}=\frac{2L_{0}}{C_{0}} \tag{1}\]4  
where \(\
u_{T}\) and \(\
u_{R}\) are transmitted and reflected velocities as measured, respectively.