# Strain Rate Dependencies and Competitive Effects of Dynamic Strength of Some Engineering Materials
## 4 Discussion
### Calculation of Critical Stress Dependencies on Strain and Loading Rates
Since quasi-brittle fracture and plastic flow occurs on a portion of the stress-strain curve, which can be approximated by a linear function, simple analytical expressions can be obtained for calculating the strain rate dependences of critical stresses. The change of stress in the material can be represented as a linear function in time \(\sigma(t)=\sigma H(t)=E\epsilon H(t)\), where \(E\) is the modulus of elasticity, \(\sigma\) is the stress rate, \(\varepsilon\) is the strain rate, \(t\) is the time and \(H(t)\) is the Heaviside step function. By substituting this function into criterion (4), one can find the onset time \(t_{cr}\) of failure or plastic deformation and the corresponding critical stress \(\sigma_{\sigma}\):  
\[t_{cr}(\varepsilon)=\left\{ \begin{array}{ll}\frac{\tau}{2}+ \frac{\sigma_{st}}{E\varepsilon}\\ \sqrt{\frac{2\sigma_{st}\tau}{E\varepsilon}}\end{array} \right.\quad \text{and}\quad\sigma_{\sigma}(\varepsilon)=\left\{ \begin{array}{ll}\sigma_ {st}+\frac{E\varepsilon\tau}{2},&t_{cr}\geq\tau,\\ \sqrt{2\sigma_{st}E\varepsilon\tau},&t_{cr}<\tau.\end{array} \right. \tag{3}\]  
It should be noted that with the exception of \(\tau\), all parameters in condition (2) can be determined from the experiments indicated in previous sections. The incubation time can be determined using an approximation of calculated curves to experimental points with a minimal deviation. In addition, the incubation time can be determined from the magnitude of the critical pulse received for the time to the process start less than the incubation time. However, the method for determining the incubation time directly from one experiment is still at a development stage. For example, some ideas are discussed in [24; 41]. Observable experimental dependencies can be calculated using Equation (3). The quasi-static strength was determined as the average value of the test results at quasi-static loading. The dynamic strength (the incubation time of failure) was determined semi-empirically using the nonlinear approximation via the Levenberg-Marquardt algorithm, which is a combination of [PERSON]'s method and a gradient descent method. The parameter \(\tau\) was selected in such a way that theoretical curves (Equation (3)) fit the experimental points with minimal deviation in all considered ranges of stress/strain rates. The calculated curves and the corresponding coefficients of determination \(R^{2}\) are shown in Figures 6-9. The material parameters used for the calculation are presented in Tables 1-3. The identified change of the ranking of the materials for the maximum critical stress during the transition from quasi-statics to dynamics coincides with the ranking of the failure incubation times of these materials (see Tables 1-3). For example, the saturated limestone has a quasi-static strength less than that of the frozen limestone, but the incubation time of failure of the saturated limestone is higher than that of the frozen material. This means that under dynamic loads, the saturated limestone is more durable than the frozen limestone. Figure 6: Critical failure stresses in quasi-brittle materials versus strain rate under compression (**a**) and stress rate under splitting (**b**). Curves are calculated by Equation (3). \(R^{2}\) is the coefficients of determination. Figure 7: Critical failure stresses in mortar (**a**,**b**) and limestone (**c**,**d**) versus strain rate under compression (**a**,**c**) and stress rate under splitting (**b**,**d**). Curves are calculated by Equation (3). \(R^{2}\) is the coefficients of determination. The arrows in Figure 8 indicate the region of the realization of the critical stress competition effect. As in the case of critical stresses for the quasi-brittle materials the ratio of yield strengths of the materials and their structural states at high strain rates can be estimated using the incubation times of plastic yield (see Table 2). As mentioned above, the quasi-brittle failure behavior of the bituminous binders was observed only for a range of dynamic loads.