# 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  
Let us consider the obtained experimental results from the point of view of the incubation time approach. In the case of relatively slow loading, the condition for the transition of a material to a new state may be a condition for achieving a critical value \(\sigma_{st}\) of a certain loading parameter \(\sigma\):  
\[\sigma\geq\sigma_{\text{st}} \tag{1}\]  
Such threshold characteristics (ultimate strength, yield strength, critical stress intensity factor and so on) are widely used in practical calculations and are included among the main characteristics of materials. Under conditions of high-speed and/or intense short-term loading, criteria like (1) become invalid. The threshold stress of failure or the transition of a material into a plastic state may exceed the quasi-static limit by several times. The question arises of how to find ways to describe the dynamic behavior of a material and, as a consequence, the search for parameters that can characterize the behavior of a material and can be determined from experiments. This way can be the incubation time approach [19; 20; 21; 22; 23; 24]. The approach allows for a smooth transition from quasi-static to dynamic loads. The main concept of this approach is that all structural processes in a material, be it failure or plastic flow, do not occur instantaneously, as is customary in quasi-static loads, but over a period of time. It is intuitively clear that the duration of these processes may be related to an intensity of a load, but the approach assumes that these durations can be characterized by a constant value, which is a parameter of the material. Figure 5: Critical stress dependencies on a strain rate under compression at room temperature (**a**) and splitting at a negative temperature (**b**) of bituminous binders. \(R^{2}\) is the coefficients of determination. The general form of the incubation time criterion, as well as its variants for various problems, can be found in [41]. To consider the critical stress competition effect in the experimental results at hand, it is enough to use the criterion in the form of:  
\[J(t)\geq J_{cr},\quad J_{cr}=\sigma_{st}\tau\quad\text{and}\quad J(t)=\int\limits _{t-\tau}^{t}\sigma(x,t\tau)dt\tau \tag{2}\]  
where \(J(t)\) is the stress pulse, \(J_{cr}\) is the critical stress pulse, \(\sigma(x,t^{\prime})\) is the stress at a given point \(x\) in the medium at time \(t\), \(\sigma_{st}\) is the quasi-static critical stress and \(\tau\) is the incubation time associated with the dynamics of structural processes preceding the observed event and characterizing, therefore, the duration of the preparatory process for this event. The time \(t_{cr}\) and place \(x_{cr}\) of the event are defined as the moment and the point (in the material volume) of the fulfilment of condition (2). Thus, condition (2) can be considered as a condition for input of a structural force pulse for a period \(\tau\) sufficient for structural transformations (for example, failure or plastic flow) in the elementary volume of a material. The parameters \(\sigma_{st}\) and \(\tau\) are parameters of the material, and \(\tau\) does not depend on the shape and duration of loading. Note again that despite the general notation, two cases are considered in this study. These cases need to be selected depending on the context of the problem. The first is when the failure occurs at the conditionally elastic stage of the stress-strain curve, that is, it is a case of brittle or quasi-brittle failure. Then, the quasi-static critical stress is the ultimate stress \(\sigma_{FS}\) and the incubation time is the incubation time of failure \(\tau_{F}\). The second is when the onset of plastic deformation is examined. In this case, the quasi-static critical stress is the yield stress \(\sigma_{YS}\) and the incubation time is the incubation time of plastic yield \(\tau_{Y}\).