PSI - Issue 39

Yuri Petrov et al. / Procedia Structural Integrity 39 (2022) 552–559 Yuri Petrov/ Structural Integrity Procedia 00 (2019) 000 – 000

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̈ + = ( ) (0) = ̇(0) = 0 ( ) ≥ ⇔

(13). Problem (13) can be solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , where and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (13) is the following: ( ) = { (1 − cos( )), ≤ (cos( ( − )) − cos( )), > (14), where = √ ⁄ is the oscillator eigen frequency. If the system failure is considered and fracture time is denoted by ∗ , equation ( ∗ ) = can be used to build dependence of the pulse duration on the fracture time ∗ for a fixed pulse amplitude. For a threshold case, when fracture takes place right when the loading ends ( = ∗ ) a point ∗ = (1 − ⁄ )⁄ is present on the ( ∗ ) graph (Fig. 2). All pulses with durations longer than the threshold result in the exact same fracture time ∗ yielding a vertical line on the ( ∗ ) graph. The fracture with a delay can be studied if > case from (2) is considered. For this case solution of the equation ( ∗ ) = yields the following ( ∗ ) dependence: = − 1 ( + ( ∗ ))+ ∗ (15). Analyzing (15) one can evaluate minimal pulse duration for a given amplitude , which corresponds to a maximal fracture time ∗ indicating maximal fracture delay: ∗ = 1 ( − ( 2 )) = 1 ( − 2 ( 2 )) (16a), (16b). Formulas (16a,b) impose natural limitation on the pulse amplitude due to restrictions on the argument which should belong the [−1,1] range. Thus, in order to cause the system failure, the pulse amplitude should be more or equal to a half of static critical force (a static force which would break the spring if static problem is considered): ≥ ⁄2 . The constructed ( ∗ ) dependence is shown in figure 2. This way, it is shown that if the linear oscillator failure is considered, some fundamental dynamic fracture effects, such as the fracture delay, can be observed. This is to attributed to inertia of the mass of the oscillator. In fact, the response of the oscillator to short pulse loads in terms of fracture is analogous to behavior of the crack subjected to pulse loading, since similar effects such as fracture delay can be observed for both systems. 4. Conclusions The incubation time approach to dynamic fracture was used to analyze crack initiation due to pulse load applied to the crack faces. It was found that the incubation time fracture criterion is able to predict particular dynamic fracture effects, such as the fracture delay and considerable variation of the starting value of the stress intensity factor. The conducted calculations show that the threshold loads are of the highest interest from the research perspective, since this type of loads help to unveil specific dynamic fracture effects.