PSI - Issue 28

Nikita Kazarinov et al. / Procedia Structural Integrity 28 (2020) 2168–2173 N. Kazarinov et al./ Struc ural Integrity P ocedia 00 ( 20) 000–000

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This way, one may conclude that fracture delay effect known from dynamic fracture of solids (Kalthoff and Shockey (1977)) can be observed on a very simple mechanical system. Depending on the loading pulse duration fracture may occur with or without delay. Moreover, minimal amplitude of the breaking pulse is twice smaller than the static ultimate force ! . 3. Dependence of strength on the loading rate In this section problem (1) will be considered for a case of a raising load namely when ( ) = , where coefficient indicates loading rate. Analogously to the previous case explicit solution can be written for the problem (1): ( ) = J − 1 ( )K , = F ⁄ (5). Analysis of equation ( ∗ ) = ! can be carried out to show that for relatively high loading rates fracture time is low and ∗ can exceed significantly static ultimate force ! and thus, dynamic branch (Bragov et al. (2017)) of the strength curve is present. On the contrary low values result in the spring break when load equals static spring strength: ∗ ≈ ! . This feature of the studied system lets one model (at least qualitatively) some known experimental data on sample strength depending on the loading rate using proper model calibration. Here experiments on crack initiation due to trapezoid pulse loading by Ravi-Chandar and Knauss (1984) will be discussed. In these experiments crack initiated and propagated in Homalite-100 specimens due to a trapezoid shape pressure pulse applied to the faces of the initial cut on the sample side. All the pulses had equal raising time – 25 µ s, but various amplitudes resulting in different loading rates. The authors measured the fracture initiation time ∗ and corresponding dynamic ultimate stress intensity factor (SIF) – 0 * and this way, 0 * ( ∗ ) dependence was experimentally constructed. This dependence can be approximately obtained using a simple linear oscillator model with appropriately calibrated parameters. The plane problem of an infinite plate with a semi-infinite cut with a uniform arbitrary load applied to the cut faces admits analytical solution. The problem was stated and solved in a number of works and here final results for plain strain conditions will be used. The tearing stress 11 on the initial crack continuation can be evaluated using Green’s function. Here a two-term asymptotic representation of the corresponding Green’s function is used (Bratov et al. (2011)): ( , ) = J √ √ + √ 2⁄3 K ( 5 − ) (6). In formula (6) variables and contain material data including velocities of elastic waves, is a Heaviside step function and 5 is the dilatational wave velocity for the studied material. Detailed expressions for and can be deduced using formulas presented in work by Bratov et al. (2011). The final values for these parameters based on the material data (table 1) are: = 9.5756, = −0.0051 . This way, normal stresses on the crack continuation can be expressed as the following convolution: 11 ( , , ) = _ ( , ) ( , − ) # 6 (7), where ( ) is load applied to the crack faces. In the considered case ( ) is a trapezoidal pulse with different amplitudes (Fig. 2). This way, force acting on a -sized square area near the crack tip is the following: ( , ) = _ 11 ( , , ) * 6 (8). If one virtually substitutes problem of strength of the cracked specimen by a problem of strength of a correspondingly calibrated oscillator model, the effect of strength – loading rate dependence can be shown using the model (figure 2). In order to calculate the virtual oscillator mass 7 a square area is virtually extruded in the crack tip vicinity and thus the mass can be calculated using expression 7 = 3 . The spring stiffness is chosen so that the model could yield acceptable results for very low loading rates (almost static loading conditions): 7 = ! ! ⁄ , where ! is a critical static force for the selected crack advancement distance , ! is the approximate crack opening (Sun (2011)) corresponding to the static crack initiation following Irwin’s static fracture criterion (Irwin (1957)) 0 = 0! and = 0.83 is a fitting parameter: