PSI - Issue 28

N.V. Mikhailova et al. / Procedia Structural Integrity 28 (2020) 2026–2031 / Structural Integrity Procedia 00 (2020) 000–000

2028

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3. Calculation results Modeling of spall fracture was carried out for the experiments performed in (Schuler et al., 2006). Spall fracture tests were carried out on cylindrical concrete rods. In experiments spalling occurs in several places, i.e. multiple spalling is realized. The properties of the material used in the modelling are presented in Table 1. The loading impulse was obtained by extrapolating the velocity of the free surface measured in (Schuler et al., 2006) that is shown in Fig. 1. The velocity was recalculated into stress using the relation ����� � � �����������

Table 1. Material parameters. Property

Value

2500 � �� � 4080 3.24 10

Density

Speed of sound

Static tensile strength

Incubation time

Fig. 1 Free surface velocity history of concrete (Schuler et al., 2006).

The spalling modelling was performed for a striker speed of 7.6 m/s where the specimen experienced multiple fracture. The length of the rod was 250 mm. Using the method presented in the work, the time of fracture condition fulfillment was calculated depending on the cross section of the sample. The minimum was reached in the section �∗ � � ����� which was taken as the first location of the fracture. To determine the next fracture location, it is necessary to take into account changes in the stress field. A new surface, formed after spalling, splits the loading impulse. Fig. 2(a) schematically shows the position of the impulse in the sample at the fracture moment. Under the x-axis there is a compression wave that moves towards the free surface. The part of the loading impulse that reflected from the free surface is located above the x-axis and impacts the sample by tensile loading. The dashed line corresponds to the spall section along which the sample divided into two parts because of fracture. Thus, parts of the impulse after fracture are located on opposite sides of the spall section In the frame of elasticity of the problem, an idealized simplified model is considered and it is assumed that no energy is lost when the impulse is separated by the spalling surface. After the fracture event, the impulse part colored in blue will move towards the loading surface of the sample. The orange part of the impulse will reflect off the new surface with a tensile effect. The gray part of the pulse remains in the spall layer and will be re-reflecting inside it. Obtained time dependences of stress are shown in Fig. 2(b) and 2(c) calculated for two sections. Here, positive value of stress corresponds to tensile loading. Dashed lines show compression and tension waves, and the solid line corresponds to the stress obtained by the addition of these waves. Fig. 2(b) shows the stress in the section located to the left of the spalling surface. The lower graph (Fig. 2(c)) demonstrates the stress at the point located in the spall layer.