PSI - Issue 35

Kai Friebertshauser et al. / Procedia Structural Integrity 35 (2022) 159–167

162

4

K. Friebertsha¨user and M. Werner and K. Weinberg / Structural Integrity Procedia 00 (2021) 000–000

3. Quantitative comparison of penny-shaped crack and pressure-induced fracture

In order to evaluate the peridynamic simulation of pneumatic fracture quantitatively, we compare here the three dimensional computational results with the analytical solution of a penny-shaped crack. The penny-shaped crack problem was first introduced by Sneddon and Mott (1946), who investigated the stress distribution in the vicinity of cracks in linear elastic solids and established a relationship between crack opening width w and internal pressure state p . According to Sneddon, the crack opening width w follows the equation

1 / 2  

1 / 2  

a 2 c 2

1 − 1 −

4 · p ( r ) · (1 − ν 2 ) π · E

c 2 − r 2

(13)

w ( r ) =

with crack radius c and pressure loaded radius a (see Fig. 2 a)). The pressure p ( r ) inside of the crack is assumed to be constant, described by p ( r ) = p 0 for 0 < r < a and p = 0 otherwise. Under the condition a = c , the crack opening width w is then

2 ) π · E

c 2 − r 2

4 · p 0 · (1 − ν

1 / 2

w ( r ) =

(14)

.

For our peridynamics calculation a cylinder with a diameter of 220 mm and a height of 100 mm is discretized with a uniform material point cloud of N = 302 946 points. In the initial configuration, the crack radius is c = 0 . 025 m. The compressive load is applied on two layers of material points by a force density b k . For this force density it applies (15) where n k is the normal direction, ∆ x the grid spacing, and the linearly increasing internal pressure p ( t ) = q 0 · t , where q 0 = 7 · 10 3 bar s − 1 with current time t [s]. The material and model parameters are given in Tab. 1. b k ( t ) = p ( t ) ∆ x n k

Table 1. Material and model parameters for penny-shaped crack computation Parameter Unit

Value

[kg m − 3 ]

Density ρ

2300

Young’s modulus E Poisson’s ratio ν Gri ffi th’s parameter G c

[MPa]

50000

[-]

0 . 2

1 ]

[Nm − [mm] [mm]

1

Grid spacing ∆ x

1 . 2

Horizon δ

6 . 820

number of points N

[-]

302946

Fig. 2 shows the results measured at a material point with local radius r = 18 mm. The analytical solution is determined according to Eq. (13) for each time step with the respective applied pressure p ( t ). The simulation result shows a very good agreement with the analytical solution for the first stages of crack growth. Due to the linear pressure increase, the crack width w also increases linearly. The deviations starting at a pressure of 10 bar can be attributed to the fact that the crack has spread to approximately twice the initial crack length, i.e., the configuration now di ff ers from the assumptions of the analytical solution. Another reason for the deviation lies in the assumptions of Eq. (13), which is determined for an infinite domain. With progressing crack growth, boundary e ff ects occur due to the finite domain, which is not considered in Sneddon’s analytical solution. It is possible to reduce the deviations by extending the model in the boundary regions, but this also significantly increases the computation time due to the higher number of material points. We also performed a small convergence study on grid spacing, where the number of points was gradually in creased while keeping the horizon constant. We found that as the number of points increases and the point families contain more members, the simulation results converge to a unique solution. The influence of the horizon concerning convergence is negligible if δ > 3 ∆ x , where ∆ x is the grid spacing. We conclude that the peridynamic model with a linear solid material allows for reliable and correct computation of pressure-driven crack growth and remark that the same results can be found for other material points.