PSI - Issue 61

Igor Gribanov et al. / Procedia Structural Integrity 61 (2024) 89–97 I.Gribanov et al. / Structural Integrity Procedia 00 (2024) 000–000

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Fig. 3. (a) The fracture (red) is initiated at the interior node of the intact material (white), splitting the center node and adding two additional nodes (grey) at the opposite edges. Four new elements are created. (b) The fracture is initiated at the edge of the mesh. Two elements are inserted. (c) The fracture path follows the existing edge by splitting the central node without inserting new elements.

Fig. 4. (a) Neighbourhood of the crack tip where substepping is performed (shown in magenta). (b) Redistribution of the bending moment a ff ecting a neighborhood around the crack during its propagation.

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Fig. 5. Evaluation of traction across a potential split. (a) Path integrals are considered on two sides of the split along the paths ∂ Ω 1 and ∂ Ω 2 . (b) In the discretized setting where each element has a constant stress tensor, the integrals are evaluated as sums. (c) In the presence of a boundary, the integration path is modified to extend from the boundary to the potential split. Reproduced after Pfa ff et al. (2014).

3.2. Tensile strength fracture criterion

The fracture algorithm is similar to the implementation by Pfa ff et al. (2014). After solving the equation of mo tion (1), a copy of the nodal displacements u n is preserved. The fracture algorithm may displace nodes, but these changes are reverted when the algorithm finishes. The net result of the algorithm is the topological change of the mesh, not the change in displacements. The decision to initiate or propagate the cracks is based on the fracture criterion, which is evaluated for each node. When the conditions are met, the surrounding elements of the node are split in the appropriate direction. Several new nodes and up to four new elements are inserted, depending on the type of split (Figure 3). After each split, the equation of motion is solved in the small neighborhood of the crack tip – within a 10-node radius (Figure 4a). Temporarily restricting the simulation to the local area allows accounting for the redistribution of stresses in the crack’s vicinity leading to accurate prediction of propagation direction. Moreover, the time step is temporarily reduced by a factor of 10 4 , which allows to resolve the rapid propagation of the crack. The corresponding redistribution of the bending moment is shown in Figure 4b. The fracture criterion is a critical part of the algorithm and is roughly based on Rankine’s (maximum tensile stress) theory. The criterion requires the evaluation of stress values at the nodes of the mesh. However, nodes connect multiple elements, each of which has its own stress tensor. Simply averaging the surrounding stress tensors yields incorrect results. Instead, the traction is evaluated as a path integral on two sides of the tentative split line (Figure 5). Referring to Figure 5a, the traction is evaluated as: 2 q k = ∂ Ω k σ · nˆ dS , (2)