PSI - Issue 61

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

93

I.Gribanov et al. / Structural Integrity Procedia 00 (2024) 000–000

5

Fig. 6. (a) Normal stress estimated with equation (5) vs. fracture angle. Red dots indicate sector boundaries. (b) Tangential vs. normal components of traction - a plot analogous to Mohr’s Circle. The shape of the plot is nearly circular due to little variation in the stress values of the surrounding elements. (c) Center node and the adjacent sectors for which the plots were created.

where k = 1 , 2 denotes the side of the split, σ is the stress tensor and nˆ is the outward-pointing normal to the integration path. In the discretized setting where σ is constant on a finite number of intervals, expression (2) is identical to the sum: 2 q 1 = σ 0 ( t 0 − s ) + N − 1 i = 1 σ i ( t i + 1 − t i ) + σ N ( − s − t N ) , (3) where t i are normals to sector boundaries, σ i denote per-sector stress tensors, and s is the normal vector to the potential split (Figure 5b). If the center node is located on the boundary (Figure 5c), the integration path is modified to extend from the boundary to the plane of the fracture, and the result is doubled to reflect that the pressure is applied only to the half-plane: q˜ k = ∂ Ω k σ · nˆ dS . (4) The tensile component of the traction is extracted by projecting the di ff erence of the tractions q 2 , q 1 onto the normal of the fracture path s ( ϕ ): σ n ( ϕ ) = ( q 2 − q 1 ) · s ( ϕ ) , (5) where ϕ is the angle of tentative fracture direction. The fracture angle ϕ is selected to maximize the adjusted tensile stress σ n ( ϕ ). For the interior points of the planar mesh the function σ n ( ϕ ) is periodic with the period π (Figure 6a). Plotting the tangential component of traction τ against the normal component σ n produces the analogue of Mohr’s Circle (Figure 6b). The fragment of the mesh from which these plots were recorded is shown in Figure 6c. Fracture is initiated when the value of σ n ( ϕ ) exceeds the tensile strength of the material. The fracture algorithm avoids creating degenerate elements by snapping the cut to the nearest edge when needed. Once the crack reaches a free end, or if the propagation stops, the time step returns back to normal and the simulation proceeds with the new mesh topology.

4. Results

The described fracture algorithm was implemented in custom C ++ code and tested in several loading scenarios. In nature, ice is typically driven by wind and water currents, and by surface waves. Accurate modeling of such forces is beyond the scope of this article, so the simplified spring-type loads are used instead. Simulation parameters that are common across all tests are summarized in Table 1. The value for Young’s modulus of ice is taken from the field test measurement (Gribanov et al. (2019)), whereas the values of density and Poisson’s ratio are taken from Sanderson (1988).

4.1. Pure tensile load

Before moving to complex bending loads, the fracture algorithm is tested in pure in-plane tension. A corresponding natural event would be the breakup of a large floe under the action of wind and water currents (but not waves). In-plane