PSI - Issue 61

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

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

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Fig. 2. Five degrees of freedom describing the displacement field.

Floe-size distribution (FSD) data is collected in laboratory experiments, field observations, and satellite imagery, and varies with season and sea conditions (Gherardi and Lagomarsino (2015); Herman et al. (2018); Perovich and Jones (2014); Toyota et al. (2011)). Such observations and analyses play an important role in the study of climate. The process of floe formation plays a crucial role in the forecasting of river ice breakup (Beaton et al. (2019); Mahabir et al. (2006)). Spring breakup of river ice is investigated from the perspective of its environmental impact, and its hazards in the form of flooding, damage to bridges and other infrastructure (Beltaos (1997)).

2.3. Modeling with plates

This work is mainly inspired by an achievement in the physics-based simulation of fracturing thin sheets by Pfa ff et al. (2014), which addresses the tearing and cracking of paper, foil, plastic, and other materials. The geometry is represented by a two-dimensional mesh, on which the equations of motion are solved implicitly. Finite elements respond as linearly elastic membranes, and plasticity can be enabled optionally. Bending sti ff ness is reproduced by linear hinges inserted between the elements (Bridson et al. (2003)). The simulation uses a unique fracture criterion that also evaluates the direction of crack propagation. Formulations of the FEM exist for plates with high sti ff ness (Gal and Levy (2006)). A renowned approach is the discrete Kirchho ff theory (DKT) element (Batoz et al. (1980)). Another useful formulation is based on the Mindlin Reissner plate theory and leads to a computationally simple yet accurate shell element. Such an element combines the properties of a plate and a membrane, allowing to account for stresses caused by bending, shear, and stretching. Comparisons of the various plate and shell formulations can be found in the literature (On˜ate (2013)). This work uses the implicit dynamic FEM formulation in the Lagrangian form, with the HHT- α integration method. The discretized equation of motion is M ¨ u n + 1 + ˜ D = α f n ( u n , ˙ u n ) + (1 − α ) f n + 1 ( u n , ˙ u n , ¨ u n + 1 ) , (1) where u is the global displacement vector, M is the mass matrix, ˜ D is the damping matrix, α is the parameter of the integration scheme, f n and f n + 1 are the external forces at steps n and n + 1. The non-linear equation (1) is solved with Newton-Raphson method to determine the displacement vector u n + 1 at time step n + 1. The nodes of the deformable sheet have five degrees of freedom – three Cartesian components u x , u y , u z , and two in-plane rotations θ x ,θ y (Figure 2). Having two rotation angles is su ffi cient, because the floe is expected to remain in horizontal orientation. The nodal displacement vector is expressed as u k = [ u x , u y , u z ,θ x ,θ y ] T , where k is the index of the node. 3. Formulation

3.1. Plate elasticity

For a detailed discussion of the Reissner-Mindlin plate theory and the derivation of strain-displacement matrices the reader is referred to the relevant literature (On˜ate (2013)). The formulation makes assumptions about the distribution of stress values in the plate. To be used in the fracture criterion, the stress components are calculated on the exterior surfaces of the plate. These tensors have di ff erent values – when a plate is bent, one side gets compressed while the other gets stretched. Additionally, bending moments are evaluated for the purpose of visualizing the results.