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Lucas Long
Lucas Long

Crack Simulation Mechanical 2006 Crack

These models provided new insight into key parameters and driving forces, however, the micro-mechanical processes involved during dynamical crack propagation are still essentially unknown. While their direct observation is so far not feasible, the discrete element method (DEM) has previously been successfully used to study the influence of snow microstructure on the mechanical behavior of snow18,19,20,21 and crack propagation in weak layers15,22. Therefore, DEM is an appealing method to study the effect of the complex and highly porous snow microstructure on the dynamics of crack propagation, which does not require the assumption of a complex macroscopic constitutive model. DEM allows the generation of highly porous samples crucial to model snow failure and was, for instance, used to perform 2-D simulations of a PST yielding good agreement with field experiments. However, the oversimplified shape (triangular structure) and the 2-D character of the weak layer employed by Gaume et al.15 prevented a detailed analysis of the internal stresses during crack propagation.

crack Simulation Mechanical 2006 crack

Our aim is therefore to numerically simulate an exemplary experimental PST with a 3-D DEM model to better understand the micromechanics involved during dynamical snow fracture. We first present a method to evaluate the location of the crack tip, which is particularly challenging due to the closure of crack faces during propagation. Our model reproduced the experimentally observed displacement field, accelerations and crack propagation speed well. Furthermore, the model provides detailed insight into the micro-mechanical processes and stresses within the weak layer and allows us to identify the main drivers of crack propagation.

To accurately determine crack speed, it is crucial to define the position of the crack tip. This is not straightforward since we do not have an opening crack, but a closing (anti-)crack. Hence, we suggest four metrics for determining the position of the crack tip: (a) bond-breaking position, (b) normal displacement thresholds, (c) maximum stress, and (d) maximum normal slab acceleration (Fig. 2). Definition (a) is based on the number of broken bonds at a given time step. Starting from the end of the column, at each time step the crack tip is defined as the position where the number of broken bonds corresponds to 70% of the maximum number of broken bonds (Fig. 2a: diamond dots). Definition (b) is based on a normal displacement threshold as suggested by van Herwijnen and Jamieson9. For every subset, the time is recorded when the displacement exceeds the threshold value; to compute the speed, \(\Delta_d\) is equal to the discretization length (\(\Delta_x = 20 \textmm\)) and \(\Delta_t\) is the recorded time difference (Fig. 2b). Figure 2e shows the temporal evolution of the crack tip position defined with three different threshold values of normal displacement (three shades of yellow). Increasing the threshold value shifts the crack tip position in time. In definition (c), the crack tip is the position of maximum total stress (\(\sigma_tot = \sqrt \sigma_zz^2 + \tau_xz^2 ;\) Fig. 2c). Finally, in definition (d), the crack tip is the position of maximum normal acceleration as a function of time (Fig. 2d). Figure 2e, f shows good agreement between the four methods of crack tip definition; the crack speed between 1 m and the end of the column shows the same behavior regardless of the type of metric used. The crack tip location based on the maximum stress definition is ahead of the locations obtained with the other methods (Fig. 2e, f) in line with the assumption that stresses induce fracture. The crack tip definition based on the bond breaking position provides the maximum of information during the cutting phase. While the metric based on acceleration (d) was used to define the crack tip position, due to noise in particle acceleration shortly after the onset of crack propagation, it could not be used to reliably compute crack speed.

Comparison of experimental and simulated PST results. (a) Experimental (top) and simulated (bottom) displacement field norm (magnitude) when the crack tip reached 3.7 m (\(\leftu\left(x,z\right)\right=\sqrtu_x^2+u_z^2\)). The displacement field is colored from no-displacement in blue to 10 mm displacement in red. (b) Temporal evolution of the normal displacement averaged over the height. The colors represent the horizontal location and the black dashed line the corresponding location for the simulated PST. The red dashed line corresponds to the time when the crack tip reached 3.7 m, i.e. corresponds to the displacement field shown in (a). (c) Crack speed evolution along the PST beam. The grey line shows the experimental crack speed with its confidence interval (grey envelope) and the blue, violet and green lines the simulated crack speed. Crack speeds are computed from different crack tip definitions: grey and blue line based on displacement threshold of 0.2 mm, green line based on maximum stress and violet line based on the position of breaking bonds. (d) Temporal evolution of the normal slab acceleration for two locations [at 2 m and 3 m; line colors as in (b)].

During the simulation, we tracked micro-mechanical quantities: the position of bond-breaking events, normal displacement and acceleration as well as shear and normal stresses at the top of the basal layer. The temporal evolution of these quantities revealed six distinct sections during the fracture process (Fig. 4, labeled at the top from 1 to 5). In the following, we describe five of the six sections for the situation shown in Fig. 4, i.e. when the crack tip is at about at 2.3 m, corresponding to a simulation time of 0.344 s.

Mechanical parameters during crack propagation: snapshot at 0.344 s (after the start of the simulation). (a) Average normal slab displacement in blue and average normal slab acceleration in green. (b) Top view of the weak layer showing the bond states: in grey broken bonds, in red bonds that are breaking at current time step. The blue line shows the breaking bond distribution along the length of the beam. (c) Normal stress \(\sigma _zz\) and (d) shear stress \(\tau _zx\) along the length of the beam. The light red box highlights the fracture process zone (FPZ). The light yellow box highlights the part of the beam where the stresses are redistributed. The orange vertical dashed lines indicate the start and end of the different sections (1, 2, 3, 4, and 5).

When the crack speed was in a steady state (orange dashed lines in Fig. 5), we observed a constant length of the fracture process zone (3) and the elastic redistribution section (4). Figure 5b shows these section lengths as function of crack tip position, the lengths (2), (3) and (4) clearly exhibit a plateau at the same time confirming a steady state stress regime during crack propagation.

We developed a 3-D discrete element model to investigate the micro-mechanical processes at play during crack propagation in snow fracture experiments. Microscopic model properties were calibrated based on macroscopic snowpack quantities using the method developed by Bobillier et al.18. The field data of a PST fracture experiment was recorded during winter 2019 and analyzed with image correlation techniques11. The experiment provides bulk snow properties and the displacement field during crack propagation and allows studying mixed-mode failure of a porous weak snow layer. Our DEM model of the PST accurately reproduced the observed dynamics of crack propagation including the structural collapse of the weak layer. Moreover, our PST model provides insight into the micro-mechanics of the failure processes before and during self-sustained crack propagation.

For closing crack faces under compression (anticrack), the crack tip is not clearly defined and no formal and well-accepted definition exists. Hence, we introduced four different metrics to determine the position of the crack tip, based on slab displacement threshold, maximum normal slab acceleration, stress maxima and the distribution of breaking bonds (Fig. 2). We then computed the crack speed based on these crack tip definitions, resulting in very similar values (Fig. 3c). Over the length of the experiment, crack speeds exhibit two phases: an initial transitional regime with rapidly increasing crack speed, a steady-state regime between 1.6 m and 4 m. These phases were observed in both the numerical as well as the experimental PST.

The DEM model of the PST experiment allows insight into the micro-mechanical behavior of weak layer failure. We suggest six sections to describe the crack dynamics during a PST experiment; (1) sawing, (2) weak layer collapse, (3) fracture process zone, (4) elastic redistribution, (5) undisturbed (initial) stress state, and slab-substratum contact (6). We looked into three of these sections in more detail: (2) The structural weak layer collapse (crushing) where the stresses remain low and only a few bonds are breaking. (3) The fracture process zone where the material softens and most of the bonds are breaking and where the stress is maximal. (4) The elastic redistribution zone where the stresses are converging to the initial undistributed stress state and no bonds are breaking. During steady state propagation, we observed that these three sections travel along the beam keeping their behavior, which is defined by the geometrical and mechanical properties (Fig. 5; Supplementary Movie 2). Frame by frame, the stresses were analyzed and the results indicated a mixed-mode bond failure with a main normal stress component (Supplementary Movie 2). We also noted (not shown) that the PST width does not influence the crack tip morphology. Before reaching a steady state speed regime, we defined a transitional regime where we observed a decrease in the normal stress and an increase of the shear stress component.


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