Computational modeling of dynamic fracture process in brittle materials

M.Sc. Sahir Nawaz Butt

Various models based on discrete and continuum numerical representations of fracture propagation at various spatial scales have been proposed since the 1970s, however a complete theory able to describe the process of dynamic fracture quantitatively still does not seem to be available. Nucleation and propagation of even a single crack couples various physical mechanisms on widely varying spatial and temporal scales. The question of crack propagation speed is of significant importance for understanding the dynamic fracture process. As the elastic energy released at the crack tip has to travel through the crack surfaces, the maximum speed at which a crack can propagate is dictated by the speed limit of surface waves, i.e. the Rayleigh wave speed. However, crack propagation experiments performed in brittle amorphous materials revealed that the crack speed does not even reach half of the theoretically predicted value. The aim of this project is to understand and model the dissipation mechanisms, such as crack surface roughening, microbranching, and macro crack branching, involved during dynamic fracture process in amorphous brittle materials.

Crack propagation in Plexiglass plates
We investigated the ability of the peridynamic model to reproduce the features observed in dynamic fracture experiments [1, 2]. These features include the onset of micro-branching instability, crack surface topology obtained at different crack speeds, material toughening with increasing crack velocity as well as the limit of crack velocity below the Rayleigh wave speed for mode-I cracks. Three dimensional peridynamic analyses of dynamic propagation of single cracks in PMMA plates subjected to quasi-static loads are carried out (Fig. 1). The results obtained from the simulations are in a good qualitative agreement with the experiments [3, 4, 5, 6].

Figure 1: (a) Crack propagation path and (b) Crack speed with respect to stored elastic energy (per unit area).

Crack surface topology
Simulations are able to reproduce the transition from a single crack propagating at lower crack speeds, to the development of micro-branches from the main crack with the increasing crack velocity. This transition from a single crack to an ensemble of micro-branches leaves distinct features on the crack surfaces, as can be seen in Fig. 2. Simulations are able to reproduce the mirror, mist and hackle transition of the topology of the fracture surfaces observed in experiments (Fig. 3).

Figure 2: (a) 3D crack propagation path and (b) Average crack propagation speed with respect to stored elastic energy (per unit area).

Figure 3: Comparison of crack surface topology obtained from simulations with experiment [7].

Influence of specimen geometry
The effect of the specimen geometry, i.e. specimen thickness and height, on crack dynamics is analyzed. These two dimensions of the specimen affect the crack dynamics in different ways. On one hand, the thickness of the specimen causes the crack dynamics to change because of the three dimensional effects. A single flat crack surface in a 3D plate has a curved crack front and the curvature of this crack front is governed by the thickness of the plate. This curvature increases the fully damaged zone, i.e. the length of the crack front, which results in a lower local dissipation rate. On the other hand, the height of the specimen affects the interaction of the stress waves reflecting from the boundaries. The fracture patterns obtained for specimens with varying height are shown in Fig.4. It can be seen that the crack propagation velocity (Fig.4b) and the microbranching frequency (Fig.4c) are increased with decreasing specimen height. This is because the number of elastic wave reflections increase (along the horizontal boundaries) as the specimen height is reduced.

Figure 4: (a) Crack paths obtained from specimens with three different heights, while loaded at a constant stored elastic energy per unit area, (b) Crack propagation speed, and (c) Correlation between the length of microbranches and their frequency along the main crack.


M.Sc. Sahir Nawaz Butt
+49 234 / 32-29062


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  • [2] Fenghua Zhou, Jean-Francois Molinari, and Tadashi Shioya. A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials. Engineering fracture mechanics, 72(9):1383–1410, 2005.
  • [3] S. Butt and G. Meschke. A rate-dependent damage model for prediction of high-speed cracks. Proceedings in Applied Mathematics and Mechanics (PAMM), 2018.
  • [4] S. Butt and G. Meschke. Peridynamic investigation of dynamic brittle fracture. PAMM, 19(10.1002/pamm.201900180), 2019.
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  • [6] Sahir N Butt and Günther Meschke. Peridynamic horizon-effects on wave dispersion and crack propagation velocity. In Proceedings of the COMPLAS 2019, Bsrcelona, Spain, 2019.
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