Create citation alert 1742-6596/1368/4/042039 Abstract The study is focused on the application of different approaches for initial crack propagation angle determination. Copper plate with the central crack under complex mechanical stresses (Mode I and Mode II loading) is studied by extensive molecular dynamics simulations using EAM potential. In addition, the complete Williams expansion for the crack tip fields containing higher-order terms is used. Crack propagation angle is obtained by 1) multi-parameter fracture mechanics approach based on three fracture mechanics criteria, MTS, maximum tangential strain and SED; 2) atomistic modeling for mixed-mode loading of plane medium with the central crack. From our simulations we can derive crack propagation directions and crack angles. MD calculations were run for three different values of the mixity parameter: 0.4, 0.5 and 0.6. Calculated values of crack angles were -51:5 deg, -46:6 deg and -42:2 deg accordingly. All the fracture criteria tested give similar values of crack growth angle for different values of the mixity parameter. It is shown that initial crack propagation angles given by both approaches are very close, especially when higher order terms of Williams series expansion for stress/displacement field description are taken into account. Thus, one can conclude that the criteria of classical continuum mechanics MTS and SED can give satisfactory predictions for crack initiation direction. Crack propagation direction angles given by conventional fracture mechanics reasonably agree with the angles obtained from molecular dynamics simulations.
We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the obtained fracture dynamics to that from cracks propagating in sintered polymethylmethacrylate (PMMA) interfaces. In previous works, it has been demonstrated that such an approach could reproduce the mean advance of fractures and their local front velocity distribution. Here, we go further by showing that the proposed model also quantitatively accounts for the complex self-affine scaling morphology of crack fronts and their temporal evolution, for the spatial and temporal correlations of the local velocity fields and for the avalanches size distribution of the intermittent growth dynamics. We thus provide new evidence that an Arrhenius-like subcritical growth is particularly suitable for the description of creeping cracks.
Complete dynamics crack
In the physics of rupture, understanding the effects that material disorder has on the propagation of cracks is of prime interest. For instance, the overall strength of large solids is believed to be ruled by the weakest locations in their structures, and notably by the voids in their bulk samples1,2. There, cracks tend to initiate as the mechanical stress is concentrated. A growing focus has been brought on models in which the description of the breaking matrix remains continuous (i.e., without pores). There, the material disorder resides in the heterogeneities of the matrix3,4,5,6,7,8,9. The propagation of a crack is partly governed by its spatial distribution in surface fracture energy, that is, the heterogeneity of the energy needed to generate two opposing free unit surfaces in the continuous matrix10, including the dissipation processes at the tip11. From this disorder, one can model a rupture dynamics which holds a strongly intermittent behaviour, with extremely slow (i.e., pinned) and fast (i.e., avalanching) propagation phases. In many physical processes, including12,13,14 but not limited15,16,17,18 to the physics of fracture, such intermittency is referred to as crackling noise19,20. In the rupture framework, this crackling noise is notably studied to better understand the complex dynamics of geological faults21,22,23,24,25, and their related seismicity.
Over the last decades, numerous experiments have been run on the interfacial rupture of oven-sintered acrylic glass bodies (PMMA)26,27,28. Random heterogeneities in the fracture energy were introduced by sand blasting the interface prior to the sintering process. An important aspect of such experiments concerns the samples preparation, which allows to constrain the crack to propagate along a weak (disordered) plane. It simplifies the fracture problem, leading to a negligible out-of plane motion of the crack front. This method has allowed to study the dynamics of rugous fronts, in particular because the transparent PMMA interface becomes more opaque when broken. Indeed, the generated rough air-PMMA interfaces reflect more light, and the growth of fronts can thus be monitored.
Different models have successfully described parts of the statistical features of the recorded crack propagation. Originally, continuous line models4,5,20,29 were derived from linear elastic fracture mechanics. While they could reproduce the morphology of slow rugous cracks and the size distribution of their avalanches, they fail to account for their complete dynamics and, in particular, for the distribution of local propagation velocity and for the mean velocity of fronts under different loading conditions. Later on, fiber bundle models were introduced6,30,31, where the fracture plane was discretized in elements that could rupture ahead of the main front line, allowing the crack to propagate by the nucleation and the percolation of damage. The local velocity distribution could then be reproduced, but not the long term mean dynamics of fronts at given loads. One of the most recent models (Cochard et al.8) is a thermally activated model, based on an Arrhenius law, where the fracture energy is exceeded at subcritical stresses due to the molecular agitation. It contrasts to other models that are strictly threshold based (the crack only advances when the stress reaches a local threshold, rather than its propagation being subcritical). A notable advantage of the subcritical framework is that its underlying processes are, physically, well understood, and Arrhenius-like laws have long shown to describe various features of slow fracturing processes26,32,33,34,35,36. In particular, this framework has proven to reproduce both the mean behaviour of experimental fronts37 (i.e., the average front velocity under a given load) and the actual distributions of propagation velocities along these fronts8, whose fat-tail is preserved when observing cracks at different scales38. It has recently been proposed39,40 that the same model might also explain the faster failure of brittle matter, that is, the dramatic propagation of cracks at velocities close to that of mechanical waves, when taking into account the energy dissipated as heat around a progressing crack tip. Indeed, if fronts creep fast enough, their local rise in temperature becomes significant compared to the background one, so that they can avalanche to a very fast phase, in a positive feedback loop39,40.
We consider rugous crack that are characterised by a varying and heterogeneous advancement a(x, t) along their front, x being the coordinate perpendicular to the average crack propagation direction, a the coordinate along it, and t being the time variable (see Fig. 1). At a given time, the velocity profile along the rugous front is modelled to be dictated by an Arrhenius-like growth, as proposed by Cochard et al.8:
where \(V(x,t)=\partial a(x,t)/\partial t\) is the local propagation velocity of the front at a given time and \(V_0\) is a nominal velocity, related to the atomic collision frequency41, which is typically similar to the Rayleigh wave velocity of the medium in which the crack propagates42. The exponential term is a subcritical rupture probability (i.e., between 0 and 1). It is the probability for the rupture activation energy (i.e., the numerator term in the exponential) to be exceeded by the thermal bath energy \(k_B T_0\), that is following a Boltzmann distribution41. The Boltzmann constant is denoted \(k_B\) and the crack temperature is denoted \(T_0\) and is modelled to be equal to a constant room temperature (typically, \(T_0=298\,\) K). Using this constant temperature corresponds to the hypothesis that the crack is propagating slowly enough so that no significant thermal elevation occurs by Joule heating at its tip (i.e., as inferred by Refs.39,40). Such propagation without significant heating is notably believed to take place in the experiments by Tallakstad et al.28 that we here try to numerically reproduce, and whose geometry is shown in Fig. 1. Indeed, their reported local propagation velocities V did not exceed a few millimetres per second, whereas a significant heating in acrylic glass is only believed to arise for fractures faster than a few centimetres per second40,44. See the supplementary information for further discussion on the temperature elevation.
In Eq. (1), the rupture activation energy is proportional to the difference between an intrinsic material surface fracture Energy \(G_c\) (in J m\(^-2\)) and the energy release rate G at which the crack is mechanically loaded, which corresponds to the amount of energy that the crack dissipates to progress by a given fracture area. As the front growth is considered subcritical, we have \(G
Finally, the average mechanical load that is applied on the crack at a given time is redistributed along the evolving rugous front, so that \(G=G(x,t)\). To model such a redistribution, we here use the Gao and Rice3 formalism, which integrates the elastostatic kernel along the front:
In this equation, \(\overlineG\) is the mean energy release rate along the front and \(\text PV\) stands for the integral principal value. We, in addition, considered the crack front as spatially periodic, which is convenient to numerically implement a spectral version of Eq. (2)45 as explained by Cochard et al.8. 2ff7e9595c
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