Influence of fracture criteria on dynamic fracture propagation in a discrete chain
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Abstract
The extent to which timedependent fracture criteria affect the dynamic behavior of fracture in a discrete structure is discussed in this work. The simplest case of a semiinfinite isotropic chain of oscillators has been studied. Two historydependent criteria are compared to the classical one of threshold elongation for linear bonds. The results show that steadystate regimes can be reached in the low subsonic crack speed range where it is impossible according to the classical criterion. Repercussions in terms of load and crack opening versus velocity are explained in detail. A strong qualitative influence of historydependent criteria is observed at low subsonic crack velocities, especially in relation to achievable steadystate propagation regimes.
Keywords
Dynamic fracture Fracture criteria Discrete chain Incubation time Tuler–Butcher1 Introduction
Studying fracture propagating in discrete structures results in a tool capable of analyzing a broad range of phenomena which would not emerge in the settings of continuum mechanics. This approach has found fruitful applications when dealing with crystals, cellular materials, cracks in fiberreinforced matrices and investigations at the atomic level (e.g. Abraham et al. 1998; Atrash and Sherman 2011; Bolander and Sukumar 2005; Tsai et al. 2010; Glassmaker et al. 2007). Lattice structure models become even more crucial in the framework of dynamic propagation. In this respect, many kinds of instabilities can be predicted by intuitive considerations without the need of ad hoc hypotheses (see Marder and Gross 1995; Bernstein and Hess 2003; Kessler and Levine 2003; Bitzek et al. 2015). Finally, discrete models can be also treated as the discretization of the corresponding continuum problems where a choice of the fracture criterion may play an important role when dynamic fracture propagation is in question. For example it may lead to different predictions on the stability of possible steadystate regimes (e.g. Morini et al. 2013; Mishuris et al. 2012).
A lattice structure, in the dynamic scenario, is composed of concentrated masses interacting via links characterized by an interaction potential. In this paper the latter is parabolic, relating only the closest neighbors, while nonlocal interactions have been studied in Gorbushin and Mishuris (2016a). The analyzed structure is monodimensional: a chain of oscillators, which is detached from a substrate that reflects the problem symmetry.
The focus of this work is to investigate the influence of the fracture criteria of the links on the dynamic fracture propagation in such medium. Exclusively, cracks which advance at constant speed, in a steadystate, are analyzed. Such regimes, indeed, have traditionally been of extreme interest in the field of dynamic fracture and have repeatedly been observed experimentally. A few classical studies can be highlighted for example in RaviChandar and Knauss (1984), Fineberg et al. (1991) and Hauch et al. (1999) while the topic gained new attention more recently in discrete structures such as bridges (Brun et al. 2014) or xyloexplosives (King https://www.youtube.com/watch?v=jiWxU3jXOFc).
A fracture event, though, in many materials turns out to be not simply determined by an instantaneous threshold value for some energy measure like the maximum elongation established above. We deal in the present work with noninstantaneous fracture criteria which nevertheless do not change the material stiffness. The rate with which a body is deformed or an integral measure of the deformation energy provided to a bond before it breaks are examined: the incubation time (IT) and the Tuler–Butcher (TB) criteria.
In order to illustrate the peculiarities of the criteria, one can imagine applying a ramp displacement of rate r at \(t=0\) to three links with the same static strength \(u_s\) but different failure behavior. The ideally brittle one would break at \(t_f=u_s/r\) as soon as its elongation reaches \(u_s\). The IT spring would break, accordingly to Eq. (2), at \(t_f=u_s/r+\tau /2\) if \(r<2u_s/\tau \) or at \(t_f=\sqrt{2u_s\tau /r}\) alternatively, thus establishing a distinction between low and high deformation rates. The criterion shows a delayed failure causing an ultimate elongation bigger than the static one in this loading condition. In the case of a nonmonotonic load, though, such delay might result in an elongation at failure which is smaller than the static one or during unloading (see Volkov et al. 2017). The TB criterion also predicts a delayed failure, but at \(t_f=u_s/r+u_s\root 3 \of {3D^2/r^2}\) according to (3). The difference with the IT case is that, now, an oscillating load which is strong enough to break the spring in statics will also do it in dynamics. Notice in fact that a constantly increasing cumulated damage would sooner or later surpass D [see lefthand side in Eq. (3)].
The rest of the paper is devoted to the model of a fracture in a structured medium subjected to the aforementioned criteria and their effects on the stable regimes of propagation. The results are expressed in terms of trapped lattice energy, applied remote force and crack tip opening.
2 Background
The assumption that the crack propagates at a constant speed also requires some additional consideration. In particular it means, for a given oscillator i sitting at \(x_i\), that it is not allowed to break before all the links situated at \(x<x_i\) do (links on the lefthand side of Fig. 1). From the propagation point of view, it must be clarified that a regime which involves nucleation of daughter cracks ahead of the mother crack tip (\(\eta >0\)) is nonadmissible. The detachment of the chain has to progress continuously. We shall discuss in the next sections how drastically the failure criteria change the admissible scenarios of stable detachment velocity.
2.1 Ideally brittle links
3 Problem and methods
3.1 Incubation time criterion
3.2 Tuler–Butcher criterion
4 Results
4.1 Incubation time criterion
In order to verify the analytical solution, we searched for the steadystate regimes via solving Eq. (8) by a finite difference scheme similarly as in Gorbushin and Mishuris (2017) where the same numerical procedure is extensively explained. A chain of 2000 masses was loaded with a distant vertical constant force. The instant of fracture for a link was identified according to the condition Eq. (2). After iterating, the next failures tended to occur at constant time interval and such an interval was used for calculating the stable crack speed for a given force. The analytical solution is perfectly matched and the numerical approach confirms that a steadystate propagation is not achievable at low velocities (results not shown).
This kind of computation is quite heavy because of the algorithm adopted to identify the time of fracture at every location \(x_i\). Before a steady fracture propagation is reached, indeed, during the transient regime, the history \(u_i(t)\) in the last interval \(\tau \) must be recorded and the integral Eq. (3) updated for every \(x_i\). This marks a principal difference with the instantaneous traditional criterion in which case the quick check \(u\le u_s\) is sufficient. We have tried to simplify the procedure via making the criterion pseudoinstantaneous. Theoretically, Eq. (17) should be valid only for steadystate fracture. Nevertheless, if one estimates the instantaneous crack velocity \(\tilde{v}(t)\) from the last two failures, the criterion can be artificially reduced to \(u\le u_0(\tilde{v},\tau )\) instead of calculating the integral Eq. (17) at all. Such an attempt has been proven to be effective beyond expectations for the particular studied problem in achieving the same steadystates as in theory and the rigorous numerical simulation for same applied load.
Another crucial effect of \(\tau >0\) on the crack propagation is that it monotonically enlarges the regions of achievable steadystates as illustrated in Fig. 4. For instance, the speed \(0.2 v_c\), that is nonadmissible for an ideally brittle material with critical elongation parameter \(u_s\), can be reached with \(\tau =3\) and bigger.
4.2 Tuler–Butcher criterion
Like for incubation time materials, as it can be seen in Fig. 4, new zones of admissibility appear in the low velocity region for larger D. Nevertheless, there is a significant qualitative difference with the incubation time situation: such admissible intervals emerge small and scattered, but then, with increasing D, expand gradually and merge until every subsonic crack speed can be obtained for D close to unity.
5 Discussion and conclusions
The dynamic fracture propagation in discrete structures has been investigated in a considerable number of possible scenarios (see references above) but the influence of failure criteria different from a threshold stress has not gained the attention that it deserves despite the fact that noninstantaneous criteria have already been shown to be reliable in continuum mechanics [e.g. as recently discussed in Alves and Lobo (2017)]. As a first step to fill this gap, two timedependent criteria have been analyzed in detail when applied to the dynamic fracture propagation of a chain of oscillators and they have been compared to the classical ideal brittle fracture. In both cases, enhanced admissibility have appeared at low crack speeds and mapped in Fig. 4. An increasing incubation time \(\tau \) enlarges the admissibility continuously but never covers all the subsonic crack speeds. More than that, a TB material which requires a larger work of the overstretch for fracture and characterized by a bigger D also creates completely new zones of achievable steadystates and it is predicted that all the subsonic range is possible if \(D\gtrsim 0.13\).
Speaking of the steadystate crack opening, the timedependent criteria cause a delay in fracture after reaching the static strength of the bonds. This means that in most cases one should expect \(u_0>u_s\) like it would happen when monotonically elongating a single spring. At low v, though, this is not the behavior caused by a constant force applied on a complex structure. Ample and rapid oscillations ahead of the crack tip cause the delayed fracture to happen at \(u_0<u_s\). While such propagation regimes are admissible at high \(\tau \) for incubation materials, the same is not true for TB ones (see Figs. 6, 7 and 8). The mathematical form of the latter failure criterion indeed excludes such steadystates on the grounds that daughter cracks would jeopardize the steadystate assumption. In short, a theoretical limit has been found which states that for a TB chain a dynamic fracture can propagate at constant speed only if \(u_0\ge u_s\).
The velocity dependent energy release rate ratio \(G/G_0\) is a solution which is irrespective of the particular fracture criterion adopted. It is also valid regardless of the way the energy is introduced into the system. In the present work we use a constant force as the external load. However, in case one prefers to implement different kind of loading, like for instance in Marder and Gross (1995) when dealing with lattices, or for example to facilitate a specific experimental procedure, the relation between the new load and the crack speed has to be evaluated, while the energy—crack speed diagram remains the same.
Two numerical integration schemes have been used to solve the set of governing equations in the IT scenario and compared with the analytical solution derived in the present paper. The first of these verified the criterion condition in its integral form Eq. (2) at every time step for all the unbroken bonds. Such an approach enables one to simulate also the transient regimes before a steadystate is reached. The steadystates were achieved only in the admissible regions of Fig. 4 and there the forcevelocity relations agreed perfectly with the ones in Fig. 5. The second test was performed by adopting a pseudostatic failure criterion: a dynamic threshold elongation \(u_0(\tilde{v},\tau )\) was established based on the instantaneous crack velocity \(\tilde{v}\) and Eq. (17). This simplified algorithm performed much faster than the first one, and still returned correct results. For this specific structure at least then, it seems that many of the conclusions drawn here can still be valid in the transient regimes.
Beyond the particular scope of this paper, various propagation regimes, in absence of crack arrest, can appear: steadystate, other regular ones (clustering or forerunning as discussed in Mishuris et al. (2008, 2009), Slepyan et al. (2010, 2015), Nieves et al. (2016, 2017) or chaotic regimes. The realization of one or another heavily depends on the loading type, its intensity and on the structure itself. However, when the problem is faced from a mathematical point of view, assumptions of steadystate regimes have always been made in order to obtain simple solutions. With the analytical results in one hand, an a posteriori examination is required which identifies where the solution fulfills the assumptions and constraints: only that part of the solution is labeled as admissible. Generally speaking, though, it does not mean that all those regimes will emerge in practice as steadystate. In most of the cases it happens; nevertheless, as it has been shown in numerical simulations, other ordered regimes of propagation may arise such as clustering. In such circumstances, the steadystate velocity predicted theoretically reveals as the average speed at which the cluster moves (see Mishuris et al. (2009), Nieves et al. (2016, 2017) on the matter).
We have not treated the problem of branching in the present settings of historydependent criteria. It has turned out already in Marder and Gross (1995), Gorbushin and Mishuris (2016b, 2017) that such instabilities can become relevant at high crack speeds. In the considered geometry, loading condition and material parameters, the branching mostly happens along the crack surfaces but not on the crack line ahead. That is evident from the solution profile \(u(\eta )\) for \(v/v_c=0.2\) in Fig. 3. If the horizontal springs show the same dynamic resistance as the vertical ones, their fracture can precede the chain detachment from the substrate, making a steadystate propagation impossible and sensibly reducing the limiting speed with respect to \(v_c\). The admissibility check would imply that, for none of the consecutive oscillators, the difference \(u_{i+1}u_i)\) does reach the condition imposed by the fracture criterion. More complex scenarios may occur with structures characterized by flexural stiffness, heterogeneities or localized feeding waves (Mishuris et al. 2009; Nieves et al. 2016, 2017). Furthermore, a steadystate regime can be unattainable, resulting in unstable or alternating velocities, when the structure is not loaded far from the crack tip, but via accumulated energy in the form of residual stresses of the bonds (AyzenbergStepanenko et al. 2014). Complications have also been the object of investigation in the framework of bridged cracks (Mishuris et al. 2008).

it is at low speed regimes that an experimental investigation should be carried out more carefully for understanding whether it is necessary to incorporate historydependent fracture criteria in the dynamic fracture model;

the energy release rate ratio and shapes of the displacement profiles as functions of the velocity are invariants, in linear theory, and can promptly be used and adapted to the most suitable fracture criterion for the analyzed problem.
Notes
Acknowledgements
N. Gorbushin and G. Vitucci acknowledge support from the EU project CERMAT2 (PITNGA 2013606878), G. Mishuris and G. Volkov are thankful to the EU project TAMER (IRSESGA2013610547). G. Mishuris also thanks the UK Royal Society Wolfson Research Merit Award, while G. Volkov received support also from RFBR (170100618, 165153077). We wish to express our gratitude to Michael Nieves for interesting discussions. The Authors recognize the usefulness of the peer review process which enabled them to clarify the key points of the article.
Supplementary material
Supplementary material 1 (mp4 2500 KB)
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