Fracture Nucleation Phenomena and Thermally Activated Crack Dynamics in Monocrystals

We explore irreversible thermally activated growth of cracks which are shorter than the Griffith length. Such a growth was anticipated in several studies [Golubović, L. & Feng, S., (1991). Rate of microcrack nucleation, Physical Review A 43, 5223. Golubović, L. & Peredera, A., (1995). Mechanism of time-delayed fractures, Physical Review E 51, 2799]. We explore this thermally activated growth by means of atomistic Monte-Carlo dynamics simulations of stressed monocrystals. This crack growth is stepwise. Each step is marked by nucleation of a microcavity close to the crack tip, and by creation of a passage connecting the microcavity and the crack. If the external tensile stress is weak, many such nucleation events occur before the crack length reaches the Griffith size. In addition to the simulations, we also present an analytic theory of the stepwise thermally activated crack growth. The theory explains surprising observation form our simulations that the thermally activated crack growth remains fairly well directed in spite of the stochastic nature of the crack growth process.


Introduction
Solids under external tensile stresses are examples of metastable states of matter similar to supercooled liquids or magnetic systems [Lifshits & Pitaevski, 1981]. The threshold of fast ("instantaneous") failure is in fact a metastability limit, i.e., spinodal point. At this point, the external stress as a function of the strain reaches its maximum, . However, stressed solids can break under stresses that are significantly weaker than . It may however take a long time before weakly stressed solids break. That is, the fracture is time-delayed with relatively long life time which is a function of temperature and the applied stress [Brenner, 1965]. Time delayed fracture is physically related to the phenomena of micro-failure nucleation and growth [Golubović & Feng, 1991;. These phenomena are similar to nucleation phenomena in metastable states of matter [Lifshits & Pitaevski, 1981]. The classical work of Griffith (1921) contains some elements suggesting a phenomenological theory of microcrack nucleation. The critical, Griffith crack is somewhat similar to critical nucleus in a metastable state: Cracks longer than the Griffith crack rapidly and irreversibly grow [Mott, 1948].
The growth of cracks shorter than the Griffith crack size may seem unlikely due to energy reasons. Yet, such a growth still does occur in realistic systems due to thermally activated processes of failure nucleation [Brenner, apr.ccsenet.org Applied Physics Research Vol. 13, No. 1;202134 1965. Recent examples of these phenomena are time-delayed fractures in stressed mechanical structures such as space elevators [Edwards & Westling, 2003;Golubović & Knudsen, 2009;Knudsen & Golubović, 2014;Golubović & Knudsen, 2019].
In the present study, we explore these thermally activated growth phenomena by means of atomistic Monte-Carlo dynamics simulations of stressed monocrystals. This growth is a sequence of steps each involving a nucleation of a microcavity in the crack tip region and creation of a passage between the microcavity and the crack. In the presence of weak external stresses, many such growth steps occur before the crack length reaches the Griffith size.
In addition to the simulations, we also discuss an analytic theory of these phenomena. The theory elucidates the step-wise character of the thermally activated crack growth. The theory explains surprising fact that thermally activated crack growth remains fairly well directed in spite of the random nature of the growth process.
To elucidate the results of the present study, it will be needed to first review earlier theories of failure nucleation phenomena in solids under external tensile stresses. First attempt in this direction is inspired by early work of Griffith (1921); see also Landau & Lifshits (1970). In this so called classical theory of failure nucleation, microcracks are analogs of the stable phase droplets in a metastable state. Consider a micro-crack of length in a solid under a uniaxial stress perpendicular to the crack in a 2d-solid. Its creation costs the energy (1) is of the order = ( ) = 2 / 2 . Therefore, the Griffith length crack is unstable: For > , this instability eventually produces the familiar fast crack growth [Mott, 1948]. For < , one has > 0 so the crack growth is disfavored. This situation is similar to that encountered in more common nucleation phenomena [Lifshits & Pitaevski, 1981]. The seems to play the role of a nucleation energy barrier . This analogy suggests the use of the standard Arrhenius law ~1~exp ( ), with = = 2 2 , to evaluate the nucleation rate -or the time needed for the failure to be thermally nucleated.
Validity of this classical failure theory (based on Griffith size crack) is however not assured. This was emphasized by Golubović & Feng (1991, GF in the following). For example, surface diffusion [Herring, 1950a]  In view of the above arguments, GF suggested that the actual energy barrier for the fracture nucleation can be estimated as ( ), that is, Thus, Therefore, the failure nucleation rate in GF picture is, for weak stresses, ≪ , substantially greater than that predicted in the classical theory based on critical Griffith size crack. Golubović & Peredera (1995, GP in the following) used atomistic dynamics simulation to readdress the problem of failure nucleation in solids. These simulations strongly indicate that microcavities rather than microcracks are major nucleated defects in the delayed fracture regime. Energy cost of a microcavity nucleation is very different in the physical character from the energy cost of the Griffith crack nucleation. In a 2d solid, with a cavity with linear size , the energy cost of cavity creation is of the form, where the first term is the cavity surface energy (~ in 2d). In addition, the cavity energy in Eq.
(2) also contains a cavity volume contribution (~2 in 2d) produced by the applied stress [GP; Herring, 1950b]. Indeed, the creation of the cavity causes an elongation of the solid along the direction of the externally applied stress. This causes a decrease the solid sample energy by an amount ~ stress×cavity volume ~ 2 . By Eq.
(2), cavity energy is at maximum at a critical cavity size which is of the order, = / , while the energy barrier for the cavity nucleation is of the order, Notably, the above energy-barrier scale for failure nucleation, proposed in GP, coincides with the aforementioned energy barrier proposed (on different grounds) in GF. Likewise, the minimal crack length introduced by GF coincides with the critical cavity size, , = , In the following we will show that these length and energy scales enter a comprehensive picture of thermally activated crack growth. An early attempt to microscopically explore this growth was done long ago by Golubović and Moldovan (1995, GM in the following) in a short conference publication.

Atomistic dynamics simulations of thermally activated crack growth
Earlier studies discussed in the Introduction support the idea of thermally activated growth of cracks with lengths in the range > > . To elucidate the character of this growth, we employ atomistic (off lattice) Monte-Carlo dynamics simulations, technically similar to the simulations in GP and .
Monte-Carlo dynamics is a close relative of overdamped Langevin dynamics. Such computational approaches are well suited for studying slow physical processes such as nucleation phenomena dominated by potential energy of particles. The use of Langevin style approaches allows for large time steps and thus enables simulating particle systems up to very long times. We note that, relative to the aforementioned GM study, our present study features simulations done with systems which are run to significantly longer times. This feature is especially significant for the present study addressing very slow nucleation phenomena occurring in the regime of time delayed fractures. By virtue of having large time steps one can simulate and explore larger particle systems. This adds a significant element of realism to the results of Langevin style simulations of physical systems. Figure 1: Preparation of the sharp (elastic) crack used in our simulations as the initial (preexisting) flaw: We start from a stressed sample without a crack. We switch off atomic interactions across a portion of a crystalline plane.
The crack subsequently elastically opens. After that we restore the interactions.
In this study, we simulate behavior of cracks shorter than the Griffith length, < in a 2d Lenard-Jones solid sample. In Figures 1 to 7 we give the time evolution of a solid sample initially containing such a crack. The sample is under an external tensile force = 225 (in units for which the standard parameters of L-J potential are unity, i.e., = 1 and = 1) acting along the vertical direction. In Figures 1 to 7 we give only the middle portion of a larger sample with 12150 = 150 × 81 atoms at relatively low temperature which is twice smaller than the melting temperature.
Our goal here is to simulate the evolution of an artificially created flaw having the form of a cut. We are motivated by realistic experimental and practical situations in which one follows evolutions of preexisting flaws. Initially sharp (elastic) crack was prepared from a stressed sample without the crack by switching off interatomic interactions across a portion of the length 0 of a horizontal crystalline plane [see Figure 1]. After the crack opening, we restored the interactions. If 0 is smaller than ≈ 10 , the restored attractive interactions quickly close the crack (leaving, sometimes, behind a few vacancies). If 0 is longer than ≈ 60 , the crack immediately proceeds to grow in a rapid fashion without any significant time delay. Hereafter, we present our results for the most interesting case when the initial crack size 0 (here = 32 ) is in the range 10 < 0 < 60 . In this range the crack growth is slowed down by the time delays related to nucleation processes underlying the crack growth. The most significant feature visible in Figures 2 to 7 is a notable nucleation of microcavities in the vicinity of the crack tips. This process goes on in a way similar to the one found in GP in the absence of an initially present (preexisting) crack. Here, however, the presence of the crack and consequent higher stress concentration at its tips strongly enhances the rate of microcavity nucleation and speeds up the subsequent sample failure that happens after about 140,000 MCC, see Figure 7.
For comparison, our simulations of the same sample without a preexisting crack indicate that the integrity of the sample is essentially intact, with only a few cavities, up to time-scales longer than one million MCC.
Slow crack growth in Figures 2 to 7 is a combination of the two basic types of processes: (i) nucleation of a microcavity close to a crack tip, followed by (ii) creation of a passage connecting the cavity with the crack tip. For example, in the vicinity of the right tip, a cavity was nucleated between 48,000 and 52,000 MCC [see Figures 3 and 4]. Subsequently, this microcavity was connected to the crack by a passage nucleated between 56,000 and 68,000 MCC [see Figure 4]. The further crack growth is dominated by a repeat of the same sequence of cavity nucleation followed by passage formation [see, for example, the events at the right crack tip between 80,000 and 88,000 MCC, Figure 5, and further crack evolution in Figures 6 and 7].
Manifestly, the time ∆ needed to complete a cavity-passage creation cycle decreases as the crack size increases.
For example, ∆ ≈ 20,000 MCC for the cavities in Figure 4, whereas ∆ ≈ 8,000 MCC for the cavities in Figure 5. Another notable feature in Figures 2 to 7 is that the size of the nucleated cavities also typically decreases as the crack size increases.
We discuss these growth features in Section 3. The overall crack growth is thus a sequence of steps each involving a cavity nucleation. This is manifest also in the shape of the crack surfaces that appears after many such steps, see Figures 6 and 7. We sketch this shape also in Figure 8 for the purposes of the discussions in Section 3.
Our simulations indicate that the creation of microcavities (vacancy clusters) is assisted by dislocations emitted by the crack tip which round (blunt) sharp crack tips. Interestingly, emitted dislocations frequently create bound states of a dislocation with a vacancy. These bound states decay by releasing a free dislocation. For example, at 20,000 MCC ( Figure 2) we see such a dislocation-vacancy pair next to the left crack tip. Between 20,000 and 24,000 MCC a dislocation was released from the pair and started to glide in the upleft direction. Strikingly, during its gliding, this dislocation produces new vacancies. At 24,000 MCC we can see two such vacancies left behind the dislocation along its glide plane [one of them is close to the left crack tip, and soon gets connected to the mother crack by a passage nucleated between 28,000 and 32,000 MCC, see Figures 2 and 3]. As another example, the cavity close to the right crack tip at 52,000 MCC (Figure 4) was also produced during a glide of a dislocation which was initially next to this tip (see Figure 3 at 48,000 MCC). After assisting creation of the cavity we see near the right tip at 52,000 MCC, this dislocation continues to glide in the down-right direction, and, at about 56,000 MCC, it transforms into a dislocation-vacancy pair, a "fat dislocation" (see Figure 4 at 56,000 MCC).
This pair is unstable and the dislocation is released, so that at 60,000 MCC we can see only a vacancy at its place.
Apparently, dislocation emissions from crack tips play a fundamental role in the nucleation of vacancies and microcavities (vacancy clusters) in the vicinity of the crack tips. In addition, dislocation emissions produce plastic deformations rounding, i.e., blunting crack tips.
Our simulations thus evidence that thermally activated crack growth in monocrystals can go as a sequence of

Qualitative analytic discussions
Our simulations in Section 2 suggest a theoretical picture of thermally activated crack growth. Such a growth was addressed in the GM conference paper mentioned in the Introduction. The GM discussions were however much too short to exhibit and discuss significant physical and mathematical details (derivations) of the underlying theoretical picture. Therefore, for the first time, we will provide detailed presentation of the theory of thermally activated crack growth. We also stress that new and more extensive simulations described in Section 2 of the present study have provided more advanced insights and some surprising new findings requiring explanation. In the following we will provide analytic theory capable to explain these new findings from our present simulations.

Figure 8:
Step-wise character of the thermally assisted crack dynamics.
The crack growth seen in Figures 2 to 7 is stepwise [as summarized in Figure 8]. Each growth step is marked by the formation of one or maybe several cavities nearly of the critical size ≈ / with energy ≈ ( ) ≈ (see Section 1). Here, is the stress in the proximity of the rounded crack tip with the rounded tip radius ≈ . The tip rounding (blunting) is due to plastic deformations caused by dislocation emissions. The stress ≈ ( ) 1/2 ; here, is the stress far away from the crack, i.e., externally applied stress. Using these facts, we find, and In the limit of weak external stresses , the crack growth is composition of many steps (as depicted in Figure 8).
The increment of the crack length during each step is comparable with the microcavity size. Thus, the n-th step size, So, and, By Eq. (7), as the crack length grows from 0 to , there are growth steps. Using here the Section 1 relations, / ≈ / ≈ / , we get, (as noted in GM, however without the above detailed derivation) as well as, Note that the number of growth steps is generally large for ≪ .
Because of cavity nucleation, the thermally activated crack growth has prominently stepwise character. Each step is marked by an increase of the crack length by the amount ∆ ≈ ( ) . For the n-th growth step, this process takes a time interval proportional to the cavity nucleation time, ) ≈ exp ( ( ) ) = ( 2 ) (11) = / , so for weak stresses , the L-dependence of the step time duration is dominantly due to the exponential term in Eq. (11). By Eq. (11), the step time duration strongly decreases, i.e., the crack growth rate substantially increases as the crack size increases. This is in agreement with the simulations of Section 2. The growth time required for a crack starting with the size 0 to reach , is given by the sum, This result can be expressed as, Thus, by Eq. (6), Therefore, by Eq. (11) This result can be expressed in a more informative form by writing the exponent therein as Above, we employed Section 1 relations, As ≈ , we have Thus, for the exponent in Eq. (12) we get, A striking finding, notable from our simulations in Figures 2 to 7, is that the crack remains nearly straight during its thermally activated growth. Its transverse deviation, perpendicular to the initial horizontal crack direction, appears to be relatively small. This is rather surprising finding, having in mind a stochastic nature of this growth.
To understand this, let us estimate the transverse displacement, , of the crack tip. After steps, the crack surface will be a zig-zagged line of segments. The length of the n-th segment is ≈ ( ) (the microcavity typical size, Eq. (3)). After steps, the crack tip exhibits both longitudinal and transverse displacements relative to its initial position: In addition to the tip's longitudinal displacement (perpendicular to the external tensile stress), the crack tip also displaces in the transverse direction (along the tensile stress), in a random manner, by the distance, = 0 ( 0 ) + 1 ( 1 ) + ⋯ + −1 ( −1 ).
Here, are uncorrelated random numbers with < >= 0 and < 2 >≈ 1. < >≈ 0, for ≠ , as there are no significant correlations between nucleation events in different steps. Therefore, Here, the average is done with respect to the realizations of the random numbers . So, As noted above, < >= , . Thus, < 2 >= 2 ( 0 ) + 2 ( 1 ) + ⋯ + 2 ( −1 ), or, by Eq. (3), By using this and Eq. (8), we find that, Calculating the above integral yields the result, By Eqs. (13) and (8) Thus, the deviation of the crack from the straight direction grows only as a square root of the logarithm of the crack's length. For all practical purposes, the logarithmic factor in Eq. (14) can be ignored, and is nearly a constant length ≈ . Indeed, because and 0 are both in the range between and , the argument of the logarithm in Eq. (14) is smaller than = . Thus, unless the external stress is astronomically small (and the sample astronomically large and old), the logarithmic factor in Eq. (14) can be ignored. Therefore, the theory predicts that cracks remain nearly straight during thermally activated growth. This theoretical finding is in agreement with our simulations of Section 2 in Figures 2 to 7. Strikingly, the simulations indeed show that thermally activated crack growth remains well directed in spite of the random character of the growth process. The zig-zagged path marked by a crack growing from an initial size to the Griffith size is thus significantly less rough than, for example, directed polymers, growing interfaces, or rapidly growing cracks longer than for which frequently grows as power of the crack length .

Summary and discussion
To summarize, in this study we used numerical simulations to study the nature of the thermally activated dynamics of cracks shorter than the Griffith length in monocrystals. This growth is a sequence of steps each involving a nucleation of a microcavity close to the crack tip, and creation of a passage between the microcavity and the crack.
If the external tensile stress is much smaller than , many such nucleation events occur before the crack length reaches the Griffith size. As the crack length grows, the size of a nucleated cavity = the increment of crack length during a steps, decreases. Because of this, the energy barrier for the cavity nucleation also decreases with increasing crack length. So, the crack growth rate increases with increasing crack length as documented by our simulations and the theory of Section 3. The crack's transverse roughness grows very slowly, as the square root of the logarithm of crack length. Thus, during the thermally activated crack growth, the crack remains essentially straight, as was indeed observed in the simulations in Figures 2 to 7. This finding is striking since thermally assisted crack growth is essentially a stochastic process.
Our study has investigated these phenomena via an atomistic dynamical simulation which allows having a closer look at what is actually going on in the crack tip region. We thus find that, in monocrystals, an important role in vacancy and cavity nucleation may be played by crack emitted dislocations [Section 2]. For example, we see, in the crack tip vicinity, nucleation of dislocation-vacancy pairs. Such an object dissociates and yields a vacancy and a released dislocation, which may again transform into a dislocation-vacancy pair, etc. Thus, we find an interesting source of vacancies active in the crack tip region. In principle, other mechanisms assisting microcavity nucleation are also possible in monocrystals. For example, the stress gradient should drive thermally (or otherwise) excited vacancies towards crack tips. Agglomeration of these vacancies may nucleate microcavities. However, in a crystalline solid, our simulations indicate that this vacancy-drift scenario is insignificant in comparison to a much faster vacancy production mediated by the crack emitted dislocations. We stress that we found this feature from simulations of a two-dimensional crystal. It remains for future simulations of 3D monocrystals to investigate similar phenomena involving dislocation lines and compare their efficiency with other mechanisms of vacancy and microcavity nucleation. Also, it would be of interest to simulate thermally assisted crack growth in amorphous, glassy materials. In glasses, standard dislocations are absent! However, some (more restricted) slip-type collective motion can be still envisioned to assist nucleation of microcavities nearby crack tips in a structural glass.
Alternatively, a mechanism similar to the vacancy-drift scenario is maybe more typical for glasses. These issues remain to be investigated in future studies.