Thermal weakening of cracks and brittle-ductile transition: a phase model

TL;DR

This paper introduces a phase model for crack propagation considering thermal effects, explaining the brittle-ductile transition and stick-slip behavior through a dual-phase thermal weakening mechanism.

Contribution

It presents a novel phase model that incorporates thermal effects into crack propagation, elucidating the brittle-ductile transition and thermal weakening phenomena.

Findings

Identification of two crack propagation phases: mechanical and thermally weakened.

Numerical simulations demonstrating dual-phase crack behavior.

Prediction of a critical temperature for the brittle-ductile transition.

Abstract

We present a model for the thermally activated propagation of cracks in elastic matrices. The propagation is considered as a subcritical phenomenon, the kinetics of which being described by an Arrhenius law. In this law, we take the thermal evolution of the crack front into account, assuming that a portion of the released mechanical energy is transformed into heat in a plastic process zone. We show that such a model leads to a two-phase crack propagation: a first phase at low velocity in which the temperature elevation is of little effect and the propagation is mainly governed by the mechanical load and by the toughness of the medium, and a second phase in which the crack is thermally weakened and propagates at greater velocity. Such a dual behavior can potentially explain the usual stick-slip in brittle fracturing, and we illustrate how with numerical simulations of mode I cracks propagating in thin disordered media. In addition, we predict the existence of a limiting ambient temperature above which the weakened phase ceases to exist and we propose this critical phenomenon as a novel explanation for the brittle-ductile transition of solids.

Figures

Equations

$$V=\alpha \nu e^{\frac{{\alpha }^2(G-G_c)}{k_{B}T}}$$

$$\frac{\partial (\Delta T)}{\partial t}=\frac{\lambda }{C}{\nabla }^2(\Delta T)+\frac{\varphi GV}{C\pi l^2}f$$

$$V=V_0{e}^{\frac{{\alpha }^2(G-G_c)}{k_B[T_0+\Delta T(V,G)]}}.$$

$$\frac{V-V^{\ast }}{V^{\ast }}\sim {\left(\frac{T_0-T_0^{\ast }}{T_0^{\ast }}\right)}_{G=G^{\ast }}^{\beta }$$

$$\frac{G-G^{\ast }}{G^{\ast }}\sim {\left(\frac{V-V^{\ast }}{V^{\ast }}\right)}_{T_0=T_0^{\ast }}^{\delta }$$

$$\frac{G^{\ast }}{V^{\ast }}\frac{\partial V}{\partial G}\sim {\left(\frac{T_0-T_0^{\ast }}{T_0^{\ast }}\right)}_{G=G^{\ast }}^{-\gamma }$$

$$\frac{V_{\text{stop}}-V_{\text{aval}}}{V^{\ast }}\sim {\left(\frac{T_0^{\ast }-T_0}{T_0^{\ast }}\right)}^{{\beta }^{\prime }}$$

$$\frac{G_{\text{aval}}-G_{\text{stop}}}{G^{\ast }}\sim {\left(\frac{V_{\text{stop}}-V_{\text{aval}}}{V^{\ast }}\right)}^{{\delta }^{\prime }}$$

$$\frac{G^{\ast }}{V^{\ast }}\frac{V_{\text{stop}}-V_{\text{aval}}}{G_{\text{aval}}-G_{\text{stop}}}\sim {\left(\frac{T_0^{\ast }-T_0}{T_0^{\ast }}\right)}^{-{\gamma }^{\prime }}.$$

$$G(t)=\frac{3E{h}^3{v_u}^2{t}^2}{8{a(t)}^4}\text{if }a\gg h,$$

Full text

Rupture dynamics, thermal weakening, statistical physics, critical phenomenon

I Introduction

Of paramount importance in engineering and geophysics, the impact of temperature in fracturing processes have since long been studied. It can simplistically be sorted into two categories: background effects where the temperature is treated as an environmental constant affecting the rates at which the defects of a medium are propagating or healing Brenner (1962); Zhurkov (1984); Brantut et al. (2013); Lawn (1993) and dynamic effects where the propagation of fractures self-induces a rise in temperature in the vicinity of the crack front Rice and Levy (1969); Fuller et al. (1975); Chapman and Walton (1983); Pallares et al. (2012); Toussaint et al. (2016). In the latter case, the heat elevation can be regarded as more than a secondary effect of the medium’s damage: it can be an active process back affecting the crack propagation. This phenomenon will be here referred to as “thermal weakening.” Such a weakening has notably been studied in earth science where it is believed to play a role in faults stability and earthquake triggering Braeck and Podladchikov (2007); Rice (2006) and it was included in the so-called rate-and-state framework Noda et al. (2009) as an explanation for rate weakening faults. Several mechanisms have been proposed to explain thermal weakening, such as the softening Marshall et al. (1974); Carbone and Persson (2005) or melting of fracture surfaces or the thermo-pressurization of fault fluids Rempel and Rice (2006); Wibberley and Shimamoto (2005); Sulem and Famin (2009). We here consider a model which disregards such effects and focuses on the statistical physics consideration of higher reactions rates (i.e., quicker fracture propagation) at higher temperatures, as implied by an Arrhenius law Hammes (1978). This model notably showed good agreement with the rupture dynamics, experimentally reported in various polymers Vincent-Dospital et al. (2019). In this work, we further discuss how, in addition, it stands as a physical explanation for the brittle-ductile transition of matter.

II The thermal weakening model

Arrhenius based models for the velocity of crack fronts have long been considered Brenner (1962); Zhurkov (1984); Santucci et al. (2004); Lawn (1993) and have recently been shown to show good agreement with experimental observables of mode I cracks slowly propagating in acrylic glass bodies Lengliné et al. (2011); Tallakstad et al. (2011); Cochard et al. (2018). The rupture is then not considered as a Griffith-like threshold mechanism Griffith (1921) where the crack only advances for $G>G_{c}$, where $G$ is the energy release rate of the crack in J m-2 (arising from the mechanical load given to the crack front) and $G_{c}$ the fracture energy of the medium (the energy barrier per surface unit to overcome molecular bonds). It is rather considered as a thermally activated subcritical phenomenon ($G<G_{c}$) for which the crack velocity is expressed as:

[TABLE]

where $\alpha$ is a characteristic size (m) of the fracturing process, that is associated with its energy barrier. $k_{B}\approx 1.38\times{10}^{-23}$ J K-1 is the Boltzmann constant, $T$ the absolute temperature at the crack tip and $\nu$ the thermal bath collisional frequency. Equation (1), as any Arrhenius law, is a continuous expression of a discrete process arising at the molecular scale. Cochard et al. (2018) have recently discussed it at length. The exponential term is the probability (i.e. $<1$) for the thermal agitation to exceed the activation energy $-\alpha^{2}(G-G_{c})$ and hence for the crack to advance by a length $\alpha$. This probability is challenged every $1/\nu$ seconds. In theory $\nu$ is also temperature dependent but this is of negligible effect compared to the exponential dependence of the probability term Hammes (1978) and we hence define $V_{0}=\alpha\nu$, the maximum crack velocity obtained when the activation energy is always reached. $V_{0}$ shall typically be in the range of the Rayleigh surface wave velocity Freund (1972). Because we consider the thermal evolution around the crack tip we also note $T=T_{0}+\Delta T$, where $T_{0}$ is the ambient temperature and $\Delta T$ any variation away from it at the tip.

Such variations are induced by the dissipation of the mechanical energy given to the elastic matrix in a plastic zone that surrounds the crack tip Irwin (1957). There are many processes responsible for such an energy loss, as the creation of new defects surfaces and the emission of mechanical waves, but we here focus on the release of heat. The model we use is based on the work of Toussaint et al. (2016): a portion $\phi$ of the energy release rate is dissipated on a cylindrical zone of radius $l$ centered around the crack tip. Such a configuration leads to a thermal evolution governed by:

[TABLE]

which is a diffusion equation including a source term. $\lambda$ is the medium’s thermal conductivity in J s-1 m-1,K-1, $C$ is the volumetric heat capacity in J K-1 m-3, $t$ is the time variable and $\nabla^{2}$ is the Laplace operator. $f$ is the support function of the heat production zone of surface integral $\pi l^{2}$ (i.e., $f=1$ in the zone and $f=0$ otherwise). Solving this equation for a crack propagating at a constant velocity and constant release rate, one can show that the thermal elevation at the tip reaches a steady state after a short transient time. Figure 1 shows the evolution of this steady state as a function of $V$ and for two values of $G$. See the supplemental material for details on its computation. In our model, we use this relation to describe $\Delta T(V,G)$, thus discarding any transient regime. Equation (1) becomes:

[TABLE]

Parameters used for illustration

Note that most of the previously introduced parameters are strongly dependent on the medium in which the crack propagates. The figures we display here use parameters that could be likely for the propagation of interfacial cracks in sintered acrylic glass bodies Lengliné et al. (2011); Tallakstad et al. (2011) and are discussed in the supplemental material: $\alpha=2.5\times{10}^{-11}$ m, $G_{c}=250$ J m-2, $T_{0}=293$ K, $C=1.7\times{10}^{6}$ J K-1 m-3, $\lambda=0.19$ J s-1 m-1 K-1, $V_{0}=1000$ m s-1, $l=20$ nm and $\phi=1$. Note that we use this set of values only to propose some likely orders of magnitude for our parameters, and not to accurately represent the rupture of a specific material, as done in Vincent-Dospital et al. (2019).

III Phase behavior

Equation (3) defines, for a given load $G$, a function $S_{G}$ such as: $V=S_{G}(V)$. To fit the model, the actual velocity at which a crack advances must be a solution of this equation (i.e., be a fixed point for the function $S_{G}$) Arrowsmith and Place (1990). Figure 2 illustrates that, depending on the value of $G$, $S_{G}$ has one to three fixed points: three possible values for the crack velocity. This finite number of solutions arises from the steady-state approximation. If we were to consider the transient regimes, $S_{G}(V)$ would be, for a front propagating at any velocity $V$ and load $G$, a target velocity. Any crack not having reached a steady state would thus accelerate or slow down to follow this function. The intermediate fixed point, when it exists, is then unstable (virtually impossible): a crack with a velocity value just above this point ($V<S_{G}(V)$) is too slow to be steady. The heat generation at the tip is higher than what the diffusion can accommodate, the temperature rises and the velocity increases to converge to the upper fixed point. On the contrary, if a crack is slightly slower than the intermediate solution ($V>S_{G}(V)$), the crack cools down to the lower fixed point. We here assume that such transitions happen in a negligible time so the steady velocities are sufficient to describe the main dynamics. The outer solutions of (3) being the only stable ones, the model displays a two-phase behavior. The lower velocity marks a slow phase. The temperature elevation at the crack tip has little effect on the propagation, as $\Delta T(V,G)\ll T_{0}$. The higher solution corresponds to a thermally weakened phase where $\Delta T(V,G)$ has reached the plateau temperature of Fig. 1. The velocity is there increased as the induced heat is potentially significant compared to the thermal background.

Notice in Fig. 2 that there are two particular values of the load $G$ for which either the lower or the higher phase ceases to exist. We denote them $G_{\text{aval}}$ and $G_{\text{stop}}$ (with $G_{\text{aval}}>G_{\text{stop}}$) as they correspond to mechanical loads at which a slow crack will have to avalanche to the thermally weakened phase or at which a fast (weakened) crack can only cool down to the slow phase. For $G$ in between these two thresholds, a hysteresis situation holds, there are several solutions for $V$ and the crack might or might not be thermally weakened, depending on the mechanical history. To $G_{\text{aval}}$ and $G_{\text{stop}}$ correspond some specific velocities $V_{\text{aval}}<V_{\text{stop}}$ in between which a crack cannot propagate, as any solution is there unstable. Figure 3 shows the possible crack velocities for various values of $G$. One can notice how similar it is to a first order phase transition Callen (1985) for the order parameter $V$ associated to avalanches (jumps in $V$) triggered by variations in the driving field $G$ at temperature $T_{0}$. Such a description compares interestingly with various ($V$, $G$) branches that are experimentally reported, for instance in the rupture dynamics of pressure adhesives Dalbe et al. (2014); Barquins and Ciccotti (1997), PMMA Hattali et al. (2012); Lengliné et al. (2011) or elastomers Morishita et al. (2016) and the model can hence be matched to actual data over decades of velocities Vincent-Dospital et al. (2019). Note that, in the hysteresis domain, we do not discriminate on the relative stability of each phase. One can however argue, by analogy with other phase transition systems Callen (1985), that one of the two solutions could only be metastable, that is, in an equilibrium which is less energetically favorable than the one of the alternative phase. In this case, when traveling though an heterogeneous medium where the variations in fracture energy are enough to get shifts from only one state to the other, one of the phase could still be preferential for the crack propagation.

IV CRITICAL POINT

Besides $G$, $T_{0}$ is the only other parameter of (3) which is not dependent on the medium’s properties. Figure 4 thus shows the predicted propagation velocities for various ambient temperatures. Notice the existence of a critical ambient temperature: $T_{0}^{*}$, at which $G_{\text{aval}}=G_{\text{stop}}=G^{*}$ and $V=V^{*}$. Beyond $T_{0}^{*}$, the Joule effect cannot overcome the thermal background enough for the crack to be weakened. Increasing the load then only leads to a smooth increase in the velocity. To relate to the theory of critical phenomena in phase transitions Callen (1985) we looked for the real numbers $\beta$, $\delta$ and $\gamma$ such that:

[TABLE]

[TABLE]

[TABLE]

where $\sim$ stands for a mathematical equivalence in the vicinity of the critical point (any pre-factor is overlooked). These exponents describe how $V$ converges towards $V^{*}$ beyond the critical point ($T_{0}\geq T_{0}^{*}$). We also characterized how the hysteresis domain shrinks, looking for $\beta^{\prime}$, $\delta^{\prime}$ and $\gamma^{\prime}$ such that:

[TABLE]

[TABLE]

[TABLE]

With a bisection, we numerically estimated the critical point, checking for the number of solutions of $V=S_{G}(V)$ (three solutions below $T_{0}^{*}$ and one above). Analyzing the shape of the velocity map in the derived vicinity we found: $\beta\approx 1/3$, $\delta\approx 3$, $\gamma\approx 2/3$, and $\beta^{\prime}\approx 1/2$, $\delta^{\prime}\approx 3$, $\gamma^{\prime}\approx 1$ (see the supplemental material). Both sets of exponents respect the scaling relation Callen (1985): $2\beta+\gamma=\beta(\delta+1)$. We hence derived critical exponents which are, along the phase co-existence domain, the same as the mean field exponents for, say, the liquid-gas transition Callen (1985), but different beyond the critical point. The mean field characteristic might arise from the statistical nature of the Arrhenius law only representing an average velocity while consecutive molecular bonds can be overcome at very different speeds. Another interpretation is that it translates the zero-dimensional character of our model. We have indeed disregarded any velocity variations and elastic interactions along the crack front, making the assumption that it is thin or symmetrical enough perpendicularly to the propagation direction.

V Simulations of 0D fronts

in disordered media

Let us finally illustrate the phase transitions with some simulations of such zero-dimensional fronts loaded in mode I. The loading geometry that we consider is shown in Fig. 5. The support body consists in two sintered elastic plates which are progressively separated at the edge. The deflection on the side, $u(t)$ (in m), is increased linearly with time: $u(t)=v_{u}t$. Using the Euler–Bernoulli beam theory Anderson (2005), one can compute the energy release rate at the tip of such a system:

[TABLE]

with $E$ the body Young modulus (in Pa), $h$ half of its thickness and $a$ the crack advancement such as: $V=\partial{a}/\partial{t}$. By inserting (10) in (3), we obtain the differential equation in $a(t)$ that governs the crack progression and that we solved with a time step adaptive Runge-Kutta algorithm Dormand and Prince (1980). We here consider a crack interface with a homogeneous background cohesion $G_{c}=G_{c_{b}}$ which is only disturbed by a single tough asperity of length $L_{a}$ ($G_{c_{a}}>G_{c_{b}}$). Figure 5 shows a schematic for this anomaly while Fig. 6 shows, for several values of $G_{c_{a}}$, the course of the crack over it and the corresponding evolution of the energy release rate. When the front reaches the asperity, the crack velocity dramatically decreases as it reaches a tougher area. Meanwhile the load $G$ increases because the far field deflection continues to build up on a now quasi-static crack. Once the anomaly finally gets passed, the simulations show two possible scenarios. If $G_{\text{aval}}(G_{c_{b}})$ (i.e., the phase shift threshold for the background $G_{c_{b}}$) was not reached over the anomaly, then the crack only accelerates back to its pre-asperity state. However, if $G_{\text{aval}}(G_{c_{b}})$ was overcome, the crack shifts phase and becomes thermally weakened: it avalanches until $G=G_{\text{stop}}$. In Fig. 6, one can read the values of $G_{\text{aval}}$ and $G_{\text{stop}}$ and remark that they match the theoretical values displayed in Fig. 3. Note that, if the load was to be quickly increased, an avalanche could be triggered without the need for any asperity. We showed, however, how the medium’s disorder can lead to some spontaneous thermal weakening of the crack course.

VI Discussion and conclusion

By combining an Arrhenius law and the heat equation, we have thus demonstrated the possibility of a thermally activated dynamic phase transition in the propagation of cracks. This phase description may have major implications for the understanding of fracture dynamics. With a rather simple subcritical model, we indeed explain both slow creep regimes and fast ruptures. We do not however strictly disregard over-critical propagations, as $G>G_{c}$ only implies that the Arrhenius activation energy is null and hence always exceeded. In this case, we predict $V\sim V_{0}$. Note that at such high velocities, crack fronts tend to complexify Ravi-Chandar and Yang (1997); Fineberg et al. (1991), and our model might not hold as such, as it only considers single fronts. We derive a tip temperature approaching the $10^{4}$ K range. Although it is high, some experimental characterisations of triboluminescense Chapman and Walton (1983); Pallares et al. (2012) have shown that fast cracks can reach such a temperature, which only stands on small volumes ($\sim l^{2}L$, where $L$ is the length of the front) and short time periods ($\sim l/V$) such that it does not imply a gigantic level of energy nor it necessary leads to local fusion or sublimation of the solid. Note that the temperature merely measures the amplitude of the atoms agitation, and that its statistical definition actually suffers for heat production zones smaller than the molecular scale. While atomic scale simulations Rountree et al. (2002) would be more appropriate to study the induced heat, such computationally demanding models are often run at given (fixed) temperatures. Yet, some occurrences Germann et al. (2004); Budarapu et al. (2015) derive a non negligible induced heat.

Besides describing the two phases, we explained the potential shifts from one to the other and point out here how compatible this is with Maugis’ reinterpretation Maugis (1985) of the Griffith criteria Griffith (1921) and so, with the usual stick-slip in brittle fracturing processes Santucci et al. (2004); Dalbe et al. (2014), when avalanches get considerably larger than the scales of the in situ quenched disorder. We also showed that above a critical ambient temperature, $T_{0}^{*}$, this phenomenon cannot occur. For materials where $T_{0}^{*}$ is lower than the melting point at a given confining pressure, a same solid then displays a different behavior under cool or hot conditions: fragile when cold, but smoother/ductile when warm, as thermal avalanches are inhibited. The model thus could stand as a novel and physical explanation for the fragile-ductile transition of matter. Of course, it might be oversimplifying that to assume that all our parameters stay constant when varying $T_{0}$. The general physical principles however remain valid. Previous theories Hirsch et al. (1989); Tanguy (2007); Hartmaier and Gumbsch (2005) actually support the importance of the crack-tip plasticity in the fragile-ductile transition, but rather relate it to the nucleation and mobility of dislocations ahead of the front. Such processes are compatible with induced thermal elevation Germann et al. (2004), but are not directly captured by our mesoscopic description of the heat production zone.

Finally, and although we presented a mode I model, we suggest that some analogy is to be made with the frictional effects induced in mode II and mode III fracturing. Notably, as frictional heating is believed to be a cause for the instability of some seismic faults, a potential earthquake triggered when overcoming a strong fault plane asperity might indeed be amplified due to thermal weakening. The existence of the critical point would then explain the disappearance of such amplifications at higher depth (i.e., where rocks are in ductile conditions Scholz (1988)) as the thermal background is there enough to make the frictional heating negligible and, hence, favors creep over brittle ruptures.

Acknowledgements

and contributions

T.V.-D. developed and analyzed the model and the simulations, and redacted the first versions of the manuscript. R.T. proposed the physical basis of the model and its mathematical formulation. A.C. set the basis for the numerical implementation of the model and the principles of the resolution algorithm. K.J.M. contributed in the interpretation of the model in fracture mechanics applications. E.G.F. contributed to analyze the model in terms of critical point characteristics. All authors participated to the redaction of the manuscript and agreed with the submitted version. The authors declare no competing financial interests in the publishing of this work and acknowledge the support of the IRP France-Norway D-FFRACT, of the Universities of Strasbourg and Oslo and of the CNRS INSU ALEAS program. Readers are welcome to comment and should address to vincentdospitalt@unistra.fr or renaud.toussaint@unistra.fr. See the Supplemental Material at [URL] for a discussion on the parameters we used, details on how to compute the temperature elevation at the crack tip and decades on which we fitted the critical exponents.