Background

Subcritical crack growth and crack healing are chemically mediated or enhanced mechanical processes that change the microstructure and macroscopic behavior of stressed rocks. In particular, subcritical crack growth is a mechanism of brittle creep that can weaken rocks and lead to failure (Brantut et al. 2013). Crack formation and propagation likely also interplays with pressure dissolution during burial diagenesis. As stress distributions evolve during creep, or as pore fluid pressures or tectonic forces change, however, the stresses on cracks may be relieved or reversed, creating the opportunity for chemically-mediated healing and rock strengthening. Almost a century of work has developed a theoretical framework for understanding crack evolution semi-quantitatively but even for the simplest case, Mode I tensile cracks, and common mineral phases, such as calcite or quartz, it remains a challenge to relate the reaction kinetics of key molecular processes to the mechanical evolution of single cracks, crack populations and granular rocks. Moreover, there is very little knowledge of crack processes in other silicate minerals and rocks studied in this proposal such as ultramafic rocks (van Noort at al. 2017).

Growth and Healing of Individual Cracks

The seminal work of Griffith (1921) proposed that crack extension of area A would occur when the strain energy released (GdA) is balanced by the surface energy of the two new crack surfaces (2𝛾dA). The Griffith criterion for crack propagation is then G-2𝛾 ≥ 0, where the inequality accounts for dissipation associated with processes (e.g., plastic deformation) other than bond breaking. Irwin (1957) introduced the stress intensity factor, K=√GE, to calculate G for simple crack geometries as a function of stress and the material through the Young’s modulus, E. Rice (1978) employed the framework of non-equilibrium thermodynamics to extend Griffith’s work in two important ways. First, in place of a constant value for 𝛾, many potential contributions to the excess interfacial Helmholtz free energy were considered, opening the way to considering dynamic chemical surface effects. Second, in place of empirical approaches such as Charles’s law, a new rate law for crack propagation was developed using the chemical kinetics concept of detailed balance. In this approach the net reaction rate (i.e., crack length) is assumed to be given by competing forward and reverse fluxes across a single rate-limiting step with a chemical-mechanical activation free energy. However, accurate and predictive use of the consequent rate law requires mechanistic knowledge of the fundamental crack-propagation pathways. Because this knowledge is rarely available, even recent studies of crack processes still invoke empirical coefficients fitted to power law functions (Atkinson 1984; Swanson 1984; Bergsaker et al; 2016). Improved molecular and mesoscale descriptions of interface interactions are needed to develop a mechanistic rate law for crack growth.

Healing mechanisms at high temperature, e.g. >200˚C for quartz, include solid-state processes such as sintering (defect diffusion from the interface to the bulk). At lower temperatures, however, a dependence on fluid composition implies a role for chemical reaction (Smith & Evans 1984; Schott et al. 1990). Certain mechanisms, e.g., hydrogen bonding, cationic bridging, and siloxane bridging, were proposed based on comparisons of bonding energies and strain energies (Michalske & Fuller 1985) but have eluded direct determination.

In summary, despite significant interest in crack processes in materials and Earth sciences the fundamental molecular and interfacial controls on single-crack evolution are not known well enough for the establishment of robust continuum models.

Proposed Work

Double-Torsion Tension and Compression Loading of Single Cracks

The double-torsion (DT) instrument is an accurate method for studying single-crack processes (Evans 1972). In this geometry, stress is introduced by point load displacement and the mode-I stress intensity factor K_I of the propagating crack is approximately constant over a wide range of the crack length. Once the loading-point displacement is stopped, the stress on the crack tip (therefore K_I and G_I) decreases continuously, together with the speed of crack propagation, v_C, enabling the relationship between v_C and K_I (and G_I) to be obtained from a single experiment. The same technique can be used to examine spontaneous healing of the crack, by slowly reducing the applied tensile stress on the crack tip. We have revised the DT design further allow compression loading on the sample that has undergone cracking for direct measurements of the healing process (Fig 3a).

In addition, the conventional optical monitoring of crack evolution will be augmented by photoelastic films, strain gauges, and conductive electric wires to track crack tips (Fig. 2b). This will provide additional measurements of the strain and stress field (Lee et al. 2004; Jankowski et al. 2009; Shin & Hawong 2010), and will offer flexibility in the design of the mechanical loading control, sample size and the specimen chamber for environmental control.

Mechanistic Single-Crack Experimental Studies

Crack propagation in aqueous solution is facilitated because the sorption of water, and possibly solutes, to the fresh mineral surface lowers the interfacial free energy. The reverse process, crack closing and healing, thus requires the expulsion of surface-bound molecules. Cyclic studies in which the external environment is manipulated, as well as temperature, will be designed to test mechanistic aspects of the limiting processes in each case.

1) A fresh crack will be formed in aqueous solution, then the stress relaxed. The environmental chamber will be drained and, a dry gas flowed while recording the rate of crack retreat and relative humidity in the system. We hypothesize that water flux from the crack, driven by the gradient in water chemical potential between the interface and the exterior, will drive spontaneous crack closure. Through independent studies of water sorption thermodynamics on calcite, the expected hydration state of the calcite surfaces and the crack status will be correlated.

2) Fresh cracks will be formed in aqueous solutions containing sorbing divalent cations, then the stress relaxed. The aqueous solution will be exchanged to establish a chemical potential gradient for cation desorption and efflux from the crack. We hypothesize that the rate of crack healing will be determined by the rates of cation desorption and surface transport modeled below and in the project on Carbonate Solubility.

3) Fresh cracks will be formed in aqueous solutions containing non-sorbing monovalent cations and the stress reversed to apply compression. We hypothesize that hydrated, compressed fresh crack surfaces may exhibit apparent healing by being forced into minima in the interaction forces that are dependent on counter-ion concentration (Israelachvili & Pashley 1983) that can be independently determined from Surface-Forces Apparatus studies.

Because we anticipate that different chemical-mechanical processes will be operational with different kinetics, the time-dependence of crack stress cycling behavior will be investigated.

4) The aging of an open crack alters the closure and healing processes and we hypothesize that the chemical evolution of fresh crack surfaces through dissolution/precipitation reactions that transport atoms from high-energy locations, could a key mechanism introducing hysteresis. Accordingly, the progress of the dissolution/precipitation reactions, which can be evaluated independently, should be inversely related to the healing strength, or proportional to the level of hysteresis. Imaging methods will be applied to investigate the morphology of fresh and aged crack surfaces (e.g., AFM) and the contact geometry of healed cracks (e.g., Cryo-FIB-TEM).

5) The aging of a closed, compressed crack affects re-opening. This will be tested by correlating the new propagation threshold, G'_th, as a function of applied stress and time. In particular, by evaluating the progress of the dominant mechanism or reaction pathway identified in the study of molecular controls, we should be able to predict the level of strength recovery, i.e. the aging effect. Similar to prior AFM studies of aging affecting the shear strength of aged contacts (Li et al. 2011), we will ask if aging time causes a log(t) trend in G'_th that will be compared to macroscopic rock core studies (below).

Influence of Surface-Forces on Crack Propagation Rates

The crack-tip environment is a nano-confined space in which the physicochemical processes affecting the thermodynamics and kinetics of crack growth—i.e., water and solute transport and surface adsorption—have never been fully resolved. In particular, the Griffith criterion for crack propagation is based upon the interfacial free energy for the mineral in bulk solution but when two surfaces are close to each (which is the case at the crack tips) there are additional strong interactions (Fig. 4a) that can be either attractive or repulsive depending on separation and solution chemistry. These interactions have been quantified by surface-forces apparatus studies (Israelachvili & Pashley 1983; Israelachvili 2011), but their effect on the stress intensity factor has been largely ignored in the literature of fracture mechanics and subcritical crack growth. In addition, the accessibility of reactive species at the crack tip is controlled by the transport process and reactions (e.g. adsorption) along the crack path, the process of which controls the spatiotemporal evolution of fluid chemistry near the crack tip and thus impacts the surface forces (Fig. 4b). Models that couple these surface processes with the mechanics of crack growth would be a significant advance upon current empirical models.

Molecular Simulation of Water Expulsion Thermodynamics

Molecular simulations will provide quantitative tests of the hypothesized limitations on the rate of crack closure and healing. Recently, Rimsza et al. (2017) used the atoms-to-continuum method with the ReaxFF reactive molecular dynamics force-field to directly predict the effect of stress on the rate of crack propagation in silica glass in a vacuum. Here, we focus on specific fluid–mineral processes that may control the response to stress. In particular, we examine the free energy controls on water expulsion from stressed interfaces.

Because knowledge of the driving force for crack healing is not sufficient to predict the dynamics of the process will also calculate the free energy for water transport various locations within the crack to the bulk phase outside of the fracture. This is a straightforward calculation with a one-dimensional constraint on the water molecule position, similar to our previous study of ion sorption energetics (Zarzycki et al. 2015). Similar calculations of water and ion transport along calcite surfaces will inform the above modeling of the rates of subcritical crack growth.

Macroscopic Studies of Crack Healing

Ultimately, we seek to demonstrate that the chemical-mechanical processes operating at the scale of individual cracks are responsible for the evolution in the stiffness of macroscopic rocks. In natural rocks subjected to time-varying triaxial stress, however, subcritical-stress crack growth and crack healing may be accompanied by frictional effects at grain boundaries and it is a challenge to untangle the relative contributions from each of these mechanisms.

Here, we propose a new triaxial-stress experiment in which microcrack damage is introduced by first increasing deviatoric stress to above the threshold for subcritical crack growth and then holding the deviatoric stress constant to allow slow crack growth to occur (this part is a standard creep test). However, we will also monitor how the elastic stiffness of the sample changes through time during the hold, which is not commonly performed during a creep test. The observed creep and the measured stiffness decrease will be attributed to the sum of individual cracks that grow according to the rules determined in the earlier single-crack studies. So long as we do not approach the localization and failure transition for the rock, crack interaction should not have a dominant influence on the response of each individual crack.

We then will decrease the deviatoric stress to levels corresponding to crack healing as determined in the earlier single-crack studies and will monitor how the elastic stiffness increases as the cracks close during a low deviatoric-stress hold. These studies will first be performed on carbonate rocks (limestone and marble) in order to test the influences of solution chemistry and isotropic stress as explored in the earlier single-crack studies. Im future work, it is important to also investigate whether various types of sandstone with varying levels of clay exhibit similar behavior.

For polycrystalline samples, pre- and post-experimental characterization using electron back-scattered diffraction (EBSD) or microfocus synchrotron X-ray diffraction (µSXRD) will be applied to evaluate changes in strain distribution.

References