Sliding Crack Model Research Articles

On the origin of a wedge crack within the icy crust of Europa

We suggest that a crack that developed into a wedge‐shaped band within the icy crust of Europa (at ∼28°S, 170°W [see Schenk and McKinnon, 1989, Figure 4; Prockter et al., 1999, Figure 1]) originated under an applied compressive stress through the operation of a frictional sliding mechanism. We analyze this suggestion using a scale‐independent, sliding crack model and obtain an estimate of the maximum compressive stress to initiate the crack that compares reasonably well with earlier estimates [Helfenstein and Parmentier, 1985; McEwen, 1986; Greenberg et al., 1998] of Europan crustal stresses that are based upon elastic deformation of a shell. We then show through an application of both power law dislocation creep and diffusion creep that colder, near‐surface Europan ice appears to be capable of supporting the deduced stresses with little relaxation over a period from 104 to 105 years.

The role of nucleation and growth of microcracks in brittle solids under compression: A numerical study

Microcrack growth and nucleation in brittle microheterogeneous materials is studied by means of the two-dimensional sliding crack model. The problem is treated numerically using a boundary element method (BEM) to solve a singular integral equation for the dislocation density along the crack contour. The evolution of crack patterns is compared for uniaxial and biaxial compression.

Analytical simulation of the dynamic compressive strength of a granite using the sliding crack model

AbstractA sliding crack model is employed to simulate rock strength under dynamic compression. It is assumed that the growth and nucleation of a sliding crack array presented results in the shear fault failure and dominate the mechanical properties of rock material. The pseudo‐tractions method is used to calculate the stress intensity factor of the sliding crack array under compression. With the utilization of a dynamic crack growth criterion, the growth of the sliding crack array is studied and the simulated strengths of a granite under dynamic compression are correspondingly obtained. It is concluded that the simulated rock strengths increase with increasing strain rates at different confining pressures, and the rising rates have a trend to decrease with increasing confining pressures. It is also indicated that the simulated rock strengths increase with increment of confining pressure at different strain rates, and the rising rates are almost identical at different strain rates. The simulation results are validated by the experimental data for the granite. Copyright © 2001 John Wiley & Sons, Ltd.

Study on Strength of Rock Material under Dynamic Triaxial Compressive Loads Based on Sliding Crack Model

Study on Strength of Rock Material under Dynamic Triaxial Compressive Loads Based on Sliding Crack Model

Modeling the stress–strain diagram for brittle rock loaded in compression

The sliding crack model introduced by Nemat-Nasser and modified by Kemeny and Cook is combined with the USR criterion of fracture to model the stress–strain diagram of brittle rocks loaded in compression. To date, the sliding crack model has been based on the stress intensity factor formulation and the fracture criterion of linear elastic fracture mechanics. The LEFM formulation is tensile-stress based and therefore it ignores the constraining effect of the finite compressive strength. The model introduced here still uses LEFM, but only to map out the stress field at the crack tip. The LEFM fracture law is replaced with the multiaxial USR criterion; which includes both the maximum and the minimum principal stress. The analytical solution for the stress–strain diagram is compared with compressive test data for Lac du Bonnet granite. The solution has the right form to capture the process of stable fracture propagation provided the deformation is produced through elasticity and axial crack propagation.

Crack growth in brittle solids under compression

The two-dimensional sliding crack model, which is motivated by the micromechanics, is applied to examine the response of brittle materials to compressive loads. The problem is treated numerically by means of a boundary element method (BEM) solving a singular integral equation for the dislocation density along the crack contour. The results are compared with experimental data and with more simplified models, illustrating the quality of the present approach. As an example, the model is applied to simulate the response of compact rock subjected to a load cycle.

The sliding crack model of brittle deformation: An internal variable approach

Rice's internal variable theory including a micro-to-macro transition is employed to formulate a micromechanical, two-dimensional damage model of brittle deformation in compression. The sliding crack model is selected as a basic dissipative mechanism underlying macroscopic inelastic deformation. Microfluxes and conjugated thermodynamic forces are identified and incremental stress-strain relations are derived in loading and unloading. Comparison with the model of Nemat-Nasser and Obata (Nemat-Nasser, S. and Obata, M. (1988) A microcrack model of dilatancy in brittle materials. Journal of Applied Mechanics55, 24–35, in which the sliding crack mechanism was analyzed from a kinematic point of view, is presented. An illustrative example is worked out showing the capability of the present model to predict experimentally observed response of a compact rock.

Microstructure effects on microcracking and brittle failure of dolomites

In this paper the influence of microstructure on crack initiation stress and ultimate strength is investigated using results and analysis of 32 triaxial compression tests performed on cylindrical cores of dolomite samples which exhibit a wide range of grain sizes and mosaic textures. All tests were performed at a constant strain rate, under confining pressures between 0 to 40 MPa. The predictive capability of conventional criteria for ultimate strength which are based on empirical fitting parameters such as cohesion and internal friction angle, or mechanical properties such as unconfined compressive strength, is shown to be quite poor, due to the influence of microstructure. Microstructure controls ultimate strength to such a degree that an assumed mechanical property such as unconfined compressive strength may vary by more than a factor of two, where two different microstructure patterns are present. The validity of published analytical expressions which predict fracture initiation stress assuming the sliding crack model is tested using both mean and maximum grain size, and inserting the measured fracture initiation stress as the remote stress. It is shown that these approximate models fail to describe true behaviour because they ignore boundary conditions which exist at the tip of the leading crack at different mosaic textures. Early attempts to discuss the influence of microstructure on rock strength have shown that ultimate strength is inversely related to mean grain size. This study demonstrates that grain size alone can not be used in correlation with ultimate strength. Rather, the combination of both grain size and porosity dominate the mechanical response of the rock. Fracture initiation stress is found to be more sensitive to the influence of grain size than ultimate strength, possibly because the length of initial cracks controls the level of stress concentration at the tip of leading cracks. However, fracture initiation stress is shown to be inversely related to both porosity and mean grain size, thus the importance of porosity in the initiation process must be recognized. Ultimate strength is influenced primarily by porosity and mosaic texture, and is less sensitive to mean grain size, possibly because once fracture propagation is initiated, grain arrangement controls fracture interaction processes which lead to macroscopic failure.

Time-dependent compressive failure around an opening

Abstract Compressive failure of rock at a free face is a time-dependent process that proceeds outward from the free face. The model for general compressive failure is the sliding crack model of Kemeny, Cook 1987, including crack interaction and subcritical propagation of wing cracks. For a rock mass with an array of propagating cracks near a free surface, moduli are different in all three orthogonal directions. Elastic constants are derived for the resulting orthorhombic symmetry. This model of identical, equally spaced interacting cracks maximizes the reduction in stiffness with crack propagation. The resulting stress reduction around an opening prevents failure even for high levels of stress. It fails to predict the commonly observed slabbing or exfoliation failures. For σ 3 /σ 1 small, failure is typically more localized than assumed in the above model. This situation is modeled as propagation of a single long crack with periodic out-of-plane jogs with frictional sliding. Such a crack can propagate in the σ 1 direction at nearly constant stress. Proximity of a free face increases K I , however, for long cracks the effect of the free face is small. A small compressive σ 3 greatly reduces K I , and with the σ 3 gradient found near an opening, subcritical propagation velocity will drop precipitously away from the opening. This is consistent with failure by progressive exfoliation of thin sheets.

The progressive fracture of Lac du Bonnet granite

The strength of intact rock is made up of two components: the intrinsic strength, or cohesion; and the frictional strength. It is generally assumed that cohesion and friction are mobilized at the same displacements such that both components can be relied on simultaneously. Damage testing of samples of Lac du Bonnet granite has shown that as friction is mobilized in the sample, cohesion is reduced. This progressive loss of intrinsic strength and mobilization of friction is modelled using the Griffith locus based on a sliding-crack model. There appears to be a maximum cohesion that can be relied on for engineering purposes, and this strength is less than half of the unconfined compressive strength measured in the laboratory.

A model for non-linear rock deformation under compression due to sub-critical crack growth

Time dependency in rock deformation under compression is modelled by considering an elastic body containing cracks that grow under compressive stresses due to sub-critical crack growth. This is considered the prime mechanism for the time-dependent deformation of brittle rocks at low temperatures. The growth of cracks under compressive stresses is formulated using the “sliding crack” model, which considers extensile crack growth due to stress concentrations around pre-existing flaws. Subcritical crack growt is included into the sliding crack model by utilizing the empirical Charles power law relation between crack velocity and the crack tip stress intensity factor. The model is able to predict the dependence of the stress-strain curve on the applied strain rate, and agrees extremely well with experimental data. Also, the model is able to predict the occurrence of both transient and tertiary creep. The transient creep behavior is derived in closed-form, and is found to give creep that depends on the logarithm of time, which is similar to many empirical formulae for creep in brittle rocks. Tertiary creep in the model is due to crack interaction, and is found to occur at a critical value of crack density. This allows time-to-failure predictions to be made, which could be useful for underground structures required to remain open for long periods of time.

Theoretical investigation of the sliding crack model of dilatancy : Stevens, J L; Holcomb, D J J Geophys Res. V85, NB12, 10 Dec 1980, p7091– 7100

Theoretical investigation of the sliding crack model of dilatancy : Stevens, J L; Holcomb, D J J Geophys Res. V85, NB12, 10 Dec 1980, p7091– 7100

The reversible Griffith crack: A viable model for dilatancy

Previous work (paper 1) on the mechanism of dilatancy has outlined the deficiencies of the sliding crack model. We have developed a new model, based on reversible Griffith cracks, that is in accord with experimental results to data. These results can be summarized as follows: (1) dilatant rock exhibits volume increases at sufficiently high increasing differential stress, (2) there is substantial hysteresis in physical properties between increasing and decreasing stress, (3) examination of stressed and unstressed samples, using a scanning electron microscope, reveals stress‐induced cracks that are preferentially oriented with the crack plane close to the maximum compressive stress axis, (4) few shear cracks (required by the sliding crack model) are observed, and (5) dilatant rock exhibits ‘memory’ of the maximum differential stress since the last complete unloading. When returned to this stress state, the physical properties return to their previous values but at other stress states there is large hysteresis between loading and unloading. A classical Griffith crack exhibits all of these properties if it is postulated to be reversible. That is, below a critical stress, the crack closes in an unstable mode and healing occurs. This behavior requires close registration of the crack faces.

A theoretical investigation of the sliding crack model of dilatancy

Laboratory measurements of volumetric strain and velocity in Westerly granite are analyzed to determine the physical nature of microcracks in the rock. The commonly used factional ‘sliding crack’ model of dilatancy is examined in detail. The characteristics of an ensemble of sliding cracks are determined from the characteristics of the individual sliding surfaces. It is found that sliding cracks which open with a stick slip motion cannot account for the observed ‘memory’ of the peak stress in cycling experiments. A system of stably sliding cracks would have the required memory but would also produce a very long dead band at the beginning of sample unloading during which no change in inelastic volumetric strain could take place. An analysis of the elastic constants at the beginning of unloading shows that this dead band is not observed in laboratory experiments. This analysis shows that the sliding crack is an unlikely mechanism for dilatancy. An alternative is developed in the following paper.

A quantitative model of dilatancy in dry rock and its application to westerly granite

A general model of dilatancy was developed based on the behavior of individual microcracks. Macroscopic effects were described by combining the effects of individual cracks using a statistical approach. The model was developed using distribution functions for crack size, crack strength, and local stress. Hysteresis in volumetric strain and stress‐strain data for partial unloading and reloading and general loading paths can be described. The model is best suited to describing rock which has been subjected to a number of loading cycles sufficient to remove the residual volumetric strain that is present in the initial cycles. This makes the rock more representative of the real earth and removes transient effects that obscure the dilatant effect. Volumetric strain data from triaxial experiments on Westerly granite were well described by the model for confining pressures from 0 to 4.11 kbar. Only one parameter was needed to describe the effect of changing the confining pressure, and this can be interpreted as an elastic effect on the crack volume. Some aspects of the crack behavior that were necessary to explain the data are incompatible with the usual sliding crack model of dilatancy. Combined with the lack of observable shear cracks, this casts serious doubts on the validity of the sliding crack model.