Simulation of time-dependent crack growth in brittle rocks under constant loading conditions
Introduction
Experiments and theoretical research on the phenomenon of fracture and damage started centuries ago. But no quantitative results were obtained until the great work of Griffith [1], which had lead to the onset of modern fracture mechanics. Griffith analyzed the stresses of a cracked plate, which was studied before by Inglis [2], and develop a theory for crack initiation and propagation. Orowan [3] modified Griffith’s theory by including the influence of plasticity into the energy balance. Irwin [4] introduced the parameter later known as the stress intensity factor K. After the fundamentals of linear elastic fracture mechanics were well established around 1960 [5], researchers had focused more on the influence of materials’ plasticity on the fracture analysis. One trend in recent fracture and damage mechanical research is related to time-dependent studies (life time prediction or time to failure prediction) for rocks (e.g. Kemeny [6], [7], [8], Mishnaevsky [9], Shao et al. [10], Li et al. [11], Rinne [12], Konietzky et al. [13]).
Based on the linear elastic fracture mechanical theory (LEFM), numerical simulations of time-dependent fracture propagation on Westerly Granite have been performed by Konietzky et al. [13]. The innovations of their research work include the simulation of specific initial cracks with certain distributions at the micro-scale, and the simulation of sub-critical and critical fracture growth to describe the time-dependent damage process until failure. It is assumed, that the damage and finally the failure of a rock specimen is the result of the partially parallel growth and coalescence of many initially existing microcracks at the grain size level rather than the growth of one or only a few single cracks. Other studies, like the experimental investigations on dynamic fracture performed by Ravi-Chandar and Knauss [14], [15], also implied this concept. As a further improvement of the work of Konietzky et al. [13], this study includes the influence of orientation distribution of the initial microcracks on the crack growth, and an adopted wing crack propagation scheme to describe the growth of initial microcracks.
Section snippets
Theoretical basis
Irwin [4] introduced stress intensity factor to describe the stress distribution and displacements near the crack tip of brittle materials. A fracture criterion based on the stress intensity factor can be described as follows: failure of the material occurs when the stress intensity factor reaches the critical value, the so-called fracture toughness KC. It was assumed by classical LEFM that the crack will propagate ultrasonically when the stress intensity factor K reaches the fracture toughness
Numerical simulations
The numerical calculations were performed with FLAC in plain strain mode using the internal program language FISH (Itasca [32]). Data from Westerly Granite were used for the numerical model, as is shown in Table 1.
Applications
Pre-cracked specimens under uniaxial load were studied utilizing the proposed modeling scheme. Size and geometry of the numerical models are shown in Fig. 11(a) and (b). The simulation results are compared with those obtained by lab tests on Hwangdeung granite specimens (Lee and Jeon [33]), which are of smaller scale, but the ratio between crack size and specimen size is the same (Fig. 11(b)). The numerical model is divided into 20,000 zones with each zone of the size 0.04 × 0.04 m2. The initial
Conclusions
A time-dependent crack propagation scheme is developed to simulate subcritical and critical crack growth of microcracks in brittle rock. The development of macroscopic crack formed by the coalescence of microcracks is simulated utilizing the proposed modeling scheme. The time-related macroscopic failure of a rock sample is studied under different loading conditions. Some preliminary studies are also performed to compare modeling results with typical in situ observations. Some conclusions are
Acknowledgment
The authors thank the anonymous reviewers for their valuable hints and recommendations for the improvement of this paper.
References (34)
A model for non-linear rock deformation under compression due to sub-critical crack growth
Int J Rock Mech Min Sci
(1991)Time-dependent drift degradation due to the progressive failure of rock bridges along discontinuities
Int J Rock Mech Min Sci
(2005)- et al.
Modelling of induced anisotropic damage in granites
Int J Rock Mech Min Sci
(1999) Subcritical crack propagation in rocks: theory, experimental results and applications
J Struct Geol
(1982)- et al.
Sub-critical crack growth in anisotropic rock
Int J Rock Mech Min Sci
(2006) Thermodynamics of the quasi-static growth of Griffith cracks
J Mech Phys Solids
(1978)- et al.
Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst
J Geophys Res
(1982) - et al.
Pillar strength in underground stone mines in the United States
Int J Rock Mech Min Sci
(2011) The phenomena of rupture and flow in solids
Phil Trans Royal Soc Lond Ser A
(1921)Stresses in a plate due to the presence of cracks and sharp corners
Trans Inst Naval Architects
(1913)
Fracture and strength of solids
Rep Prog Phys
Analysis of stresses and strains near the end of a crack traversing a plate
J Appl Mech
Fracture mechanics
The time-dependent reduction of sliding cohesion due to rock bridges along discontinuities: a fracture mechanics approach
Rock Mech Rock Engng
Determination for the time-to-fracture of solids
Int J Fracture
Crack growth time dependence analysis of granite under compressive-shear stresses state
J Coal Sci Engng (China)
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