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Deformation of host rocks and flow of magma during growth of minette dikes and breccia-bearing intrusions near Ship Rock, New Mexico

Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the crack tip that is needed to balance the Herring torque arising from the anisotropic surface energy. This force-balance condition can be interpreted physically based on energetic considerations in the traditional framework of continuum fracture mechanics, in support of its general validity for real systems beyond the scope of phase-field models. The obtained law of crack motion reduces in the quasistatic limit to the principle of local symmetry in isotropic media and to the principle of maximum energy-release-rate for smooth curvilinear cracks in anisotropic media. Analytical predictions of crack paths in anisotropic media are validated by numerical simulations. Interestingly, for kinked cracks in anisotropic media, force-balance gives significantly different predictions from the principle of maximum energy-release-rate and the difference between the two criteria can be numerically tested. Simulations also show that predictions obtained from force-balance hold even if the phase-field dynamics is modified to make the failure process irreversible. Finally, the role of dissipative forces on the process zone scale as well as the extension of the results to motion of planar cracks under pure antiplane shear are discussed.

http://www.lps.ens.fr/~hakim/09jmpsvhak.pdf

Laws of crack motion and phase-field models of fracture

Various brittle geological structures form from the tip of preexisting joints. Experiments have been carried out to investigate their mechanical origin and to find out whether a planar defect can propagate in its ownplane under mode II geometrical conditions (displacement parallel to the defect plane and perpendicular to the edge) as supposed by the classical rupture mechanics model, and in particular, whether mode II can be an elementary fracture mechanism, which is not clear in the model and has not been experimentally demonstrated. A biaxial loading device (plane strain) allowed direct observation of fracture propagation from an angled slot in plates of Polymethylmethacrylate (PMMA, Altuglas) and sandstone (low and high porosity). Under uniaxial load, mode I propagation of the classical branch fracture occurred in both materials, but its development was inhibited under biaxial loading (low σ3/σ1 ratio). At higher stress/strain levels, shear zones (clearly distinguishable from the branch fracture) developed from the slot tip. In PMMA they consisted of the prograding development of en echelon shallow microcracks formed in mode I and more or less associated with plastic deformation. In sandstone a narrow shear zone propagated in the prolongation of the defect (slowly in the high-porosity samples and abruptly in the low-porosity ones) accompanied by conjugate shear fractures. The deformation mechanism was cataclasis, implying tensile (mode I) microfractures. These experiments suggest that mode II cannot exist as an elementary (primary) fracture mechanism but can only be a macroscopic fracture phenomenon which must necessarily involve tensile (mode I) microcrack formation. The latter is linked to stress concentrations in the prolongation of the existing joint. As such mode II initiated shear zones have not been generated in glass, a high density of small defects which can be mobilized under shear stress would seem essential. These ideas can be used to interpret some types of natural faults.

Can natural faults propagate under Mode II conditions

A review of mechanical models of dike propagation: Schools of thought, results and future directions

Interpretation of abundant seismic data suggests that Kilauea's primary conduit within the upper mantle is concentrically zoned to about 34-km depth. This zoned structure is inferred to contain a central core region of relatively higher permeability, surrounded by numerous dikes that are in intermittent hydraulic communication with each other and with the central core. During periods of relatively high magma transport, the entire cross section of the conduit is utilized. During periods of relatively low to moderate transport, however, only the central core is active. As the conduit penetrates the oceanic crust and enters the volcanic shield, it simultaneously supplies the deeper sections of the rift zones (6-to 10-km depth) and the roots of the summit reservoir with picritic magma. The rift zones at depth are inferred to be almost wholly molten and to possess a high degree of fluid continuity from Heiheiahulu in the East Rift Zone, 45 km westward through the roots of the summit magma reservoir, and well into the Southwest Rift Zone. Higher in the shield, the subcaldera magma reservoir and the shallow rift zones occupy the 2-to 4-km depth interval. Summit-differentiated olivine tholeiite (ρ ≈ 2.62 g cm−3) is periodically injected laterally along a horizon of neutral buoyancy within the rift zones, where the density of the magma is just balanced by the in situ density of the shield (Ryan, 1987a, b). Deep rift zone intrusions push seaward the deep tectonic blocks of the volcano's south flank. Shallow rift intrusions build a sheeted dike complex, inferred to be in isostatic equilibrium with the higher-density deep rift cores below. General finite element analyses are presented for the deformation and stress fields surrounding such dikes in the horizontal and vertical planes. The dike tip in two and three dimensions is surrounded by a tubular core of tensile (σ1, σ2) and shear stress (τmax). The displacement field is characterized by counterrotating cells on either side of the dike tip which, in vertical orientation, produce the characteristic subsidence above the dike complex, with uplift on either side, forming a ridge-trough-ridge structure. A finite element model of Kilauea's shield computes the displacement fields and principal stress (σ1) distributions resulting from intrusive activity on each or both of the rift zones. Within the summit region, tensile stress lobes produced by the three-dimensional upward extension of the intrusions superpose constructively to produce calderawide regimes of tensile stress, conducive to caldera development. Parametric studies of (1) intrusion in the East Rift Zone only, (2) intrusion in the Southwest Rift Zone only, and (3) intrusion in both rift zones demonstrate their unique kinematic contributions. For case 1, the caldera undergoes a counterclockwise rotation (torque up state) conducive to the development of rightstepping en echelon eruptive fissures, as exemplified by the August 14, 1971, eruption. For case 2, the caldera undergoes a clockwise rotation (torque down state) conducive to the development of left-stepping eruptive fissures, as occurred during the December 31, 1974, eruption. For case 3, the caldera substructure is driven due southward, producing the southward migration of the upper portions of the summit magma reservoir.

E. Sommer

Papers

Formation of fracture ‘lances’ in glass

An optical method for determining the crack-tip stress intensity factor