Elastic reciprocity and symmetry constraints on the stress field due to a surface-parallel distribution of dislocations

Viesca, Robert C.; Rice, James R.

doi:10.1016/j.jmps.2011.01.011

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…Taking v ≡ x − ξ , the kernals are [e.g., Head , 1953]

k_{1} (v) \equiv \frac{- v}{4 h^{2} + v^{2}} + \frac{8 h^{2} v}{{true(4 h^{2} + v^{2} true)}^{2}} + \frac{4 h^{2} v^{3} - 48 h^{4} v}{{true(4 h^{2} + v^{2} true)}^{3}}

k_{2} (v) \equiv \frac{24 h^{3} v^{2} - 32 h^{5}}{{true(4 h^{2} + v^{2} true)}^{3}}

For a symmetric distribution of slip near the free surface, their contribution is an antisymmetric change in normal stress and a symmetric change in the shear stress. (As summarized by Viesca and Rice [2011], misprints in early reported results for such kernels have unfortunately propagated through the literature. )…”

Section: Rupture Nucleation By Local Increases In Pore Pressurementioning

confidence: 99%

Nucleation of slip‐weakening rupture instability in landslides by localized increase of pore pressure

Viesca¹,

Rice²

2012

J. Geophys. Res.

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108

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[1] We model landslide initiation as slip surface growth driven by locally elevated pore pressure, with particular reference to submarine slides. Assuming an elastic medium and friction that weakens with slip, solutions exist in which the slip surface may dynamically grow, without further pore pressure increases, at a rate of the order of the sediment shear wave speed, a situation comparable to earthquake nucleation. The size of the rupture at this transition point depends weakly on the imposed pore pressure profile; however, the amount of slip at the transition depends strongly on whether the pore pressure was broadly or sharply elevated. Sharper profiles may result in pore pressures reaching the total slope-normal stress before dynamic rupture is nucleated. While we do not account for modes of failure other than pure slip on a failure surface, this may be an indication that additional modes involving liquefaction or hydraulic cracking may be factors in the initiation of shallow slope failure. We identify two length scales, one geometrical (h, depth below the free surface) and one material (', determined by the frictional weakening rate) and a transition in nucleation behavior between effectively "deep" and "shallow" limits dependent on their ratio. Whether dynamic propagation of failure is indefinite or arresting depends largely on whether the background shear stress is closer to nominal peak or residual frictional strength. This is determined in part by background pore pressures, and to consider the submarine case we simplify a common sedimentation/consolidation approach to reflect interest in near-seafloor conditions. Citation: Viesca, R. C., and J. R. Rice (2012), Nucleation of slip-weakening rupture instability in landslides by localized increase of pore pressure,

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“…Taking v ≡ x − ξ , the kernals are [e.g., Head , 1953]

k_{1} (v) \equiv \frac{- v}{4 h^{2} + v^{2}} + \frac{8 h^{2} v}{{true(4 h^{2} + v^{2} true)}^{2}} + \frac{4 h^{2} v^{3} - 48 h^{4} v}{{true(4 h^{2} + v^{2} true)}^{3}}

k_{2} (v) \equiv \frac{24 h^{3} v^{2} - 32 h^{5}}{{true(4 h^{2} + v^{2} true)}^{3}}

Section: Rupture Nucleation By Local Increases In Pore Pressurementioning

confidence: 99%

Nucleation of slip‐weakening rupture instability in landslides by localized increase of pore pressure

Viesca¹,

Rice²

2012

J. Geophys. Res.

Self Cite

108

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“…To solve for the crack opening in a manner which solves the elasticity equations in 2D plane strain and meets the traction-free surface boundary conditions at z ¼ H, the numerical integral equation formulation of Erdogan et al [10] (with correction of a misprinted kernel as noted in Ref. [11]) was used, which relates the pressure distribution p(x, t) to the crack opening gap h(x, t) at each time t (with inertia neglected because of the slowness of fracture propagation speeds relative to elastic wave speeds). Also, in view of the low fracture toughness of ice, K Ic % 0.1 MPa, it was judged that toughness became unimportant, in the sense quantified by Garagash and Detournay [12], once crack half-length L(t) was greater than % 10 m. Effectively, over the long length scale of fracture growth (L > 1 km), the problem of fracture becomes asymptotically indistinguishable from the problem of lift-off along a nonadhering (zero K Ic ) interface.…”

Section: Discussionmentioning

confidence: 99%

“…0 þ , the upper side of the fracture plane (i.e., the base of the ice sheet), is given by Erdogan et al [10], although a misprint as identified in Ref. [11] must be corrected. Fig.…”

Section: Discussionmentioning

confidence: 99%

Time Scale for Rapid Draining of a Surficial Lake Into the Greenland Ice Sheet

Rice

Tsai

Fernandes

et al. 2015

Journal of Applied Mechanics

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A 2008 report by Das et al. documented the rapid drainage during summer 2006 of a supraglacial lake, of approximately 44 Â 10 6 m 3 , into the Greenland ice sheet over a time scale moderately longer than 1 hr. The lake had been instrumented to record the time-dependent fall of water level and the uplift of the ice nearby. Liquid water, denser than ice, was presumed to have descended through the sheet along a crevasse system and spread along the bed as a hydraulic facture. The event led two of the present authors to initiate modeling studies on such natural hydraulic fractures. Building on results of those studies, we attempt to better explain the time evolution of such a drainage event. We find that the estimated time has a strong dependence on how much a pre-existing crack/crevasse system, acting as a feeder channel to the bed, has opened by slow creep prior to the time at which a basal hydraulic fracture nucleates. We quantify the process and identify appropriate parameter ranges, particularly of the average temperature of the ice beneath the lake (important for the slow creep opening of the crevasse). We show that average ice temperatures 5-7 C below melting allow such rapid drainage on a time scale which agrees well with the 2006 observations.

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“…While the stress r ij11 (r, z; r 0 , z 0 ) due to D rr , deduced from potentials (24) and explicitly given in Appendix C, satisfies the symmetry relation (33), the corresponding stress in Korsunsky (1995) solution does not. Interestingly, Viesca and Rice (2011) also use elastic reciprocity as a means to establish symmetries in dislocation solutions for a half-plane. Actually, with provision for a misprint in the expressions of the potentials, Korsunsky's solution is consistent with the application of the DD derivation rules (22) (Paynter et al, 2007).…”

Section: Discussionmentioning

confidence: 99%

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

Gordeliy

Detournay

2011

International Journal of Solids and Structures

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a b s t r a c tThis paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper-and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.

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Elastic reciprocity and symmetry constraints on the stress field due to a surface-parallel distribution of dislocations

Cited by 4 publications

References 14 publications

Nucleation of slip‐weakening rupture instability in landslides by localized increase of pore pressure

Nucleation of slip‐weakening rupture instability in landslides by localized increase of pore pressure

Time Scale for Rapid Draining of a Surficial Lake Into the Greenland Ice Sheet

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

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