Buoyancy-driven crack propagation: A mechanism for magma migration. (English) Zbl 0602.73112
A solution has been obtained for steady propagation of a two-dimensional fluid fracture driven by buoyancy in an elastic medium. The problem is formulated in terms of an integro-differential equation governing the elastic deformation, coupled with the differential equation of lubrication theory for viscous flow in the crack.
The numerical treatment of this system is carried out in terms of an eigenfunction expansion of the cavity shape, in which the coefficients are found by use of a nonlinear constrained optimization technique. When suitably non-dimensionalized, the solution appears to be unique. It exhibits a semi-infinite crack of constant width following the propagating fracture. For each value of the stress intensity factor of the medium, the width and propagation speed are determined. The results are applied to the problem of the vertical ascent of magma through the earth’s mantle and crust. Values obtained for the crack width and ascent velocity are in accord with observations. This mechanism can explain the high ascent velocities required to quench diamonds during a Kimberlite eruption. The mechanism can also explain how basaltic eruptions can carry large mantle rocks (xenoliths) to the surface.
The numerical treatment of this system is carried out in terms of an eigenfunction expansion of the cavity shape, in which the coefficients are found by use of a nonlinear constrained optimization technique. When suitably non-dimensionalized, the solution appears to be unique. It exhibits a semi-infinite crack of constant width following the propagating fracture. For each value of the stress intensity factor of the medium, the width and propagation speed are determined. The results are applied to the problem of the vertical ascent of magma through the earth’s mantle and crust. Values obtained for the crack width and ascent velocity are in accord with observations. This mechanism can explain the high ascent velocities required to quench diamonds during a Kimberlite eruption. The mechanism can also explain how basaltic eruptions can carry large mantle rocks (xenoliths) to the surface.
MSC:
| 74L05 | Geophysical solid mechanics |
| 74R05 | Brittle damage |
| 76R99 | Diffusion and convection |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 76T99 | Multiphase and multicomponent flows |
| 74G70 | Stress concentrations, singularities in solid mechanics |
| 65R20 | Numerical methods for integral equations |
| 86A99 | Geophysics |
Keywords:
buoyancy-driven; Chebychev polynomials; two-dimensional fluid fracture; integro-differential equation; elastic deformation; differential equation of lubrication theory; viscous flow; eigenfunction expansion of the cavity shape; nonlinear constrained optimization technique; semi-infinite crack of constant width; stress intensity factor; propagation speed; vertical ascent of magmaCitations:
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