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Rudnicki 1984
Rudnicki, J.W. (1984). Effects of dilatant hardening on the development of concentrated shear deformation in fissured rock masses. Journal of Geophysical Research 89: doi: 10.1029/JB080i011p09259. issn: 0148-0227.

This study examines the effects of dilatant hardening on the development of concentrated shear deformation. Specifically, the analysis considers the shear of an inelastically deforming rock mass containing a weakened layer of thickness h. The presence of the weakened layer causes localization instability, characterized by an unbounded ratio of a strain increment in the weakened layer to that in the far field, to occur earlier than it would be predicted from the response of the material surrounding the embedded layer. The development of instability in time depends on the ratio of the rate of imposed shear strain &ngr; to that for fluid mass exchange between the layers, c/h3, where c is an effective diffusivity. In the limit &ngr;h2/c⋅0, the pressure in the weakened layer is equal to that in the surrounding material and localization instability occurs at the peak of the drained stress-strain curve. For finite &ngr;h2/c, instability is delayed until the material in the weakened layer has been driven to the peak of its undrained, dilatantly hardened stress-strain curve. The time delay between final instability and the time at which the weakened layer passes the peak of its drained stress-strain curve is called the precursor time tpr because rapid straining of the weakened layer occurs during this period. For small &ngr;h2/c, as appropriate for tectonic applications and most laboratory experiments, a nonlinear asymptotic analysis predicts that tpr∞(αh2/c)2/3(λ/&ngr;)1/3 Δ-1/6, where λ is the half width of the peak of the stress-strain curve, Δ is the difference in the peak stresses of the rock mass and the weak layer divided by λ times the elastic shear modulus, and α is a nondimensional measure of the strength of dilatant hardening. For a wide range of numerical values the precursor times are very short; less than a few hours for tectonic strain rates and less than a few tens of seconds for typical laboratory strain rates.

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Journal
Journal of Geophysical Research
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American Geophysical Union
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