The ring shear box is analyzed using an isotropic strain hardening/softening model for granular media, permitting an estimate to be made of the stresses developed under conditions of simple shear, at the critical void ratio. Observation of the radial stress on the inner or outer walls of the test chamber would provide a measure of the relative value of the otherwise unknown intermediate principal stress. In a series of five tests on a quartz sand, average pressures exerted by the sand on the outer wall of the test chamber reached well defined, repeatable levels. As interpreted by the theory, the tests showed that the intermediate principal stress was equal to the direct stress on the Coulomb friction planes during simple shear:
$(1−sinφcv)σ1=σ2=(1+sinφcv)σ3$
where σ1 > σ2 > σ3 (compression positive), and φcv is the angle of internal friction at the critical void ratio, ie, during continuing displacement at constant volume. Similar observations are impossible for plane strain in general because displacement cannot be controlled so completely, but it is reasonable to conjecture that the same relationship holds for all such deformations, including those associated with the active and passive Rankine states.
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