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Shear experiments on soils have revealed the effects of shear rate, confining pressure, and grain size on the residual shear strength, but their nature is not well understood. To interpret these behaviours, a single dimensionless inertial number  I  from granular physics is introduced. A linear relationship between coefficient of residual friction  μ   r   and the natural logarithm of  I  was found by analysing geotechnical test data from other literature and helps to resolve the  μ ( aI )-rheology, which was proved invalid in the quasi-static regime. A method is proposed that introduces two three-dimensional yield criteria for soils to classify the frictional properties between grains in the quasi-static regime. The empirical coefficient of start-up friction is replaced by strength parameters of the soil. When compliant with the Mohr–Coulomb yield criterion, this coefficient is positively correlated with the internal angle of friction but negatively correlated with the Lode angle. Moreover from further analysis, the calculated strength is smallest in the pure tension state, largest in the pure compression state, and intermediate in the pure shearing state. This result is consistent with the properties of compressive endurable and tensive intolerable for natural geomaterials.

Physical interpretation of shear-rate behaviour of soils and geotechnical solution to the coefficient of start-up friction with low inertial number