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MR41C-06:
Determination of Shear Strength of Rocks from Scratch Tests: Theoretical Justification

##### Abstract:

There is considerable empirical evidence that the specific energy ε determined from shallow scratch tests on sedimentary rocks is about equal to the uniaxial compressive strength*q*, see Figure showing the correlation between ε and

*q*for 334 rocks (Richard et al, J Engineering Geology, 2012). The specific energy ε, defined as the energy expended per unit volume of fragmented rock, corresponds to the ratio of the average cutting force over the cross-sectional area of the groove created by the motion of the cutter; it is virtually independent of the depth of cut provided that the cutter is sufficiently sharp so that frictional dissipation beneath the cutter is negligible and also provided that the depth of cut is below a critical value

*d*that is proportional to the intrinsic length scale (

_{c}*K*

_{Ic}/

*q*)

^{2 }with

*K*

_{Ic}denoting the rock toughness. Indeed, the critical depth of cut

*d*

_{c}_{ }separates two regimes of cutting, ductile and brittle. In the ductile regime (depth of cut

*d*smaller than

*d*

_{c}but larger than the grain size), the rock is intensively sheared ahead of the cutter and the specific energy is constant. On the other hand, in the brittle regime (

*d*larger than

*d*

_{c}) chipping takes place and the specific energy decreases as the inverse square root of

*d*. In sedimentary rocks,

*d*

_{c}is typically less than 1 mm.

The apparent correlation between the specific energy ε in the ductile regime and the uniaxial compressive strength *q* can be explained from an analysis of results of plane strain compression tests conducted in an apparatus that does not inhibit the development of shear bands (Labuz and Dai, J. of Geotechnical and Reoenvironmental Eng., 2000). This analysis indicates that the residual strength is reached on the shear band for slip distance of about 1~2 grain size and that the thickness of the shear band is also equal to about 2 grain size. Furthermore, the ratio of the energy required to destroy the cohesive links between the grains inside the shear band over the volume of the shear band can be shown to be indeed comparable to *q*. The specific energies pertaining to the compression test and the scratch test are thus comparable, not only because there is complete de-cohesion of the constitutive matrix in both cases, but also because the strains are of order 1 in both tests.