The Delft Sand, Clay & Rock Cutting Model, 2019a

The Delft Sand, Clay & Rock Cutting Model.
It should be mentioned that the layer thickness was very small in these experiments, resulting in a crushed zone
reaching to the surface. In other words, the rock was crushed completely.
8.2.1. Some Relations.
The relation between shear strength (cohesion) c, internal friction angle φ and the minimum and maximum
principal stresses can be derived according to, using the basic Mohr-Coulomb relations:
c tan
(8-1)
And:
max min max min max min
cos
sin
(8-2)
2 2 2
This gives:
max min max min max min
cos c
sin tan
2 2 2
(8-3)
Multiplying with cos(φ) and reorganizing gives:
max min 2 max min
2
sin
2 2
cos
c cos
2
max min
sin
max min max min
c cos
2 2
sin
(8-4)
max
max
1 sin
2 c cos 1 sin
min
1
sin
min
2ccos 1 sin
This equation can also be written as:
2
max min tan 2 c tan
4 2
4 2
(8-5)
This relation is valid for all linear failure criteria with a cohesion c and an internal friction angle φ. Now if two
Mohr circles are found with index 1 and 2. Index 1 for the smallest circle and index 2 for the largest circle, the
following relation is valid in relation to the failure curve and internal friction angle:
max,2 max,1
min,2
min,1
1 sin
1 sin
r
(8-6)
This gives:
r 1 2 r r
1
sin and cos and tan
r 1 r 1 2
r
(8-7)
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