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

Rock Cutting: Atmospheric Conditions.
8.2.6. Parabolic Envelope UTS and UCS.
Based on the UTS or BTS and the UCS a parabole can be constructed of the normal stress as a function of the
shear stress, with boundary conditions that the parabole goes through the UTS or BTS point (shear stress equals
zero, normal stress equals -UTS or -BTS and derivative dσ/dτ equals zero) and touches the UCS Mohr circle as a
tangent. With m=UCS/UTS or m=UCS/BTS the equation of this parabole is:
1
UTS m 1 1
2
2
UTS
(8-27)
It is more convenient to write this equation in the form where the shear stress is a function of the normal stress,
giving:
2
2
UTS m 1 1 UTS
(8-28)
Figure 8-13 shows the resulting parabole. Although still based on UTS or BTS and UCS and not on two tests with
shear failure, the resulting failure curve seems more realistic. The Mohr circles for UTS, BTS and UCS are
determined for UCS=100 MPa, UTS=BTS=15 MPa. The transition brittle-ductile according to Mogi (1966) is at
a normal stress of 104 MPa. Taking an average internal friction angle from a normal stress of zero to a normal
stress of 240 MPa gives φ=18.6º.
140
120
100
80
Shear Stress τ vs. Normal Stress σ
Sigma Axis
Tau Axis
Shear Stress τ (MPa)
60
40
20
0
-20
-40
-60
-80
Mohr Circle UCS
Test
Mohr Circle UTS
Test
Mohr Circle BTS
Test
Parabole
-100
-120
-140
-100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
Normal Stress σ (MPa)
© S.A.M.
Figure 8-13: The parabolic failure criterion.
Mogi Criterion
Brittle Ductile
Copyright © Dr.ir. S.A. Miedema TOC Page 255 of 454