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

The Delft Sand, Clay & Rock Cutting Model.
8.3.1. The Model of Evans.
For brittle rock the cutting theory of Evans (1964) and (1966) can be used to calculate cutting forces (Figure
8-21). The forces are derived from the geometry of the chisel (width, cutting angle and cutting depth) and the
tensile strength (BTS) of the rock. Evans suggested a model on basis of observations on coal breakage by wedges.
In this theory it is assumed that:
1. A force R is acting under an angle δ (external friction angle) with the normal to the surface A-C of the wedge.
2. A resultant force T of the tensile stresses acting at the center of the arc C-D, the line C-D is under an angle β
(the shear angle) with the horizontal.
3. A third force S is required to maintain equilibrium in the buttock, but does not play a role in the derivation.
4. The penetration of the wedge is small compared to the layer thickness hi.
The action of the wedge tends to split the rock and does rotate it about point D. It is therefore assumed that the
force S acts through point D. Along the fracture line, it is assumed that a state of plain strain is working and the
equilibrium is considered per unit of width w of the wedge.
The force due to the tensile strength σT of the rock is:
T
T
(8-41)
T r cos d w 2 r sin w
Where r·d is an element of the arc C-D making an angle with the symmetry axis of the arc. Let hi be the depth
of the cut and assume that the penetration of the edge may be neglected in comparison with hi. This means that the
force R is acting near point C. Taking moments on the chip cut about point D gives:
hi
R cos
T r sin 2 T
r sin w r sin
sin
(8-42)
From the geometric relation it follows:
h i
r sin (8-43)
2sin
Hence:
T hi
w
R (8-44)
2sin cos
The horizontal component of R is R·sin(α+δ) and due to the symmetry of the forces acting on the wedge the total
cutting force is:
c T i
sin
F 2 Rsin h w
sin cos
(8-45)
The normal force ( on cutting force) is per side:
Fn Rcos Thi w
2sin cos
cos
(8-46)
The angle β can be determined by using the principle of minimum energy:
dF c
d
0
(8-47)
Giving:
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