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

The Delft Sand, Clay & Rock Cutting Model.
h n s
h
F F sin F cos
sin sin cos
(2-47)
The equilibrium of forces in the vertical direction:
v n s
v
F F cos F sin
cos cos sin
(2-48)
Equations (2-47) and (2-48) form a system of two equations with two unknowns σ and τ. The normal stresses σh
and σv are considered to be known variables. To find a solution for the normal stress σ on the plane considered,
equation (2-47) is multiplied with sin(α) and equation (2-48) is multiplied with cos(α), this gives:
h
sin sin sin sin cos sin (2-49)
v
cos cos cos cos sin cos (2-50)
Adding up equations (2-49) and (2-50) eliminates the terms with τ and preserves the terms with σ, giving:
v
sin
2 2
h
cos
(2-51)
Using some basic rules from trigonometry:
2
1
cos 2
cos
(2-52)
2
2
1
cos 2
sin
(2-53)
2
Giving for the normal stress σ on the plane considered:
v h v h
cos2
2 2
(2-54)
To find a solution for the shear stress τ on the plane considered, equation (2-47) is multiplied with -cos(α) and
equation (2-48) is multiplied with sin(α), this gives:
h
sin cos sin cos cos cos (2-55)
v
cos sin cos sin sin sin (2-56)
Adding up equations (2-55) and (2-56) eliminates the terms with σ and preserves the terms with τ, giving:
v hsincos
(2-57)
Using the basic rules from trigonometry, equations (2-52) and (2-53), gives for τ on the plane considered:
v
h
sin2
2
(2-58)
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