268
8 Elastic Solutions in Geomechanics
Fig. 8.5 Vertical tractions
acting on the hemispherical
surface
The above stress distribution satisfies the boundary conditions, since σ z = τ rz =
0 for z =0.
To Determine the Constant B
Consider the hemispherical surface of radius “a” as illustrated in Fig. 8.5. For any
point on this surface, let R = a = constant. Also, ψ be the angle between a radius of
the hemisphere and the z-axis.
The unit normal vector to the surface at any point can be written as
ˆ
n =
⎡
⎣
sin ψ
0
cos ψ
⎤
⎦
while r and z components of the point are
z = a cos ψ, r = a sin ψ
The traction vector that acts on the hemispherical surface is,
T =
⎡
⎣
T r
T θ
T z
⎤
⎦ =
⎡
⎣
σ r 0 τ r z
0 σ θ 0
τ r z 0 σ z
⎤
⎦
⎡
⎣
sin ψ
0
cos ψ
⎤
⎦ =
⎡
⎣
σ r sin ψ + τ r z cos ψ
0
τ r z sin ψ + σ z cos ψ
⎤
⎦
Considering the component of stress in the z-direction on the hemispherical
surface,
we can write
T z = −(τ r z sin ψ + σ z cos ψ)
Substituting the values of τ rz , z , sinψ and cosψ, in the above expression, we get