Stress-Strain Behavior of Jointed Rock - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 86

(3.104)

When the discontinuities are fi lled, the yield functions need to be formulated in terms
of the shear parameters of the fi lling material

φ F * and c F * (Fig. 3.19, column 2).
Rough and uneven discontinuities may be described by a bilinear failure criterion for the
peak shear strength (Fig. 3.19, column 3). In this case an adaptation of the corresponding
yield function is required. For non-persistent discontinuities the yield functions can be
adjusted to the corresponding shear failure criteria as well (Fig. 3.19, columns 4 and 5).
The plastic potential Q D for shear failure is obtained by replacing the friction angle

φ F , c F ,

φ D

in (3.102) by the dilatancy angle ψ D :

(3.105)

In the case of tensile failure perpendicular to the discontinuity Q D = F D applies.
Inserting (3.105) into (3.100) and (3.101) yields:

if F D > 0, (3.106)

if F D > 0. (3.107)

Integration of (3.106) and (3.107) with respect to time yields the following relationship
between viscoplastic normal strain and shear strain:

ε n,D vp = - tan γ res,D vp .

ψ D ⋅ (3.108)

Thus, with increasing viscoplastic shear strain, linearly increasing dilatant viscoplastic
normal strain occurs. This is represented in the δ n,D - δ s,D diagram in Fig. 3.27 (lower

right) for normal stress levels σ n2 with blue lines. However, at large shear defor-

mations a constant angle of dilatancy, which is not equal to zero, leads to an unrealistic-
ally large volumetric strain. Therefore

σ n1 and

ψ D should be set equal to zero when a certain

shear displacement δ s0 is exceeded (Fig. 3.27, lower right).

In a rock mass with one discontinuity set, the discontinuities are not necessarily oriented
parallel to one of the axes of the global coordinate system (x,y,z). Then σ n and τ res are

functions of the stress components τ zx . The strain rates

resulting from viscoplastic deformations on the discontinuities are then calculated using
the chain rule:

σ x , σ y , σ z , τ xy , τ yz and

(3.109)