Stress-Strain Behavior of Jointed Rock - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 64

The shear strength of discontinuities with clayey, silty or sandy fi llings whose aperture is
considerably greater than the amplitude of their roughness or unevenness is largely de-
termined by the peak and residual shear strength of the fi lling. These may be described
in a simplifi ed manner by Mohr-Coulomb failure criteria (Wittke 1990):

τ res = σ n ⋅ tan φ F + c F , (3.55)

τ res * = φ F * + c F * , σ n ⋅ tan (3.56)

φ F * and c F * are the shear parameters for the peak and residual strengths
of the discontinuity's fi lling, respectively.
Ladanyi & Archambault (1977) found that the shear strength of rough discontinuities
fi lled with clay and sandy silt decreases with increasing thickness of the fi lling. Not
before the thickness exceeds the height of irregularities of the joint surfaces by 50% will
the shear strength of the joints approach the shear strength of the fi lling. Consequently,
(3.56) and (3.57) may be considered as lower bounds for the shear strength of rough,
fi lled discontinuities.
In the following, the shear strength of a non-persistent discontinuity, that is, a disconti-
nuity that is interrupted by rock bridges, will be discussed.
Fig. 3.15 (upper) shows a one-dimensional model of a non-persistent discontinuity with
an infi nite number of regularly ordered, open discontinuity sections separated by rock
bridges. The open discontinuity sections are modeled by cracks with elliptic cross-sec-
tions with lengths of 2a and a maximum aperture of 2b. The length of the rock bridges
is denoted with ℓ. To study the shear strength of such a discontinuity it is suffi cient to
consider only a section of length 2a + ℓ, as illustrated in Fig. 3.15 (upper) since the ver-
tical boundaries of this section represent symmetry planes.
Generally the major semi-axes “a” of the cracks are large compared with the mi-
nor semi-axes “b”. In the following, the limiting case b

where φ F , c F ,

0, which is known as a
Griffi th crack, will be considered. The Griffi th crack is assumed to be located in an
infi nite disc of an isotropic elastic continuum loaded at its boundaries by a normal
stress and a shear stress , respectively. The stress distribution along the axis
of the Griffi th crack for plane strain conditions is then given by the following ana-
lytic solution (Stevenson 1945):

→

if x ≥ a, (3.57)

if x ≥ a. (3.58)

In (3.57) and (3.58) it is assumed that the origin of the coordinate x is located in the
crack's center. Consequently the stresses σ n and τ res at the tips of the crack (x = a) are

infi nitely large (Fig. 3.15).