must not be tensile. Then, separating the principal stresses we find:
(9.52)
Rearranging (9.50) to put it in a similar form:
(9.53)
These two expressions must coincide at the point
in stress space where the failure surface and the
restriction on the stress state are both satisfied.
Eliminating 3 by adding (9.52) and (9.53), and
then substituting for 1 in (9.52) we find:
(9.54)
Thus, the restriction n Յ 0 on the potential shear
fracture truncates the failure surface where 3 is
equal to the negative of half the uniaxial compressive strength (Fig. 9.25). For greater values of 3 , it
has been suggested that the uniaxial tensile
strength would limit the stress state such that
(Paul, 1961):
(9.55)
In this way the intersection of the failure surface
with the ( 1 , 3 )-plane is composed of two straight
lines. The lightly shaded region (Fig. 9.25) below
and to the right of these lines is off limits because
of shear or tensile failure.
A number of laboratory studies have compared
data on the stress state at failure during triaxial
tests to the Coulomb criterion. These studies
include those that focus on the effects of
confining pressure, temperature, deformation
rate, and pore pressure on the strength of a wide
variety of rocks (Handin and Hager, 1957, 1958;
Handin et al., 1963). As an example, we return to
the data selected from a study of the mechanical
properties of sedimentary rocks from Tertiary
basins of Japan (Hoshino et al., 1972). Recall that in
this study the rocks were deformed in triaxial
compression tests at room temperature with no
pore pressure. When plotted in principal stress
space (Fig. 9.12) these data approximate a linear
relationship of the form (9.50). Using the effective
principal stresses,
in
Ј 1 ϭ 1 ϩ P p , Ј 3 ϭ 3 ϩ P p ,
1 ϭ T u , for 1 Ն 3 Ն Ϫ
1
2 C u
3 Յ ϪS 0 [(1 ϩ 2
i ) 1ր2 ϩ i ] Յ Ϫ
1
2 C u
Ϫ 3 [(1 ϩ 2
i ) 1ր2 Ϫ i ] ϭ 2S 0
1 [(1 ϩ 2
i ) 1ր2 ϩ i ]
1 [(1 ϩ 2
i ) 1ր2 ϩ i ] ϩ 3 [(1 ϩ 2
i ) 1ր2 Ϫ i ] Յ 0
(9.50) to account for pore fluid pressure, the relationship remains linear and compares favorably
to data (Fig. 9.13b) from triaxial compression tests
on Berea Sandstone (Handin et al., 1963).
9.4 Brittle failure in a field of
heterogeneous stress
The Coulomb criterion, reviewed in the previous
section, asserts that solids subject to a homogeneous stress state lose their load-carrying capacity
when a certain combination of shear and normal
stress acting on potential shear fractures reaches
a critical value. The Coulomb criterion is calibrated using laboratory measurements of the uniaxial compressive strength and the coefficient of
internal friction. Although this criterion provides
a reasonable fit to some laboratory data and has
been applied to numerous practical problems
with some degree of success, it does not explicitly
address development of the fractures themselves
in the heterogeneous stress state induced by the
fractures. Here we introduce methods for calculating the perturbed stress field around holes and
cracks and use these to investigate brittle failure
in a field of heterogeneous stress (Anderson, 1951;
Hubbert, 1951; Chinnery, 1961, 1963, 1966; Cooke
and Underwood, 2001; Bourne, 2003).
9.4.1 The boundary value problem of
C. E. Inglis
One of the most cited articles in the literature of
fracture mechanics was presented by C. E. Inglis in
1913 to the Royal Institute of Naval Architects in
England (Inglis, 1913). The title of his article,
“Stresses in a plate due to the presence of cracks
and sharp corners,” might seem somewhat
abstract for a group of naval architects, but one of
the principal causes of ship disasters in those days
was the growth of fractures in plates making up
the hulls of ships. To address this problem Inglis
solved the elastic boundary value problem for an
elliptical hole with major diameter, 2a, and minor
diameter, 2b, in an elastic plate (Fig. 9.26).
At the time Inglis took up this problem, engineers knew that holes (e.g. those cut in deck plates
to make hatches) would alter the local state of
364
BRITTLE BEHAVIOR
(9.52)
Rearranging (9.50) to put it in a similar form:
(9.53)
These two expressions must coincide at the point
in stress space where the failure surface and the
restriction on the stress state are both satisfied.
Eliminating 3 by adding (9.52) and (9.53), and
then substituting for 1 in (9.52) we find:
(9.54)
Thus, the restriction n Յ 0 on the potential shear
fracture truncates the failure surface where 3 is
equal to the negative of half the uniaxial compressive strength (Fig. 9.25). For greater values of 3 , it
has been suggested that the uniaxial tensile
strength would limit the stress state such that
(Paul, 1961):
(9.55)
In this way the intersection of the failure surface
with the ( 1 , 3 )-plane is composed of two straight
lines. The lightly shaded region (Fig. 9.25) below
and to the right of these lines is off limits because
of shear or tensile failure.
A number of laboratory studies have compared
data on the stress state at failure during triaxial
tests to the Coulomb criterion. These studies
include those that focus on the effects of
confining pressure, temperature, deformation
rate, and pore pressure on the strength of a wide
variety of rocks (Handin and Hager, 1957, 1958;
Handin et al., 1963). As an example, we return to
the data selected from a study of the mechanical
properties of sedimentary rocks from Tertiary
basins of Japan (Hoshino et al., 1972). Recall that in
this study the rocks were deformed in triaxial
compression tests at room temperature with no
pore pressure. When plotted in principal stress
space (Fig. 9.12) these data approximate a linear
relationship of the form (9.50). Using the effective
principal stresses,
in
Ј 1 ϭ 1 ϩ P p , Ј 3 ϭ 3 ϩ P p ,
1 ϭ T u , for 1 Ն 3 Ն Ϫ
1
2 C u
3 Յ ϪS 0 [(1 ϩ 2
i ) 1ր2 ϩ i ] Յ Ϫ
1
2 C u
Ϫ 3 [(1 ϩ 2
i ) 1ր2 Ϫ i ] ϭ 2S 0
1 [(1 ϩ 2
i ) 1ր2 ϩ i ]
1 [(1 ϩ 2
i ) 1ր2 ϩ i ] ϩ 3 [(1 ϩ 2
i ) 1ր2 Ϫ i ] Յ 0
(9.50) to account for pore fluid pressure, the relationship remains linear and compares favorably
to data (Fig. 9.13b) from triaxial compression tests
on Berea Sandstone (Handin et al., 1963).
9.4 Brittle failure in a field of
heterogeneous stress
The Coulomb criterion, reviewed in the previous
section, asserts that solids subject to a homogeneous stress state lose their load-carrying capacity
when a certain combination of shear and normal
stress acting on potential shear fractures reaches
a critical value. The Coulomb criterion is calibrated using laboratory measurements of the uniaxial compressive strength and the coefficient of
internal friction. Although this criterion provides
a reasonable fit to some laboratory data and has
been applied to numerous practical problems
with some degree of success, it does not explicitly
address development of the fractures themselves
in the heterogeneous stress state induced by the
fractures. Here we introduce methods for calculating the perturbed stress field around holes and
cracks and use these to investigate brittle failure
in a field of heterogeneous stress (Anderson, 1951;
Hubbert, 1951; Chinnery, 1961, 1963, 1966; Cooke
and Underwood, 2001; Bourne, 2003).
9.4.1 The boundary value problem of
C. E. Inglis
One of the most cited articles in the literature of
fracture mechanics was presented by C. E. Inglis in
1913 to the Royal Institute of Naval Architects in
England (Inglis, 1913). The title of his article,
“Stresses in a plate due to the presence of cracks
and sharp corners,” might seem somewhat
abstract for a group of naval architects, but one of
the principal causes of ship disasters in those days
was the growth of fractures in plates making up
the hulls of ships. To address this problem Inglis
solved the elastic boundary value problem for an
elliptical hole with major diameter, 2a, and minor
diameter, 2b, in an elastic plate (Fig. 9.26).
At the time Inglis took up this problem, engineers knew that holes (e.g. those cut in deck plates
to make hatches) would alter the local state of
364
BRITTLE BEHAVIOR