Griffith suggested that the appropriate radius of
curvature in (9.61) would be the intermolecular
spacing for the solid, so r c Ϸ 5 ϫ 10
Ϫ10 m, and he
determined the value of the product
by
experimentation in order to estimate the intrinsic
tensile strength.
Griffith introduced a small scratch of known
length, 2a, onto the surface of cylindrical tubes or
spherical bulbs of glass with a glass cutter. Then
he pressurized the vessels with gas until they
burst, and recorded the ultimate pressure. The
relationship between the gas pressure and the
remote stress component acting across the
scratch,
was determined from the boundary
value problem for internal pressure in elastic
tubes and spheres. Griffith discovered that the
product,
was nearly
constant for the scratch lengths he tested, from 4
to 23 mm. Substituting into (9.61) Griffith evaluated the stress at the tip of the incipient fracture
as 2.3 ϫ 10
4 MPa and suggested this was an estimate for the intrinsic strength, T i . However, this
estimate is very great relative to measured uniaxial tensile strengths which typically are of the
order 10
1 to 10
2 MPa (Table 9.1).
To understand this puzzling result, Griffith
divided the estimated intrinsic strength by
Young’s modulus for glass, E ϭ 6.2 ϫ 10
4 MPa, to
estimate a strain of 0.37 at failure. He realized that
Hooke’s Law probably would not hold at these
levels of strain and he knew that the concepts of
an elastic continuum were on shaky ground at the
molecular scale. Therefore he inferred that the
intrinsic tensile strength would be somewhat less
than the value calculated above. Griffith suggested that a reasonable order-of-magnitude value
was:
(9.62)
This value is greater than the uniaxial tensile
strengths for glass specimens, the strongest of
which Griffith measured at T u ϭ 1.7 ϫ 10
2 MPa, and
many specimens had uniaxial tensile strengths as
much as two orders of magnitude less than the
intrinsic strength. These results inspired the conclusion mentioned above: brittle solids such as
glass must contain a myriad of flaws, too small to
be detected by the optical microscope, but very
yy (x ϭϮ a, y ϭ 0) ϵ T i Ϸ
E
10
Ϸ 6 ϫ 10 3 MPa
r
yy a 1ր2 Ϸ 2.63 ϫ 10 5 N m 3ր2
r
yy ,
r
yy a 1ր2
large compared to molecular dimensions. These
Griffith flaws serve to concentrate the stress and
thereby weaken the material so that the macroscopic measures of strength are much less than
the intrinsic strength. Fossils, clasts of different
lithologies, pore cavities, grain boundaries, and
microcracks are some of the many possible flaws
that can provide the stress concentration necessary to initiate fractures in rock.
9.4.3 Griffith’s criteria for brittle failure
In the second of his two classic papers on fracture
Griffith explored the problem of fracture initiation and failure under a biaxial remote stress,
using the solution of Inglis for an elliptical cavity
in an elastic material (Inglis, 1913; Griffith, 1924).
Here we review Griffith’s analysis, but, to make it
more relevant to geological applications, we
include fluid pressure acting on the cavity walls.
The limiting case of an elliptical cavity with very
great aspect ratio, a/b Ͼ Ͼ 1, is used to model a
crack-like flaw. Flaws of all possible orientations
relative to the remote principal stress axes are
examined, and the one with the greatest induced
tensile stress is identified as the “most dangerous.” When this tensile stress equals the uniaxial
tensile strength the initiation of a tensile fracture
is predicted and this condition is assumed to be
coincident with failure of the material. This procedure is used to define a failure surface in principal stress space and that surface is compared to
laboratory data.
The boundary conditions at an infinite distance from the elliptical hole (Fig. 9.27) consist of
uniformly distributed normal stresses that are the
principal stresses, 1 and 3 , directed along the xand y-axes, respectively:
(9.63)
Recall from the discussion of elliptical coordinates in Chapter 2 that ϭ constant defines a
family of confocal ellipses with 0 being the particular ellipse designated as the surface of the
hole. The boundary conditions there consist of a
uniform normal traction of magnitude P and no
shear traction:
(9.64)
BC: on ϭ 0 , t n ϭ ϪP, t s ϭ 0
xy ϭ 0, yy ϭ 3
BC: at √x 2 ϩ y 2 ϭ ϱ, xx ϭ 1 ,
9.4 BRITTLE FAILURE IN A FIELD OF HETEROGENEOUS STRESS
367
curvature in (9.61) would be the intermolecular
spacing for the solid, so r c Ϸ 5 ϫ 10
Ϫ10 m, and he
determined the value of the product
by
experimentation in order to estimate the intrinsic
tensile strength.
Griffith introduced a small scratch of known
length, 2a, onto the surface of cylindrical tubes or
spherical bulbs of glass with a glass cutter. Then
he pressurized the vessels with gas until they
burst, and recorded the ultimate pressure. The
relationship between the gas pressure and the
remote stress component acting across the
scratch,
was determined from the boundary
value problem for internal pressure in elastic
tubes and spheres. Griffith discovered that the
product,
was nearly
constant for the scratch lengths he tested, from 4
to 23 mm. Substituting into (9.61) Griffith evaluated the stress at the tip of the incipient fracture
as 2.3 ϫ 10
4 MPa and suggested this was an estimate for the intrinsic strength, T i . However, this
estimate is very great relative to measured uniaxial tensile strengths which typically are of the
order 10
1 to 10
2 MPa (Table 9.1).
To understand this puzzling result, Griffith
divided the estimated intrinsic strength by
Young’s modulus for glass, E ϭ 6.2 ϫ 10
4 MPa, to
estimate a strain of 0.37 at failure. He realized that
Hooke’s Law probably would not hold at these
levels of strain and he knew that the concepts of
an elastic continuum were on shaky ground at the
molecular scale. Therefore he inferred that the
intrinsic tensile strength would be somewhat less
than the value calculated above. Griffith suggested that a reasonable order-of-magnitude value
was:
(9.62)
This value is greater than the uniaxial tensile
strengths for glass specimens, the strongest of
which Griffith measured at T u ϭ 1.7 ϫ 10
2 MPa, and
many specimens had uniaxial tensile strengths as
much as two orders of magnitude less than the
intrinsic strength. These results inspired the conclusion mentioned above: brittle solids such as
glass must contain a myriad of flaws, too small to
be detected by the optical microscope, but very
yy (x ϭϮ a, y ϭ 0) ϵ T i Ϸ
E
10
Ϸ 6 ϫ 10 3 MPa
r
yy a 1ր2 Ϸ 2.63 ϫ 10 5 N m 3ր2
r
yy ,
r
yy a 1ր2
large compared to molecular dimensions. These
Griffith flaws serve to concentrate the stress and
thereby weaken the material so that the macroscopic measures of strength are much less than
the intrinsic strength. Fossils, clasts of different
lithologies, pore cavities, grain boundaries, and
microcracks are some of the many possible flaws
that can provide the stress concentration necessary to initiate fractures in rock.
9.4.3 Griffith’s criteria for brittle failure
In the second of his two classic papers on fracture
Griffith explored the problem of fracture initiation and failure under a biaxial remote stress,
using the solution of Inglis for an elliptical cavity
in an elastic material (Inglis, 1913; Griffith, 1924).
Here we review Griffith’s analysis, but, to make it
more relevant to geological applications, we
include fluid pressure acting on the cavity walls.
The limiting case of an elliptical cavity with very
great aspect ratio, a/b Ͼ Ͼ 1, is used to model a
crack-like flaw. Flaws of all possible orientations
relative to the remote principal stress axes are
examined, and the one with the greatest induced
tensile stress is identified as the “most dangerous.” When this tensile stress equals the uniaxial
tensile strength the initiation of a tensile fracture
is predicted and this condition is assumed to be
coincident with failure of the material. This procedure is used to define a failure surface in principal stress space and that surface is compared to
laboratory data.
The boundary conditions at an infinite distance from the elliptical hole (Fig. 9.27) consist of
uniformly distributed normal stresses that are the
principal stresses, 1 and 3 , directed along the xand y-axes, respectively:
(9.63)
Recall from the discussion of elliptical coordinates in Chapter 2 that ϭ constant defines a
family of confocal ellipses with 0 being the particular ellipse designated as the surface of the
hole. The boundary conditions there consist of a
uniform normal traction of magnitude P and no
shear traction:
(9.64)
BC: on ϭ 0 , t n ϭ ϪP, t s ϭ 0
xy ϭ 0, yy ϭ 3
BC: at √x 2 ϩ y 2 ϭ ϱ, xx ϭ 1 ,
9.4 BRITTLE FAILURE IN A FIELD OF HETEROGENEOUS STRESS
367