The stress components are functions of the polar
coordinates, r and , and the constants A, B, and C
in these equations are related to the elastic constants as:
(8.103)
Consider the two points just outside the inclusion, where the r ϭ R
ϩ , and the polar angles are
ϭ0 and ϭ/2. There we calculate the circumferential stress using (8.101). Again, consider the case
where the two Poisson’s ratios are equal to , so
the relevant constants are
. Using these coordinates
and constants we find:
(8.104)
(8.105)
Because the coefficients that multiply the remote
stresses are just interchanged for the two points,
consider only a uniaxial stress, 1 , acting along
the x-axis:
(8.106)
(8.107)
If the inclusion is much softer than the surroundings, G i Ͻ Ͻ G s , the circumferential stress
component is similar to that for an open circular
hole: on the interface that is perpendicular to the
applied stress the local stress is equal in magnitude but opposite in sign, and there is a stress concentration factor of 3 on the interface that is
parallel to the applied stress.
If the inclusion is much stiffer than the surroundings, G i Ͼ Ͼ G s , there is a diminution factor of
1/2 on the interface that is perpendicular to the
r ϭ R ϩ , ϭ
2 → Ά
3 r
1 , for G i ϽϽ G s
r
1 , for G i → G s
0, for G i ϾϾ G s
(r ϭ R ϩ , ϭ 0) →
Ά
Ϫ r
1 , for G i ϽϽ G s
0, for G i → G s
1
2 r
1 , for G i ϾϾ G s
(r ϭ R ϩ , ϭ ր2) ϭ
3
1 ϩ 2k r
1 Ϫ
1 Ϫ k
1 ϩ 2k r
2
(r ϭ R ϩ , ϭ 0) ϭ Ϫ
1 Ϫ k
1 ϩ 2k r
1 ϩ
3
1 ϩ 2k r
2
B ϭ
1
2 A, and C ϭ Ϫ
1
2 A
A ϭ 2(1 Ϫ k)ր(1 ϩ 2k),
1
4
C ϭ
k Ϫ 1
k s ϩ 1
A ϭ
2(1 Ϫ k)
k s ϩ 1
; B ϭ
i Ϫ 1 Ϫ k( s Ϫ 1)
2k ϩ i Ϫ 1
;
applied stress, and the stress is zero on the interface that is parallel to the applied stress. In this
sense the stiff inclusion creates a stress shadow in
the surrounding softer material. The stiffer inclusion carries more of the applied load than adjacent regions by up to a factor of 3/2. The adjacent
regions carry less of the applied load by up to a
factor of 1/2. These results provide a useful benchmark for assessing the effect of stiffer or softer
rock masses on the state of stress.
Distributions of the Cartesian stress components are illustrated in Fig. 8.29 for a uniaxial
stress in the remote field ϭ 1 and a softer inclusion with shear modulus G i ϭ 10 GPa and Poisson’s
ratio i ϭ embedded in stiffer surroundings with
shear modulus G s ϭ 30 GPa and Poisson’s ratio s ϭ
. Note that the stress components are uniform
within the inclusion and that both the y-component of normal stress and the shear stress are zero.
The x-component of normal stress within the
softer inclusion is diminished in value relative to
the remote stress. The contour patterns outside the
inclusion are quite complex, but they clearly illustrate the facts that perturbations due to the inclusion are symmetric, they decrease with distance,
and they are negligible at radial distances r Ͼ 5R.
8.6.2 Generalized Hooke’s Law for
anisotropic rocks and minerals
The most general linear relationship among the
components of strain and stress is one in which
each of the nine components of strain, ij , is linearly related to the nine components of stress, ij .
For example, given a Cartesian coordinate system
with orthogonal axes x, y, and z, the equation for
the normal strain component, e xx , may be written
as (Nye, 1985, Chapter VIII):
(8.108)
The s ijkl are constants of proportionality and the
four subscripts on each s correspond to the two subscripts of the strain component e, followed by the
two subscripts of the stress component , with subscripts (1, 2, 3) corresponding to (x, y, z), respectively. There are eight more equations of this form,
each one linearly relating a component of strain to
ϩ s 1131 zx ϩ s 1132 zy ϩ s 1133 zz
ϩ s 1121 yx ϩ s 1122 yy ϩ s 1123 yz
xx ϭ s 1111 xx ϩ s 1112 xy ϩ s 1113 xz
1
4
1
4
r
1
8.6 ELASTIC HETEROGENEITY AND ANISOTROPY
325
coordinates, r and , and the constants A, B, and C
in these equations are related to the elastic constants as:
(8.103)
Consider the two points just outside the inclusion, where the r ϭ R
ϩ , and the polar angles are
ϭ0 and ϭ/2. There we calculate the circumferential stress using (8.101). Again, consider the case
where the two Poisson’s ratios are equal to , so
the relevant constants are
. Using these coordinates
and constants we find:
(8.104)
(8.105)
Because the coefficients that multiply the remote
stresses are just interchanged for the two points,
consider only a uniaxial stress, 1 , acting along
the x-axis:
(8.106)
(8.107)
If the inclusion is much softer than the surroundings, G i Ͻ Ͻ G s , the circumferential stress
component is similar to that for an open circular
hole: on the interface that is perpendicular to the
applied stress the local stress is equal in magnitude but opposite in sign, and there is a stress concentration factor of 3 on the interface that is
parallel to the applied stress.
If the inclusion is much stiffer than the surroundings, G i Ͼ Ͼ G s , there is a diminution factor of
1/2 on the interface that is perpendicular to the
r ϭ R ϩ , ϭ
2 → Ά
3 r
1 , for G i ϽϽ G s
r
1 , for G i → G s
0, for G i ϾϾ G s
(r ϭ R ϩ , ϭ 0) →
Ά
Ϫ r
1 , for G i ϽϽ G s
0, for G i → G s
1
2 r
1 , for G i ϾϾ G s
(r ϭ R ϩ , ϭ ր2) ϭ
3
1 ϩ 2k r
1 Ϫ
1 Ϫ k
1 ϩ 2k r
2
(r ϭ R ϩ , ϭ 0) ϭ Ϫ
1 Ϫ k
1 ϩ 2k r
1 ϩ
3
1 ϩ 2k r
2
B ϭ
1
2 A, and C ϭ Ϫ
1
2 A
A ϭ 2(1 Ϫ k)ր(1 ϩ 2k),
1
4
C ϭ
k Ϫ 1
k s ϩ 1
A ϭ
2(1 Ϫ k)
k s ϩ 1
; B ϭ
i Ϫ 1 Ϫ k( s Ϫ 1)
2k ϩ i Ϫ 1
;
applied stress, and the stress is zero on the interface that is parallel to the applied stress. In this
sense the stiff inclusion creates a stress shadow in
the surrounding softer material. The stiffer inclusion carries more of the applied load than adjacent regions by up to a factor of 3/2. The adjacent
regions carry less of the applied load by up to a
factor of 1/2. These results provide a useful benchmark for assessing the effect of stiffer or softer
rock masses on the state of stress.
Distributions of the Cartesian stress components are illustrated in Fig. 8.29 for a uniaxial
stress in the remote field ϭ 1 and a softer inclusion with shear modulus G i ϭ 10 GPa and Poisson’s
ratio i ϭ embedded in stiffer surroundings with
shear modulus G s ϭ 30 GPa and Poisson’s ratio s ϭ
. Note that the stress components are uniform
within the inclusion and that both the y-component of normal stress and the shear stress are zero.
The x-component of normal stress within the
softer inclusion is diminished in value relative to
the remote stress. The contour patterns outside the
inclusion are quite complex, but they clearly illustrate the facts that perturbations due to the inclusion are symmetric, they decrease with distance,
and they are negligible at radial distances r Ͼ 5R.
8.6.2 Generalized Hooke’s Law for
anisotropic rocks and minerals
The most general linear relationship among the
components of strain and stress is one in which
each of the nine components of strain, ij , is linearly related to the nine components of stress, ij .
For example, given a Cartesian coordinate system
with orthogonal axes x, y, and z, the equation for
the normal strain component, e xx , may be written
as (Nye, 1985, Chapter VIII):
(8.108)
The s ijkl are constants of proportionality and the
four subscripts on each s correspond to the two subscripts of the strain component e, followed by the
two subscripts of the stress component , with subscripts (1, 2, 3) corresponding to (x, y, z), respectively. There are eight more equations of this form,
each one linearly relating a component of strain to
ϩ s 1131 zx ϩ s 1132 zy ϩ s 1133 zz
ϩ s 1121 yx ϩ s 1122 yy ϩ s 1123 yz
xx ϭ s 1111 xx ϩ s 1112 xy ϩ s 1113 xz
1
4
1
4
r
1
8.6 ELASTIC HETEROGENEITY AND ANISOTROPY
325