Soil behaviour in shear respectively. The yield function is written in terms of stress components or principal stresses, and defines the yield point as a function of current effective stresses and stress history. The Mohr–
Coulomb criterion, which will be described in Section 5.3, is one possible (simple) yield function if perfectly plastic behaviour is assumed. The hardening law represents the relationship between the increase in yield stress and the corresponding plastic strain components, i.e. defining the gra- dient of Y′P in Figure 5.3(d). The flow rule specifies the relative (i.e. not absolute) magnitudes of the plastic strain components during yielding under a particular state of stress. The remainder of this book will consider simple elastic–perfectly plastic material models for soil, as shown in Figure 5.3(b), in which elastic behaviour is isotropic (Section 5.2) and plastic behaviour is defined by the Mohr–Coulomb criterion (Section 5.3).
It is therefore only necessary to know any two of the elastic properties; the third can always be found using Equation 5.6. It is preferable in soil mechanics to use ν and G as the two properties.
From Equation 5.4 it can be seen that the behaviour of soil under pure shear is independent of the normal stresses and therefore is not influenced by the pore water (water cannot carry shearing stresses). G may therefore be measured for soil which is either fully drained (e.g. after consolidation is completed, see Chapter 4) or under an undrained condition (before consolidation has begun), with both values being the same. E, on the other hand, is dependent on the normal stresses in the soil (Equation 5.5), and therefore is influenced by pore water. To determine response under imme- diate and long-term loading it would be necessary to know two values of E, but only one of G.
Poisson’s ratio is also dependent on the drainage conditions. For fully or partially drained conditions, ν< 0.5, and is normally between 0.2 and 0.4 for most soils under fully drained condi- tions (see Section 5.10). Under undrained conditions, the soil is incompressible (as no pore-water drainage has yet occurred). The volumetric strain of an element of linearly elastic material under normal stresses σx, σy and σz is given by
DV
V =ex+ey+ez = 1 2-E n s s s
(
¢ + ¢ + ¢x y z)
where V is the volume of the soil element. Therefore, for undrained conditions ΔV/V = 0 (no change in volume), hence ν = νu = 0.5. This is true for all soils provided conditions are completely undrained.
In some 3-D problems, there are only substantial strains in a 2-D plane, with strains in the third direction being negligible. This is true of problems which are ‘long’ in one direction, such as strip footings, long retaining walls and slopes/embankments. Such conditions are described as plane strain. If the y-direction is taken as that in which the strains are negligible, then εy = γxy = γyz = 0.
From the second line of Equation 5.5
¢ =
(
¢ + ¢)
sy n sx sz
from which it is possible to reduce Equation 5.5 to e
e
g n
n n n
n n n
n
x z xz
é ë êê ê
ù û úú ú
=
(
+)
- -
(
+)
-
(
+)
-(
+)
1 2 1
1 1 0
1 1 0
0 0 2 1
2
2
G éé
ë êê ê
ù
û úú ú
¢
¢ é ë êê ê
ù û úú ú s s t
x z xz
(5.7)
The elastic constants G, E and ν can further be related to the constrained modulus (Eoed¢ ) described in Section 4.2.2. In the oedometer test, the soil strains in the z-direction under drained conditions, but the lateral strains are zero in both directions and there are no applied shear stresses and no resultant strains (εx = εy = γxz = γxy = γyz = 0). The two lateral stresses are also equal in magnitude (i.e.
¢ = ¢
sx sy). This is therefore an example of further reduction to 1-D conditions. From Equation 5.5:
0
0 1
2 1 1
1
e n 1
n n
n n
n n
s s t
z
x y xz
=
(
+)
− −
− −
− −
′
′
G
From either the first or second line:
¢ = - ¢
s n
ns
x z
1 (5.8)
Soil behaviour in shear from which
e n n
n s
z = z
′
− ′ − ′
− ′
′
1 1 2
1
2
E
where E′ and ν′ are the Young’s modulus and Poisson’s ratio for fully drained conditions. Then, from the definition of E¢oed (Equations 4.3 and 4.5):
¢ = ¢
= ¢ - ¢
( )
- ¢ - ¢
\ ¢ = ¢
(
+ ¢) (
- ¢) (
- ¢)
E E
E E
oed z
z
oed
s e
n
n n
n n
n 1
1 2
1 1 2
1
2
(5.9)
Substituting Equation 5.6
2 1 1 1 2
1 1 2 2 1
G E
G E
(
+ ¢)
= ¢(
+ ¢) (
- ¢) (
- ¢)
\ = ¢
(
- ¢) (
- ¢)
n n n
n n n
oed
oed
(5.10)
Therefore, the results of oedometer tests can be used to define the shear modulus.
Non-linear elasticity
In reality, the shear modulus of soil is not a material constant, but is a highly non-linear function of shear strain and effective confining stress, as shown in Figure 5.4(a). At very small values of strain, the shear modulus is a maximum (defined as G0). The value of G0 is independent of strain, but increases with increasing effective stress. As a result, G0 generally increases with depth within soil masses. If the shear modulus is normalised by G0 to remove the stress dependence, a single non-linear curve of G/G0 versus shear strain is obtained (Figure 5.4(b)). Atkinson (2000) has sug- gested that this relationship may be approximated by Equation 5.11:
G
G0 0
1
1 1 0
= -
( )
-
(
g gg gp)
£B
p
B . (5.11)
where γ is the shear strain, γ0 defines the maximum strain at which the small-strain stiffness G0 is still applicable (typically around 0.001% strain), γp defines the strain at which the soil becomes plastic (typically around 1% strain) and B defines the shape of the curve between G/G0 = 0 and 1, typically being between 0.1–0.5 depending on the soil type. This relationship is shown in Figure 5.4(b).
For most common geotechnical structures, the operative levels of strain will mean that the shear modulus G<G0. Common strain ranges are shown in Figure 5.4(c), and these may be used to estimate an appropriate linearised value of G for a given problem from the non-linear relation- ship. The full non-linear G–γ relationship may be determined by:
1 undertaking triaxial testing (described in Section 5.4) on modern machines with small-strain sample measurements; this equipment is now available in most soil testing laboratories;
2 determining the value of G0 using seismic wave techniques (either in a triaxial cell using specialist bender elements, or in-situ as described in Chapters 6 and 7) and combining this with a normalised G/G0 versus γ relationship (e.g. Equation 5.11; see Atkinson, 2000 for further details).
Of these methods, the second is usually the cheapest and quickest to implement in practice. In principle, the value of G can also be estimated from the curve relating principal stress difference and axial strain in an undrained triaxial test (this will be described in Section 5.4). Without small-strain sample measurements, however, the data are only likely to be available for γ> 0.1%
(see Figure 5.4(d)). Because of the effects of sampling disturbance (see Chapter 6), it can be prefer- able to determine G (or E) from the results of in-situ rather than laboratory tests.