1. Monday-6
Stress paths, State Paths and Use of Voids Ratio

2. This section deals with the stress paths under different type of conditions. The state parameters and the state paths and the use of voids ratio or water content as a state variable in addition to the two stresses q, and p.

3. Also, specific volume v can be used instead of voids ratio or water content. 
v = 1 + e

4. Only the behaviour of saturated clay will be discussed in this Course of lectures. The concepts are also eextended to partially saturated soils by other authors. Initially, the applied stress path is defined as applied on a soil element and this stress path becomes the drained path in the case of high permeability soils where the drainage of water is instantaneous.

5. However for low permeability soil such as clays, the undrained stress path deviates from the applied stress path
depending on the magnitude of the pore pressure that is developed. In the case of normally consolidated clays,
which have the loosest packing, the maximum pore pressure develops and the sample tends to decrease in volume due to shear.

6. That is the same as for very loose sand
That is the same as for very loose sand. However in the case of heavily over consolidated clay and very dense sand, negative pore pressures develop with a tendency for the soil to expand in volume under drained condition.

7. Isotropic Consolidation

8. When all the three principal stresses are equal in magnitude
then the stress system is said to be isotropic.

9. The application of a cell pressure of 20 kPa or 20 KN/m2 to a specimen of clay in the triaxial apparatus causes the sample to be subjected to an isotropic stress of 20 kN/m2. If the sample is now allowed to consolidate and when the consolidation is 100 pc complete we
say that the triaxial specimen is under an isotropic consolidation pressure of 20 kN/m2.

10. =
20 kN/m2

11. The deviator stress q is zero during isotropic consolidation.
Thus the effective stress path during isotropic consolidation lie along the p-axis in the (q,p) plot. If the behaviour of soils is isotropic, then isotropic stress increments cause isotropic strain increments.

12. Thus all the strain increments are equal. For isotropic consolidation
q = 0 and p = p0 =
Thus the effective stress path during isotropic consolidation lie along the p axis as shown in Fig. 2.1

14. The strain increments are such that
and the shear strain increment is zero.

15. For re-sedimented clay in the normally consolidated state, the voids ratio- mean normal stress relationship during isotropic consolidation is linear. Some time there can be a small concavity, but a linear assumption is often justified. In the early days, a logarithm to the base ten is used. 

16. log10 (10) = 1 
However when we deal
with differential algebra it
become useful to use
Napier logarithm and thus  
lne (10) = In using the
logarithm to the base ten log is
used as abbreviation and
for logarithm to the base e, ln
is used as abbreviation.

17. Fig. 2.2 illustrates the consolidation and swelling under isotropic stress
with linear mean normal stress scale. In Fig. 2.2, a number of swelling curves are shown as the swelling is done from isotropic stresses of 50, 100, 250, 500 and 1000 kN/m2.

21. The relationships shown in Fig. 2 and 3 can be expressed as
e0 is the initial voids ratio under an isotropic stress p0 and e, and p are the current voids ratio and mean normal stress respectively.

22. In terms of the natural logarithm, this relation becomes

23. Similarly, the swelling relations can be expressed as
and

24. Anisotropic consolidation
We have already defined consolidation as a phenomenon during which the stress ratio is maintained a constant.On the compression side, the major principal stress is the axial stress and the minor principal stress is the radial stress.

25. If we denote,
Then,
The k values also include the at rest K0 conditions..

26. Such that
During anisotropic consolidation
the ratio is found to be a
constant.

27. When is a constant
is also a constant
Thus,

28. During constant stress ratio paths, the strain ratios are also
constant as stated before. These values are given in Table 2.2.
The dilatancy ratio is plotted in Fig It also contained a point
corresponding to the K0 condition when the dilatancy ratio is 1.5.

32. This dilatancy ratio was used in the prediction of strains in Drained tests by Roscoe & Poorooshasb in an empirical manner.
Subsequently various stress strain theories have appeared with elasto-plastic concepts to model the dilatancy behavior. These aspects will be discussed in later Chapters.

33. Drained stress paths with
Conventional drained triaxial test carried out in the laboratory is such that the cell pressure is maintained constant after the consolidation.

34. If we do not involve the back pressure in our discussion, then the consolidation pressure is the same as the cell pressure and remains constant during the conventional drained test.

36. Differentiating for increments
and
Since
= constant

37. This stress path is shown in Fig. 2.6
Thus a conventional drained triaxial compression test from isotropic or anisotropic conditions can be described by
and
This stress path is shown in Fig. 2.6

39. If we now want to carry our a drained triaxial compression test with
Then the deviator stress and the cell pressure be adjusted.

40. Solving these equations, we have
Therefore in this test if the deviator stress is increased by 1 kN/m2, then the cell pressure must also be increased by a sixth of a kN/m2.

41. If we want to follow a K0 consolidation path with a K0 value of 0
If we want to follow a K0 consolidation path with a K0 value of 0.7 say, then the
and
The dq and the ds3 values be now adjusted appropriately such that

42. The common triaxial tests carried out are Then, and dp > 0
i. Compression loading test
for which ds3' = 0, ds1' = 0
Then,
and dp > 0

43. Compression unloading test
ds1' = 0, ds3' < 0
and dp < 0

44. iii. Extension loading test ds1' = 0, ds3' > 0
Then,
and dp > 0

45. iv. Extension unloading test
ds1' < 0, ds3' < 0
Then,
and dp < 0

46. In the case of drained test the applied stress path and the drained stress path are coincident as the excess pore
pressure due to shear is fully dissipated and is assumed to be zero. 

47. Undrained behavior
Unlike metals granular materials when subjected to shear change in volume.
The normally consolidated clay and lose sand tend to reduce in volume
and the heavily over-consolidated and dense sand tend to dilates in volume.
In the drained case such change in volume tales place.

48. However, in the undrained case when the tendency is to reduce
in volume, positive excess pore pressures develop. That is in the
case of normally consolidated clays and loose sand positive pore
pressure develops under undrained shear.

49. On the other hand, if the samples tend to expand in volume during
shear and this expansion is prevented then the samples tend to
develop negative pore pressures. Thus the heavily over consolidated
clays and dense sand tend to develop negative pore pressures during undrained shear.

50. Fig. 2.7 illustrates the undrained and applied stress path for a constant cell pressure condition when a sample is sheared from the isotropic stress in compression.

52. The undrained stress path shown in Fig. 2.7 illustrates
that the sample develop positive pore pressure and the state path
in (v, p) plot move from A'' to B'' as the positive pore pressure is
developed.

53. Normally Consolidated and Overconsolidated States
The definition of normally consolidated and overconsolidated
clays should be applicable for both the consolidation mode and shear with increasing stress ratios on the compression and extension sides.

54. Almost all the natural deposits of clays are at least lightly over consolidated. Normally consolidated state without
shear under isotropic condition correspond to the loosest packing of the clay particles with the highest voids ratio for a given value of the mean normal stress.

55. In Fig. 2.8 the line ABC correspond to the loosest packing in the normally consolidated state. In this figure all the possible states and impossible states are also shown. Except for B (Fig. 2.9)
all other states on the line BD correspond to the overconsolidated state.

58. The x-axis in Fig. 2.9 correspond to exponential logarithm and as such the slopes of consolidation and swelling lines are denoted as l and k. The specific volume is defined as
v = 1 + e

59. N corresponds to the specific volume when the mean normal stress p is unity on the normally consolidated state. For a maximum past pressure of pm and a current stress of unity except for the value of N all other specific volumes between N and vk correspond to the overconsolidated state since N is the
loosest specific volume.

60. It should be noted that while the consolidation and swelling curves in
1-D consolidation are parallel to the isotropic consolidation and swelling lines in the semi-log stress plot, the 1-D consolidation states correspond to a sheared mode and is with denser packing as shown in Fig For isotropic conditions, the normally and overconsolidated states are quite clear.

62. For example, consider a sample of clay subjected to the maximum past pressure of 100 kN/m2. Then at 100 kN/m2 of mean normal stress the sample is in a normally consolidated state. However, if this specimen is subsequently subjected to a stress release, such that the current stress is 50 kN/m2, then the specimen is now in
an overconsolidated state.

63. The degree of overconsolidation,
In the above definition only isotropic stresses are considered and the deviator stress is zero (see Fig. 2.11). In Fig. 12, let the point A corresponds to a normally consolidated state.

64. If now the specimen is subjected to a constant volume stress path (undrained stress path), it would follow a path of the form AC. All states of stress to the left of curve , AC and bounded by the failure envelope and the p-axis will correspond to the overconsolidated state. Those on the curve AC and to the right outside the curve will be in the
normally consolidated state.

66. Hence a specimen at state D in the normally consolidated state can be brought to the overconsolidated state, provided it is subjected to a stress path of the form DE as shown in Fig The specimen at state E in the over consolidated state, can only be brought to the normally consolidated state (say F) by the application of any stress path EF which crosses the boundary AC or lie on it.

68. Hence associated with the normally consolidated state is always a boundary (which is the constant volume path-- undrained path) separating the over consolidated states.

69. State Paths in (q,p), (v,p), (v,q) Planes
The major contribution made by Roscoe, Schofield and Wroth is to look at the consolidation and shear in the (q,p), (v,p) and v,q) planes. Some time and perhaps most of the time log(p) and log(q) are used instead of q and p when the specific volume, water content or voids ratio variation
is studied. The Napian logarithm is used. The drained state paths of normally
consolidated clay is shown in Fig

71. Fig. 2.14 contains the state paths under undrained condition bounded
by the isotropic states and the critical state line. The projection of the critical state line in the (q,p) plane is a straight line and can be expressed as,
q = Mp. Also the projection in the
(v, lnp) plot can be expressed as

73. The projection in (v, lnq) plot is also a straight line for normally
consolidated clays. It can be shown that the slope M in compression is given by
and
in extension by

74. These expressions can be used to determine the angle of internal
These expressions can be used to determine the angle of internal
friction, f. The results of the undrained test are shown in Fig
In the case of overconsolidated samples dilation takes place during
drained shear and negative excess pore pressures develop during undrained shear.

76. The overconsolidated samples were found to fail on a Hvorslev type failure. Roscoe, Schofield and Wroth made an idealistic postulate that for a given specific volume, there is one Hvorslev
failure envelope and the samples are ideally expected to reach the critical state line. The normally consolidated clays are well studied now and the strains in shear can be calculated from the Cambridge theories.