1. Critical state soil mechanics in prediction of strains
Tuesday 5th December--2
Critical state soil mechanics in prediction of strains

2. Elements of Critical State Soil Mechanics in Prediction of Strains
1. Definition of stress and strain parameters
2. State parameters and the use of (q, p, e) and
applied stress paths and effective stress paths
3. Normally consolidated and over-consolidated states
4. Stress strain behavior and strength

3. Elements of Critical State Soil Mechanics in Prediction
of Strains (contd.)
5. Elementary concepts in theory of elasticity and
plasticity
6. Theories for normally consolidated clays
7. Natural deposits and over-consolidated states
8. Numerical analysis

4. Classical contributions
1. Rendulic (1936)
(on constant voids ratio contours)
2. Hvorslev (1936)
(on mean equivalent pressure)
3. Roscoe, Schofield & Wroth (1958)
(on state boundary surface)
4. Drucker (1959)
(Stability criterion

5. 5. Poorooshasb (1961) and
Roscoe & Poorooshasb (1963)
(Incremental drained strain as incremental
undrained and anisotropic consolidation
strains)
6. Thurairajah (1961)
(Energy Balance equation)
Calladine (1963)
(on elastic wall concept)
Roscoe, Schofield &Thurairajah (1963)

6. 7. Burland (1965)
Roscoe & Burland (1967)
( Modified energy balance equation
& constant q yield loci)
8. Wroth & Loudon (1967)
Pender (1970)
(on modeling over-consolidated clays )

7. Undrained tests Roscoe, Schofield & Wroth way of Interpretation Use of
Failure
Use of
(q,p) plot - stress
path &
Voids ratio or water
content or specific
volume with q and p
plots
Failure

8. Roscoe Schofield & Wroth contribution State boundary surface
in (q,p, e or v or w

9. equivalent pressure,p e
Hvorslev’s mean
equivalent pressure,p e
Densest state for
NC clays
Loosest packing
When sample A is sheared
to B, at B the mean equivalent
pressure is the same as the
consolidation pressure
corresponding to point D.
DBE is an undrained stress path

10. State Surface in 2-D Fundamental theories only for states on AB
Not necessarily
a straight
line
Same as
Rendulic’s
work
overconsolidated
State Surface in 2-D

11. Volumetric strain in drained test and excess pore
pressure in undrained test (NC state)

12. Volumetric dilation in heavily overconsolidated
clays in drained case and large excess negative
pore pressure in undrained case

13. Basic Philosophy of Models
1. Calculate volumetric strain increment dev from voids ratio change
2. Obtain dilatancy ratio

14. 3. Calculate des

15. Basic Philosophy of Models (contd.)
1. Above procedure valid for volumetric strains
in normally consolidated clays for all normally consolidated states.
2. For shear strain the method is valid for radial
stress paths for normally consolidated clays.
3. For non-radial stress paths undrained strain
increment be added to that determined from
(2) above.

16. Soil as a Dilatant Material with Dilatancy Dependent on Stress History
1. Normally consolidated state
2. Lightly over-consolidated state
3. Heavily over-consolidated state

17. Dilatancy during shear a. Drained condition ( High permeability)
1. An increase or decrease in volume
during shear
a. Drained condition
( High permeability)
b. Undrained condition
( Extremely low permeability )

18. Stress paths Undrained behavior p’ = p0 + q/3 -u Drained behavior
AB - applied stress path
drained path
A’B’ - undrained path
Drained behavior
p’= p0 + q/3

19. 2. Under drained condition
a. Normally consolidated clay
highest reduction in volume
due to shear
b. Heavily over-consolidated clay,
initially small reduction in
volume, followed by very large
increase in volume at the latter
stage in shear close to failure

20. 3. Under undrained condition
a. Normally consolidated clay develops
highest positive excess pore pressure
due to shear
b. Heavily over - consolidated clay
initially develops small positive
excess pore pressure due to shear
and subsequently very large
negative pore pressure

21. Volumetric strain & excess pore pressures
Normally consolidated clay
Heavily over-consolidated clay

22. Elasto-plastic Behavior
All strains are splitted into elastic and
plastic component
e1 = e1e + e1p ev = e1 + 2 e3
e3 = e3e + e3p es = ( e1 - e3 )
Elastic strains can be calculated from theory of elasticity and plastic strains are calculated separately using established concepts in theory of plasticity. The elastic shear strain is taken as zero for simplicity

23. Volumetric strain Volumetric strain can be calculated from the
voids ratio change. In an incremental form
where V is the volume and e is the voids ratio

24. Integrating where e0 and V0 are the initial voids ratio
and volume respectively

25. v = 1+ e Isotropic consolidation and swelling Consolidation Slope l
Cc = 2.303l
Swelling
Slope k
Cs = 2.303k

26. In order to calculate the volumetric strain from
voids ratio change, the undrained stress path is
needed. For normally consolidated clays the
undrained stress paths are geometrically similar
for various consolidation pressures. Also, the
difference in shape between the undrained stress
path and the volumetric yield locus is small.
Normally for clays
is less than 10 to 20 percent. If this value is taken as zero, then the undrained stress path coincides with the volumetric yield locus.

27. Volumetric Yield Locus and Undrained Stress Path
The undrained stress path is a reflection of the pore pressure development, which in turn is a refection of the dilatancy in the soil. Thus it is important that the
dilatancy is modeled correctly. The Roscoe & Burland equation seem to fit dilatancy well.

28. Roscoe & Burland -- Energy Balance
Thus

29. The above equation is in terms of plastic strains
The above equation is in terms of plastic strains. It can be easily shown that
The dilatancy ratio above in terms of total strain can be verified during isotropic and anisotropic consolidation and is found to be
in agreement with experimental data.

30. Drucker's Stability Criterion
Drucker from Brown University
postulated the stability crierion for
stable materials obeying the normality
rule as

31. Plastic Potential normal to plastic strain rate direction
For associated flow rule
plastic potential coincide
with yield curve
Plastic
strain
rate

32. Normally
consolidated
clay
Stable behavior
dq des + dp dev > 0

33. Drucker Unstable beyond C dq des + dp dev < 0 Stable up to C

34. This will then give
Thus, the gradient of the yield locus
is normal to the plastic strain ratio

35. Volumetric Yield Locus
Using the energy balance equation
and stability criterion
The differential equation of the yield locus can be obtained as

36. Over-consolidated clays - current studies
Inside SBS

37. No theory developed in a fundamental
way for covering over-consolidated states

38. State paths of
over-
consolidated
specimens
sheared with
different
applied
stress path in
(q,p) and
(v,p) planes

39. over- consolidated clays
Natural deposits and
over- consolidated clays
1. The undrained stress path is dependent on
the OCR values and cannot be determined in a fundamental way
2. The dilatancy ratio also depend on the
stress ratio and p values changes in magnitude from positive to negative as the OCR values increase from lightly over- consolidated state to heavily
over- consolidated state

40. Undrained stress path of Overconsolidated clays
Empirical approach to modeling undrained stress path
Data from Loudon, 1967
Modeling by Pender

41. Plastic strain rates in Over-consolidated clay
Pender’s empirical modeling

42. Pender’s Plastic strain increment ratio

43. Experimental work at AIT
Isotropic stress conditions

44. Experimental work at AIT

45. Experimental work at AIT
Constant p compression
Constant p extension

46. Stress probing inside
the state boundary
surface from four
stress states

47. Boundary of yielding
and collapse

48. A unified outer
boundary with
yielding on the
state boundary
surface and
failing on the
curved envelope.
No need to
divide the limit
state to friction
and cohesion.