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

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

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

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. 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)

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 )

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

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

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

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

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

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

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

3. Calculate des

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.

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

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 )

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

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

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

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

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

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

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

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

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.

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.

Roscoe & Burland -- Energy Balance Thus

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.

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

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

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

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

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

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

Over-consolidated clays - current studies Inside SBS

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

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

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

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

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

Pender’s Plastic strain increment ratio

Experimental work at AIT Isotropic stress conditions

Experimental work at AIT

Experimental work at AIT Constant p compression Constant p extension

Stress probing inside the state boundary surface from four stress states

Boundary of yielding and collapse

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.