CIVL 272 Soil Mechanics
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Topic V: A Critical State Model to Interpret Soil
behavior
1. Introduction
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We are going to build a mosaic to provide a simple framework
to describe, interpret, and anticipate soil responses to various
loadings. The framework is essentially a theoretical model
based on critical state soil mechanics  critical state model
(Roscoe, Schofield, and Wroth, 1968). Laboratory and field
data, especially results from soft normally consolidated clays.
The critical state model (CSM) we are going to study is a
simplification and an idealization of soil behavior. However,
the CSM captures the behavior of soils that are of greatest
importance to geotechnical engineers. The central idea in the
CSM is that all soils will fail on a unique failure surface in (q,
p’, e) space. Thus, the CSM incorporates volume changes in its
failure criterion.
The CSM is a tool to make estimates of soil responses when
you cannot conduct sufficient soils tests to completely
characterize a soil at a site or when you have to predict the
soil’s response from changes in loading during and after
construction. Although there is a debate on the application of
the CSM to real soils, the ideas behind the CSM are simple.
What will you learn?
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Estimate failure stresses for soil
Estimate strains at failure
Evaluate possible soil stress states and failure if the
loading on a geotechnical system were to change.
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2. Parameter Mapping
Mohr-Coulomb failure envelope in the p’-q space
Critical state line (CSL)
q f  M  p 'f (f denotes failure)
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Normal consolidation line (NCL)
NCL (isotropic normal consolidation line)
Cc
C
 c  0.434C c
ln 10 2.3
C
C
  r  r  0.434C r
ln 10 2.3

Both  and  are positive for compression. For many
soils, / has values within the range 1/10 to 1/5.
CIVL 272 Soil Mechanics
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3 Failure line
The fundamental concept in critical state modeling is
that a unique failure surface exists in (q, p’, e) space,
which defines failure of a soil irrespective of the history
of loading or the stress paths followed. Note that the
critical sate is a constant stress state characterized by
continuous shear deformation at constant volume.
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Critical state line in p’-q-e space
CIVL 272 Soil Mechanics
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Critical state line (CSL) in p’-q space and e-lnp’ space
Note that
q = Mc p’
6 sin 'cs
Mc 
3  sin 'cs
The CSL is parallel to
the NCL and has a
slope .
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4. Soil yielding (soil plasticity, primarily based on
Dr. Li’s notes)
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Soil yielding in the triaxial test (normally consolidated
or slightly consolidated clay).
(1) Isotropic compression
ESP
(2) Drained compression (CD test)
ESP
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(3) Undrained compression (CU test)
ESP
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Plasticity
There are three essential features of the theory of plasticity:
yielding, hardening, and flow
(1) Yielding surface
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(2) Hardening
The yield stress in soil is not a constant.
The yield surface is assumed to be an ellipse (modified
Cam clay model) and its initial size or major axis is
determined by the preconsolidation stress, pc’ (the
maximum historical stress). The higher preconsolidation
stress, the larger the initial ellipse. In addition, the yield
surface only changes its size but not shape and location in
the q-p’ space. This assumption is called isotropic
hardening.
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The yielding surface in the (p’-q-e space)
The surface represents the state boundary. Inside the
surface, only the elastic deformation can take place. On the
surface, soils start to yield and plastic deformation can be
observed.
The function of yield surface on the p’-q space for the
modified Cam clay model is
( p' )
2
 p' p 'c

q2
M
2
0
CIVL 272 Soil Mechanics
The yield surface on p’-q and e-lnp’ space
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(3) Flow rule
When the state of stress reaches the yield surface, the
plastic strain increment vector is normal to the plastic
potential curve. A common assumption, knows as the
associated flow rule or normality principle, considers that
the yield surface coincides with the plastic potential curve,
therefore, the plastic strain increment is normal to the yield
surface.
CIVL 272 Soil Mechanics
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Prediction of the behavior of normally consolidated clay
(1) Isotropic compression
(2) Drained compression (CD test)
(3) Undrained compression (CU test)