Stresses in triaxial specimens

1
Stresses in triaxial specimens
F Deviator load
?r
?r
?r Radial stress (cell pressure)
?a Axial stress
From equilibrium we have
2
Typical triaxial results
q
Increasing cell pressure
?a
3
Mohr Circles
To relate strengths from different tests we need
to use some results from the Mohr circle
transformation of stress.
?
c
?
?1
?3
The Mohr-Coulomb failure locus is tangent to the
Mohr circles at failure
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Soil Strength Which Parameters to Use?Angle of
Internal Frictions f- the fundamental strength
property of non cemented soils
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Soil Strength Which Parameters to Use?But
unlike the sliding block problem soils present
dilation
Constant N test
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Soil Strength Which Parameters to Use?But
dilation is a factor of density and stress state
D, M, L Dense, Medium, Loose
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Soil Strength Which Parameters to Use?But
dilation is also a factor of stress state
CS fcs!!!
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Soil Strength Which Parameters to Use?Critical
State Concept
When sheared, state of soil tends towards a
unique line in t - s' - e space. Or
alternatively, the q p e space This is
called the critical state line (CSL).
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Soil Strength Which Parameters to Use?undrained
loading of clays
Shear stress t
Drained strength sd
f'cv
Undrained strength su
Undrained strength su
Drained strength sd
s'n
Void ratio e
Positive pore pressure reduces effective stress
Suction increases effective stress
Dilation
Undrained testÞ no volume change allowed
Loose states
eo
Dense states
Contraction
CSL
Normal effective stress s'n (or mean effective
stress p')
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Discussion What is su and Why Can we Use it for
Clays?
Facts

• In undrained loading of clays, for the same
  density the critical state shear stress is
  constant (regardless the initial stress).
• Which means that if eo is constant with depth we
  can idealized the Mohr-Coulomb failure envelope
  by a simpler Tresca (csu, fu0) failure
  envelope. This is what we called the Total Stress
  Analysis (TSA).

But remember su (or cu) is not a fundamental
strength property!!!
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Relation between effective and total stress
criteria
Because each sample is at failure, the
fundamental effective stress failure condition
must also be satisfied. As all the circles have
the same size there must be only one effective
stress Mohr circle
?
?
?1
?3
We have the following relations
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Soil Strength Which Parameters to Use?undrained
loading of claysTOWARDS Critical State Soil
Mechanics
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critical state soil mechanicsSchofield Wroth,
1968
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compression experiments on sandNakata, Kato,
Hyodo, Hyde, Murata, 2001

• Observation
• the projection of the virgin compression line on
  the e-scale is not unique
• l is not unique either

• So what is really going on?
• McDowell-Bolton-Robertson (JMPS, 1996) elegantly
  justified hardening via crushing
• what about particle size effect?
• how to explain variations in py0?
• what about the CSL- why is it not unique?
• can we avoid curve fitting?

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Breakage Mechanics
Soil
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breakage definition Hardin, 1985
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breakage definition revisitedEinav, JMPS,
2007a,b
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ultimate grain size distribution
Sammis-King-Biegel, Pageoph 1987
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ultimate grain size distribution Oded Ben-Nun,
2007
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fractional breakage Einav, JMPS, 2007a
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Discussion on stored energyWhat makes it so
useful?
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statistical average Einav, JMPS, 2007a
so, we can evaluate the gsd p(d) using B, and the
initial and ultimate gradings
for example, the overall stored energy Y is the
average of the stored energy y(d) in fraction
size d then
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Stored energy per fraction Einav, JMPS, 2007a
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Stored energy per fraction Einav, JMPS, 2007a
Hypothesis fraction energy is scaled
proportional to the grains surface area
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modelling schematisation
damage mechanics
breakage mechanics
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breakage energy Einav, JMPS, 2007a
Breakage Energy
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postulate of breakage growth fracture
propagation in granular matter Einav, PRSA,
2007
breakage dissipation
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connection to well established theory
fracture mechanics (Griffith, 1921)
breakage mechanics (Einav, 2007)
Ec critical breakage energy constant J
normalised surface area
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breakage and plasticity as active and
passive mechanisms

• In compression force chains are jammed prior to
  crushing.
• After crushing the force chains get unjammed,
  followed by rearrangement (i.e. plasticity) as
  the passive mechanism.
• In shear plasticity evolves from active sliding
  (and rotation) of particles ? breakage is passive
  from abrasion.

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Breakage and Mohr-Coulomb Einav, 2007
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students model of breakage Einav, 2007d, Phil
Trans A
CLAYS critical state soil mechanics
SANDS breakage soil mechanics