10 Stress, Strain, Elasticity and Plasticity

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???????10? Stress, Strain, Elasticity and
Plasticity

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• ahirooka_at_civil.kyutech.ac.jp
• http//geo.civil.kyutech.ac.jp

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Stress, Strain, Elasticity and Plasticity
model physical representation in the sense of a
scale model ? a conceptual idea or a number
of mathematical equations
For Soils
mathematical models
based on theories of elasticity, plasticity, and
friction
Critical State Model
Schofield and Wroth, 1968 Atkinson and Bransby,
1978
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Special case Solid cylinders of soil in triaxial
tests s2 s3 de2 de3
Soil an aggregate of mineral grains with a
fluid filling the pore spaces dry pore fluid
is air alone saturated pore fluid is water
alone
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• effective stress s s u controls soil
  behavior

The principle of effective stress (stated by
Terzaghi 1936) ?All measurable effects of a
change of stress, such as compression,
distortion, and a change of shearing resistance,
are due exclusively to changes of effective
stress.
Mathematical models for soil behavior must be
written in terms of effective stress, not total
stress.
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• 1.3 Analysis of Deformation and Strain

dsv, dsh stress increment dv, dh small
change of dimensions
Mode of deformation. (a) Stresses on a plane
element. (b) Continuous straining. (c)
Discontinuous slipping.

• no discontinuities in the element. the behavior
  in each and every part of the element is the
  same.
• all deformations are due to very large strain
  within the slip zone.
• ? Slip Plane ( in the limit , the zone zero
  thickness)

(b) and (c) often occur simultaneously, but (b)
or (c) usually predominates.
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Increment of plane displacement
the engineers shear strain
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• Increment of strain
• increments of strain in a plane element
  (positive)
• principal strain

counterclockwise shear strains positive
Mohrs circle of plane strain corresponding to
the increments of strain above
P pole
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Planes of zero strain increment.
y an angle of dilation
eg maximum shear strain ev volumetric strain
(if e20)
P pole
the increment of minor principal strain is
negative and tensile
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, in the limit,
, in the limit,
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Directions of zero strain Increment
From the geometry, the tangent to the point A
makes an angle y to the de axis.
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all deformations and strains are contained within
slip line ? discontinuous slipping
Analysis of discontinuous slipping
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• 1.4 Analysis of State of Stress

State of stress in two dimensions
Mohrs circle of stress corresponding to the
state of stress above
P pole
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stress parameters t
s
mobilized angle of shearing resistance r
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Stress on planes for which tangents from the
origin touch the Mohrs circle
the planes for discontinuous slipping
P pole
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Mohrs circles of total and effective stress
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• 1.5 Relationships between stress and strain

principal plains of stress and principal plains
of strain and strain increment coincide
Coaxiality condition
(??)
not a general theoretical requirement of a
material behavior a good approximation for
material behavior
increment of energy transferred from the external
forces
dE
Unit Work
a part of energy stored and recovered on unloading
dU
a part of energy dissipated in the material
dW
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? continuous strain
? discontinuous slipping
? determinant of C is non-zero
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for axial symmetry
for plane strain
Incremental Model
? alternative formulation which relates
increments of strain to states, rather than
increments, of stress
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• 1.6 Elastic and Plastic Deformations

Mathematical model for material behavior
states and increments of stress ? states and
increments of strain
based on the theories of elasticity and plasticity
Examine the behavior of an ideal soil-like
material subjected to principal effective stress
sa , sb and sc ? the principle of effective
stress Strains depends on effective, not total,
stresses.
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• Stress-strain curves for an ideal soil-like
  material for uniaxial compression
• Strain hardening

OY linearly elastic, strains are fully
recovered
for states beyond Y irrecoverable
plastic strains occur
Y yield point
the stress is reduced from G linearly
elastic in the range BG
G yield point ? Yield stress increases with
plastic strain
? Strain hardening
F failure point
G?F suffering elastic and plastic strains
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• Stress-strain curves for an ideal soil-like
  material for uniaxial compression
• Strain softening

OY linearly elastic, strains are fully
recovered
for states beyond Y irrecoverable
plastic strains occur
Y yield point
the stress is reduced from G linearly
elastic in the range BG
G yield point ? Yield stress decreases with
plastic strain
? Strain softening
G?F suffering elastic and plastic strains
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• 1.7 Yielding, Hardening and Plastic Flow

various combinations of sa and sc at yield and
at failure for a strain hardening material
YaYc yield curve FaFc failure
envelope YaYc FaFc geometrically similar or not
YaYc ? GaGc Strain Hardening change of yeild
curve ? Hardening law plastic strain
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The state of the material can be on or within the
yield surface but cannot lie outside it.
for a strain hardening material
for a strain softening material ?????
yield surface
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Hardening material
loading path A?B A?C A?D traverse the
yield surface and plastic strains occur
unloading path A?E traverse an elastic
wall and plastic strains are zero purely elastic
strains occur
Paths for loading and unloading
unloading path A?E Impossible!!!!
OgGaGc elastic wall
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• Flow rule of plasticity theory
• Plastic potential
• Normality condition

plastic potential yield curve ? associated flow
rule ? normality condition
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• 1.8 Ideal Elastic Behavior

? Hookes law
the behavior of isotropic elastic material
E Youngs modulus n Poissons ratio
reversible recoverable ?conservative
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on the principal stress (strain) plane
(s2 s3 , de2 de3)
for the special case of axial symmetry
K bulk modulus G shear modulus
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for plane strain
(de2 0)
Kps bulk modulus appropriate for plane
strain Gps shear modulus appropriate for
plane strain
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for the special case of axial symmetry des
dependent only on dq increment of shear
stress ? increment of shear strain dev
dependent only on dp increment of normal
stress ? increment of volumetric strain
for plane strain deg dependent only on
dt dev dependent only on ds
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? plasticity theory (1.7)
the relationship between two ratios ? the
value of Poissons ratio independent of
Youngs modulus
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• 1.9 Elasto-Plastic Behavior

total strains elastic component plastic
component
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• 1.10 Equilibrium and Compatibility

a complete solution for stresses and displacement
of a soil structure ? necessary to satisfy
conditions of equilibrium and compatibility
for plane strain
in the xz plane
g the unit weight of the soil
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in terms of effective stresses
no seepage pore pressure are hydrostatic
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the compatibility condition
and hence
valid for drained and undrained loading
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• 1.11 Relationships between Stress and Strain for
  Plane Strain

the stress-strain relationship for plane strain