11 Stress, Strain, Elasticity and Plasticity An Introduction to the Mechanics of Soils

1
???????11? Stress, Strain, Elasticity and
Plasticity An Introduction to the Mechanics of
Soils

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

2
?????????-1

• Analysis of Deformation and Strain

• Mode of deformation.
• Stresses on a plane element.
• Continuous straining.
• Discontinuous slipping.

3
?????????-2
A
P pole

• 45-y / 2

b 45 y / 2

• Planes of zero strain increment.

Directions of zero strain increment
4
?????????-3
Analysis of discontinuous slipping
counterclockwise shear strains positive
P pole
5
?????????-4
Stress on planes for which tangents from the
origin touch the Mohrs circle
P pole
Mohrs circle of stress corresponding to the
state of stress above
the planes for discontinuous slipping
P pole
6
?????????-5
Mohrs circles of total and effective stress
7
?????????-6

• Relationships between stress and strain

principal plains of stress and principal plains
of strain and strain increment coincide
Coaxiality condition
Unit Work
? dissipative ex. plastic material
? conservative ex. elastic material
? continuous strain
? discontinuous slipping
for axial symmetry
for plane strain
8

• 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.
9

• 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
10

• 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
11

• 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
12
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
13
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
14
(No Transcript)
15

• Flow rule of plasticity theory
• Plastic potential
• Normality condition

plastic potential yield curve ? associated flow
rule ? normality condition
16

• 1.8 Ideal Elastic Behavior

? Hookes law
the behavior of isotropic elastic material
E Youngs modulus n Poissons ratio
reversible recoverable ?conservative
17
on the principal stress (strain) plane
(s2 s3 , de2 de3)
for the special case of axial symmetry
K bulk modulus G shear modulus
18
for plane strain
(de2 0)
Kps bulk modulus appropriate for plane
strain Gps shear modulus appropriate for
plane strain
19
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
20
? plasticity theory (1.7)
the relationship between two ratios ? the
value of Poissons ratio independent of
Youngs modulus
21

• 1.9 Elasto-Plastic Behavior

total strains elastic component plastic
component
22
(No Transcript)
23
(No Transcript)
24

• 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
25
in terms of effective stresses
no seepage pore pressure are hydrostatic
26
the compatibility condition
and hence
valid for drained and undrained loading
27

• 1.11 Relationships between Stress and Strain for
  Plane Strain

the stress-strain relationship for plane strain
28
(No Transcript)
29
(No Transcript)
30
An Introduction to the Mechanics of Soils

• 2.1 Introduction

the mechanical behavior of small elements of soil
? a number of mathematical models
a stress strain relationship valid for all
cases of loading and unloading
elastic behavior ? plastic behavior ? the state
boundary surface (a yield surface)
the critical state model
31

• 2.2 Drained and Undrained Loading

loading unloading
pore pressure response ? excess pore pressure
generated
consolidation effective stress change
volume change
drained loading
the rate of loading ? the rate of consolidation
undrained loading
32

• 2.3 The State Boundary Surface

33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38
(No Transcript)