nb: uniaxial strain is a type a non-rotational transformation uniaxial strain
TRANSCRIPT
NB: Uniaxial strain is a type a non-rotational transformation
Uniaxial strain
Pure Shear
NB: Pure shear in is a type a non-rotational transformation
Simple Shear
NB: Simple shear is rotational
Progressive pure shear
Progressive pure shear is a type of coaxial strain
Progressive simple Shear
Progressive simple shear is non coaxial
III. Strain and Stress
• Strain
• Stress
• RheologyReadingSuppe, Chapter 3Twiss&Moores, chapter 15
Additional References :Jean Salençon, Handbook of continuum mechanics: general concepts,
thermoelasticity, Springer, 2001Chandrasekharaiah D.S., Debnath L. (1994) Continuum Mechanics
Publisher: Academic press, Inc.
Stress
Stress is force per unit area
– Spreading out the weight reduces the stress with the same force.
F=mg
Normal Stress is skier’s weight distributed over skis surface area.
Thought experiments on stress…
The “flat-jack” experiment…
What if we rotate the slot???
Two stress components…
n
n
1
2
The stress (red vector) acting on a plane at M is the force exterted by one side over the other side divided by plane area…
Stress acting on a plane at point M…Let n be the unit vector defining an oriented surface with elementary area da at point M. (n points from side A to side B)
Let dT be the force exerted on the plane by the medium on side B. It can be decomposed into a normal and shear component parallel to the surface. The stress vector is:
2 2
n n n
n n n
n
Normal stress
Shear stress
,( ) i j j
dTn n n
da
Side B
Side A
Convention: positive in compression
The state of stress at a point can be characterizes from the stress tensor defined as …
i, j 11 12 13 21 22 23
31 32 33
The stress tensor
i, j 11 12 13 21 22 23
31 32 33
Symmetry…
i, j j ,i
Principal stresses
11 1
, 22 2
33 3
0 0 0 0
0 0 0 0
0 0 0 0i j
1 2 3
Engineering sign convention tension is positive,Geology sign convention compression is positive…
Plane perpendicular toprincipal direction has no shear stress…
Because the matrix is symmetric, there is coordinate frame such that….
The deviatoric stress tensor…
Stress tensor = mean stress + deviatoric stress tensor
i, j m 0 0
0 m 0
0 0 m
1 m 0 0
0 2 m 0
0 0 3 m
i, j m i, j
11 22 33 1 2 3
3 3m
mean stress:
Expressed in a reference frame defined by the principal directions:
Do not confuse the deviatoric stress tensor with the ‘differential stress’, often noted , defined as
1 3
Sum of forces in 1- and 2-directions…
2-D stress on all possible internal planes…
The Mohr diagram
Sum of forces in 1- and 2-directions…
2-D stress on all possible internal planes…
Rearrange equations yet again…
Get more useful relationship betweenprincipal stresses andstress on any plane….
Rearrange equations…
The Mohr diagram
[1] What does a point on the circle mean?
[2] What does the center of the circle tell you?
[3] Where are the principle stresses?
[4] What does the diameter or radius mean?
[6] Where is the maximum shear stress?
Any point on the circle gives coordinates acting on the plane at an angle to
Maximum shear stress max occurs for =45°; then max = (
(is the mean or hydrostatic stress= that which produces change in volume
(is the maximum possible shear stress= that which produces change in shape
In direction of and = 0; hence and are on the abscissa axis of Mohr graph
Pole of the Mohr circle
n
A
B
2
P
Poles of the Mohr circle
n
A
P
Poles of the Mohr circle
n
A
P
A represent the state of stress on a facet with known orientationThe geometric construction, based on the pole of the facet (P), allows to infer the state of stress on any orientation
Representation of the stress state in 3-D using the Mohr cirles.
n
The state of stress of a plane with any orientation plots in this domain
This circle represent the state of stress on planes parallel to
This circle represent the state of stress on planes parallel to
This circle represent the state of stress on planes parallel to
Classification of stress state
– General tension
– General compression
– Uniaxial Compression
– Uniaxial tension
– Biaxial stress
Pure Shear(as a state of stress)
The exression ‘Pure shear’ is also used to characterize the a particular case of biaxial stress
Do not confuse with pure shear as a state of strain
n
III. Strain and Stress
• Strain
• Stress
• RheologyReadingSuppe, Chapter 3Twiss&Moores, chapter 15
Additional References :Jean Salençon, Handbook of continuum mechanics: general concepts,
thermoelasticity, Springer, 2001Chandrasekharaiah D.S., Debnath L. (1994) Continuum Mechanics
Publisher: Academic press, Inc.
• A rheological law relates strain to stress and time.
A review of simple rheological models…
(f) Elasto-plastic
F
(e) Perfect plastic
F
Elastic rheology
Elastic deformation is recoverable
In linear elasticity stress is ‘proportional’ to strain
In linear elasticity stress is ‘proportional’ to stress (‘Hooke’s law’)
, , , , ,i j i j k l k l
C
C
C is the elasticity tensor. It is a symmetric tensor (21 elasticity coefficients)
Elastic rheology
For an isotropic linear elastic medium there are only 2 elasticity coefficients, e.g.: the Young modulus and Poisson ratio:
r
l
1 lE Shear modulus
Bulk modulus
Young Modulus
Poisson ratio
Viscous rheology
For Newtonian rheology the strain rate is ‘proportional’ to stress.
Viscous deformation is non recoverable
Plastic rheology
For Newtonian viscous rheology the strain rate is ‘proportional’ to stress.
Plastic deformation is non recoverable
(e) Plastic
F
t1 t2
t1 t2
NB: The strain evolution depends on the experimental conditions (ex: stiffness of the apparatus)
s
s
Coupled rheological models…
Recoverable, with hysteresis loop
Non-recoverable
F
t1 t2
…Coupled rheological models
(f) Elasto-plastic
Non-recoverable