iii. strain and stress basics of continuum mechanics, strain basics of continuum mechanics, stress...
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III. Strain and Stress
• Basics of continuum mechanics, Strain
• Basics of continuum mechanics, Stress
ReadingSuppe, 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.
Force• Force is the cause of deformation and/or motion
of a body.
• 2 kinds of force: –Contact forces - involve physical contact between objects. Examples: the force involved in kicking a ball, pulling a wagon–Field forces - don't involve physical contact between objects. Examples: the gravitational force and the electromagnetic force
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.
Sign convention for n
• Geomechanics: normal stress is positive in compression
(convention used here)
• Continuum mechanics:normal stress is positive in extension
n
1
2
n
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…
13
23
33
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
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
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
Rearrange equations yet again…
Get more useful relationship betweenprincipal stresses andstress on any plane….
Rearrange equations…
[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 = (
( the mean or hydrostatic stress which produces change in volume is (
(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
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 sometime used to characterize the a particular case of biaxial stress
Do not confuse with ‘pure shear’ as a state of strain
n
Applications
• Dip angle of a normal fault
• Dip angle of a thrust fault
• Stress ‘refraction’ across an interface.