stress, strain, and viscosity
TRANSCRIPT
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Stress, Strain, and Viscosity
San Andreas FaultPalmdale
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Solids and Liquids
Solid Behavior:- elastic- rebound- retain original shape- small deformations are temporary- (e.g. Steel, concrete, wood,
rock, lithosphere)
Liquid Behavior:- fluid - no rebound- shape changes- permanent deformation- (e.g. Water, oil,
melted chocolate, lava)
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Solids and Liquids
Linear Viscous Fluid:
- Rate of deformation is proportional to the applied stress
- A “linear” viscous fluid (w.r.t. stress and strain rate)
- Also known as a “Newtonian Fluid”
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Solids > ? > Liquids
Is there anything that behaves in a way between solid and liquid ?
Plastic Material- solid- but deforms permanently- malleable- ductile
Ductile or malleable materials are “non-linear”
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Solids and Liquids
Are all materials either a solid or a fluid, all the time ?
Applied heat can cause solid materials to behave like a fluid
Some material may be elastic when small forces are applied
but deform permanently with larger applied forces
Elastic: Deformed material returns to original shape
Ductile: Stress exceeds theelastic limit and deforms material permanently
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This rock responded to stress by folding and flowing by ductile deformation.
Occurs under high heat and high pressure
This rock fractured under stress by brittle deformation.
Occurs under low heat and at shallow depths in the Earth's crust.
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Unstressedcube of rock
Figure 9.2
© 2008, John Wiley and Sons, Inc.
Three types of stress
TENSIONALSTRESS
COMPRESSIONALSTRESS
SHEARSTRESS
Stress: is the force acting on a surface, per unit area – may be greater in certain directions.
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Strain
Consider a layer of fluid between 2 plates
The top plate moves with velocity, V
V
h
D'B'
A
B
C
D
The shape change can be written as V t / H
The rate of shape change is d/dt(shape change)
rate of deformation = = V/H (units of 1/time).
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Stress
V
h
D'B'
A
B
C
D
Pressure is applied to move the fluid
Stress is described as force per unit area (units of pressure, Pa)
= force/area = F/LW
L
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Viscous FluidV
h
D'B'
A
B
C
D
A viscous fluid is defined by the relationship of stress to strain rate
L
2.
Viscosity () is the constant of proportionality (units of Pas) This “constitutive” equation describes the mechanical properties
of A material
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Viscous Fluid
If material has a high viscosity (),
it will strain less for a given applied stress ()
2.
= / 2
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Physical Laws of Motion
Newton's 1st Law: Object in motion stays in motion; Action and reaction,
Velocity motion imparted by the top plate induces a reaction of the fluid below
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Physical Laws of Motion
Newton's 2nd Law: Acceleration of fluid is proportional to the net force
What does this mean?
- If there is no acceleration, then forces balance, that isthings move but don't accelerate. In this case, forces balanceand there is “no net force”
For all viscous fluids, the net force = 0 Velocities in the Earth's mantle are small and accelerations
are negligible
Momentum is also negligible in slow viscous fluids
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Conservation of Mass
Mass is conserved in fluid flow (density changes are negligible)
Fluid is “incompressible” .
Rate of fluid flow into box = flow out of box
No net accumulation of material
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Stress >
Strain >
Viscosity >
See Class notes – Formal treatment of stress and strain
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Hydrostatic Stress
Hydrostatic stress is defined as confining pressure
Normal stresses acting on a particle are equal on all sides
Known as “isotropic”
With no tangential components
11
= 22
= 33
= -P
12
=
= 3
= 0
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Deviatoric Stress
Mantle flow is not driven by hydrostatic pressure
But is driven by deviations from it, known as deviatoric stress
Let's consider an average normal stress
ave
11
+ 22
+ 33
= ii/ 3
3 Then deviatoric stress is given by
dev
ij
ave
the applied stress - average normal stress = difference (deviatoric stress)
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Deviatoric Stress
dev
ij
ave
If deviatoric stress is non-zero, than fluid flow proceeds
11
– ave
12
13
21
22
-ave
23
31
32
33
- ave
If the diagonals are all equal, then there is no deviatoric stress And there is no fluid flow
Minimumstress
Maximumstress
Deviatoricstress
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Lithostatic Stress A special case of hydrostatic stress
Hydrostatic stress increases with depth in the Earth
But at each depth, normal stresses are balanced
As a slab sinks into the Earth's interior, it experiences progressivelyincreasing hydrostatic stress with depth known as lithostatic stress.
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Measuring Lithostatic Stress
Near the surface, stress measurements are strongly effected by faults, joints, etc.
At deeper depths, pressure closes faults and fractures and stresses are transmitted across faults
Activity: calculate deviatoric stress in a continental block
Normal faults in El Salvador Jointing in a rock
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Measuring Lithostatic Stress
Dep
th
Lin et al., 2006
Fracture density measured in the field in south Africa Boreholes show reduced hydraulic conductivity at deeper depths Indicating that fractures close with increasing lithostatic stress
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Strain
Strain is a measure of deformation
When deformation occurs, different parts of a body are displaced by different amounts
p1
p2 x1
x2
p3
p4
u1 = b x
2
Displacements (u) in x1 direction increase with increasing x
2 .
u2 = 0 No displacement in x
2 direction:
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Strain
p1
p2 x1
x2
p3
p4
Displacement of p1 and p2 are not equal This implies a gradient of displacement
d u1 = b = - tan
d x2
Where is the angle through which the body deforms Shearing deformation changes along the x
2 direction.
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Strain: Solid Body Rotation
p1
p2 x1
x2
If 1 =
2 , then we have rotation only and no deformation
In this case, the gradient in deformation can be written as:
tan
d x1
d u2
tan d x
2
- d u1
Solid body rotation suggests the gradient of deformations are equal
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Strain: How do we distinguish deformation and rotation ?
Think about how 1 and
2
relate to each other
( Rotation only
(nonzero Deformation
(0 Pure shear
(0
Simple shear
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Strain: How do we distinguish deformation and rotation ?
We generally write this in terms of the tangent of the rotation angle
d x1
d u2
12
= ½ ( tan tan ½ ( - )
d x2
d u1
Rotation:
d x2
d u1
Deformation:
12
= ½ ( tan tan ½ ( + )
d x1
d u2
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The Strain Tensor
The strain tensor can them be written
d xj
d ui
ij =½ ( + )
d xi
d uj
And the rotation tensor can them be written
d xi
d uj
ij =½ ( - )
d xj
d ui
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The Strain Tensor : Test
The strain tensor can them be written
d xj
d ui
ij =½ ( + )
d xi
d uj
Let's try a test: In the case of a box that is “stretched”
What is i and j in ij
?
Then find the solution
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The Strain Tensor : Case of Volume Change
If a cube expands (or shrinks)
V – Vo = 11
+ 22
+ 33
Vo
ij =
d xi
d ui u
u Indicates volume change, dilitation, divergence
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The Strain Tensor : Case of Volume Change
u = 0 Means there is no volume changeand fluid is incompressible
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Strain Rate
Strain rate describes deformation change over time
How fast can a material deform ?
How fast can fluid flow ?
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Strain Rate
Strain rate describes deformation change over time
How fast can a material deform ?
How fast can fluid flow ?
Rates are concerned with velocity
We can use velocity gradient to measure the rate of fluid flow or shear
d xj
d Vi
ij =½ ( + )
d xi
d Vj
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Viscous Fluids
Viscous fluids resist shearing deformation
Have a linear relationship between stress and strain rate (known as Newtonian fluids)
Fluid examples:- air, water viscosities are low
- honey, thick oil viscosities are high
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Viscous Fluids
Viscous fluids resist shearing deformation
Have a linear relationship between stress and strain rate (known as Newtonian fluids)
See Class Notes:
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Viscous Fluids
Viscous fluids resist shearing deformation
Have a linear relationship between stress and strain rate (known as Newtonian fluids)
V
h
D'B'
A
B
C
D
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