Transcript Rheology

Rheology
Different materials deform differently under the same state of
stress. The material response to a stress is known as rheology.
Ideal materials fall into one of the following categories:
o Elasticity.
o Viscosity.
o Plasticity.
Mechanical analogues
  C
Elastic

Viscous
Plastic
Stress-Strain relations

ds
s  
dt
slip  0 if
 s   n
Recoverable versus permanent:
The eformation is recoverable if
the material returns to its initial
shape when the stress is
removed.
Deformation is permanent if
the material remains
deformed when the stress
is removed.
While elastic deformation is recoverable, viscous and plastic
deformations are not.
Real materials exhibit a variety of behaviors:
from ic.ucsc.edu/~casey/eart150/Lectures/Rheology/13rheology.htm
The behavior of real materials is better described by
combining simple models in series or parallel. For
example:
A visco-elastic (or
Maxwell) solid:
An elasto-plastic
(Prandtl) material:
A visco-plastic (or
Bingham) material:
A firmo-viscous
(Kelvin or Voight)
material:
It turns out that rocks subjected to small strains (seismic
waves, slip on faults, etc.) behave as linear elastic
materials.
Elasticity:
The one-dimensional stress-strain relationship may be
written as:
  C
n
where C is an elastic constant.
Note that:
o The response is instantaneous.
o Here the strain is the infinitesimal strain.
The material is said to be linear elastic if n=1.
Hooke’s law:
  C
In three dimensions, Hooke’s law is written as:
 ij  Cijkl kl
where Cijkl is a matrix whose entries are the stiffness coefficients.
It thus seems that one needs 81(!) constants
in order to describe the stress strain
relations.
 11  C111111  C111212  C1113 13  C1121 21  C1122 22  C1123 23  C1131 31  C1132 32  C1133 33
 12  C1211 11  C121212  C121313  C1221 21  C1222 22  C1223 23  C1231 31  C1232 32  C1233 33
 13  C1311 11  C131212  C131313  C1321 21  C1322 22  C1323 23  C1331 31  C1332 32  C1333 33
 21  C211111  C2112 12  C211313  C2121 21  C2122 22  C2123 23  C2131 31  C2132 32  C2133 33
 22  C221111  C221212  C221313  C2221 21  C2222 22  C2223 23  C2231 31  C2232 32  C2233 33
 23  C231111  C231212  C231313  C2321 21  C2322 22  C2323 23  C2331 31  C2332 32  C2333 33
 31  C3111 11  C311212  C311313  C3121 21  C3122 22  C3123 23  C3131 31  C3132 32  C3133 33
 32  C321111  C321212  C321313  C3221 21  C3222 22  C3223 23  C3231 31  C3232 32  C3233 33
 33  C331111  C331212  C331313  C3321 21  C3322 22  C3323 23  C3331 31  C3332 32  C3333 33
Thanks to the symmetry of the stress tensor,
the number of independent elastic constants is
reduced to 54.
 11  C1111 11  C1112 12  C1113 13  C1121 21  C1122 22  C1123 23  C1131 31  C1132 32  C1133 33
 22  C2211 11  C2212 12  C2213 13  C2221 21  C2222 22  C2223 23  C2231 31  C2232 32  C2233 33
 33  C3311 11  C3312 12  C3313 13  C3321 21  C3322 22  C3323 23  C3331 31  C3332 32  C3333 33
 12  C1211 11  C1212 12  C1213 13  C1221 21  C1222 22  C1223 23  C1231 31  C1232 32  C1233 33
 13  C1311 11  C1312 12  C1313 13  C1321 21  C1322 22  C1323 23  C1331 31  C1332 32  C1333 33
 23  C2311 11  C2312 12  C2313 13  C2321 21  C2322 22  C2323 23  C2331 31  C2332 32  C2333 33
Thanks to the symmetry of the strain tensor, the
number of independent elastic constants is
further reduced to 36.
 11  C111111  C111212  C1113 13  C1122 22  C1123 23  C1133 33
 22  C221111  C2212 12  C221313  C2222 22  C2223 23  C2233 33
 33  C3311 11  C3312 12  C3313 13  C3322 22  C3323 23  C3333 33
 12  C1211 11  C1212 12  C121313  C1222 22  C1223 23  C1233 33
 13  C1311 11  C1312 12  C1313 13  C1322 22  C1323 23  C1333 33
 23  C2311 11  C2312 12  C231313  C2322 22  C2323 23  C2333 33
The following formalism is convenient for problems in which
the strains components are known and the stress
components are the dependent variables:
 ij  Cijkl kl
In cases where the strain components are the dependent
parameters, it is more convenient to use the following
formalism:
 ij  Sijkl kl
Where Sijkl is a matrix whose entries are the compliance
coefficients.
The case of isotropic materials:
o A material is said to be isotropic if its properties are
independent of direction.
o In that case, the number of non-zero stiffnesses (or
compliances) is reduced to 12, all are a function of only 2
elastic constants.
Young modulus:
y
E
y
Shear modulus (rigidity):
shear stress
G 
shear strain
Bulk modulus (compressibility):

pressure
K 
volumetric change

Poisson’s ratio:
x
 
y

Poisson’s ratio of incompressible isotropic materials equals 0.5:
Real materials are compressible and their Poisson ratio is
less than 0.5.
All elastic constants can be expressed as a function of only 2
elastic constants. Here is a conversion table:
Generalized Hooke’s law, i.e. Hooke’s law for
isotropic materials:
1


 11    11   22   33
E
E
E

1

 22    11   22   33
E
E
E


1
 33    11   22   33
E
E
E
 12 
 13 
 23 
 12
G
 13
G
 23
G
o Normal stresses along X, Y and
Z directions do not cause shear
strains along these directions.
o Tensions along one direction
cause shortening along
perpendicular directions.
o Shear stresses at X, Y and Z
directions do not cause strains at
perpendicular directions.
o Shear stresses at one direction
do not cause shear strains at
perpendicular directions.
Triaxial experiments: Increasing temperature weakens rocks.
Weakening of fine-grained
limestone with increasing
temperature. Vertical axis is
stress in MPa.
ic.ucsc.edu/~casey/eart150/Lectures/Rheology/13rheology.htm
Triaxial experiments: Increasing pressure strengthen rocks.
Strengthening of finegrained limestone with
increasing Pressure.
Vertical axis is stress in
MPa. This effect is much
more pronounced at low
temperatures (less than
100o) and diminishes at
higher temperatures
(greater than 100o)
ic.ucsc.edu/~casey/eart150/Lectures/Rheology/13rheology.htm
Triaxial experiments: In the Earth both temperature and
confining pressure increase with depth and temperature
overcomes strengthening effect of confining pressure
resulting in generally ductile behavior at depths.
ic.ucsc.edu/~casey/eart150/Lectures/Rheology/13rheology.htm
Triaxial experiments: Pore pressure weakens rocks.
o Fluid pressure
weakens rocks,
because it reduces the
effective stresses.
o Water weakens
rocks, by affecting
bonding of materials.
ic.ucsc.edu/~casey/eart150/Lectures/Rheology/13rheology.htm
Triaxial experiments: Rocks are weaker under lower strain
rates.
Slow deformation allows
diffusional crystal-plastic
processes to more closely keep
up with applied stresses.
ic.ucsc.edu/~casey/eart150/Lectures/Rheology/13rheology.htm
Deformed
marble bench
that sagged and
locally fractured
under the
influence of
gravity (and
occasionally
also users).
ic.ucsc.edu/~casey/eart150/Lectures/Rheology/13rheology.htm