Strain - Ductile Deformation
fault
rocks
Change material behavior with depth: brittle - ductile
Deformed trilobites
Deformation - strain
elastic behavior:
• rigid body translation
• rigid body rotation
plastic strain (here “distortion”):
• internal deformation:
points inside a body move
relative to each other
Coaxial strain
pure shear
instantaneous
stretching axes
Non - coaxial strain: simple shear
constant
height
Simple Shear
a
b
Simple shear transformation applied to a sheep. a) Undeformed sheep. b) Sheared sheep.
angular shear
Plane Strain
Hydrostatic Stress
Deviatoric Stress
Elastic behavior
Deformation:
Force = spring const. x dist.
F=kx
(Hooke’s law)
strain: instantaneous - recoverable
Elastic behavior: Hooke’s law
F = kx
σ
σ=Eε
E
θ
θ = tan−1 E
ε
Strain
Stress
step function application of stress
t1
t2
Time
t1
t2
Time
9
Elastic behavior of solids
(b)
(a)
b0
(b)
Fig.6 Interatomic forces in an ionic crystal of NaCl stru
(Turcotte & Schubert Figs. 7.4 and 7.5)
The volume
and energy per ionpositions
pair are respectively
elastic moduli are a consequence
of equilibrium
of
V = 2 r3 potentials/ forces
atoms in a crystal lattice due to atomic
(Coulomb, Pauli
and exclusion)
U = -C0/r + D0/rn = -C(V/V0)-1/3 + D(V/V0)-n/3
where C0 = z2e2A/4!"0 with Madelung constant A, D0 > 0 w
interatomic distance. The first term is the reduction of potenti
charges are brought together from an initially infinite separatio
energy) to a finite spacing r. The second term is the short-rang
which prevents excessive overlap between the electronic char
two ions. The equilibrium value of r (or V) will be that fo
Fig.6 Interatomic forces in an ionic crystal
of NaCl
minimised,
i.e.structure.
that for which dU/dr (or dU/dV) = 0. Applicatio
Viscous behavior: leaky dashpot
σ
θ
Increasing strainrate results in
increasing stress. Viscosity is
constant of proportionality: a
material property.
θ = tan−1 η
ε·
Stress
Strain
η
σ = η ε·
t1
t2
Time
t1
t2
Time
Maxwell body
·
σ
σ
ε· = +
η G
η
Strain
Stress
E
t1
t2
Time
t1
G
for constant strain: ε· = 0 ⟹ σ = σ0 exp − t
( η )
t2
Time
η
exponential decay of stress with Maxwell relaxation time τM =
G
Maxwell relaxation time: how quickly does stress decay in the Earth?
Small strain deformation of olivine
Stress
Time
What does the viscosity of a rock depend on?
• stress
• temperature, pressure
• crystalline defects*
strain rate as a function of stress is described by semi-empirical ‘ ow laws’
·ε = Aσ nd −p exp − E + PV
(
RT )
where A constant, n stress exponent, p grain size exponent, E
activation energy, V activation volume, R gas constant, P
pressure, T temperature
fl
*point defects, dislocations, grain boundaries
Di usion creep: n = 1, p = 3
also called Newtonian creep, with linear dependence of strainrate on stress
·ε = Aσd −3 exp − E + PV
(
RT )
ff
Parameters A and E and V are determined by experiment.
They depend on rock type:
quartz (upper crust), feldspar (lower crust), olivine (mantle).
Pressure dependence is neglected for the crust (i.e. V = 0).
Dislocation creep: n = 3 - 3.5, p = 0
·ε = Aσ n exp − E + PV
(
RT )
Parameters are again dependent on rock type.