State Variable Equations
ε ij = ε
t
E
ij
+ ∫ ε ijI dτ + ε TH ij
Kinematic
0
ε ijI = ε oI (σ kl , T , P , f p , ς r ,...)
ij
ς m = ς m (ε ijI ,σ kl , P , T , f p , ς s ,...)
Kinetic
Evolution
Stouffer and Dame, “Inelastic deformation of metals”, 1996
Apparent (Empirical)
Activation Parameters
ΔH
)
ε = εoσ exp( −
RT
∂ ln ε
Stress exponent
n=
∂ ln σ T ,P ,...
n
∂ ln ε
V* ∂P T ,σ ,...
Activation Volume
∂ ln ε
H* ∂ (1 / T ) σ ,P ,,...
⎧Energy
Activation ⎨
⎩Enthalpy
1 ∂ ln ε
A* b ∂ (σ ) P ,T ,...
⎧ Volume
Activation ⎨
⎩ Area
Orowan’s Equation
(Kinetic Equation)
Generation
Recovery
Recrystallization
ε = ρm b v
Lattice Friction
Precipitates
Dislocation Interactions
Source/Sink
Frank-Read Source
• Force balance gives
σ=
μb
2R
• The dislocation density is
−1 / 2
L=ρ
• Then
μbρ 1 / 2
σ crit =
2
Work Hardening and
Recovery
dσ
d
∂σ
∂σ
⎡
⎤
σ (ε (t ) ,t )⎦ =
ε+
=
⎣
dt dt
∂ε t
∂t ε
dσ
Bailey-Orowan Eq.
= hε + r
dt
• At steady state, hardening = recovery
• If square root of disl. dens. proportional to mean
free path, λ = a ρ 1 / 2
and if density is a function of stress ,σ 2 = αμ b ρ 1 / 2
then can show that h = α
μ
• Qtz
1
Olivine
1-3
a
W
50
FCC Easy Glide
350
104
Thermodynamics of
Glide and Climb
ΔG2(σ)
• Line glide resistance
Thermal activation
σb
Applied Stress
τ b
ΔG(σ)
store
Δa(σ)
ao
Kochs et al., 1975
a1’
a2
a2’
• Work done on mat’l.
Dissipated energy
Force on dislocation
Velocity Kinetics
⎧ ΔL is distance tonext obstacle
⎪
ΔL
v =
where ⎨t g is time to glide
t g + to
⎪
⎩to is time to overcome obstacle
• If obstacle can be overcome by thermal activation
(e.g. lattice friction, cutting of soft ppts., unraveling attractive junctions)
ΔL>>a1a1’ then ΔL=λ (where λ is the dislocation spacing)
tg << to e.g. Cross slip in FCC metals
ΔL≅a1a1’, obstacle met as soon as overcome,
Glide-controlled Creep
• Obstacles can’t be overcome but may be avoided by
climb,
Recovery-controlled Creep
Recovery Processes
• Recovery-removal of defects
– Recovery processes
•
•
•
•
•
•
Collapse of dipoles
Loop collapse
Annihilation
Sub-grain boundary annihilation
Climb and glide to grain boundary, surface
Sub-grain boundary coarsening
• Recrystallization-creation and motion of high angle grain
boundaries
– Recrystallization Processes
•
•
•
•
Grain growth
Static recrystallization
Dynamic recrystallization
Chemically induced gb migration
Power Law Creep
n
⎛ σ ⎞
⎛ ΔH ⎞
ε = ε oυ o
f ( μi ) ⎜
⎟ exp ⎜ −
⎟
kT
⎝ kT ⎠
⎝ μshear ⎠
μΩ
pi
• n combined effect of density and mobility terms
• Fugacity and chemical potential of phases may affect
dislocation mobility
• Debye frequency, shear modulus and molecular volume
• For specific models see Kohlstedt et al., and Al et Kohlstedt
95.
Competition between Diffusion
and Dislocation Creep
• Dislocation Creep:
⎡ Q xs ⎡⎣(1 − Cxsσ ) ⎤⎦ ⎤
ε = Axsσ exp- ⎢
⎥
RT
⎢⎣
⎥⎦
σ
⎛ Qdiff ⎞
ε = Adiff m ⋅ exp ⎜ −
⎟
d
⎝ RT ⎠
2
• Diffusion Creep
• Composite Flow (Ter Heege et al.)
ε = εcsv cs + εdiff v diff
• State Variable: Grain size
Low Temperature
High Stress Laws
(Other Creep)
⎡ Δg ⎛ σ ⎞ ⎤
ε = εo exp ⎢ −
⎜1 − ⎟ ⎥
τ ⎠ ⎥⎦
⎢⎣ kT ⎝
p
q