#1
#2
Ti (ii) “Sequential”
1 T
(a) “Series-alternating” - The two mechanisms are constrained to operate alternately.
& ln ε s
#1
#2
1 1 1 = + & s ε1 ε 2 & & ε
At ant T, observed creep rate is; “independent” ⎛ Q ⎞
mechanism
& ε s ∝ exp⎜ − ⎟ RT ⎠ ⎝
Q : apparent
activation energy
“Series-alternating” B (Sequential)
“Series-dependent”
Transition behavior in creep
Consider two mechanisms with true activation energies Q1 and Q2 (i) “Independent” [Mechanisms occur independently of each other]
t 1 ∑i 1 1 1 1 = i = + + ⋅⋅⋅ + = ∑ & & & & & εs ε ε1 ε 2 εi i εi 1 1 1 = + for two mechanisms & & & ε s ε1 ε 2
(b) Other definition: “Series dependent” [Both mechanisms occur together, but slowest mechanism controls creep.] We will use series-alternating to represent sequential.
(ii) “Sequential” (a) Series-alternating [two mechanisms depend on each other. one cannot occur without the other.] e.g. Polycrystal Dislocations The two mechanisms are constrained to operate alternately. move in grains σ The time length during which each process operates are additive; t = t1 + t2 + t3 + · · · · + tm To maintain integrity in material;
& For i mechanism; steady-state creep rate is ε s = & & & Thus, for two mechanisms; ε s = ε1 + ε 2
“Apparent activation energy” comes from
& ∑ε
i
i
⎛ Q ⎞ & ε s = ω exp⎜ − ⎟ (ω: constant) ⎝ RT ⎠ & & ∂ (ln ε s ) R ∂ε s Q = −R =− & ∂ (1 / T ) ε s ∂ (1 / T ) For two mechanism having Q1 and Q2; ⎛ Q ⎞ ⎛ Q ⎞ & & & ε s = A exp⎜ − 1 ⎟ + B exp⎜ − 2 ⎟ = ε1 + ε 2 (A,B : constant) ⎝ RT ⎠ ⎝ RT ⎠ & ∂ε s Q ⎛ − Q1 ⎞ Q 2 ⎛ −Q2 ⎞ = − 1 A exp⎜ − B exp⎜ ⎟ ⎟ ∂ (1 / T ) R RT ⎠ R RT ⎠ ⎝ ⎝ & & & Q1 Q Q ε + Q 2ε 2 R ∂ε s &1 + 2 ε 2 = 1 1 & ∴Q = − ⋅ = ε & & & & & ε s ∂ (1 / T ) ε s εs ε1 + ε 2 Q − Q1 λ = Simplify to & & where λ = ε 2 / ε1 Q 2 − Q1 1 + λ
GBS
σ
• grain boundary sliding • Accommodated in grain
1 1 =∑ & & εs i εi 1 1 1 = + For two processes; & & & ε s ε1 ε 2
ε s = ε1 = ε 2 = ⋅ ⋅ ⋅⋅ = ε m
& & ε1 ⋅ ε 2 & & ε1 + ε 2
Measure Q as a function of T : polycrystalline Al
AB ─ Low temperature behavior BC ─ Transition CD ─ High temperature behavior
Q varies with T below 250 K. Q ≅ 27.5 kcal/mol (=115 kJ/mol) from 250-375 K Q ≅ 35 kcal/mol (=146 kJ/mol) from 500-800 K
& & Ti: Intersection temperature where ε1 = ε 2
∴ Values of Q may increase or decrease as T is increased.
Ti
1 T
Transition behavior in creep Consider two mechanisms with true activation energies Q1 and Q2 (i) “Independent” [Mechanisms occur independently of each other]
Transition behavior in creep
Consider two mechanisms with true activation energies Q1 and Q2 (i) “Independent” [Mechanisms occur independently of each other]
(b) “Series dependent” - Both mechanisms occur together, but slowest mechanism controls creep. 1
Ti
T
Transition behavior in creep
Consider two mechanisms with true activation energies Q1 and Q2 (i) “Independent” [Mechanisms occur independently of each other]
& ln ε s
−
Q1
R
ln2
A
−
Q2
R
#1
#2
Ti
1 T
⎛ Q ⎞ ⎛ Q ⎞ & & & ε s = A exp⎜ − 1 ⎟ + B exp⎜ − 2 ⎟ = ε1 + ε 2 ⎝ RT ⎠ ⎝ RT ⎠ (A,B : constant)
& & & At Ti , ε s = 2ε1 = 2ε 2 (point A)
& Let ε i = ti ⋅ ε i , thus
& ∴ εs =
Apparent activation energy for two mechanisms;
⎛ Q ⎞ ⎛ Q ⎞ A exp⎜ − 1 ⎟ ⋅ B exp⎜ − 2 ⎟ ⎝ RT ⎠ ⎝ RT ⎠ & εs = ⎛ Q ⎞ ⎛ Q ⎞ A exp⎜ − 1 ⎟ + B exp⎜ − 2 ⎟ ⎝ RT ⎠ ⎝ RT ⎠
& For i mechanisms; steady-state creep rate is ε s = & & & Thus, for two mechanisms; ε s = ε1 + ε 2
“Apparent activation energy” comes from
& ∑ε
i
i
⎛ Q ⎞ & ε s = ω exp⎜ − ⎟ (ω: constant) ⎝ RT ⎠ & & ∂ (ln ε s ) R ∂ε s Q = −R =− & ∂ (1 / T ) ε s ∂ (1 / T ) For two mechanism having Q1 and Q2; ⎛ Q ⎞ ⎛ Q ⎞ & & & ε s = A exp⎜ − 1 ⎟ + B exp⎜ − 2 ⎟ = ε1 + ε 2 (A,B : constant) ⎝ RT ⎠ ⎝ RT ⎠ & ∂ε s Q ⎛ − Q1 ⎞ Q 2 ⎛ −Q2 ⎞ = − 1 A exp⎜ − B exp⎜ ⎟ ⎟ ∂ (1 / T ) R RT ⎠ R RT ⎠ ⎝ ⎝ & & & R ∂ε s Q1 Q Q ε + Q 2ε 2 &1 + 2 ε 2 = 1 1 & ε ∴Q = − ⋅ = & & & & & ε s ∂ (1 / T ) ε s εs ε1 + ε 2 Q − Q1 λ = Simplify to & & where λ = ε 2 / ε1 Q 2 − Q1 1 + λ