Effects of Cold Work

Stress-strain curvesfor a material with progressively increasingcold work

What happens to theenergy to failure (toughness)with increasing cold work?

• Results forpolycrystalline iron:

• σy and TS decrease with increasing test temperature.• %EL increases with increasing test temperature.• Why? Vacancies

help dislocationspast obstacles.

1. disl. trapped by obstacle

2. vacancies replace atoms on the disl. half plane

3. disl. glides past obstacle

obstacle

Adapted from Fig. 6.14, Callister 6e.

σ-ε Behavior vs Temperature

00 0.1 0.2 0.3 0.4 0.5

200

400

600

800

Str

ess

(M

Pa

)

Strain

-200°C

-100°C

25°C

• 1 hour treatment at Tanneal...decreases TS and increases %EL.

• Effects of cold work are reversed!

• 3 Annealingstages todiscuss...

Adapted from Fig. 7.20, Callister 6e. (Fig.7.20 is adapted from G. Sachs and K.R. van Horn, Practical Metallurgy, Applied Metallurgy, and the Industrial Processing of Ferrous and Nonferrous Metals and Alloys, American Society for Metals, 1940, p. 139.)

Effect of Heating After %CWte

nsi

le s

tre

ng

th (

MP

a)

du

cti

lity

(%E

L)

Annealing Temperature (°C)

300

400

500

600 60

50

40

30

20Recovery

Recrystallization

Grain Growth

ductility

tensile strength300 700500100

Annihilation reduces dislocation density.

• Scenario 1

• Scenario 2

atoms diffuse to regions of tension

extra half-plane of atoms

extra half-plane of atoms

Disl. annhilate and form a perfect atomic plane.

1. dislocation blocked; can’t move to the right

obstacle dislocation

2. grey atoms leave by vacancy diffusion allowing disl. to “climb”

4. opposite dislocations meet and annihilate

3. “Climbed” disl. can now move on new slip plane

τR

Recovery

• New crystals are formed that:--have a small disl. density--are small--consume cold-worked crystals.

33% coldworkedbrass

New crystalsnucleate after3 sec. at 580C.

Adapted from Fig. 7.19 (a),(b), Callister 6e.(Fig. 7.19 (a),(b) are courtesy of J.E. Burke, General Electric Company.)

0.6 mm 0.6 mm

Recrystallization

• All cold-worked crystals are consumed.

After 4seconds

After 8seconds

Adapted from Fig. 7.19 (c),(d), Callister 6e.(Fig. 7.19 (c),(d) are courtesy of J.E. Burke, General Electric Company.)

0.6 mm0.6 mm

Further Recrystallization

Recrystallization Temperature

Larger deformation → Higher stored elastic energyLarger deformation → Higher dislocation densityLarger deformation → Lower recrystallization temperature

• At longer times, larger grains consume smaller ones. • Why? Grain boundary area (and therefore energy)

is reduced.

• Empirical Relation:

After 8 s,580C

After 15 min,580C

dn − do

n = Ktelapsed time

coefficient dependenton material and T.

grain diam.at time t.

exponent typ. ~ 2

0.6 mm 0.6 mmAdapted from Fig. 7.19 (d),(e), Callister 6e.(Fig. 7.19 (d),(e) are courtesy of J.E. Burke, General Electric Company.)

Grain Growth

Grain Growth Data for Brass (Cu-Zn)

Notes:D ~ tn, n < 1Temperature always wins out over time!!!(Thermally activated processes are almost always Arrhenius)(Kinetic processes often show t0.5 behavior)

Atomic Scale Processes in Grain Growth

Grain Growth

Local processes involve short range atomic migrationacross the boundary

Large grains consumesmall grains

Driving force for grain growthReduction in grain boundary area

Summary of Annealing of Metals

• Dislocations are observed primarily in metalsand alloys.

• Here, strength is increased by making dislocationmotion difficult.

• Particular ways to increase strength are to:--decrease grain size--solid solution strengthening--precipitate strengthening--cold work

• Heating (annealing) can reduce dislocation densityand increase grain size.

Summary

ISSUES TO ADDRESS...

• How does diffusion occur?

• Why is it an important part of processing?

• How can the rate of diffusion be predicted forsome simple cases?

• How does diffusion depend on structureand temperature?

Chapter 5:Diffusion in Solids

• Glass tube filled with water.• At time t = 0, add some drops of ink to one end

of the tube.• Measure the diffusion distance, x, over some time.• Compare the results with theory.

to

t1

t2

t3

xo x1 x2 x3time (s)

x (mm)

Diffusion Demo

100%

Concentration Profiles0

Cu Ni

• Interdiffusion: In an alloy, atoms tend to migratefrom regions of large concentration.

Initially After some time

100%

Concentration Profiles0

Adapted from Figs. 5.1 and 5.2, Callister 6e.

Diffusion: The Phenomena (1)

• Self-diffusion: In an elemental solid, atomsalso migrate.

Label some atoms After some time

A

B

C

DA

B

C

D

Diffusion: The Phenomena (2)

Diffusion Mechanisms

Vacancy-Assisted Diffusion

Motion of host or substitutionalImpurity into a vacant site

Energetics:Two factors

Energy to form vacancyEnergy to move vacancy

Host atom motion: Self-diffusionImpurity atom motion: Impurity diffusion

Interstitial Diffusion

Motion of interstitial atom fromIntersticial site to interstial site

Energetics:Main factors

Energy to move atom# of interstitial atoms(self-diffusion)

• Case Hardening:--Diffuse carbon atoms

into the host iron atomsat the surface.

--Example of interstitialdiffusion is a casehardened gear.

• Result: The "Case" is--hard to deform: C atoms

"lock" planes from shearing.--hard to crack: C atoms put

the surface in compression.

Fig. 5.0, Callister 6e.(Fig. 5.0 iscourtesy ofSurface Division, Midland-Ross.)

Processing Using Diffusion (1)

• Doping Silicon with P for n-type semiconductors:• Process:

1. Deposit P richlayers on surface.

2. Heat it.

3. Result: Dopedsemiconductorregions.

silicon

siliconmagnified image of a computer chip

0.5mm

light regions: Si atoms

light regions: Al atoms

Fig. 18.0, Callister 6e.

Processing Using Diffusion (2)

• Flux (#/area/time):

J =

1A

dMdt

⇒kg

m2s

⎡

⎣ ⎢

⎤

⎦ ⎥ or

atoms

m2s

⎡

⎣ ⎢

⎤

⎦ ⎥

• Directional Quantity

• Flux can be measured for:--vacancies and interstitials--host (A) atoms--impurity (B) atoms

Jx

Jy

Jz x

y

z

x-direction

Unit area A through which atoms move.

Modeling Diffusion: Flux

• Concentration Profile, C(x): [kg/m3]

• Fick's First Law:

Concentration of Cu [kg/m3]

Concentration of Ni [kg/m3]

Position, x

Cu flux Ni flux

• The steeper the concentration profile,the greater the flux!

Adapted from Fig. 5.2(c), Callister 6e.

Jx = −D

dCdx

Diffusion coefficient [m2/s]

concentration

gradient [kg/m4]

flux in x-dir.

[kg/m2-s]

Concentration Profiles & Flux

• Steady State: the concentration profile doesn't change with time.

• Apply Fick's First Law:

• Result: the slope, dC/dx, must be constant(i.e., slope doesn't vary with position)!

Jx(left) = Jx(right)

Steady State:

Concentration, C, in the box doesn’t change w/time.

Jx(right)Jx(left)

x

Jx = −D

dCdx

dCdx

⎛

⎝ ⎜

⎞

⎠ ⎟ left

=dCdx

⎛

⎝ ⎜

⎞

⎠ ⎟ right

• If Jx)left = Jx)right , then

Steady State Diffusion

• Steel plate at 700 °C withgeometryshown:

13

• Q: How much carbon transfers from the rich to the deficient side?

J = −DC2 − C1

x2 − x1

= 2.4 × 10−9 kg

m2s

Adapted from Fig. 5.4, Callister 6e.

C1 = 1.2kg/m3

C2 = 0.8kg/m3

Carbon rich gas

10mm

Carbon deficient

gas

x1 x205m

mD=3x10-11m2/s

Steady State = straight line!

Ex: Steady State Diffusion

• Concentration profile,C(x), changesw/ time.

• To conserve matter: • Fick's First Law:

• Governing Eqn.:

Concentration, C, in the box

J(right)J(left)

dx

dCdt

= Dd2C

dx2

−dx

= −dC

dtJ = −D

dC

dxor

J(left)J(right)

dJ

dx= −

dC

dt

dJ

dx= −D

d2C

dx2

(if D does not vary with x)

equate

Non Steady State Diffusion

• Copper diffuses into a bar of aluminum.

• General solution:

"error function"Values calibrated in Table 5.1, Callister 6e.

C(x, t) − Co

Cs − Co

= 1− erfx

2 Dt

⎛ ⎝ ⎜

⎞ ⎠ ⎟

pre-existing conc., Co of copper atoms

Surface conc., Cs of Cu atoms bar

Co

Cs

position, x

C(x,t)

tot1

t2t3 Adapted from

Fig. 5.5, Callister 6e.

Ex: Non Steady State Diffusion

• Copper diffuses into a bar of aluminum.• 10 hours at 600 °C gives desired C(x).• How many hours would it take to get the same C(x)

if we processed at 500 °C?

(Dt)500ºC =(Dt)600ºCs

C(x,t) −CoC − Co

= 1− erfx

2Dt

⎛

⎝ ⎜

⎞

⎠ ⎟

• Result: Dt should be held constant.

• Answer:Note: valuesof D areprovided here.

Key point 1: C(x,t500°C) = C(x,t600°C).Key point 2: Both cases have the same Co and Cs.

t500= (Dt)600

D500

= 110hr

4.8x10-14m2/s

5.3x10-13m2/s 10hrs

Processing Question

• The experiment: we recorded combinations oft and x that kept C constant.

to

t1

t2

t3

x o x 1 x 2 x 3

• Diffusion depth given by:

xi ∝ Dti

C(xi, t i ) − CoCs − Co

= 1− erf xi2 Dt i

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = (constant here)

Diffusion Demo: Analysis

• Experimental result: x ~ t0.58

• Theory predicts x ~ t0.50

• Reasonable agreement!

BBBBBBBBBBBBB

B

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3

( )

ln[t(min)]

Linear regression fit to data:ln[x(mm)] = 0.58ln[t(min)] + 2.2

R2 = 0.999

Data From Diffusion Demo

• Diffusivity increases with T.

• Experimental Data:

1000K/T

D (m2/s) C in α-Fe

C in γ-Fe

Al in Al

Cu in C

u

Zn in CuFe in α-Fe

Fe in γ-Fe

0.5 1.0 1.5 2.010-20

10-14

10-8T(C)1

50

0

10

00

60

0

30

0D has exp. dependence on TRecall: Vacancy does also!

pre-exponential [m2/s] (see Table 5.2, Callister 6e)activation energy

gas constant [8.31J/mol-K]

D= Doexp −QdRT

⎛

⎝ ⎜

⎞

⎠ ⎟ diffusivity

[J/mol],[eV/mol] (see Table 5.2, Callister 6e)

Dinterstitial >> Dsubstitutional

C in α-FeC in γ-Fe Al in Al

Cu in Cu

Zn in Cu

Fe in α-FeFe in γ-Fe

Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)

Diffusion and Temperature

Diffusion and Temperature

xi ∝ Dti

Make simple estimatesfor ‘diffusion lengths’ xi

1 sec100 sec (1.667 min)10,000 sec (2.7 hrs)100,000 sec (1.15 days)1,000,000 sec (11.6 days)

Diffusion FASTER for...

• open crystal structures

• lower melting T materials

• materials w/secondarybonding

• smaller diffusing atoms

• cations

• lower density materials

Diffusion SLOWER for...

• close-packed structures

• higher melting T materials

• materials w/covalentbonding

• larger diffusing atoms

• anions

• higher density materials

Summary:Structure & Diffusion