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Elastic Strain Recovery σ y0 = Yield Strength (typically defined at ε = 0.002) Initial loading σ yi = Yield Strength After unloading and reloading Note: for many metals YSafter plastic deformation

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Page 1: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Elastic Strain Recovery

σy0 = Yield Strength(typically definedat ε = 0.002)Initial loading

σyi = Yield StrengthAfter unloadingand reloading

Note: for many metals YS↑ after plastic deformation

Page 2: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Tensile Test

Page 3: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Hardening

• An increase in σy due to plastic deformation.

• Curve fit to the stress-strain response:

σ

ε

large hardening

small hardeningu

nlo

ad

relo

ad

σy 0

σy 1

σT = C εT( )n“true” stress (F/A) “true” strain: ln(L/Lo)

hardening exponent: n=0.15 (some steels) to n=0.5 (some copper)

Page 4: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

CHAPTER 7: DISLOCATIONS AND STRENGTHENING

ISSUES TO ADDRESS...• Why are dislocations observed primarily in metals

and alloys?

• How are strength and dislocation motion related?

• How do we increase strength?

• How can heating change strength and other properties?

Page 5: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Dislocations & Materials Classes

+ +

+

+ + + + + + + +

+ + + + + +

+ + + + + + +

• Metals: Disl. motion easier.-non-directional bonding-close-packed directions

for slip. electron cloud ion cores

• Covalent Ceramics(Si, diamond): Motion hard.-directional (angular) bonding

• Ionic Ceramics (NaCl):Motion hard.

-need to avoid ++ and --neighbors.

+ + + +

+ + +

+ + + +

- - -

- - - -

- - -

Page 6: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

What happens when bonds snap?

Two possible results:

Page 7: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

How does plastic deformation happen?

Calculating the shear stress for the mechanism above gives a result of ~ G/6 (with G = shear modulus)

Typical experimental values are however ~ 10-4 - 10-6 G

Plastic deformation does NOT happen this way!

Crystals contain dislocations.

Most plastic deformation happens by the movement of dislocations across the material.

Page 8: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Dislocation Motion

Plasticallystretchedzincsinglecrystal.

• Produces plastic deformation,• Depends on incrementally breaking

bonds.

Adapted from Fig. 7.9, Callister 6e.(Fig. 7.9 is from C.F. Elam, The Distortion of Metal Crystals, Oxford University Press, London, 1935.)

Adapted from Fig. 7.8, Callister 6e.

Adapted from Fig. 7.1, Callister 6e. (Fig. 7.1 is adapted from A.G. Guy, Essentials of Materials Science, McGraw-Hill Book Company,New York, 1976. p. 153.)

• If dislocations don't move,deformation doesn't happen!

Page 9: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Dislocation Motion

Edge DislocationMotion

Screw DislocationMotion

Page 10: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Dislocation Motion

Dislocation Motion:Response to shear stress

Analogy between caterpillar motion and dislocation motion

Page 11: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Arrangement of Atoms Around an Edge Dislocation

extra half-plane

• Dislocation “Core”– Region where the bonding is “wrong”

• Dislocation “Strain Field”– Planes near the dislocation are bent, or strained

• Bonds are stretched, compressed, or “bent” but not “wrong”

Page 12: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Characteristics of Dislocations

Schematic of strain fields foran edge dislocation

Dislocation repulsion(same sense of dislocations)

Dislocation attraction(opposite sense dislocations)

Page 13: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

The Tangent Vector or Line Vector

Line VectorIdentifies the orientation of the core in the crystal

Can change along the dislocationDislocations don’t have to be “straight”

ScrewEdge

t

t

tt

Page 14: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Geometry and Nomenclature of Dislocation Glide:Glide Planes

• Glide Plane– Definition: a plane on which a specific dislocation can glide– Physically: the glide plane must contain b and t– Mathematically: n = b x t(note b = Burgers vector and t = dislocation line tangent vector)

Screw Dislocationb || t ⇒

MANY Glide Planes!

Edge Dislocationb ⊥ t ⇒

Only ONE Glide Plane!

BUT...

Page 15: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

“REAL” Glide Planes for Screw Dislocations

• What Really Happens:– Glide of Screw Dislocations TENDS to occur on Close-Packed

Planes

• Common Glide Planes:– (111) in fcc metals: True Close-packed Plane– (0001) in hcp metals: True Close-packed Plane– (110) in bcc metals: Closest Packed Plane

Page 16: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Nomenclature for Dislocation Glide

Definitions

Slip Deformation of crystalline materials by dislocation Glide(also called yielding)

Slip Direction is the direction of the shear displacement incurred• || b, NOT the direction of dislocation motion!• Burgers vectors like to be as short as possible

– ⇒ TENDS to be a closely-packed direction

Slip Plane is the plane on which the dislocations glide

Slip Systems are combinations of slip planes and slip directions in which slip can occur by the glide of a dislocation (b,n) pairs

Page 17: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Slip Systems

• To specify a slip system, need:– Slip plane, n

• Usually a closely-packed plane

– A slip direction (Burgers vector!), b• Usually a close-packed direction• b lies in the slip plane

» If b of a screw dislocation isn’t in the plane, the dislocation isn’t either

» If b of an edge dislocation isn’t in the plane, the dislocation can’t glide

Page 18: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Slip Systems in FCC Metals

Slip DirectionsCommon b: a/2<110>

Slip PlaneClose-packed planes: {111}

Slip Systemsb must lie in the slip plane (b ⊥ n)

(111)

[110]

(111)

[110]

There are TWO {111} planes that contain each possible <110> Direction∴ Six <110> type directions leads to 12 slip systems!

Page 19: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Slip Systems in BCC Metals

Slip DirectionsCommon b: a/2<111>

Slip PlaneClosely-Packed Planes are {110}

Slip Systemsb must lie in the slip plane (b ⊥ n)

[111] <111>(110)

Two <111> type Directions in Each {110} Plane∴ Six {110} type planes yield 12 slip systems of this type

Page 20: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Slip Systems in HCP Metals

Slip DirectionsCommon b: <11-20>

Slip PlaneClose-packed planes: {0001}

Slip Systemsb must lie in the slip plane (b ⊥ n)

There is only ONE {0001} plane!∴ Three most favored slip systems!

AND they all lie in a plane!

Page 21: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Slip Systems

(Red arrow ← indicates most common slip systems)

Page 22: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Example: Dislocations in Iron

Two different grains

Dislocations are lined up!

Page 23: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Even More on Shear Stress

θθσθθ

θ

θτ

θ

θ

σ

cossincossin

cos

sin'

sin

cos'

00

0

0

==⎟⎠⎞

⎜⎝⎛

==

=

=

=

AF

AF

AF

FF

AA

AF

S

S

A0

A’

For uniaxial deformation: maximum shear stress at 45°But … this does not take crystallography into account!!!

Page 24: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Even More on Shear Stress

• Crystals slip due to a resolved shear stress, τR. • Applied tension can produce such a stress.

τR= σcos λcos φ

Relation between σ and τR

τR=Fs/As

Fcosλ A/cosφ

φns

AAs

Applied tensile stress: σ = F/A

FA

Fslip

directio

n

Resolved shear stress: τR=Fs/As

As

τR

τR

Fs

slip

directio

n

slip plane

normal, ns

λF

Fsslip

directio

n

Page 25: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Critical Resolved Shear Stress

• Condition for dislocation motion: τR > τCRSS

• Crystal orientation can makeit easy or hard to move disl.

10-4G to 10-2G

typically

τR= σcos λcos φ

τR = 0

φ=90°

σ

τR = σ/2λ=45°φ=45°

σ

τR = 0

λ=90°

σ

Page 26: Elastic Strain Recovery - Materialsmatclass/101/pdffiles/Lecture_9.pdfElastic Strain Recovery σ y0 = Yield Strength (typically defined at ε= 0.002) Initial loading σ yi = Yield

Dislocation Motion in Polycrystals

• Slip planes & directions(λ, φ) change from onecrystal to another.

• τR will vary from onecrystal to another.

• The crystal with thelargest τR yields first.

• Other (less favorablyoriented) crystalsyield later.

σ

Adapted from Fig. 7.10, Callister 6e.(Fig. 7.10 is courtesy of C. Brady, National Bureau of Standards [now the National Institute of Standards and Technology, Gaithersburg, MD].)

300 µm