lecture slides - 27 - berkeley · pdf file• plasticity initiates at “yield...

J.W. Morris, Jr. University of California, Berkeley

MSE 200A Fall, 2008

Mechanical Properties

•  Elastic deformation

•  Plastic deformation

•  Fracture


MSE 200A Fall, 2008

Mechanical Properties: The Tension Test

•  For materials properties, replace load-deflection by stress-strain –  Engineering stress, s = P/A –  Engineering strain, e = ΔL/L0

•  Mechanical properties are revealed in the tension test I.  Elastic behavior II.  Plastic deformation III.  Plastic instability and fracture

P

ΔL

L

e

s y

u s

I II III

s


MSE 200A Fall, 2008

The Engineering Stress-Strain Curve

•  Properties measured: –  Elastic modulus (slope of elastic curve) –  Yield strength (sy) –  Ultimate tensile strength (su) –  Uniform elongation (eu) –  Total elongation (eT) –  Reduction in area (RA = [A0-Af]/A0)

For a typical ductile metal: I. Elastic deformation II. Stable plastic deformation III. Unstable deformation IV. Fracture

e

s y

u s

I II III

eu eT S


MSE 200A Fall, 2008

I. Elastic Deformation

•  Stress-strain relation is linear (Hooke’s Law)

•  Strain is recoverable –  relaxation along the load line to zero stress

s

e

sy

us

I II III

eu eT


s

e

€

s = Ee


MSE 200A Fall, 2008

II. Stable Plastic Deformation

•  Stress-strain relation is non-linear

•  Strain is non-recoverable –  Relaxation is elastic ⇒ permanent plastic strain –  Strain is uniform and stable - “work hardening”

•  Plasticity initiates at “yield strength”, sy –  In ductile material, sy is not obvious –  sy is usually defined by “0.2% offset strain”

•  Yield strength = stress that produces a plastic strain of 0.2%


s

eep ee

s

e

s

e0.2%

sy


MSE 200A Fall, 2008

III. Ultimate Tensile Strength

•  Engineering stress is maximum at su = ultimate tensile strength

•  At su plastic deformation becomes unstable –  Strain localizes in a “neck” in the length of the specimen –  Under load control, sample fails at su –  Under strain control sample elongates at lower load until failure

•  Two common measures of tensile strain –  Uniform elongation - eu - to ultimate strength –  Total elongation - eT - to fracture (depends on “gage length”)


L

P

P

L

P

P

s

e

sy

us

I II III

eu eT


MSE 200A Fall, 2008

Forms of the Engineering Stress-Strain Curve

s

e

sy

us

I II III

eu

eT s

e

s

e0.2%

sy

s

e

su

s

e

su

s

e

su

s

e

su

Ductile metal “Yield point”

Brittle solid Elastomer


MSE 200A Fall, 2008

€

ε = ln ΔLL0

+1

⇒ε = ln 1+ e( )

•  True strain is defined from its differential

–  (L/L0) applies when strain is uniform –  (A0/A) can be used for non-uniform strain

•  True stress

True Stress and Strain

L

P

P

L

P

P

€

σ =PA

=PA0

A0A

= s

LL0

€

dε =dLL

= −dAA

€

dV = 0 = AdL + LdA

⇒dLL

= −dAA

€

ε =dLLL0

L

∫ ⇒ ε = ln LL0

= ln

A0A

€

σ = s(1+ e)

V = constant (plastic deformation):

e = engineering strain

€

e =ΔLL0

=LL0−1

€

V = A0L0 = AL


MSE 200A Fall, 2008

Stress-Strain Relations

•  Stress-strain relations

–  stresses equal for small strain

•  Why use engineering curve? –  Clearly shows tensile strength –  su is an important design value –  No difference in E, sy=σy

ß

‰

ßy

ßf

‰f

s

e

sy

us

I II III

€

σ = s(1+ e)

€

ε = ln(1+ e)

€

s =σ exp(−ε)

€

e = exp(ε) −1


MSE 200A Fall, 2008

Influence of Temperature on Strength

•  Three regimes: –  Decrease in strength with T at low T/Tm –  Relatively constant strength at intermediate T –  Rapid loss of strength as T approaches Tm

•  Increasing strain rate is like decreasing T –  Larson-Miller parameter:

•  T* = Tln(de/dt)

ß y

T/T m


MSE 200A Fall, 2008

High Temperature Creep

•  Creep is deformation under constant load –  Primary creep –  Steady state –  Tertiary creep to fracture

•  Thermally activated process –  Steady state strain rate governed by “Dorn equation”

‰

t

I II III€

ln( ˙ γ )

€

ln(τ)

I: n=3, Q=QD

II: n=2, Q=QGB

III: n=4-9, Q=QD

€

˙ ε = Aσ n exp − QkT

€

ln(˙ ε )

€

ln(σ)


MSE 200A Fall, 2008

Fracture

•  Fracture may result from: –  Elastic instability (buckling, flutter) –  Plastic instability (necking) –  Crack instability (fracture)

•  Propagation of pre-existing flaw

•  Crack instability –  Due to stress concentration at crack tip –  Critical stress required to drive crack

L

P

P

L

P

P

ßa

ßa

ß

r

ßT

ßa

a

€

σ c ≥Q−1 KIc

a

Kic = “plane strain fracture toughness” = material property governing fracture

(Q-1 = geometric factor)


MSE 200A Fall, 2008

The Influence of Temperature on Toughness

•  Ductile-brittle transition –  Prominent in bcc metals –  High toughness, ductile at T>TB –  Low toughness, brittle at T<TB –  TB near room T in common steels

•  The transition temperature TB –  Increases with strength (sy) –  Increases with strain rate (impact) –  Increases with thickness

(constraint)


MSE 200A Fall, 2008

Fatigue

•  Fracture under cyclic load

•  Fatigue cracks –  Nucleate –  Grow to critical size –  Propagate to failure –  Even at s < sy

•  Mechanism –  Cyclic plastic deformation –  No obvious sign of growth

s

sm

t

sm-s

log(n)

su

s l


MSE 200A Fall, 2008

Stress Corrosion Cracking

•  Crack –  Grows by corrosion –  Reaches critical size –  Propagates to failure

•  May cause catastrophic failure –  No obvious sign of growth

s

su

s c

t


MSE 200A Fall, 2008

Mechanical Properties

•  Elastic deformation

•  Plastic deformation

•  Fracture


MSE 200A Fall, 2008

I. Elastic Deformation

•  Stress-strain relation is linear (Hooke’s Law)

•  Strain is recoverable –  relaxation along the load line to zero stress


€

s = Ee

e

s y

u s

I II III

eu eT S

e

S


MSE 200A Fall, 2008

Importance of Elastic Behavior

•  Engineering design –  Most structures are designed to remain below yield –  For example,

•  s < sy/3 (boiler and pressure vessel code) •  s << sy (turbine blades, springs)

•  Elastic failure –  Utlimate strength –  Buckling –  Flutter

•  Bridges, other structures •  Aircraft


MSE 200A Fall, 2008

The Tensile Moduli: Young’s Modulus and Poisson’s Ratio

•  Assume isotropic material –  Need two elastic moduli, E and ν

•  Young’s modulus:

•  Poisson’s ratio:

–  Volume change: €

sz = Eez (E = Young’s modulus)

€

ex = ey = −νez (ν = Poisson’s ratio)

€

ΔVV0

= (1+ ex )(1+ ey )(1+ ez)[ ] −1≅ ex + ey + ez[ ]

€

ΔVV0

= ez (1− 2ν)

L0(1+ez)

D0(1+ey)


MSE 200A Fall, 2008

Multiaxial Deformation

•  Linear elastic stresses are additive

–  Get ey, ez, sy, sz by interchanging x, y, z.

hydrostatic pressure

uniaxial tension

simple shear

balanced shear

€

ex =1E

sx −ν (sy + sz )[ ]

€

sx =E

(1− 2ν)(1+ ν )(1−ν)ex + ν(ey + ez)[ ]


MSE 200A Fall, 2008

The Physical Moduli: Bulk and Shear Moduli

•  Fundamental properties are reflected in the response to –  Change of volume (pressure) –  Change of shape (shear)

€

E = E0(V ) + Econf (Econf from atom arrangement at given V)

volume shear

hydrostatic pressure

uniaxial tension

simple shear

balanced shear

bulk modulus (β) shear modulus (G)


MSE 200A Fall, 2008

Physical Basis of Elastic Moduli

•  Elastic moduli balance –  β = resistance to volume change –  G = resistance to shape change

•  Four cases: –  β >> G ⇒ incompressible solid

•  E ~ 3G •  ν ~ 0.5

–  β ~ 3G ⇒ normal metal •  E ~ 2.7G •  ν ~ 0.35

–  β ~ G ⇒ strong directional covalent •  E ~ 2.25G •  ν ~ 0.125

–  β << G ⇒ bond angles preserved •  E ~ 9 β •  ν ~ -1 (No natural solid has ν < 0)

compression shear

€

E =9βG

(G + 3β)

€

ν =3β − 2G2(G + 3β)


MSE 200A Fall, 2008

Engineering the Elastic Modulus

•  Elastic properties reflect atomic bonding –  Microstructure manipulation has little effect –  Small composition changes cause ΔE ∝ c(E2-E1) –  One exception is Li in Al

•  Adding Li to Al ⇒ E↑, ρ↓ ⇒E’↑ significantly •  Al-(1-2.5)Li alloys of interest for aircraft

•  Composite materials –  High-modulus materials are usually brittle

•  Add high-modulus fiber or particle to ductile matrix •  Combine high modulus with useful toughness

–  Examples: •  Fiberglass (glass-epoxy) •  Graphite-epoxy •  SiC-Al


MSE 200A Fall, 2008

Composite Materials

•  Fiber composites: ~ uniform strain (e1 ~ e2 = e)

•  Particulate composites: ~ uniform stress (s1 ~ s2 = s) €

s =PA

= s1A1A

+ s2

A2A

= (E1 f1 + E2 f2)e

€

E = E1 f1 + E2 f2

€

e =ΔLL0

=ΔL1L1

L1L0

+ΔL2L2

L2L0

= e1 f1 + e2 f2 =f1E1

+f2E2

s

€

1E

=f1E1

+f2E2

Fiber:

€

E = E1 f1 + E2 f2

Particulate:

€

1E

=f1E1

+f2E2

- stiffer element dominates

- softer element dominates


MSE 200A Fall, 2008

Composite Materials

•  Fiber composites: –  High modulus but directional properties –  Use in applications where loading is uniaxial –  Or, use 2-d or 3-d configurations

•  Less directionality, but lower modulus

•  Particulate composites: –  Lower modulus effect, but

•  More isotropic •  Relatively tough and formable

–  Used for multiaxial loads

Fiber:

€

E = E1 f1 + E2 f2

Particulate:

€

1E

=f1E1

+f2E2

- stiffer element dominates

- softer element dominates


MSE 200A Fall, 2008

Elastic Failures

•  Lattice instability –  The ultimate strength of a solid (nanoindentation) –  Failures in tension and shear (“inherent” ductile-brittle transition)

•  Buckling –  Beams and buildings –  Sheets (dents and crushing in automobiles)

•  Vibration and flutter –  Bridge failures (flutter - Tocoma Narrows) –  Aircraft (flutter, “whirl mode” - the Lockheed Electra) –  Hovercraft (flagellation)

lecture slides - 27 - berkeley · pdf file• plasticity initiates at “yield...

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