재료의 기계적 거동

2021-06-11

(Mechanical Behavior of Materials)

FRACTURE & FATIGUE Myoung-Gyu Lee (이명규)

Department of Materials Science & Engineering Seoul National University

Seoul 151-744, Korea email : myounglee@snu.ac.kr

TA: Chanmi Moon (문찬미) 30-521 (Office)

chanmi0705@snu.ac.kr (E-mail)

WHAT IS FRACTURE?

- The separation or fragmentation of a body under stress.

- Fracture proceeds via two processes:

1. crack initiation 2. crack propagation

-We can loosely categorize fractures as brittle (fast) or ductile.

1. Brittle: failure with little or no plastic deformation 2. Ductile: failure with appreciable plastic deformation

Ductile fracture Brittle fracture

Fracture vs. Stress-Strain

Ductile fracture surface observed in a nickel base Brittle fracture in

polycrystalline

Structure of Crack Surface

Types of Engineering Fractures

Liquid metal embrittlement of Zn-coated steel

Historical background in fracture

- Mid-1930’s : concept of dislocation theory, why real solids do not possess a theoretical strength? Rather than shearing full planes of atoms simultaneously, the motion of dislocation, or line defect demands far lower stresses to bring about the slip.

- Even earlier, A.A. Griffith proposed the first explanation:

solids must contain very fine cracks (or flaws) no matter how much care is applied in the processing. Then, at the end of crack tip, the stress concentration effect causes the theoretical cohesive strength to be reached even though the apparent

nominal (applied) stress is far much lower.

Historical background in fracture

- Under this condition, the crack begins propagate and new surfaces are created (with increase in surface energy). The source of this crack growth (or surface energy increase) is the elastic strain energy that was stored in the body prior to the onset of crack propagation.

- Griffith Theory: a crack will propagate when the decrease in elastic strain energy is at least equal to the energy

needed to create the new surfaces associated with the crack.

THEORETICAL FRACURE STRENGTH

2 0

sin 2 ^th 2 ( )

th

xdx surface energy

l p ls

s g

l ⁼ p ⁼

ò

2 0 2 10

th

E E E

a s l

p p

= @ @

1/ 2

0 th

E a s _{= ç}^æg ^ö_÷

è ø

THEORETICAL FRACURE STRENGTH

Why/how do materials fail?

DEFECTS concentrate the stress locally to levels high enough to rupture the bonds.

Stress Concentration

Surface

&

internal cracks

c b

b c

nom 2

max 1 2

=

÷ø ç ö

è æ +

=

r

s s

for extremely flat ellipse or very narrow crack

s r s r

s

_{max nom} 1 2 ^c ÷÷ » 2 _nom ^c ø

ö çç

è æ +

=

Infinite plate under uniform tension with an elliptical hole (plane stress)

1/ 2

0 th

E a s _{= ç}^æg ^ö_÷

è ø

1/ 2 1/ 2

0 0

2

4 3

F

E E

a c a c

g r g r

s

^æç_ç ^ö÷_÷ ^æç_ç p ^ö÷_÷

ç ÷ ç ÷

è ø è ø

=

= s r

s r

s_{max nom} 1 2 ^c » 2 _nom ^c

÷÷ ø ö çç

è æ +

=

Critical stress intensity approach assumes that fracture occurs when maximum crack tip stress equals the

theoretical strength.

Stress Concentration

Energy approach (Griffith criterion)

(plane stress, linear elastic)

Energy approach (Griffith criterion) (plane stress, linear elastic)

As the crack extends, new surface area (dc) is created.

Correspondingly, the potential energy of the system increases (surface energy γ). This increase is balanced by a recovery of elastic strain energy in front of the crack tip during crack propagation.

4 g c

2 2

: 2 : 2 area c energy

E p

s

2 2 Tot 4

U c c

E

g ps

= -

Surface energy

Energy approach (Griffith criterion) (plane stress, linear elastic)

2 2

4 0

F

dUTot d c

dc dc c E _{s s}

g ps

=

é ù

= ê - ú =

ë û

2 1/ 2 F

E c s g

p

æ ö

= ç ÷

è ø

By Inglis, Griffith, the elastic strain energy released is

Due to presence of the crack, the surface energy is increased

Any change in total energy of the system that comes about due to crack propagation can be described

Griffith proposed that a crack will begin to propagate when the surface energy balances with the elastic strain energy released or the rate of U with respect to

Note:

Strain energy released is viewed as work being done by the system (Positive) An increase in surface energy is work done on the system (Negative)

Energy approach (Griffith criterion) (plane stress, linear elastic)

- This is the GRIFFITH equation.

- With it, we can calculate the maximum tolerable crack dimension for a given state of stress.

- Or, the maximum allowable stress if the maximum crack dimension is known.

- These equations apply only to brittle elastic solids.

- We must develop other relationships for plastic solids.

2

1/ 2 F

E c

s g

p

æ ö

= ç ÷

è ø

ELASTO PLASTIC FRACTURE MECHANICS

- The fracture strength of a

material that undergoes plastic deformation before fracture is greater than that for a perfectly brittle elastic solid.

- Plastic deformation at the root of the crack increases the radius of curvature at the crack tip (blunted), which reduces the stress concentrating effect of the crack.

ELASTO PLASTIC FRACTURE MECHANICS

toughness material

G

c E G

c

c F

:

2 / 1

÷ ø ç ö

è

= æ s p

- IRWIN (1952) suggested that a plastic work term and strain energy release rate should be added to the Griffith equation to make it applicable to metallic materials.

c G U

¶

= ¶

ELASTO PLASTIC FRACTURE MECHANICS

Fracture toughness

• Crack tip stress scales with stress and the square root of crack tip length.

• The stress intensity approach states when attains a critical value ( - a property called the fracture toughness), fracture takes place.

• For plane stress

• For plain strain

( )^{1/ 2}

c F

K s c

= p

(

^{G E}

)

^1/2

K_c = _c

2 / 1 2) 1

( ú

û ù êë

é

= -

n E K_c G^c

The critical value of G that makes the crack propagate to fracture is G_c. G_c is a material parameter called the “critical strain energy release rate”, “toughness”, or “crack resistance force”.

Now the conditions for crack growth can be represented as:

ELASTO PLASTIC FRACTURE

MECHANICS

ELASTO PLASTIC FRACTURE

MECHANICS

Fracture toughness of materials

Fatigue - general characteristics

• Fatigue can occur under axial (tension/compression), torsional stresses/strains. Cyclical strain (stress) leads to fatigue failure.

• Failure can occur at stress less than YS or UTS for static load.

• Failure is sudden, without warning, and catastrophic! (90%

of metals fail in this mode.)

• Fatigue failure is brittle in nature, even in normally ductile

S-N Curve: stress amplitude vs number of cycles

[Hertzberg]

S-N Curve: stress amplitude vs number of cycles

Note the presence of a fatigue limit in many steels and its absence in aluminum alloys.

log N_f

s_a

s

_{mean 1}

s

_{mean 2}

s

_{mean 3}

s_{mean 3} > s_{mean 2} > s_{mean 1}

The greater the number of

cycles in the loading history, the smaller the stress that

the material can withstand without failure.

S-N Curve: stress amplitude vs number of cycles

S-N behavior

Fe Fe- and Ti-based alloys, fatigue limit is 35-60% of TS.

For non-ferrous alloys (e.g., Al, Cu, Mg) do not have fatigue limit!

S-N Curve: stress amplitude vs number of cycles

n Some materials exhibit endurance limits or fatigue limits, i.e. a stress below which the life is infinite.

u Steels typically show an endurance limit, = 40% of yield;

this is typically associated with the presence of a solute (carbon, nitrogen) that pines dislocations and prevents dislocation motion at small displacements or strains (which is apparent in an upper yield point).

u Aluminum alloys do not show endurance limits; this is related to the absence of dislocation-pinning solutes.

n At large N_f, the lifetime is dominated by nucleation.

S-N Curve: stress amplitude vs number of cycles

Fatigue fracture surface

[Hertzberg]

Fatigue crack

[Dieter]

- Crack Nucleation ® stress intensification at crack tip.

Cyclic Strain control

n

Constitutive relation for cyclic stress-strain:

n

n’ ≈ 0.1-0.2

n

Fatigue life: Coffin Manson relation:

n

e

_f

’ ~ true fracture strain; close to tensile ductility

n

c ≈ -0.5 to -0.7, c = -1/(1+5n’); large n’ ® longer life.

D s = K ¢ ( D e ) n ¢

D e

_p

2 = ¢ e

_f

( 2 N

_f

)

^c

Palmgren-Miner’s rule

1

å =

i i

N n

Where N_i is fatigue life at a stress level and n_i is the number of cycles at this stress.

Coaxing