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Module #26 FATIGUE OF MATERIALS TOPICS & REFERENCES Mechanisms for Fatigue Fatigue Crack Propagation Ch. 14 in Meyers & Chawla S. Suresh, Fatigue of Materials, 2nd Ed., Cambridge (1998)

Page 793

Fatigue of Materials • Many materials, when subjected to fluctuating stresses, fail. • The stresses required to cause failure are far below those needed to cause fracture on single application of load. • Fatigue failure is failure under dynamic loading. • Fatigue is the cause for more than 90% of all service failures in structural materials. It is something that you would like to avoid. • Fatigue failures generally occur with little or no warning (with catastrophic results).

Page 794

Fatigue Fractures • They generally appear to be brittle. Stress concentrators PSBs Inclusions Etc…

[Meyers & Chawla] Page 795

[Hertzberg]

Page 796

Stress Cycles σmax

STRESS

σa

All fluctuating stress cycles are made up of a σm and a σa

Δσ = σr σm

0

TIME

σmin

Stress Range:

Δσ = σ r = σ max − σ min

Stress Amplitude (alternating stress):

σa =

Mean Stress: Stress Ratio: Amplitude Ratio:

σ max − σ min

2 σ + σ min σ m = max 2

=

σr 2

σ min σ max 1− R σ A= a = σm 1+ R

R=

Page 797

Stress Cycles σmax

STRESS

σ max ≠ σ min

0

σm TIME

σmin

• Real stress cycles are far more complex and unpredictable than the ideal one showed on the prior viewgraph. • Thus, fatigue failures are statistical in nature. • Now we need to consider how fatigue parameters influence fatigue failure. Page 798

Fatigue Results: the S-N curve • Engineering fatigue data is generally presented on S-N (or ε-N) curves. – S = applied stress

Ferrous metals and other strain aging materials

– N = # cycles to failure

S or ε

– ε = applied strain

Examples: Low carbon steel Stainless steel Titanium Etc…

SL Example: Al alloys

Non-ferrous metals

# cycles to failure (log scale) SL = fatigue/endurance limit Page 799

σa

σa

Fatigue Results: The S-N Curve

Endurance Limit at 107 Cycles

Fatigue Limit

102

104

106

Nf

108

102

104

106

108

Nf

• In materials exhibiting a fatigue limit, cyclic loading at stresses below the fatigue limit cannot result in failure. In steels SL/UTS = 0.4 to 0.5). • For non-ferrous materials we will generally define the fatigue strength as the stress that will cause fracture at the end of a specified number of cycles (usually 107).

Page 800

Categories of Fatigue

(1)

(2)

(3)

M.F. Ashby and D.R. Jones, Engineering Materials 1, 2nd Edition, Butterworth-Heinemann, Boston (1996), p. 146

Page 801

Categories of Fatigue High-strain

Low-strain

LOW CYCLE

HIGH CYCLE Finite life Infinite life

SUTS SYS

Stress, Sa

Plastic deform. of bulk

Elastic deform. of bulk 100

101

102

103

104

105

105

Number of cycles to failure, N

107

108

Adapted from J.E. Shigley, C.R. Mischke, and R.G. Budynas, Mechanical Engineering Design, 7th Edition, McGraw-Hill, Boston (2004), p. 314 Page 802

Stress-Life or Strain-Life? • Most data is presented on S-N curves. This is typical of the stresslife method which is most suitable for high-cycle applications [i.e., in the high-cycle fatigue (HCF) regime].

• However, it is easier to implement strain-life experiments.

• The strain-life method is more applicable where there is measurable plastic deformation [i.e., in the low-cycle fatigue (LCF) regime].

Δεe

σ

Δεp B

Δσ 2

A

Δε O 2

C

Δε 2

ε

O-A-B reflects initial loading in tension.

Δσ 2 Page 803

Strain-Life Method • It’s convenient to consider elastic and plastic strains separately. Δε = Δε elastic + Δε plastic

• Elastic strain amplitude is determined from a combination of , Nf and Hooke’s law. Δε e σ a ⎛ σ ′f ⎞ = =⎜ 2N f ( ⎟ 2 E ⎝ E ⎠

)

b

Δε e = elastic strain amplitude 2 σ a = true stress amplitude σ ′f = fatigue strength coefficient b = fatigue strength exponent Page 804

Strain-Life Method (2) • The plastic strain component is described by the Manson-Coffin equation. Δε p σ a c = = ε ′f ( 2 N f ) 2 E Δε p 2

= plastic strain amplitude

σ a = true stress amplitude ε ′f = fatigue ductility coefficient c = fatigue ductility exponent

• The Manson-Coffin equation describes LCF.

Page 805

Strain Life Method (3) Δε = Δε elastic + Δε plastic ∴ Δε Δε e Δε p ⎛ σ ′f ⎞ 2N f = + =⎜ ( ⎟ 2 2 2 ⎝ E ⎠

)

b

+ ε ′f ( 2 N f

)

c

• This is the Manson-Coffin relationship between fatigue life and total strain. We can use it to determine the fatigue strength.

LCF

HCF

∆εP < ∆εe

Strain Amplitude Δε/2 (log Scale)

• At high temperatures, the Manson Coffin equation breaks down because Nf decreases as temperature increases. In addition, Nf depends on cyclic frequency.

ε ′f

∆εP > ∆εe

c

σ ′f

Total strain

E ELASTIC

b

103 to 104 cycles

PLASTIC

2Nf (log Scale) Page 806

Cyclic Strain-Controlled Fatigue • Strain amplitude is held constant during cycling. • Strain-controlled cyclic loading is often found during thermal cycling (when a component expands and contracts due to fluctuations in temperature). • This is particularly dangerous when a component is made from materials exhibiting different coefficients of thermal expansion. • It is also found during reversed bending between fixed displacements. • Local plastic strains at notches subjected to cyclic loading can also result in strain-controlled conditions near the root of the notch. This is due to constraint placed on the material near the root by the surrounding mass of material.

Page 807

Trends for Engineering Metals

[Dowling]

• Constant strain-amplitude cycling: – High-strength materials are desirable for HCF. – High ductility materials are desirable for LCF. – High-strength materials have low values of ε′f and low ductility. – High-ductility materials have low values of σ′f and low strength. Page 808

Schematic stress-strain hysteresis loop • O-A-B reflects initial loading in tension. • On unloading, curve B-C, yielding begins in compression at a lower value than was observed in tension. This is due to the Bauschinger effect. • Re-loading in tension completes the hysteresis loop.

Δεe

σ

Δεp B

Δσ 2

A

Δε O 2

C

Δε 2

ε Δσ 2

• A hysteresis loop is described by its width, Δε, the total strain range, and its height Δσ, the stress range.

Page 809

Response of Materials to Strain Cycles •

Metals can harden or soften during fatigue depending upon their initial state. Strain

Control Condition +

1

3

5

Time 2

4

σ

Strain

Cyclic Hardening +

5

3

1

ε

Time 2

2 4

4

σ

Cyclic Softening

Strain

5 3 1

+

1

3

5

ε

Time 2

4

1 3 5

4 2 Page 810

0

R = -1.0 Δσ

σm = 0

TIME

σmin

STRESS

σmax σa

0

R = -0.3

Δσ

σm

TIME σmin σmax σa

STRESS

Variation of σm (and R) will cause the endurance limit to change

σa

Δσ

R = 0.0

σm

0

σmin = 0 TIME σmax σa

STRESS

Effects of Mean Stress and Stress Ratio

STRESS

σmax

Δσ

σm

0

R = 0.0

σmin

TIME Page 811

Effects of Mean Stress and Stress Ratio • As σm increases, the fatigue life decreases! •

As R increases, the fatigue life increases!

[Dieter] Page 812

Effects of Mean Stress and Stress Ratio •There are several empirical equations to relate the alternating stress to the mean stress. –Goodman 1 ⎡ ⎤ σ σ a = σ o ⎢1 − m σ ⎥ UTS ⎣ ⎦

(

–Gerber

)

R = negative

Linear

R = -1

R=0

σo

(

)

2 ⎡ ⎤ σ m σ a = σ o ⎢1 − σ UTS ⎥⎦ ⎣

R = positive

Parabolic

–Soderberg

(

)

⎡ ⎤ σ a = σ o ⎢1 − σ m σ ⎥ YS ⎣ ⎦ 1

Linear

σYS

σUTS

R=1

Mean Stress

Most experimental data lies between the Goodman and Gerber values. The Goodman relation is more conservative and is safer for design purposes. Page 813

Cumulative Damage and Life Exhaustion • Most engineering structures are subjected to variable amplitude loading as is illustrated below. [Dowling]

Figure 9.43

• As you can see, a certain stress amplitude (σa1) is applied for a number of cycles (N1). The number of cycles to failure for σa1 is Nf1. • The fraction of life used is N1 /Nf1. There will be additional expressions for regions of loading with different stress amplitudes. • The total life used can be expressed as: N1 N N N + 2 + 3 +"+ i N f1 N f 2 N f 3 N fi Page 814

Cumulative Damage and Life Exhaustion • The Palmgren-Miner rule can be used to determine whether or not the fatigue life is exhausted. [Dowling]

Figure 9.43

N N N1 N + 2 + 3 +" = ∑ i = 1 N f1 N f 2 N f 3 i N fi • Fatigue failure is expected when the life fractions sum to unity (i.e., when 100% of the fatigue life is exhausted).

Page 815

Cumulative Damage and Life Exhaustion • If the variable amplitude loading cycle is repeated a number of times, it is convenient to sum cycle ratios over one repetition of the history and to then multiply that fraction by the number of repetitions required to reach unity. • As before, fatigue failure is expected when the repeated life fractions sum to unity (i.e., when 100% of the fatigue life is exhausted). Figure 9.44

[Dowling]

⎡ Ni ⎤ B f ⎢∑ =1 ⎥ N fi ⎦ one repetition ⎣ i B f = number of repetitions to failure Page 816

The Fatigue Process σ I

II

III

σ Page 817

THE FATIGUE PROCESS

σ

• There are 3 stages: I.

Crack initiation

II.

Crack propagation or stable crack growth

I

II

III

III. Unstable crack growth or failure

σ Page 818

Fatigue Crack Growth (1) • Regardless of the mechanism that controls fatigue at elevated temperatures, we are concerned with how long it will take for a crack to grow to a critical length. • Everything starts with crack initiation! • Crack initiation is not well understood. Most cracks initiate at free surfaces. However, in those instances where cracks initiate within a solid, some sort of interface is usually involved.

Page 819

Fatigue Crack Growth (2)

•

Where “slip” fits in. Crystalline solids generally deform by slip, which leaves slip bands on the surface.

•

In unidirectional deformation, slip occurs uniformly through a grain. dislocations

Slip planes

[Proc. Roy. Soc. A., v. 242 (1959) p.211-213]

•

In fatigue, some grains show slip lines while others will not.

•

Additional deformation produces additional slip lines. Fewer slip lines are produced than the actual number of fatigue cycles.

•

In many materials, slip rapidly reaches a saturation value, resulting in distorted regions of heavy slip. Page 820

Fatigue Crack Growth (3) • THUS, slip lines in fatigue tend to be grouped into distinct bands (i.e., regions of locally heavy deformation). • Cracks are generally found to occur in these regions of heavy deformation. Some bands are more “persistent” than others. We call them persistent slip bands.

Page 821

CRACK INITIATION

[Reed-Hill & Abbaschian]

[Reed-Hill & Abbaschian] Page 822

•

Persistent Slip Bands and Crack Initiation

PSBs are embryonic fatigue cracks that open when small tensile strains are applied.

[Dieter]

[Reed-Hill & Abbaschian]

• •

Once formed cracks will initially propagate along slip planes (Stage I). Later they assume directions perpendicular to the applied stress (Stage II). Page 823

•

Persistent Slip Bands

PSBs are embryonic fatigue cracks that open when small tensile strains are applied. Figure 4.4. (a) The critical annihilation distance for screw and edge dislocations. (b) Mechanism of extrusion formation by combined glide and dislocation annihilation. (c) Irreversible slip in the PSB creating effective interfacial dislocations which put the slip band in a state of compression. (d) The combined effects of applied stresses and internal stresses. Bigger arrows indicate repulsive forces on interfacial dislocations and smaller arrows denote forces caused by the applied load (After Essman, Gösele and Mughrabi, Phil. Mag. A, v. 44, 1981, pp. 405-426).

This is just a visualization of how they can form.

[Suresh, p. 138] Page 824

[Felbeck & Atkins, p.431]

Page 825

Stage I • • •

Cracks propagate via crystallographic shear modes. A few cracks nucleate along crystallographic slip planes. The rate of crack propagation is low (on the order of a few Å/cycle). The fracture surface in Stage I is nearly featureless.

[Meyers & Chawla] Page 826

Stage II •

Stage II crack propagation shows ripples/striations.

From R.M.N. Pelloux, “Fractography,” in Atomistics of Fracture, edited by R.M. Latanision and J.R. Pickens, Plenum Press (1981) pp. 241-251

• •

Each striation represents the position of the advancing crack. The crack propagation rate is high (on the order of a μm/cycle). Striations are the result of a combination of crack propagation and blunting.

[Dieter]

Page 827

Stage III • The fatigue crack becomes too large.

a (crack length)

• The Kc of the material is exceeded resulting in fast fracture. Fatigue Life

ac Incr. Δσ

da ao

dN

N (# cycles)

Nf

Variation in fatigue crack length with # cycles to failure

Page 828

Stage III • The fatigue crack becomes too large.

a (crack length)

• The Kc of the material is exceeded resulting in fast fracture. Fatigue Life

ac

Incr. Δσ

da ao

dN

N (# cycles)

Nf

Variation in fatigue crack length with # cycles to failure Page 829

Fatigue Crack Propagation (1) • The crack growth rate can be expressed as: da = Cσ am a n dN C = constant

σ a = alternating stress a = crack length m = a constant ranging from 2 to 4 n = a constant ranging from 1 to 2

• Is equation can be re-written in terms of the total accumulated strain: da = C1ε m1 dN

C1 = constant

ε = total strain m1 = a constant ranging from 2 to 4

• This is known as the Paris equation. It predicts the crack growth rate in the region of stable crack growth (Stage II). Page 830

Linear-Elastic Fracture Mechanics Method •

We can make fatigue crack propagation more useful by relating it to linear elastic fracture mechanics (LEFM).

•

Consider a thin sheet specimen of width w with a crack already in it. The crack could result from the presence of a manufacturing defect. Applied Stress

y

σij θ x

a w Applied Stress

•

The stress near the crack tip is:

σ ij =

K Fij (θ ) + ... 2π r Page 831

Linear-Elastic Fracture Mechanics Method σ ij =

K Fij (θ ) + ... 2π r

• The K is the stress intensity factor that we defined previously as fracture toughness. K = F ( a, σ ) = σ π a ⋅ F ( a w) = σ π a ⋅Y

Shape factor; depends on specimen shape and crack geometry

• Values of K have been tabulated for materials with different crack geometries. • Of course, we are interested in the critical K values (i.e., Kc) .

Page 832

Linear-Elastic Fracture Mechanics Method • At Kc an incremental increase in the crack length (da) results in a small change in the elastic strain energy release rate (i.e., πσ2a/E) which is equaled by the energy required to extend a crack. • If you exceed Kc the crack opens up. If you are below Kc it does not.

Δσ = σ max − σ min K = Yσ π a ∴ ΔK = K max − K min = Y Δσ π a • For constant Δσ, ΔK correlates with the fatigue crack growth rate. This is illustrated on the next page. Page 833

a (crack length)

Linear-Elastic Fracture Mechanics Method Incr. Δσ Incr. ΔK

a

da

ao

Variation in fatigue crack length with # cycles to failure

dN

N (# cycles) • Obviously

da = F (ΔK ) dN

• The variation of the fatigue crack growth rate with ΔK is shown on the next page. Page 834

I

II

III

K max → K c

p 1 ΔK th No crack growth

Crack Growth Rate (log scale)

Linear-Elastic Fracture Mechanics Method

da p = A ( ΔK ) dN

Stress intensity factor range ΔK Variation in fatigue crack growth rate with stress intensity factor

I.

Crack initiation – little or no crack growth

II. Crack propagation – Paris law region III. Unstable crack growth – accelerated crack growth Page 835

Linear-Elastic Fracture Mechanics Method • Paris, Gomez and Anderson (1961) showed that the fatigue crack growth rate could be related to the stress intensity factor range by the relationship:

da p = A ( ΔK ) dN where A and p are constants that depend upon material, environment, and test conditions. • The influence of the mean stress, written in terms of the stress ratio R, is given by:

A ( ΔK ) da = dN (1 − R ) K c − ΔK p

• An increase tends to increase crack growth rates in all portions of the crack growth curve.

Page 836

Linear-Elastic Fracture Mechanics Method Region

Comments

I.

Growth rates are controlled by microstructure, σm, and environment.

II.

Growth rates are controlled by microstructure, environment, and frequency. In this region, p=3 for steels and 3-4 for Al alloys.

III.

Growth rates are controlled by microstructure, σm, and thickness.

Page 837

FATIGUE CRACK GROWTH RATES ARE STRUCTURE INSENSITIVE

(a) dc/dN vs. ΔK for several Ti, Al, and steel alloys. (b) The same data replotted as dc/dN vs. ΔK/E. Normalizing ΔK by dividing by the modulus (E) produces a curve (with some scatter about it) in which crack-growth rates in several materials cannot be as clearly differentiated as they are in (a). The results indicate that fatiguecrack growth rates are not structure-sensitive; this is in contrast to most mechanical properties. This comes from Bates and Clark, Trans. ASM, 62, 380 (1969).

[Courtney, p. 590] Page 838

Creep-Fatigue Interactions • Materials are generally used in “hostile” environments where diffusional processes can operate and/or where they are subject to environmental attack. – Ex. High-temperature turbine blade in an aircraft engine. Temperatures are in the creep regime. The environment is “toxic.” Rotational stresses and differences in thermal expansion result in cyclic loading.

How do we correlate everything that is going on? – Are we talking about creep enhanced by fatigue environment? – Are we talking about fatigue enhanced by creep? – What is the answer?

Page 839

Creep-Fatigue Interactions • When the cyclic stress or strain amplitude is small compared to the mean stress (i.e., σa

View more...
Page 793

Fatigue of Materials • Many materials, when subjected to fluctuating stresses, fail. • The stresses required to cause failure are far below those needed to cause fracture on single application of load. • Fatigue failure is failure under dynamic loading. • Fatigue is the cause for more than 90% of all service failures in structural materials. It is something that you would like to avoid. • Fatigue failures generally occur with little or no warning (with catastrophic results).

Page 794

Fatigue Fractures • They generally appear to be brittle. Stress concentrators PSBs Inclusions Etc…

[Meyers & Chawla] Page 795

[Hertzberg]

Page 796

Stress Cycles σmax

STRESS

σa

All fluctuating stress cycles are made up of a σm and a σa

Δσ = σr σm

0

TIME

σmin

Stress Range:

Δσ = σ r = σ max − σ min

Stress Amplitude (alternating stress):

σa =

Mean Stress: Stress Ratio: Amplitude Ratio:

σ max − σ min

2 σ + σ min σ m = max 2

=

σr 2

σ min σ max 1− R σ A= a = σm 1+ R

R=

Page 797

Stress Cycles σmax

STRESS

σ max ≠ σ min

0

σm TIME

σmin

• Real stress cycles are far more complex and unpredictable than the ideal one showed on the prior viewgraph. • Thus, fatigue failures are statistical in nature. • Now we need to consider how fatigue parameters influence fatigue failure. Page 798

Fatigue Results: the S-N curve • Engineering fatigue data is generally presented on S-N (or ε-N) curves. – S = applied stress

Ferrous metals and other strain aging materials

– N = # cycles to failure

S or ε

– ε = applied strain

Examples: Low carbon steel Stainless steel Titanium Etc…

SL Example: Al alloys

Non-ferrous metals

# cycles to failure (log scale) SL = fatigue/endurance limit Page 799

σa

σa

Fatigue Results: The S-N Curve

Endurance Limit at 107 Cycles

Fatigue Limit

102

104

106

Nf

108

102

104

106

108

Nf

• In materials exhibiting a fatigue limit, cyclic loading at stresses below the fatigue limit cannot result in failure. In steels SL/UTS = 0.4 to 0.5). • For non-ferrous materials we will generally define the fatigue strength as the stress that will cause fracture at the end of a specified number of cycles (usually 107).

Page 800

Categories of Fatigue

(1)

(2)

(3)

M.F. Ashby and D.R. Jones, Engineering Materials 1, 2nd Edition, Butterworth-Heinemann, Boston (1996), p. 146

Page 801

Categories of Fatigue High-strain

Low-strain

LOW CYCLE

HIGH CYCLE Finite life Infinite life

SUTS SYS

Stress, Sa

Plastic deform. of bulk

Elastic deform. of bulk 100

101

102

103

104

105

105

Number of cycles to failure, N

107

108

Adapted from J.E. Shigley, C.R. Mischke, and R.G. Budynas, Mechanical Engineering Design, 7th Edition, McGraw-Hill, Boston (2004), p. 314 Page 802

Stress-Life or Strain-Life? • Most data is presented on S-N curves. This is typical of the stresslife method which is most suitable for high-cycle applications [i.e., in the high-cycle fatigue (HCF) regime].

• However, it is easier to implement strain-life experiments.

• The strain-life method is more applicable where there is measurable plastic deformation [i.e., in the low-cycle fatigue (LCF) regime].

Δεe

σ

Δεp B

Δσ 2

A

Δε O 2

C

Δε 2

ε

O-A-B reflects initial loading in tension.

Δσ 2 Page 803

Strain-Life Method • It’s convenient to consider elastic and plastic strains separately. Δε = Δε elastic + Δε plastic

• Elastic strain amplitude is determined from a combination of , Nf and Hooke’s law. Δε e σ a ⎛ σ ′f ⎞ = =⎜ 2N f ( ⎟ 2 E ⎝ E ⎠

)

b

Δε e = elastic strain amplitude 2 σ a = true stress amplitude σ ′f = fatigue strength coefficient b = fatigue strength exponent Page 804

Strain-Life Method (2) • The plastic strain component is described by the Manson-Coffin equation. Δε p σ a c = = ε ′f ( 2 N f ) 2 E Δε p 2

= plastic strain amplitude

σ a = true stress amplitude ε ′f = fatigue ductility coefficient c = fatigue ductility exponent

• The Manson-Coffin equation describes LCF.

Page 805

Strain Life Method (3) Δε = Δε elastic + Δε plastic ∴ Δε Δε e Δε p ⎛ σ ′f ⎞ 2N f = + =⎜ ( ⎟ 2 2 2 ⎝ E ⎠

)

b

+ ε ′f ( 2 N f

)

c

• This is the Manson-Coffin relationship between fatigue life and total strain. We can use it to determine the fatigue strength.

LCF

HCF

∆εP < ∆εe

Strain Amplitude Δε/2 (log Scale)

• At high temperatures, the Manson Coffin equation breaks down because Nf decreases as temperature increases. In addition, Nf depends on cyclic frequency.

ε ′f

∆εP > ∆εe

c

σ ′f

Total strain

E ELASTIC

b

103 to 104 cycles

PLASTIC

2Nf (log Scale) Page 806

Cyclic Strain-Controlled Fatigue • Strain amplitude is held constant during cycling. • Strain-controlled cyclic loading is often found during thermal cycling (when a component expands and contracts due to fluctuations in temperature). • This is particularly dangerous when a component is made from materials exhibiting different coefficients of thermal expansion. • It is also found during reversed bending between fixed displacements. • Local plastic strains at notches subjected to cyclic loading can also result in strain-controlled conditions near the root of the notch. This is due to constraint placed on the material near the root by the surrounding mass of material.

Page 807

Trends for Engineering Metals

[Dowling]

• Constant strain-amplitude cycling: – High-strength materials are desirable for HCF. – High ductility materials are desirable for LCF. – High-strength materials have low values of ε′f and low ductility. – High-ductility materials have low values of σ′f and low strength. Page 808

Schematic stress-strain hysteresis loop • O-A-B reflects initial loading in tension. • On unloading, curve B-C, yielding begins in compression at a lower value than was observed in tension. This is due to the Bauschinger effect. • Re-loading in tension completes the hysteresis loop.

Δεe

σ

Δεp B

Δσ 2

A

Δε O 2

C

Δε 2

ε Δσ 2

• A hysteresis loop is described by its width, Δε, the total strain range, and its height Δσ, the stress range.

Page 809

Response of Materials to Strain Cycles •

Metals can harden or soften during fatigue depending upon their initial state. Strain

Control Condition +

1

3

5

Time 2

4

σ

Strain

Cyclic Hardening +

5

3

1

ε

Time 2

2 4

4

σ

Cyclic Softening

Strain

5 3 1

+

1

3

5

ε

Time 2

4

1 3 5

4 2 Page 810

0

R = -1.0 Δσ

σm = 0

TIME

σmin

STRESS

σmax σa

0

R = -0.3

Δσ

σm

TIME σmin σmax σa

STRESS

Variation of σm (and R) will cause the endurance limit to change

σa

Δσ

R = 0.0

σm

0

σmin = 0 TIME σmax σa

STRESS

Effects of Mean Stress and Stress Ratio

STRESS

σmax

Δσ

σm

0

R = 0.0

σmin

TIME Page 811

Effects of Mean Stress and Stress Ratio • As σm increases, the fatigue life decreases! •

As R increases, the fatigue life increases!

[Dieter] Page 812

Effects of Mean Stress and Stress Ratio •There are several empirical equations to relate the alternating stress to the mean stress. –Goodman 1 ⎡ ⎤ σ σ a = σ o ⎢1 − m σ ⎥ UTS ⎣ ⎦

(

–Gerber

)

R = negative

Linear

R = -1

R=0

σo

(

)

2 ⎡ ⎤ σ m σ a = σ o ⎢1 − σ UTS ⎥⎦ ⎣

R = positive

Parabolic

–Soderberg

(

)

⎡ ⎤ σ a = σ o ⎢1 − σ m σ ⎥ YS ⎣ ⎦ 1

Linear

σYS

σUTS

R=1

Mean Stress

Most experimental data lies between the Goodman and Gerber values. The Goodman relation is more conservative and is safer for design purposes. Page 813

Cumulative Damage and Life Exhaustion • Most engineering structures are subjected to variable amplitude loading as is illustrated below. [Dowling]

Figure 9.43

• As you can see, a certain stress amplitude (σa1) is applied for a number of cycles (N1). The number of cycles to failure for σa1 is Nf1. • The fraction of life used is N1 /Nf1. There will be additional expressions for regions of loading with different stress amplitudes. • The total life used can be expressed as: N1 N N N + 2 + 3 +"+ i N f1 N f 2 N f 3 N fi Page 814

Cumulative Damage and Life Exhaustion • The Palmgren-Miner rule can be used to determine whether or not the fatigue life is exhausted. [Dowling]

Figure 9.43

N N N1 N + 2 + 3 +" = ∑ i = 1 N f1 N f 2 N f 3 i N fi • Fatigue failure is expected when the life fractions sum to unity (i.e., when 100% of the fatigue life is exhausted).

Page 815

Cumulative Damage and Life Exhaustion • If the variable amplitude loading cycle is repeated a number of times, it is convenient to sum cycle ratios over one repetition of the history and to then multiply that fraction by the number of repetitions required to reach unity. • As before, fatigue failure is expected when the repeated life fractions sum to unity (i.e., when 100% of the fatigue life is exhausted). Figure 9.44

[Dowling]

⎡ Ni ⎤ B f ⎢∑ =1 ⎥ N fi ⎦ one repetition ⎣ i B f = number of repetitions to failure Page 816

The Fatigue Process σ I

II

III

σ Page 817

THE FATIGUE PROCESS

σ

• There are 3 stages: I.

Crack initiation

II.

Crack propagation or stable crack growth

I

II

III

III. Unstable crack growth or failure

σ Page 818

Fatigue Crack Growth (1) • Regardless of the mechanism that controls fatigue at elevated temperatures, we are concerned with how long it will take for a crack to grow to a critical length. • Everything starts with crack initiation! • Crack initiation is not well understood. Most cracks initiate at free surfaces. However, in those instances where cracks initiate within a solid, some sort of interface is usually involved.

Page 819

Fatigue Crack Growth (2)

•

Where “slip” fits in. Crystalline solids generally deform by slip, which leaves slip bands on the surface.

•

In unidirectional deformation, slip occurs uniformly through a grain. dislocations

Slip planes

[Proc. Roy. Soc. A., v. 242 (1959) p.211-213]

•

In fatigue, some grains show slip lines while others will not.

•

Additional deformation produces additional slip lines. Fewer slip lines are produced than the actual number of fatigue cycles.

•

In many materials, slip rapidly reaches a saturation value, resulting in distorted regions of heavy slip. Page 820

Fatigue Crack Growth (3) • THUS, slip lines in fatigue tend to be grouped into distinct bands (i.e., regions of locally heavy deformation). • Cracks are generally found to occur in these regions of heavy deformation. Some bands are more “persistent” than others. We call them persistent slip bands.

Page 821

CRACK INITIATION

[Reed-Hill & Abbaschian]

[Reed-Hill & Abbaschian] Page 822

•

Persistent Slip Bands and Crack Initiation

PSBs are embryonic fatigue cracks that open when small tensile strains are applied.

[Dieter]

[Reed-Hill & Abbaschian]

• •

Once formed cracks will initially propagate along slip planes (Stage I). Later they assume directions perpendicular to the applied stress (Stage II). Page 823

•

Persistent Slip Bands

PSBs are embryonic fatigue cracks that open when small tensile strains are applied. Figure 4.4. (a) The critical annihilation distance for screw and edge dislocations. (b) Mechanism of extrusion formation by combined glide and dislocation annihilation. (c) Irreversible slip in the PSB creating effective interfacial dislocations which put the slip band in a state of compression. (d) The combined effects of applied stresses and internal stresses. Bigger arrows indicate repulsive forces on interfacial dislocations and smaller arrows denote forces caused by the applied load (After Essman, Gösele and Mughrabi, Phil. Mag. A, v. 44, 1981, pp. 405-426).

This is just a visualization of how they can form.

[Suresh, p. 138] Page 824

[Felbeck & Atkins, p.431]

Page 825

Stage I • • •

Cracks propagate via crystallographic shear modes. A few cracks nucleate along crystallographic slip planes. The rate of crack propagation is low (on the order of a few Å/cycle). The fracture surface in Stage I is nearly featureless.

[Meyers & Chawla] Page 826

Stage II •

Stage II crack propagation shows ripples/striations.

From R.M.N. Pelloux, “Fractography,” in Atomistics of Fracture, edited by R.M. Latanision and J.R. Pickens, Plenum Press (1981) pp. 241-251

• •

Each striation represents the position of the advancing crack. The crack propagation rate is high (on the order of a μm/cycle). Striations are the result of a combination of crack propagation and blunting.

[Dieter]

Page 827

Stage III • The fatigue crack becomes too large.

a (crack length)

• The Kc of the material is exceeded resulting in fast fracture. Fatigue Life

ac Incr. Δσ

da ao

dN

N (# cycles)

Nf

Variation in fatigue crack length with # cycles to failure

Page 828

Stage III • The fatigue crack becomes too large.

a (crack length)

• The Kc of the material is exceeded resulting in fast fracture. Fatigue Life

ac

Incr. Δσ

da ao

dN

N (# cycles)

Nf

Variation in fatigue crack length with # cycles to failure Page 829

Fatigue Crack Propagation (1) • The crack growth rate can be expressed as: da = Cσ am a n dN C = constant

σ a = alternating stress a = crack length m = a constant ranging from 2 to 4 n = a constant ranging from 1 to 2

• Is equation can be re-written in terms of the total accumulated strain: da = C1ε m1 dN

C1 = constant

ε = total strain m1 = a constant ranging from 2 to 4

• This is known as the Paris equation. It predicts the crack growth rate in the region of stable crack growth (Stage II). Page 830

Linear-Elastic Fracture Mechanics Method •

We can make fatigue crack propagation more useful by relating it to linear elastic fracture mechanics (LEFM).

•

Consider a thin sheet specimen of width w with a crack already in it. The crack could result from the presence of a manufacturing defect. Applied Stress

y

σij θ x

a w Applied Stress

•

The stress near the crack tip is:

σ ij =

K Fij (θ ) + ... 2π r Page 831

Linear-Elastic Fracture Mechanics Method σ ij =

K Fij (θ ) + ... 2π r

• The K is the stress intensity factor that we defined previously as fracture toughness. K = F ( a, σ ) = σ π a ⋅ F ( a w) = σ π a ⋅Y

Shape factor; depends on specimen shape and crack geometry

• Values of K have been tabulated for materials with different crack geometries. • Of course, we are interested in the critical K values (i.e., Kc) .

Page 832

Linear-Elastic Fracture Mechanics Method • At Kc an incremental increase in the crack length (da) results in a small change in the elastic strain energy release rate (i.e., πσ2a/E) which is equaled by the energy required to extend a crack. • If you exceed Kc the crack opens up. If you are below Kc it does not.

Δσ = σ max − σ min K = Yσ π a ∴ ΔK = K max − K min = Y Δσ π a • For constant Δσ, ΔK correlates with the fatigue crack growth rate. This is illustrated on the next page. Page 833

a (crack length)

Linear-Elastic Fracture Mechanics Method Incr. Δσ Incr. ΔK

a

da

ao

Variation in fatigue crack length with # cycles to failure

dN

N (# cycles) • Obviously

da = F (ΔK ) dN

• The variation of the fatigue crack growth rate with ΔK is shown on the next page. Page 834

I

II

III

K max → K c

p 1 ΔK th No crack growth

Crack Growth Rate (log scale)

Linear-Elastic Fracture Mechanics Method

da p = A ( ΔK ) dN

Stress intensity factor range ΔK Variation in fatigue crack growth rate with stress intensity factor

I.

Crack initiation – little or no crack growth

II. Crack propagation – Paris law region III. Unstable crack growth – accelerated crack growth Page 835

Linear-Elastic Fracture Mechanics Method • Paris, Gomez and Anderson (1961) showed that the fatigue crack growth rate could be related to the stress intensity factor range by the relationship:

da p = A ( ΔK ) dN where A and p are constants that depend upon material, environment, and test conditions. • The influence of the mean stress, written in terms of the stress ratio R, is given by:

A ( ΔK ) da = dN (1 − R ) K c − ΔK p

• An increase tends to increase crack growth rates in all portions of the crack growth curve.

Page 836

Linear-Elastic Fracture Mechanics Method Region

Comments

I.

Growth rates are controlled by microstructure, σm, and environment.

II.

Growth rates are controlled by microstructure, environment, and frequency. In this region, p=3 for steels and 3-4 for Al alloys.

III.

Growth rates are controlled by microstructure, σm, and thickness.

Page 837

FATIGUE CRACK GROWTH RATES ARE STRUCTURE INSENSITIVE

(a) dc/dN vs. ΔK for several Ti, Al, and steel alloys. (b) The same data replotted as dc/dN vs. ΔK/E. Normalizing ΔK by dividing by the modulus (E) produces a curve (with some scatter about it) in which crack-growth rates in several materials cannot be as clearly differentiated as they are in (a). The results indicate that fatiguecrack growth rates are not structure-sensitive; this is in contrast to most mechanical properties. This comes from Bates and Clark, Trans. ASM, 62, 380 (1969).

[Courtney, p. 590] Page 838

Creep-Fatigue Interactions • Materials are generally used in “hostile” environments where diffusional processes can operate and/or where they are subject to environmental attack. – Ex. High-temperature turbine blade in an aircraft engine. Temperatures are in the creep regime. The environment is “toxic.” Rotational stresses and differences in thermal expansion result in cyclic loading.

How do we correlate everything that is going on? – Are we talking about creep enhanced by fatigue environment? – Are we talking about fatigue enhanced by creep? – What is the answer?

Page 839

Creep-Fatigue Interactions • When the cyclic stress or strain amplitude is small compared to the mean stress (i.e., σa

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