Engineering Conferences InternationalECI Digital ArchivesInternational Workshop on the EnvironmentalDamage in Structural Materials Under Static Load/Cyclic Loads at Ambient Temperatures

Proceedings

6-1-2016

Mechanical & chemical driving force affectingcrack nucleationK. SadanandTDA,Inc, Falls Church, VA,USA, kuntimaddisada@yahoo.com

Asuri K. VasudevanRetired,Navy, Reston, VA , USA, akruva@gmail.com

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Recommended CitationK. Sadanand and Asuri K. Vasudevan, "Mechanical & chemical driving force affecting crack nucleation" in "International Workshop onthe Environmental Damage in Structural Materials Under Static Load/Cyclic Loads at Ambient Temperatures", A.K. Vasudevan,Office of Naval Research (retired), USA Ronald Latanision, Exponent, Inc., USA Henry Holroyd, Luxfer, Inc. (retired) Neville Moody,Sandia National Laboratories, USA Eds, ECI Symposium Series, (2016). http://dc.engconfintl.org/edsm/15

SUBCRITICAL CRACK GROWTH AND

CRACK TIP DRIVING FORCES

K. Sadananda, A.K. Vasudevan,

TDA,Inc.,Falls Church, VA

&

K.N. Solanki

School for Engineering of Matter, Transport, and Energy, Arizona

State University, Tempe, AZ

1. Examination of Some Basic Concepts

2. Discreet Dislocation Modelling

&

3. Some Applications to

Notches, Short Cracks

A Simplified Method to Quantify Chemical Forces

Cyclic

Thresholds

σmax,th

Δσth

σmax

Δσ NF1

σth

Conc

Saturation

Threshold

Inert

dσth/dc

Static Load –

Inert Environment

Cyclic Loads

Inert Environment

Static Load –

Aggressive Environment

Cyclic Loads

Aggressive Environment

Time/Cycles

Str

ess

/

Lo

ad

Thresholds

C1

C1

KIC

KISCC

Chemical

force

Force-

cyclic-

strain

Reduction in

Mechanical Force

Fatigue: 2-Load Parameters Required

3-D reduced to 2D by taking cuts at constant da/dN(s).

R - is not a load-parameter – No threshold in R.

da/dN

Or

da/dt

ΔK or Kmax

A B C D E

A – Inert Environment-

negligible R effects

B –Static Load – Creep-SCC

C – Low R- Air-Cyclic

D-High R- Air Cyclic

E- Low R-Corrosion Fatigue

da/dN1

da/dN2

R3 R2 R1

da/d

N

da/dN = f(ΔK,Kmax)

For a given

Crack growth

rate

R= const.

is a plane

Crack Tip Driving Force

Force = (-ve) Potential Energy Gradient with respect to

Crack length increment = (– dE/dX)

Crack-tip Driving Forces

Elastic Crack - Griffith condition, Fσ ≥ FR-material (gamma) Rc

σ

Rc

ET

γInert

γCh

σ= const.

σ 0

Cases 1 and 2 – constant stress

Case 3 – large stress gradient

Case 4 – Decreasing & increasing

stress gradient.

Internal Stresses & Gradients

**Crack nucleation and growth

require stress magnitude and

Gradient**

Lo

g σ

Log R

Griffith’s Stress

Stress

Gradient

Chemical

Force

σ= √{2Eγ/(1-ν2)πR)}

Non-Propagating

Cracks

Lo

g σ

Log R

1

2

3

4 Internal Stresses

Application

Kitagawa

ET

Rc

Inert

Env

Log σ

Log R

Inert Env

Stress Gradients

Chemical Force

Non-Propagating

Cracks

ET

Continuous

Plastic-Elastic

Cracks

Elastic

g2 < g1 g1

a

Elastic-Plastic Crack

BCS model

T

T

T

T

T

Continuous Elastic-Plastic Crack

ac

(d)

αc depends on gamma

Elastic case:Gc = (π σapl a)/E, and σapl = {2Eγ/πa}0.5 ..(1)

Elastic-Plastic Case G ≥ 2γ + P ------- (2).

Under some limiting Conditions; G – P ≥ 2γ ------ (3),

P, is reinterpreted not as dissipating energy term but as a contributing factor to

the net mechanical driving force needed to overcome the material resistance.

Ref: J.R. Rice,, J. Appl. Mech. 1961, Vol. 23, PP. 544-50

e) Chemical Stress concentration factor: Kch = σsmooth/σnotch ≥ 1 ----- (9),

Some generalization of the crack tip driving forces

---------------

K2/E’ ≥ 2 γ + P ------ (4) and

K2/E’ – P ≥ 2γ ----- (5), Kapl - KInt ≥ (γ) or Kth --------- (6)

Kapl ± KInt ≥ Kth -------- (7)

{Kapl ± KInt ± Kch} ≥ Kth -------- (8)

Linear Approximation – small scale plasticity

How far These Linear Approximations

Are valid for Crack Nucleation and Growth

in Engineering Materials?

y-

Normalized Kitagawa-TakahashiDiagram

0.1

1

10-2 10-1 100 101 102

S20c-366- Tanaka,1981

S20C-194-Tanaka,1981

MildSteel-289-Frost,1959

G40-11 - -Haddad,1979

SM41- 251- Ohuchida,1975

SM41-251-Usami,1979

SM50-373-Kitagawa,1976

HT80-726-Kitagawa,1979

13Cr-castSteel-760-Usami,1979

13Cr-castSteel-769-Usami,1979

Copper- - Frost,1963

Al- 30.4-Frost,1964

Relative Crack Length a/a*

Reploted from K. Tanaka, Y. Nakai and M. Yamashita, Int. J. Fatigue, Vol. 17, 1981, pp. 519-533.

Slope 0.5

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10 100 1000 104

R = 0.8

R = 0

R = 0.5

R = -1.0

R elative crack length a/a*c

13Cr - Steel

Data from Usami 1987

Inc lu sion size

Cas tin gpo ro sity

Internal Stress

Consideration of Elastic-Plastic Cracks

Discrete Dislocation Models

Ref. I. Adlakha, K. Sadananda, K.N.

Solanki -Discrete dislocation modeling

of stress corrosion cracking in an iron –

From Vehoff and Rothe, 1983

αc depends on gamma

ET

Continuous

Plastic-Elastic

Cracks

Elastic

g2 < g1 g1

a

a

T

T T

Ductile extension

T

T

T

T

a + da Brittle extension

Initial Crack y

(a)

(b) (c)

T

T

T

T

T

Continuous Plastic-Elastic Crack

ac

(d) (e)

Discrete Dislocation Model

al stress

T

T

T

T

T a

c

(d)

Notice Log-Log coordinates

T

T

T

T

T

Continuous Elastic-Plastic Crack

ac

(d)

Crack Grows by Alternating glide and cleavage

HOW DO I APPLY THESE CONCEPTS

FOR ENGINEERING APPLICATIONS?

Application to Sub-Critical Crack Growth

with thresholds and overload fracture

Engineering Problem –

Diagnostics and Prognostics

a)Fatigue Crack Growth

b)Stress-Corrosion Crack Growth

c)Corrosion-Fatigue Crack Growth

Use of Failure Diagram

ac

∆σ = ∆Kth /{y(πa)0.5}

Log

(∆σ)

Log(Crack length)

∆σe

Non-Propagating

Cracks

Original Kitagawa-Takahashi

Diagram

Experimental

Data and curve

Application to Fatigue

Notice Log-Log Coordinates

Our explanation in the short

Crack regime differs from

Conventional explanation.

Internal Stress

σIN

σmax,e.

Log (cl)

Lo

g (

σ)

A

B

C

D

a*

E σf

σ = Kmax,th /{y(πa)0.5}

σ = KIC /{y(πa)0.5}

Non-Propagating

Cracks

Modified Kitagawa-

Takahashi Diagram

Crack Growth at

Constant σ

a

c Concept of Internal Stress

Two factors: Maximum Stress and Gradient.

Nucleation and Propagation

KF ≤ Kt

Fatigue Stress

Con. Factor

σNf – Smooth

σNf - Notch =

10

100

1000

10-4 10-3 10-2 10-1 100 101 102

Kitagawa Diagam for

various initial defect-sizes

and Yield Strengths

Defect Depth, mm

1000 MPa

500 MPa

250 MPa

113 MPa

Yield Stress

Usami, 1981

Usami S. In: Blacklund I, Bloom A, Beevers CJ, editors. Symposium

on fatigue threshold. Stockholm, Sweden: EMAS; 1981. p. 205-38.

Internal

Stress

contribution.

Generalization of Kitagawa Diagram

Non-Propagating Cracks

INTERNAL STRESSES

ACCUMULATE BY LOCALISED

PLASTICITY

Short Crack is Long Crack which is Short!

Application to Short-Crack Problem

Overload

Underload

Short

cracks

Long

Crack

DK

da/d

N

DKint

crack

arrest

dl

Perturbations from

Long Crack Growth

da/dN1

Deviations from Long crack growth Behavior

All Deviations arise due to Local Internal Stresses

10-9

10-8

10-7

10-6

10-5

1 10

Ti-6Al-4V

D MPa(m)0.5

Long Crack

Data collected by Airforce

Microstructurally

Short Cracks

1

2

3

4

5

Crack tip driving forces are defined for a given crack growth rate, da/dN

Overload

Underload

Short

cracks

Long

Crack

DK

da/d

N

DKint

crack

arrest

dl

Perturbations from

Long Crack Growth

da/dN1

Deviations from Long crack growth Behavior

All Deviations arise due to Local Internal Stresses

10-9

10-8

10-7

10-6

10-5

1 10

Ti-6Al-4V

D MPa(m)0.5

Long Crack

Data collected by Airforce

Microstructurally

Short Cracks

1

2

3

4

5

Crack tip driving forces are defined for a given crack growth rate, da/dN

Crack growth is insured only when total force (applied + internal) exceeds the

material crack growth resistance in a given environment – expressed as thresholds

101

102

103

10-5

10-4

10-3

10-2

10-1

Ti-6Al-4V

Crack Length, mm

Threshold K

Curve

Modified Kitagawa Diagram

Crack Growth

Regime

Microstructually

short cracks

Localized

Internal Stress

Regime

e Endurance

Crack Growth Under

Remote Stress

KAppl

+KInt.S

> = Kth

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

1 10 100

da/d

N,

m/c

ycle

D MPa(m)0.5

Al-2024 T3Long and Short Crack Data

Smax

= 105 MPa

R = -1

80 MPa

70 MPa

60 MPa

Long Crack

Data Collected by Newman

101

102

0.0001 0.001 0.01

Short Crack to Long Crack

Crack Length, m

e,max

70 MPa

60 MPa

Short Crack Growth

Long Crack Growth

Internal Stress

Regime

Long Crack

Threshold

Merging with Long Crack

ac

Al 2024-T3R = -1

Crack

Arrest

Application to Corrosion & Corrosion Fatigue

Refs.

1. K. Sadananda and A.K. Vasudevan, Review of

Environmentally Assisted Cracking, Met. Trans.A,

42A, pp. 279-295, 2011

2. K. Sadananda and A.K. Vasudevan, Failure Diagram

for Chemically Assisted Cracking, Met. Trans. A, 42A,

pp. 296-303, 2011

Initiation

C1

(a)

σ

Time

σth

σPR ≈ σf

Concentration

C1

(b)

Initiation

Failure

σth

Concentration, c

σth*

σPR* ≈ σf*

Saturation

Effect

(c)

Failure

Smooth Specimen

Crack Initiation & Propagation

K

Time

KISCC

KPR ≈ KIC

Concentration

C1

(b)

Initiation

Failure

K

Concentration

KISCC*

KPR* ≈ KIC*

Saturation

Effect

(c)

Initiation

Failure

(a)

Fracture Mechanics Specimen

Crack Propagation/ Arrest

Stress Corrosion Crack Growth

σ

Time

Concentration

C1

(b)

Kt1

Kt2

Changing ρ

and/or Kt

σ

Kt

σPR

σIN

Non-Propagating

Cracks

No-cracks (c) (a)

Behavior of a Notched Specimen with Non-propagating Cracks

Defining a new concept – Mechanical Equivalent of Chemical Internal Stress

σth*

Log (Crack Length)

Log

(S

tre

ss)

σPR*

σf*

Internal stress

Non-Propagating

Cracks

c1

KISCC/{√(πa)}

KIC/{√(πa)

Failure Diagram

A

B

C D

cS

σth ac

Long crack threshold is steady-state

property of the material/environment.

Deviations are due to fluctuating local

Forces.

Diagram demonstrates that chemical force acts similar to Mechanical force.

σth = Threshold for smooth specimen for concentration C1

σth* = Saturation Threshold for given chemical Environment.

Failure Diagram for Stress – Corrosion

σth-Smooth

σth-Notch KCH

Chemical Str.Con

Factor KCH =

For any Kt

σ th

Concentration

CS σth*

σinert

σch

σ

t

σth C1

10

100

1000

10-5 10-4 10-3 10-2 10-1 100

Crack length, m

4340 Steel

Exptal data

KIC

-Line

KISCC

-Line

Non-Propagating Cracks

f = 1880 MPa

th = 175 MPa

Chemical

Stress

A

B

C

D

Experimental data from: Y. Hirose and T. Mura, Growth Mechanism of Stress Corrosion Cracking in

High Strength Steel, Eng. Frac. Mech. 1984, Vol. 19, pp. 1057-1067.

Critical Experiments are needed to evaluate

The crack initiation at sharp notches under

Stress Corrosion.

Transition from Corrosion Pit to Fatigue crack

Pit Growth Kinetics to Crack Growth Kinetics

101

102

103

100

101

102

103

104

Pit to Fatigue Crack (X10-6

meters)

Kitagawa Diagram

Transition from Pit to Fatigue Crack

10 Hz1 Hz

Data extracted

from Genkin, 1996

R=0.05

1M NaCl - 660mV

Pit-sizes Measured

from Fracture Surfaces

Size depends on

Frequency and Stress

J.M. Genkin, Ph.D. Thesis - MIT, 1996 -"Corrosion Fatigue Crack Initiation in 2091-T351 Alclad".

Chemical &

Mechanical

forces

1Hz 1 Hz 10 Hz

Pit to Crack

0

5

10

15

10-1

100

101

102

103

104

105

Pressure (H2) - Pa

343 K

313 KVehoff & Rothe, 1983

Fe-3%Si

aN= cot(a)

Crack Grows at a constant α for a given H. 2. Cot(α) reaches

a plateau with H

Effect of Hydrogen Pressure

α

Γ depends on H

Analysis of Crack Nucleation of ahead of a notch

Using Discrete Dislocation modelling

Summary and Conclusion

1. Crack tip driving forces are defined based on simple thermodynamic

Principles for elastic and elastic-plastic crack growth.

2. Internal stresses and stress gradients are required for both crack

Initiation and propagation.

3. Propagating and non-propagating conditions are defined

4. Kitagawa-Takahashi diagram is modified to include internal stresses

And their gradients to define crack growth or crack arrest conditions.

5. Mechanical equivalent of Chemical driving forces are defined that

Augment applied stresses in initiation and propagation.

6. Chemical stress concentration and stress intensity can be defined

Using crack growth behavior in inert conditions as reference.

THE END

R-ratio effects are intrinsic to Fatigue – Kitagawa Diagram

S-N Fatigue

•Two load-parameter Requirement (a) σmax & ∆σ – for S-N

(b) Kmax & ∆K – for Crack

Growth LEFM

(c) Cyclic Plasticity Governs

∆σ

NF

R

∆σe

UTS

∆σe

σmax,e

NF

107 cycles

∆σe*

σmax,e* σmax,e* > ∆σe*

Two Endurance limits

σ

time

Kmax,th*

∆K

Two Limiting Thresholds

da/dN

∆Kth

*

Kmax

Threshold

Kmax,th*> ∆Kth*

Fatigue Crack Growth

R

da/d

N

∆K

∆K*

Kmax*

A

B

C A

B C

AB = AC + CB

Kma

x

Same crack growth rates form a

basis to determine crack tip driving forces

Crack Growth

Trajectory

0

200

400

600

800

1000

1200

200 400 600 800 1000 1200 1400

max

, MPa

Ti-6Al-4V, Bimodal

RT, Air

3X104 cycles

1X105

1X106

1X107

1X108

max,e

De

Two Load parametric description of S-N Fatigue