Treatment of Constraint in Non-Linear Fracture Mechanics · Treatment of Constraint in Non-Linear Fracture Mechanics ... effect of T-stress on plastic zone size ... Application of

Treatment of Constraint in Non-Linear Fracture Mechanics

Noel O’DowdDepartment of Mechanical and Aeronautical Engineering

Materials and Surface Science InstituteUniversity of Limerick

Ireland

Bristol UK, June 20th 2008

Acknowledgements: C. Fong Shih, S. Kamel. J. Sawyer, H. McGillivray, T. Tkazcyk

2

Agenda

Motivation and Background

Discussion of higher order terms in crack tip fields

Application to idealised materials and geometries

Application to real conditions

3

Variation in measured fracture toughness for ductile materials

Toughness ( JC) can depend on specimen geometry and size

10 mm

50 mm

25 mm

ASTM A515 from ECB specimens Kirk et al. 1991

50

150

200

250

300

350

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6

100

Motivation

ECB = edge cracked benda: crack lengthW: specimen width

4

Motivation

J does not uniquely characterise material toughness

In standard practice a unique toughness value is ensured by

following size and geometry requirements for testing

However, requirements can lead to a very conservative toughness

and this motivated research into extending the J-based approach

5

Hutchinson, 1968; Rice and Rosengren, 1968

HRR field is first term in the asymptotic solution for a power law plastic material

Amplitude of HRR stress field is J

HRR field

( )θσσαε

σσ ij

n

nij rI

J ~/1/1

000

+

⎟⎟⎠

⎞⎜⎜⎝

⎛=

( )n00 σσαεε =

6

MotivationTwo term Williams Mode I crack tip field:

For a linear elastic material stress term is parallel to the crack face so has weak effect on crack tip driving force

T stress can effect stability of crack path (Cotterell and Rice, 1980)

( )σπ

θ δij I ijK

rf T= +

2

TT

7

Slip line field solutions for rigid-perfectly plastic behaviour

Can identify regions of intense plastic slip—slip bands

Slip line fields and hence near tip stresses are considerably different

Slip Line Field Solutions

ECB CCT DECT

8

Slip line fields indicate that bend specimen has higher near tip stress distribution

Slip Line Field Solutions

ECB CCT

σ σ2223

= yσ π σ222

3=

+y

9

Crack tip stresses lower in tension than in bending

For a rigid plastic material can arbitrarily superimpose a hydrostatic stress (provided boundary conditions are satisfied))

Variation in stress associated with a variation in hydrostatic stress

This difference in hydrostatic stress is a measure of the difference in ‘constraint’ between different specimen geometries

Constraint in Elastic-plastic Fracture Mechanics

BackSector

Forward Sector

10

Crack tip fields studied numerically through “boundary layer” finite element analyses

Larsson and Carlsson, 1973, used such an approach to examine the effect of T-stress on plastic zone size

Similar studies by Bilby et al., 1986 and Betegón and Hancock, 1990

Material model: Elastic-plastic rate independent power law material:

Constraint in Elastic-plastic Fracture Mechanics

( ) ( )n0000

000

σσσσσεε

σσσσεε

+−=

<=

Power law plasticity

11

Effect of geometry introduced through T stress term applied at the boundary

When T =0 the analysis is often referred to as a small scale yielding analysis (SSY) and the resultant stress field is the SSY field

Finite Element Boundary Layer Analysis

( )σπ

θ δij I ijK

rf T= +

2

K -T field

12

Finite notch modelled (rather than sharp crack tip) to allow investigation of crack blunting


( )σπ

θ δij I ijK

rf T= +

2

13

The T stress plays the role of a geometry parameter

By varying T/σ0 can generate a range of crack tip fields

T/σ0 = 0 corresponds (almost) to the HRR field

Finite strain analysis carried out to account for large deformations in the vicinity of the crack tip


14

Stresses normalised by σ0; Distances normalised by J/σ0

J/σ0 is a measure of the crack tip opening and the crack blunting zone


Blunting zone

T/σ0 = 0

T/σ0 = -1

T/σ0 = 1

15

Similar behaviour for radial stress, σrr


Blunting zone

T/σ0 = -1

T/σ0 = 0

16

Small strain analysis (no crack blunting)

θrSSY, T = 0

T/σ0 = −1

( )σπ

θ δij I ijK

rf T= +

2

17

Construction of Q-stress fieldsFrom numerical results can construct the form of the second order elastic-plastic stress field

Define

σREF is the reference (high constraint) distribution—in this case the SSY (or HRR) field

Q is a dimensionless amplitude parameter of the second order fields

Q gives the angular and radial distribution of the fields (not yet known)

( )Qf r ij ijFE REF~ ,θ

σσ

σσ

= −0 0

18

Construction of Q-stress fieldsFrom numerical results can construct the form of the second order elastic-plastic stress field

Qf~θθ

Qfrr~

19

Angular distribution:

To a good approximation:

Construction of Q-stress fields

0),(~1),(~),(~

=

==

θ

θθ

θ

θθ

rf

rfrf

r

rr

Qf~θθ Qfrr~ Qfr

~θ

20

Two parameter elastic-plastic stress fields

or, more generally

The parameter Q is a hydrostatic stress term determined from finite element analysis for the particular geometry and load level

ijref

ij Qδσ

σσσ +⎟⎟

⎠

⎞⎜⎜⎝

⎛=

00/

( )σ σαε σ

σ θ δijn

n

ij ijJ

I rQ/ ~

/

00 0

1 1

=⎛⎝⎜

⎞⎠⎟ +

+

21

SSY, T = 0

T/σ0 = −1

SSY, T = 0

T/σ0 = −1

Q will generally be negative—HRR or SSY field is the upper bound ‘high constraint’ crack tip field

Q reduces the stress amplitude relative to the reference field

Two parameter elastic-plastic stress fields

22

Two Parameter Fracture Mechanics

Stress and strain fields depend on J and Q

Fracture toughness expressed in terms of JC(Q)

JIC is the standard high constraint fracture toughness value

corresponding to Q = 0 and is generally the lower bound

ijrefij Qδσσσσ += 00 //

23

Asymptotic solution, (Yang et al., 1993)

1 1 11 1 1

(1) 2 (2)

0 0 0 0 0 0 0

s tn n nij HRR

ij ij ijn n n

J J r J rA AI r I L L I L L

σσ σ σ

σ αε σ αε σ αε σ

+ + +⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

% % %

ijσ%

Analytical solution for power-law material model.

First three terms of crack tip field:

Exponents s and t depend on n

L is a characteristic length (e.g. unity, a, W)

are functions of n and θ

Crack tip stress fields are described by J and A.

A is a measure of the loss of constraint (equivalent to Q)

24

Two Parameter Fracture Mechanics

Analysis so far has been for idealised geometry, boundary layer analysis

Now consider ‘real’ geometries

25

FE models

M(T)

Crack depths modelled: a/W = 0.1→0.7

Gives expected range of constraint in practice.

θr

2aH

2W

P

PAll models discussed are two dimensional

SEN(B)

aS P

θ

W

r

26

Constraint drops with increasing load—Q becomes more negative

Results for Centre-Cracked Tension

σσ

θθ

0

r/(J/σ0)

Load increasing

n= 10

P

P

27

Dependence of Q on normalised J

Q is evaluated at rQ = 0.004 J/(ε0σ0 )

corresponds to rQ = 2J/σ0 for ε0 = 0.002 and α = 1

28

Analysis of M(T), a/W = 0.1

J-Q gives a somewhat better prediction for n = 10 and 5

Representative of M(T), a/W=0.4 and 0.7 and SEN(B), a/W=0.1

43norm normbP σ=

29

Analysis of SEN(B), a/W = 0.4, n = 10

Both J-Q and J-A give excellent agreement with FE prediction at this deformation.

21.409 normnorm

bPS

σ=

30

Analysis of SEN(B), a/W = 0.4, n = 10

At higher deformation both J-Q and J-A give poor prediction due to effect of global bending on crack tip fields.

Global bending may be characterised by an additional parameter (Chao et. al., 2004; Zhu & Lei, 2006).

31

Q vs Load

Tension geometries and shallow cracked bend geometry experience ‘loss of constraint’ at low load

Slope of Q vs load is approx. constant at high deformation

32

Evaluation of QWithin the boundary layer analysis there is a one-to-one relationship between T and Q for a given material

Q

T/σ0 = -1

T/σ0 = 0

33

Evaluation of Q

T is an elastic parameter and is relatively easily determined

Can thus estimate Q through an elastic analysis for T

34

Evaluation of QT stress can give a reasonable estimate of Q

Centre Cracked Tension Edge Cracked Panel

35

Power law estimates for Q

T-stress can be used to estimate Q under ‘small scale yielding’

conditions—when size of the plastic zone is small

Consider a pure power law material:

For such a material, stress at a point varies linearly with remote load

( )ε ε α σ σ/ /0 0= n

( ) ( )σ σ σ σ σij ij x n/ / ,0 0= ∞

36

Power law estimates for QIt can be shown using the HRR field that

h1 is a function which depends only on geometry and n

Similarly it can be shown for a pure power law material

Q varies linearly with load

An approximation scheme based on T under small scale yielding

and h2 under extensive plasticity may then be used

( ) ( )Q h n= ∞σ σ/ 0 2

( ) ( )Ja

h nn

αε σσ σ

0 00

11= ∞ +

/

37

Power law estimates for Q

38

Treatment of ‘real’ materials: X100 ferritic pipeline steel

Low strain fit with n =15 gives good fit to test data up to 5% strain.

High strain fit with n =25 gives close fit at strains > 5%

σ0.2=640MPa, ε0 = 0.0030

low strain fit(n = 15, α = 0.56)

high strain fit(n = 25, α = 0.062)

39

Results for X100: M(T), a/W =0.1

J-Q gives best agreement with FE.

J-A prediction based on high strain fit gives better agreement than the low strain fit.

Kamel et al., 2007

40

Results for X100: SEN(B), a/W =0.4

J-Q gives best agreement with FE.

J-A prediction based on low strain fit gives better agreement than the high strain fit.

41

A Simple Fracture Toughness Curve—RKR Model

Examine fracture criterion based on the attainment of a critical

stress at a critical distance

One parameter fracture toughness gives a single number JIC

Using two parameter fracture mechanics generate a fracture

toughness curve, Jc(Q)

r = rc

σ = σc

crack

42


r = rc

σ = σc

crack

1

0

0

1/1

000

1/1

00022

//

0

/

/

+

+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⇒

=⇒=

+⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

c

c

IC

C

IC

n

cn

Cc

n

n

QJJ

JJQ

QrI

J

QrI

J

σσσσ

εσσσ

εσσσ

43


ASTM A515 from ECB specimensKirk et al.1991

50

150

200

250

300

350

00.0 0.1 0.2 0.3 0.4 0.5 0.6

100

ASTM A515 from ECB specimensKirk et al.1991

50

150

200

250

300

350

00.0 0.1 0.2 0.3 0.4 0.5 0.6

100

44


Cleavage toughness data for mild steel tested at-50oC, Sumpter and Forbes, 1992

45

Application of J-Q approach

Toughness curve determined in laboratory

J-Q loading trajectory for component from finite-element analysis

J

Q0 -1.5

JIC

Toughness curve

Loadtrajectory

46

Alternative approach: Constraint matching

Estimate Q value at fracture for component

Test laboratory specimen with similar constraint level

Treat this fracture toughness as the ‘constraint’ matched toughness

E.g. For shallow cracked pipes under tension (Q ≈ −1) use fracture toughness Jc (or CTOD) from edge crack tension geometry

Approach widely used in offshore industry

47

Validation for ‘real’ materialsAnalysis of X65/X70/X100 ferritic pipeline steels

48

Approach for Steel Pipelines

Full 3D analysis

Crack tip mesh

10.75” pipe; X65(406.4 mm × 19.1 mm)

49

Approach for Steel Pipelines, X100

J-Q curve determined from testing small specimens

Kc = 340(1 - 0.53Q)

Kc = 270(1 - 0.63Q )

200.0

300.0

400.0

500.0

600.0

700.0

800.0

-1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

Q

Kc

(MP

am1/

2 )

Δ a = 0.5 mm

Δ a = 1.0 mm

CT data

CCT data

O’Dowd and McGillivray, 2003

50


Constraint variation in pipe from a 3D finite element analysis

a/c =0.2, a/t = 0.2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Strain = 4.03%Strain = 2.71%Strain = 0.754%Strain = 0.38%Strain = 0.259%

plane strain

plane stress

σ/σ y

r/(J/σy)

Q

R/t = 22

51


Additional safety margin from a constraint based analysis

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

L r

Kr

Level 2B FAD

Constraint modified Level 2B FAD

α = 0.5; β = -0.8

cutoff line

SAFE UN-SAFE

52

Conclusions

Crack tip driving force for an elastic-plastic material can be described by two parameters, J and Q

Q is a hydrostatic stress term, motivated by the form of the crack tip fields

Allowance for constraint can increase the safety margin or increase allowable load

Treatment of Constraint in Non-Linear Fracture Mechanics · Treatment of Constraint in Non-Linear Fracture Mechanics ... effect of T-stress on plastic zone size ... Application of

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