fracture mechanics experimental tests (astm standards...

Task 6 - Safety Review and LicensingOn the Job Training on Stress Analysis

Pisa (Italy)June 15 – July 14, 2015

Fracture Mechanics experimental tests (ASTM standards and data) 2/2

Davide Mazzini – Ciro Santus

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Content

• Fracture Toughness KIc

- Plane strain condition

- ASTM standard E399

• High Toughness JIc

- Limitation of the KIc

- ASTM standard E1820

• Measurement of Fatigue Crack Growth Rates

- Paris curve experimental determination, ASTM standard E647

Standard for Fracture Mechanics

Pisa, June 15 – July 14, 2015

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Wide plasticity CT specimen

Large Fracture Toughness

A very large specimenwould be required to test according to KIc

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Definition

Rice 1968, J - integral

Let us assume an alternative material with Elastic behavior, not linear (hyperelastic material) and the same Stress/ Strain curve than the actual Elastic-Plastic one

For a monotonic loading the Stress/ Strain history is the same

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Definition

Rice 1968, J - integral

J is an integral along a path.It does not depend on the path that can be arbitrary provided that it is around the tip of the crack from side to side.

x is the crack direction

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Definition

Rice 1968, J - integral

J does not depend on the path that can be arbitrary provided that it is around the tip of the crack from side to side.

Zero integral for a closed path

Zero integral for free surfaces

*1 2 3 4

*1 2

1 2

+ + + 0

+ 0=

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Energy Release Rate

J - integral equivalences

For an Elastic behavior, even not linear, the J integral equals the Energy Release Rate

Potential energy:

U is the stored strain energyF is the work done by external forces

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Energy release rate:same as the G (elastic) parameter,A is the crack area.

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The Griffith problem

J - integral equivalences

Infinite plate with a crack, with linear elastic material, and plane stress,Griffith found:

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The Griffith problem

J - integral equivalences

Under these conditions:2d

daJ

A E

Being for this problem:

I , 1.0K F a F

Then it follows (that’s the reason of π):2IKJ

E

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General case

J - integral equivalences

2

where:planestress

planestrain1

Shear modulus2(1 )

E EEE

EG

Only if the material is linear:

2 2 2I II III

2K K KJ

E G

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CTOD – Crack Tip Opening Displacement

CMOD – Crack Mouth Opening Displacement

COD - Crack Opening Displacement

CTOD

CMOD is just the clip gauge measurement

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CTOD – Crack Tip Opening Displacement

COD - Crack Opening Displacement

From ASTM standard E1820

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CTOD – definition

COD - Crack Opening Displacement

Alternative way to define the CTOD:2×45° lines

Crack tip blunting(approx. circumferential)

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CTOD can be related to the SIF under Small Scale Yielding

CTOD under SSY hypothesis

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CTOD – can also be related to the J integral well beyond the validity

limits of LEFM

CTOD under large plasticity

Path around the strip-yield zone ahead of a crack tip

Plane stress conditions and a nonhardeningmaterial. More generally:

m is a dimensionless constant that depends on the stress state and material properties

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CTOD – similar value under SSY hypothesis

CTOD under large plasticity

2I

YS

2I

YS

YS

Planestress:4

being:4

approximately:4 1 ( 1)

KE

K JJE

J m

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J and CTOD

ASTM standard E1820

J and CTOD are still representative even over the 80% of the fully plastic load (plastic collapse)

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ASTM standard E1820

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From J back to K

ASTM standard E1820

2

where:planestress

planestress1

Shear modulus2(1 )

E EEE

EG

2 2 2I II III

2K K KJ

E G

ASTM plane strain assumption, mode I only

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Similar to the previous standard…

ASTM standard E1820

The displacement gage

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Similar to the previous standard…

ASTM standard E1820

Types of preparation notches

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Similar to the previous standard…

ASTM standard E1820

The CT specimen

The DCT specimen

The SEB specimen

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The CT specimens

ASTM standard E1820

Two CT specimen types,the increased size for the pin seat is for very high tension loads

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Displacement gage fixture details

ASTM standard E1820

Previous standard

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J calculation for CT specimen

ASTM standard E1820

Elastic term

Plastic term

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J calculation for CT specimen

ASTM standard E1820

These values are for the initial crack, just after the (fatigue) precrack

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J-R curve determination

ASTM standard E1820

a

The suggested procedure is to determine a series of J values for increasingly crack size, and then obtain a (fit) Resistance curve

a

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J-R curve determination

ASTM standard E1820

For each crack increment (i) there is an unload cycle to determine the crack size through the compliance

( ) ( ),i iJ a

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

Total displacement, mmLo

ad, k

N

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J-R curve determination

ASTM standard E1820

(i) stands for the updated values for the crack increments

(i=0) is just the initial crack

K(i) is calculated exactly as the previous standard

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J-R curve determination

ASTM standard E1820

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J-R curve determination

ASTM standard E1820

Compliance Load Line calculation:

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J-R curve determination

ASTM standard E1820

Crack size calculation (compliance method):

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J-R curve determination

ASTM standard E1820

Compliance correction

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J-R curve

ASTM standard E1820

Power law regression line

( ) ( )

, ,

are the regression coefficientsfor fitting the points

,

oq

i i

a B C

J a

Least squares fit procedure to find the regression coefficients

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J-R curve

ASTM standard E1820

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J-R curve

ASTM standard E1820

Ic is defined near the initiation of stable crack growth. The precise point is usually ill-defined an offset (similar to yield strength) is required

J

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J-R curve

ASTM standard E1820

2

Y

Y

2

Ic Ic

Example:kJ N N200 200000 200 200MPa mmm m mm

800MPa

10 2.5mm ...easy to be satisfied!

Conversion:/ (1 ) 220GPa

6630MPa mm 210MPa m

Q

Q

J

JB

E E

K E J

Yield and Ultimateaverage

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Alternatively the δ-R curve (CTOD-R curve)

ASTM standard E1820

CTOD instead of J

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J-R curve

ASTM standard E1820

IcIc Ic

Ic is just theonset of fracture,thespecimen can sustain highervaluesof , indeed thecurvecontinues with a (stable)increaseof thecrack size.

QJJ J KE

J

J

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J-R curve

The Resistance curve

IcRising curveafter J

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J-R curve

The Resistance curve

IcRising curveafter J

Stable further propagation

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J-R curve

The Resistance curve

1 2

3

1 5

4 unst

paths: Load control, stablecrack growthlimit stable crack

crack

paths: Displacementcontrol paths

stable crack g

able

rowth

i

i

PP

P

PP

4P

Load control is usually less stable than displacement control,in most structures the conditions are between the extremes of load and displacement control

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J-R curve

The Resistance curve

Three stages of crack growthin an infinite body

Steady State

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R curve, single fracture toughness value

The Resistance curve

Unstable,K J

Ic Ic,K J

crack size a0a

( ), ( )K a J aStable

With a single value the curve is just asymptotic

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ASTM standard E1820

Homework:

Apply the ASTM E1820 procedure extracting data from the test file:Test J_Ic.xlsx

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

Total displacement, mm

Load

, kN

Then estimate JQ and verify if it can be converted into JIc

?

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The damage tolerant approach

Crack (stable) propagation under fatigue

The presence of a crack (actually detected or just postulated) can be tolerated if the propagation rate is reliably estimated.

The size of the postulated crack is the minimum detectable of the inspection method.

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Paris’ law (also known as the Paris-Erdogan law)

Fatigue crack propagation

max min

min

max

dd

where:

, are parameters depending on:- the material

-the load ratio

ma C KN

K K K

C m

KRK

Paul C. Paris

I(usually )K K

Time

1N 2N ...N

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Paris’ law validity range

Fatigue crack propagation

thKcK

Paris validity

Near threshold

Sudden(unstable) fracture

th is the thresholdstress intensity factor rangebelow this amplitude thecrack remains same sizethough fatigue loaded

K

max

c

Ic

is the thresholdstress intensity factor rangefor which Approaching this valuethecrack turns intounstable propagation

K K

K

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Paris’ law integration

Fatigue crack propagation

1-2 /2 ( 2)/2 ( 2)/21 2

d ( )d

after assuming does notchange(at least not significantly)

the dependececan beintegrated:

2 1 1( 2) ( )

with 2(usually 2)

m m

m m m m m

a C K C F aN

F

a

Nm CF a a

m m

1a 2a

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Fatigue crack propagation

Example:

1 2

14

1-2 /2 ( 2)/2 ( 2)/21 2

5

1mm, 5mm,for example 100mm

1.12100 MPa

3.25mm/cycle7.481 10

(MPa mm)

2 1 1( 2) ( )

4.620 10 cycles

m

m m m m m

a aa b bF

m

C

Nm CF a a

1a 2a

b

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Fatigue crack propagation

Example:

1 2

i

ii iii iv

51-2 i ii iii iv

1mm, 5mmlimited width, 10 mmthecalculation can be donestepwisedividing the crack range in small steps:

1.38( 1 2mm)1.65, 2.1, 2.7

1.700 10 cycles

a ab

F aF F F

N N N N N

1a 2a

b

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Fatigue crack propagation

Example:

1 2

51-2

51-2

1mm, 5mmlimited width, 10 mmParis' law (numerical) integrationMATLAB

1.843 10 cycles

(previous result 1.700 10 cycles)

a ab

N

N

100cycles(small)stepsintegration

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Fatigue crack propagation

AFGROW software for crack propagation calculation

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Fatigue crack propagation

ASTM E647

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ASTM standard E647

DefinitionsUsually tests are performedat positive load ratiose.g.: 0.1R

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ASTM standard E647

Specimen geometry

For fracture toughness B = W/2, while for fatigue this requirement is less demanding.The KI values experienced by the specimen under fatigue propagation are much lower.

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ASTM standard E647

Specimen geometry

LEFM validity condition, plane stress which is more demanding

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ASTM standard E647

Specimen notch preparation

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ASTM standard E647

Delta SIF calculation

Same relation to find the Stress Intensity Factor for the CT specimen

Currentvalue?

a

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ASTM standard E647

Compliance method for

the crack size

Different positions of the crack gage clip

Here is theclipdisplacement

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Crack size determination

Alternative ways for accurate crack size measurement during the test

- Potential drop

- Fractomat(Crack Gage)

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Crack size determination

Potential drop

Calibration procedure

2

Examplecoefficients:

26.85 9.317 0.1367X XaY Y

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Crack size determination

Fractomat crack gage

Crack range

As the crack propagates the foil resistance increases.Having a predefined geometry the calibration equation is already given by the manufacturer

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0

5

10

15

20

25

ch1ch2

a' [m

m]

Crack size determination

Possible not parallel crack propagation, especially at the beginning

Application ofFractomatat the two sides

Number of cyclesChannels 1 and 2

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ASTM standard E647

Near threshold/

high propagation rate

Near threshold

High rate

510 mm/cycle

th

7

:d 10 mm/cycled

KaN

thIc

The final part of the curve is usually not of interestonce is knownK

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ASTM standard E647

Near threshold – Decreasing procedure

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ASTM standard E647

Propagation rate calculation

A large number of cycles is recommendedbetween steps the and 1(instead of just two consecutive cycles)to have a significant, still small,crack size increment

i i

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Data example

Q+T steel, similar to AISI 4340

Several tests with Fractomat and Potential Drop

th 9.4 MPa mK

3

3.6mm/cycle0.4 10

(MPa m)m

m

C

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Data example

Homework:

Compare the experimental data with the alternative Paris model:

th( )m mda C K KdN

th

3

9.4 MPa m3.6

mm/cycle0.4 10(MPa m)m

Km

C

0.01

0.1

1

10

100

1000

5 50

POTENTIAL DROP

CRACK GAGE

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