Fracture Mechanics: Linear Elastic Fracture Mechanics...

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

Pisa (Italy)June 15 – July 14, 2015

Fracture Mechanics: Linear Elastic Fracture Mechanics 1/2

Davide Mazzini – Ciro Santus

2

Content

• Stress singularity

- Notch degenerating into a crack

- Multi-axial stress at notch root/ crack tip

- The Williams problem

• Linear Elastic Fracture Mechanics (LEFM)

- The Westergaard stress function

- Definition and calculation of the Stress Intensity Factors (SIFs)

- LEFM Validity limitations

Table of content – Class VI.b.1

Pisa, June 15 – July 14, 2015

3

Books on Fracture Mechanics

T.L. Anderson, Fracture Mechanics: Fundamentals and Applications,

third edition. CRC Press 2005.

D. Broek. The Practical Use of Fracture Mechanics. Kluwer 1989.

… and many many others

Books


4

• Experities and similitude (up to 1700)

• Elastic evaluations (nominal solutions) (Eulero, Cauchy, De SaintVenant, 1800)

• Stress concentrations (Kirsch, Inglis, 1900)

• Theory of plasticity (Prandtl, 1920)

• Sharp tip defects (Griffith, 1922)

History of “Strength of Materials”


5

• Griffith’s energy approach for brittle materials (1930)• Practical relevance (1940-1950)• Definition of K, extension to metallic materials, complete develpment

of the Linear Elastic Fracture Mechanics (LEFM) (Williams, Irwin,1950)

• Application of the LEFM to Fatigue (Paris, 1960)• Extension to ductile materials (Elatic Plastic Fracture Mechanics

EPFM) (Irwin, Dugdale, Baremblatt, Wells, Landes, Rice, 1960)• Dynamics and crack arrest (DFM), viscous and (NLFM) (AA.VV. 1980)• Engineering applications, standards for design and testing, NDT,

corrosion, anisotropic materials, Damage Tolerant approaches, ..(ASTM, ASME, ESIS, BS)

History of “Fracture Mechanics”


6

History of “Fracture Mechanics”


7

2500 liberty ships, hull assembled by the innovative process of welding

Liberty ships – World War II


8

700 experienced heavy structural damage, 145 completely destroyed,many lost (complete breakage of the hull)



9


Post-failure analysis

• Failure at low stress (sometimes with the ship in the arbor)

• Quite “brittle” fractures

• Failure more frequent in winter time (ductile to brittle transition

temperature)

• Effect of the technological process (metallurgical, geometrical: weld

crack-like defects)

Fracture mechanics was born to understand these failure!


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Circular hole in a flat plate

Complete analytical solution

Plane stress solution if a>>B

Plane strain if a<<B

Extension to other problems

Kirsch 1898

a

B

0

0

Far boundaries


11


Kirsch 1898

Far boundariesx

y

r

2 20

2 2

2 40

2 4

2 20

2 2

1 1 1 3 cos 22

1 1 3 cos 22

1 1 3 sin 22

rr

r

a ar r

a ar r

a ar r


12


Kirsch 1898

x/a

0

y/a

-1+3

rr 0

+1

x/a

y/a

Why rr at these points?


13

Circular hole in a flat plate, bi-axial loading

Kirsch 1898

Uniaxial Kt = 3

Equibiaxial Kt = 2

Pure shear Kt = 4~


14

Elliptical hole in a flat plate

Inglis 1913

Problem definition:Geometry

,

Load, nominalstress(far field stress)

a b

Far boundaries


15


Inglis 1913

A

At

t

Stress concentration:21

21

Kirsch solution for central hole

3

ab

aKb

b aK

Far boundaries


16


Inglis 1913

Far boundaries

2

Moresignificant, local radius:ba

a

a


17


Inglis 1913

t

2

t

t

21

being:

then:

1 2

, are more properly definingthe localgeometrywhen:

2

aKb

ba

aK

a

a

aK

Far boundaries


18


Inglis 1913

t0 0lim lim 2

and the power of singularity ist

Limit:

square rhe of the locao lo radiut s

aK

Far boundaries0

lim( , ) 0b


19

Stress components in-plane, plane stress

Multi-axial stress at notch root

x/a

y/a

a

B<<a

0

0

x

z

y

tK


20

Plane stress


Transversal stressfree surfaces

Almost zero stressat interior points


21

Stress components in-plane, plane strain (approx.)


x/a

y/a

a

B>a

0

( )

x

y

z x y

tK


22

Plane strain


Zero transversaldisplacement:

0z

11 1

1

After imposing 01 ( ) 0

( )

x x

y y

z z

z

x y z

z x y

E

E


23

Inglis notch-like, plane stress

ANSYS Wb


5mm

2 40 mma

t201 2 1 2 55

aK

t 5.5Why a different value here?K


24

Inglis notch-like, plane stress

ANSYS Wb


0 2 4 6 8 10-100

0

100

200

300

400

500

600

x coordinate, mm

Stre

ss c

ompo

nent

s, M

Pa

y

x

zPath on the geometry


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Inglis notch-like, plane strain

ANSYS Wb


Exercise:

Calculate the Stress components, with ANSYS Workbench, at the notch tip for the large thickness geometry, and then verify the plain strain assumption

Repeat same calculation with imposed (exactly) plain strain constraint


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The Williams problem

Williams 1957

• local geometry :

• governing parameters:

• local polar coordinates:

• useful angular variable: s

s =

=0

, r

r

s


27

Williams 1957

2 2 2

2 2Airy function: , xx yy xyx yx y x y

2 2 2 22 2

2 2 2 2

4 4 4

4 2 2 4

Governing equation: 0

2 0

x y x y

x x y y

2 2 2 22 2

2 2 2 2 2 2

4 2 4 3 4 3

4 2 2 4 4 3 2 2 2 3 2

2 2 2

2 2 2

Polar coordinates:

1 1 1 1 0

1 1 1 1 12 2 2 0

1 1 1+ ; ; rr r

r r r r r r r r

r r r r r r r r r r

r r r r r r

2

1r

Stress components


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Williams 1957

• Williams hypothesis for the Airy function:

1. General parameters: c1, c2, c3, c4 and exponent (a dimensionless real number)2. Airy equation fulfilled in the domain for any combination of c1, c2, c3, c4 and

• Corresponding stress field:

• Strain and displacement:

11 2 3 4sin 1 cos 1 sin 1 cos 1r c c c c

1

1

1

1

1

rr

r

F F

F

F

r

r

r

1ij ir u r

r

s

1r F


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r

s

Williams 1957

• In order to keep the displacements bounded:

• Local boundary conditions:

• Boundary conditions in explicit form:

• Homogeneous linear system with unknowns: c1, c2, c3, c4 and the parameter • Typical outcome of several problems: instability, free vibrations, etc.• We are interested in not trivial solutions (eigenvalue problem)• Let’s put the determinant of the system matrix to zero• Characteristic equation with as unknown (infinite solutions)

Tractionfree edges

0 2 0

0 2 0s

r r s

2 4

1 3

1 2 3 4

1 2 3 4

01 1 0

sin 2 1 cos 2 1 sin 2 1 cos 2 1 0

1 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1 0s s s s

s s s s

c cc c

c c c c

c c c c

0iu r


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Williams 1957

where 1, 2,3,......2nn n

12

3 41

2sin 1 sin 1 cos 1 cos 12 2 2 2 2

n

n nn

n n n n nr c cn

11

2 23 4

1

1

2, , , ....n

i ij jij n n ij ijn

r n c c B C rA r

Square root singular term !

s


Crack as the special case with ψ = 0

• For this case the eigensolutions are

• and the corresponding Airy’s function becomes:

• The infinite couples c3n, c4n are determined by the other boundary conditions (remote geometry of the body, applied loads, constraints)

• Final general expression for the stress components:

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Williams 1957

1, and ij ij iu rr


General conclusions of the Williams analysis

• Among the (usually) infinite terms of the stress expansion at the notch tip, only the first is unbounded (it goes to infinite as r approaches zero)

• The other terms are bounded or tends to zero approaching the notch tip

• The power of the singular term is a function of the angle 2 of the notch

• The strength of the singularity is the highest when = 0: the crack is the most severe notch

• The power of the leading singular term is universal (the same for any crack), the asymptotic terms of the elastic fields at the tip are:

32

Williams 1957


Exercise – MATLAB:

Implement a parametric calculation for the Willams problem and find the λ solution in the range of angles ψ = 0° - 89°

r

s

2 4

1 3

1 2 3 4

1 2 3 4

01 1 0

sin 2 1 cos 2 1 sin 2 1 cos 2 1 0

1 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1 0

Then thesystem can be put in matrix fo

s s s s

s s s s

c cc c

c c c c

c c c c

1

2

3

4

rm:0 1 0 1 0

1 0 1 0 0sin 2 1 cos 2 1 sin 2 1 cos 2 1 0

01 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1s s s s

s s s s

cccc

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Williams 1957


Write the determinant of the matrix, impose it to zero and solve to find

0 1 0 11 0 1 0

0sin 2 1 cos 2 1 sin 2 1 cos 2 1

1 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1

...

s s s s

s s s s

r

s


34

Williams 1957


r

s

0 0.5 1 1.5 2-4

-2

0

2

42 = 60

= 0.51222


35

Williams 1957


r

s

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

1-

Power-law singularity exponent

11

1ij r

r


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