linear elastic fracture mechanics - espci paris · 2017-06-21 · 7 center for mechanics of solids,...

Center for Mechanics of Solids, Structures and Materials

Department of Aerospace Engineering and Engineering Mechanics

“Measure what is measurable, and make measurable what is not so.” - Galileo GALILEI

Linear Elastic Fracture Mechanics

Krishnaswamy Ravi-Chandar

Lecture presented at the

University of Pierre and Marie Curie

April 30, 2014

CENTER FOR MECHANICS OF SOLIDS, STRUCTURES AND MATERIALS 2

The plan

• Introduction to fracture mechanics

– April 30, 2014

• Quasi-static instability problems (JB Leblond)

– May 6, 2014

• Introduction to dynamic fracture mechanics

– May 15, 2014

• Dynamic instabilities during fast fracture

– May 21, 2014


How strong is a solid? – 1. An atomic point of view

0max

2 ( )sin

a aF F

0 0max

0

2 ( )sin ,

a a a

a

0 0max

0 0

0max

2 2cos

2

a adE

d

a

max

0

~2

E E

a

Interatomic

distance, a

En

er

gy

, E

Re

pu

lsio

n

Att

ra

cti

on

Interatomic

distance, a

Fo

rc

e,

F T

en

sio

n

Co

mp

re

ssio

n

Material Modulus

GPa

Strength

GPa Strength/Modulus

Steels 200 0.10 – 2 0.0005 – 0.01

Glass 70 0.17 0.0025

Carbon fibers 400 4 0.01

Glass fibers 70 11 0.16

Macroscopic strength is significantly smaller than the theoretical strength

0a

0a


How strong is a solid? – 2. The role of defects

22 ( ,0) 1 2a

ab

1

22

22 ( ,0) 2 where /a

a b a

max

~ 1 10 m~ 0.005

~ 1 6 mf

E

a E

max

1

2f a constant

2a

2b x1

x2

Griffith’s experiments

1. Used experiments on glass tubes and glass bulbs loaded under internal pressure to

show that 𝜎 𝑎 was constant

2. Manufactured fresh glass fibers with diameters in the range of 1 mm to 3 microns to show that small fibers had strength of about 11 GPa


The continuum view of fracture

• Process of fracture can be cleavage, intergranular/ transgranular fracture (polycrystalline materials), cavitation (ductile metals), disentanglement (polymers), microcracking (glasses,…), fiber breakage,…

• Details of processes within Lp are not important; only the total energy needed for the fracture process is assumed to play a role in the development of the fracture

• Lp is small – “small-scale process zone” – what does this mean?

Lp


The energy balance – a continuum view

potential energy of the bodyR RW U

Total energy of the system: sE U

Energy Release Rate( )

define

fracture resistances

dG a

da

dUR

da

At equilibrium, 0E a

Fracture criterioncG a R

work done by the external forces on the body

strain energy stored in the body

R

R

W

U

where surface energy (or fracture energy)sU

2a x1

x2

Stable if 0cE a


Remarks 1:

• ac is the equilibrium crack length (reversible)

• Fracture resistance R(a) includes the effect of all dissipative fracture processes and is typically calibrated from experiments.

• For Linearly Elastic Fracture Mechanics (LEFM), the region outside Lp must exhibit linear elastic behavior, but this is not a general requirement.

• Other than Lp being “small”, there is no length scale here! The theory works at length scales from the atomic to the tectonic.

• Need methods to calculate G(a) for specific crack problems



Remarks 2:

• R varies over several orders of magnitude

– True surface energy is ~ O(1) J/m2

– Glasses and ceramics~ 10 J/m2

– Polymers ~1 kJ/m2

– Metals ~ 100 kJ/m2

• Differences arise due to different mechanisms of deformation and failure

• Must be determined through calibration experiments, such as the pioneering work of Obreimoff (1930)



Calculation of G(a)

a

, TP

,P

MC

compliance of the loading system

0 fixed displacement

fixed load

M

M

M

C

C

C

( ) / compliance of the specimenC a P

total displacement (fixed)T

( )T M MC P C a C P

21( ) ( )

2T

dG a P C a

da

2 21 1Strain Energy: ( )

2 2R MU C P C a P

21( )

2

equilibrium crack length

P C a R

Stable if 0cE a

2

( ) ( ) ( ) MC a C a C a C

Stability depends on !MC

Can cause stick-slip and other unstable

crack growth effects


Example 1: G(a) for a double cantilever beam

MC

, TP

,P

a

2d

,P

a

d

3 2

3 3

8 24( ) ; ( )

a aC a C a

Ed Ed

13 2 43

8c

EdG a R a

R

2 2

312c

ERG a R a P

d

2

( ) ( ) ( ) MC a C a C a C unstable stable

3 2

3 3

4 12( ) ; ( )

a aC a C a

Ed Ed


Example 2: G(a) for an infinite strip specimen

21

EG a

h

stable

a 2h


Fracture mechanics – the global point of view

• The global point of view works quite well for a number of problems.

• It circumvents detailed calculation of stress/strain states in the vicinity of the crack.

• Has been applied successfully in a number of structural applications

• Difficulty in calculating the compliance, C(a)

• Difficulty in calibrating the fracture energy, R

• Difficulty in selecting/identifying fracture path

• Modern numerical simulations incorporate the energy approach through the phase-field methodology.


Fracture mechanics – why a local point of view?

• Provides a systematic way of calculating G(a)

• Provides a method for analyzing different loading symmetries

• Local approach based on stress and strain fields permits decoupling of path selection from failure characterization


Loading symmetries

Mode I or Opening mode

Mode II or In-plane shear

Mode III or Anti-plane shear


Linear elasticity

1( ) ( )

2

T ε x u u

( ) 2kk σ x 1 ε

σ f 0

1

2

( ) *( ) on

( ) ( ) *( ) on

R

R

u x u x

s x σ x n s x

( ) ( ) y x x u x

- boundary conditions

1 2 3 3( , ); u u x x u x

1 2 , ,

33 3

1( , )

2

const; 0

x x u u

1 2

33 1 2

3 1 2

( , ) 2

( , )

( , ) 0

x x

x x

x x

Plane strain

11 ,22 22 ,11 12 ,12

4

If ; ;

then 0

3 3 1 20; ( , ) u u u x x

Anti-plane shear

2

3 0u


The J-integral

1

1

R

uJ U n n ds

x

1x

2x

R

n1

1. The integral is zero if contour is closed inside the body without enclosing singularities

2. If the contour goes from below to above the crack surface as indicated, the integral is independent of the path

3. This integral can be interpreted in terms of the energy release rate:

4. Path independence implies that and therefore,

1~ rσ : ε

( ) /J G a d da

1/ 2 1/ 2~ ; ~r r σ σ ε ε


Anti-plane shear

Mode III or Anti-plane shear

3 3( )2

IIIKh

r

320

lim 2 ,0IIIr

K r r

mode III stress intensity factor

1. Anti-plane shear can exist only in specimens without bounding planes – in axisymmetric geometries or infinitely thick plates

2. Free surfaces in finite thickness plates introduce coupling to mode II

3. Connection to J and G obtained by using the path independent integral:

4. Failure criterion for mode III is still being debated (more on this later!)

21( )

2IIIJ G a K


In-plane loading symmetries

Mode I or Opening mode

Mode II or In-plane shear

1 1,2

2

II

IIII

Kr F T

r

KF

r

220

lim 2 ,0Ir

K r r

mode I stress intensity factor mode II stress intensity factor

120

lim 2 ,0IIr

K r r

represents nonsingular stress

and plays a role in crack path stability

T


Calculation of the stress intensity factors

• Elastic boundary value problem to be solved

– Numerous examples exist in handbooks

– Robust numerical methods based on FEM, BEM, available

– Considered a solved problem: Given a geometry, loading, etc, there is no difficulty in determining KI, KII, and KIII.

• Example: Single-edge-notched specimen

I

P aK a f

tW W

displacementP a

gtWE W

a

w

,P load, crack length, geometryI IK K


Fracture criterion for in-plane loading

1. Connection to J and G obtained by using the path independent integral:

2. For pure mode I loading, the crack grows along the line of symmetry and the energy based fracture criterion can be restated in terms of the stress intensity factor

3. Residual strength (load carrying capacity) can be determined for structural applications

4. Stability of structures can be evaluated

2

2 21( ) I IIJ G a K K

E

21I IC

ERK K


Snap-back instability

I

P aK a f

tW W

displacementP a

gtWE W

a

w

,P

1a

2 1a a

Crack is unstable in load control and displacement control


1. For combined modes I and II, we need other criterion (criteria?) that dictates the crack path selection

a. Maximize energy release rate

b. Maximum “hoop” stress

c. Principle of local symmetry:

2. Maximum hoop stress criterion is simplest to use

3. Experimental scatter is large and unable to discriminate between the different criteria

Fracture criterion for mixed mode I + II crack

, 0I IC IIK K K Crack tilting

Crack kinking Principle of Local Symmetry: Goldstein and Salganik, Int J Fract, 1974


Crack path evolution under Mode I + II

Principle of Local Symmetry works very well for this problem

Yang and Ravi-Chandar, J Mech Phys Solids, 2001

Photograph Courtesy of Dov Bahat Ben Gurion University Tectonofractography, Springer


Possible fracture criteria for mixed mode I + III

0

, 0

II

I III

K

f K K

Criterion I: Goldstein and Salganik, Int J Fract, 1974

, 0, 0I IC II IIIK K K K

Criterion II: Lin, Mear and Ravi-Chandar, Int J Fract, 2010

1tan 2

2III

I

K

K

Hull, Int J Fract, 1995 Cooke and Pollard, J Geophy Res, 1996


Mixed mode I + III crack problem

Sommer, Eng Frac Mech, 1970

Below a threshold of KIII/KI, the crack front twists

Above the threshold, crack front fragments cr= 3.3°


Mixed mode I + III crack problem

Knauss, Int J Fract, 1971


Summary and plan

• Energy based method can provide a simple way of analyzing fracture problem (with some residual difficulty regarding the path selection)

• Stress-intensity factor based method provides an effective way of designing fracture critical structures – residual strength diagram

linear elastic fracture mechanics - espci paris · 2017-06-21 · 7 center for mechanics of solids,...

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