14 Fracture Mechanics

Code_Aster, Salome-Meca course materialGNU FDL licence (http://www.gnu.org/copyleft/fdl.html)

Fracture mechanics

2 - Code_Aster and Salome-Meca course material GNU FDL Licence

Outline

Introduction to fracture mechanicsObjectivesCrack vocabulary

Main criteria in fracture mechanics

Linear fracture mechanics in Code_AsterComputation of K by Displacement Jump Extrapolation methodComputation of G by G-theta method

Accounting for plasticity : limit of classical methods

References


Fracture mechanics: objectives and generalities

Objectives of fracture mechanics: determine the speed of propagation of an existing crack and its shape change.Under given loading conditions and boundary conditions, is the crack able to propagate? if yes, at what propagation rate?

The prediction of the crack initiation is not a focus of fracture mechanics (for this application, damage mechanics may be more appropriate)

Applications:Design (computation of fatigue life)Safety (for existing defaults)


Cracking in aeronautical industry

Ductile failure in fatigue of fuselage shells

Contribution of fracture mechanics: better estimation of service life of structures


Cracking in civil engineering

Cracking on a surface of a dam

Contribution of fracture mechanics: assessment of crack and repairing


Vocabulary for a mechanical problem with crack

Crack : macroscopic geometrical discontinuities of matter

Crack front : zone where matter sees its continuity (dimension N-2)

Crack faces : parts of crack on which discontinuity occurs (dimension N-1)

In 2 dimensions In 3 dimensions

Superior crack face

Inferior crack face

Crack front (or crack tip)

Cube with a circular crack

Crack face

Crack front


Crack in elastic solids

3 cracking modes

[ ][ ][ ][ ][ ][ ]

=≠=

0

0

0

z

y

x

u

u

u [ ][ ][ ][ ][ ][ ]

==≠

0

0

0

z

y

x

u

u

u [ ][ ][ ][ ][ ][ ]

≠==

0

0

0

z

y

x

u

u

u

x

y

z

Singular stress (crack = geometrical singularity)

( ) ( )[ ]

( ) ( )[ ]

∞

→

∞

→

θ

θ

graK

fr

aK

ir

i

r

,

,

~

~

0

0

σu

σσ

with polar coordinates

With respect to crack front,

and the crack size. Von-Mises stress at the

crack front

r

x

y

q( )θ,r

a


Expressions of displacement and stress fields

2D-plane

2D-antiplane

Displacement field:

Stress field:

Displacement field: ( )4(1 )sin

22z III

ru K

E

ν θπ

+=

sin22

cos22

IIIxz

IIIyz

K

r

K

r

θσπ

θσπ

= −

=

Stress field:


Criteria for crack propagation in LEFM

Stress intensity factors (Westergaard): local criterion

Contour integral (Rice): global criterion

Energy release rate (Griffith): global criterion

Relation between parameters

Aside: Fatigue (Paris’s law)


Stress intensity factors K

K ’s dimension: MPa√√√√m

Characterisation of stress intensity factors

( )( ) ( ) [ ][ ]

−==

→→ 2200

2

18lim20,lim u

r

ErrK

ryy

rI

πν

πσ

( )( ) ( ) [ ][ ]

−==

→→ 1200

2

18lim20,lim u

r

ErrK

rxy

rII

πν

πσ

( )( ) ( ) [ ][ ]

+==

→→ 300

2

18lim20,lim u

r

ErrK

ryz

rIII

πν

πσ


R

t

Stress intensity factor K: some analytical solutions

K depends on:- crack geometry- structure geometry- loading conditions

Codified approaches: RCC-M and RSE-M for pipesExample: semi-elliptical crack with a/b ratio = 0.3 in a pipe with R/t ratio = 10

with

2 classical examples

2a

α ∞σααπσ

απσsincos

cos2

aK

aK

II

I∞

∞

==

2a b∞σ 2

1

cos−

∞

=b

aaK I

ππσ

+

+

+

+=4

44

3

33

2

221100 t

ai

t

ai

t

ai

t

aiiaK I σσσσσπ

( )4

4

3

3

2

210

+

+

+

+=t

x

t

x

t

x

t

xx σσσσσσ


Contour integral: Rice

Characterization of stress singularityInduced from energy conservationIndependent of the considered contour

For a plane cracked solid subjected to

a mixed-mode load (modes I et II):

With the elastic energy density.

1x

2xn

ds1C

∫

∂∂−=

1

ds1

1C

ijije x

unnwJ σ

εσ :=ew


Energy release rate: G (Griffith)

Griffith’s hypothesisCracking energy is proportional to separated surface (material properties…)Total energy = Potential energy + Cracking energy

Minimum total energy principle

2D example

l l + dl ?

( ) ( ) ( ) ( ) ( )dlldllPdllEllPlE tottot +++=++= γγ 2,2

( ) ( ) ( ) ( ) γ2<−+⇔<+dl

lPdllPlEdllE tottot

γ2>∂∂−=

l

PG

Definition of G : variation of potential energy per (virtual) crack advance

Minimum total energy principle:


Generalization and illustration: G (Griffith)

Generalisation:

Illustration:

A

PG

∂∂−=

Potential energy

Cracking energy

F

UPrescribed load

G

F

UPrescribed displacement

GG’s dimension: J/m² or N/m


Relation between parameters (Irwin)

Linear elasticity

In Code_AsterComputation of K

Computation of G

Usual values (limit criterion)

( )( ) 222

2

222

11

11

IIIIII

IIIIII

KE

KKE

G

KE

KKE

G

νν

ν

+++−=

+++=

Plane strain, 3D

Plane stress

JG = Plane elasticity (plane strain + plane stress)

Propagation if

≥≥

γ2G

KK Ici

Aluminium alloy

Titanium Alloy

Hardened Steel

Polymer

Wood

Concrete

mMPaK Ic 30≈mMPaK Ic 100≈




Aside: fatigue’s law (Paris)

Principle of fatigue:Crack propagation by repetition of a weak load

Paris’ fatigue propagation law

(c, m material parameters)

– Stage A : ∆K weak, slow or non propagation

– Stage B : ∆K moderate, propagation with a constant velocity– Stage C : ∆K high, sudden failure

mKcdNda ∆= .


Cube containing circular crack

Regular mesh near crack front Crack face

Crack front

Study of a fracture mechanics problem in Code_Aster

Mandatory steps to perform crack analysis:Step 1: Meshing cracked structures (except for X-FEM method)

Step 2: Thermo-mechanical computationStep 3: Post-processing : computation of fracture mechanics parameters

Possible Thermo-mechanical computationsThermo-Elastic (linear or non linear)

Residual stresses (linear or non linear elasticity)Thermo-elastoplastic : need to use specific tools of crack analysis


FOND = DEFI_FOND_FISS (MAILLAGE = MA,FOND_FISS =_F( GROUP_NO = … / GROUP_MA =…) ,/ CONFIG_INIT =/’COLLEE’ /’DECOLLEE’/ LEVRE_SUP = _F(GROUP_MA = …),

LEVRE_INF = _F(GROUP_MA = …), / NORMALE = (x, y, z),

);

Crack definition in Code_Aster

General case :

Crack front definition

Crack surface definition

Superior face

Inferior face

Crack front

nr Warning:

- 3D front (orientation)- notch

n

n

p

pnΓprrr

∧=

2D/3D 3D

If CONFIG_INIT = ‘DECOLLEE’


Computation of K and G in Code_Aster

Two possibilities in Code_Aster: K or G (elasticity)

Computation of stress intensity factors K: operator POST_K1_K2_K3

: easy to use, relatively precise : quite sensitive to mesh quality near the front, only quasi-planar cracks

Computation of energy release rate: operator CALC_G

: theoretically more precise and less mesh sensitive

: regularity of results along crack front in 3D


Usage of the POST_K1_K2_K3operatorTABL_K = POST_K1_K2_K3 (

MODELISATION = "3D","AXIS","D_PLAN" or "C_PLAN",

/FOND_FISS = FOND,

/FISSURE =FISS

MATER = …,

RESULTAT ( or TABL_DEPL_SUP / TABL_DEPL_INF)

ABSC_CURV_MAXI= …,

)

Crack front (pre-defined with DEFI_FOND_FISS)

Model definition

Results of mechanics computation

Maximal distance from the crack front for extrapolation

X-FEM crack (pre-defined with DEFI_FISS_XFEM)


Displacement Jump Extrapolation Method (1)

0,0E+005,0E-08

1,0E-071,5E-072,0E-072,5E-07

3,0E-073,5E-074,0E-074,5E-07

5,0E-07

0E+00 1E-05 2E-05 3E-05 4E-05 5E-05 6E-05

Curvilinear co-ordinate

Dis

plac

emen

t jum

p

Computed displacement jump

function K.sqrt(r)

Extraction of node displacements along the crack front (normal direction)

ABSC_CURV_MAXI

Analytical model:

with:

Operator POST_K1_K2_K3

ABSC_CURV_MAXI ( ) [ ][ ]

−=

→ 220

2

18lim u

r

EK

rI

πν[ ][ ]nu2

[ ][ ] ( )N.LEVRE_INFLEVRE_SUP2 UU −=u

N



3 methods to extrapolate the displacement:

Method 1

Prolongation until r = 0 for the right segments

One value of K for each consecutive node couple

[ ][ ]r

u 2

With quarter-node elements

Without quarter-node elements



Slope of the line One value of K of each node of crack front

Method 2: [ ][ ]2u

Printed results:- in a table (resu file): only the max values of method 1,- in a table (resu file): an estimation of the relative difference between the 3 methods,- in the mess file (if INFO=2): computing details

Method 3

( )∫ −=dmax

0

2)]([

2

1)( drrkrUkJ

Minimisation by least square error of J(k):

One value of K

Without quarter-node elementsWith quarter-node elements


Some advices about the usage of POST_K1_K2_K3

Remarks and advices: Operator limited to plane or quasi-planar cracks (possibility to define only one normal)Choice of ABSC_CURV_MAXI: in general so that the extrapolation is made on 3 to 5

elements

Interesting verification: the relative error should be small enoughPrecision of computation: error < 10 % for validation tests;

precision is tremendously increased by using quadratic ¼ node elements (Barsoum elements, operator MODI_MAILLAGE).

Mesh type: free or structured ? If possible, use structured (one extrapolation avoided, better precision a priori) ;

Computation on an unstructured meshComputation on a structured mesh


Computation of K and G in Code_Aster

Two possibilities in Code_Aster: K or G (elasticity)

Computation of stress intensity factors K: operator POST_K1_K2_K3

: easy to use, relatively precise : quite sensitive to mesh quality near the front, only quasi-planar crack

Computation of energy release rate : operator CALC_G

: theoretically more precise and less mesh sensitive

: regularity of results along crack front in 3D


Linear fracture mechanics in Code_Aster:G-theta method

Operator CALC_G

G-theta method:Lagrangian derivation of the global energy of the system

Properties: G, local energy release rate, is solution of the following variational equation

Derivative difficult to compute directly

( ):F M M Mη ηθ→ + Family of transformations from reference configuration Represent virtual crack propagation

A

PG

∂∂−=

( ) ( )( )0

0

dW uG m ds

dη

ηθ θ

ηΓ=

⋅ = Γ = −∫

crack front normal to front

Ω∂∀ totangentθ


G-theta method: introduction of the theta field

0

θ

θ 0

Rinf

n

||||θ |||| =0

||||θ |||| =||||θ0 ||||

Rinf Rsup Rsup

Remark: computation is made between Rinf and Rsup

Geometrical definition of theta field

Basics of test functions for theta field

Spatial discretization of G

, 1,...,i i PΘ = ∈Θ =% θθθθ

In 2D:

In 3D:


G-theta method: 3D case

( ) ( ) ( ) ( ) Θ∈∀Γ=∫Γ

θθmθ ,.

0

dssssG

( ) ( ) ( ) [ ]PidssspG iiN

jjj ,1,.

0 1

∈∀Γ=

∫ ∑Γ =

θmθ

[ ]

( ) ( )

( )

Γ=

=

∈=

∫

∑

Γ

=

ii

ijij

i

N

jjij

b

dssmspa

PibGa

θ

θ

0

.with

,11

Two families of smoothing:utilisation of LEGENDRE polynomials with degree from 0 to 7

utilisation of shape functions of elements of crack front: ‘LAGRANGE’

Linear Quadratic


TAB_G = CALC_G (

RESULTAT = resu,THETA = _F( FOND_FISS = FOND,

(or FISSURE = ...),R_INF = ri,R_SUP = rs,

),

SYME_CHAR = "SYME" or "SANS "

EXCIT =_F( CHARGE = charmeca, FONC_MULT = ff,)

LISSAGE = _F(LISSAGE_THETA = …LISSAGE_G = …DEGRE = 0 7

Operator CALC_G

Loading may influence G.Advice: do not use this keyword, by default all loads are taken into account

Results from mechanical computation

Definition of theta field

Smoothing options in 3D

Non mandatory


Operator CALC_G: computational options

OPTION =

/ ‘CALC_G’

/ ‘CALC_G_GLOB’

/ ‘CALC_K_G’

Computation of G in 2D & 3D (local)

Computation of G in 3D (global)

Computation of K in 2D & 3D (local)

Usual options

mandatory keyword: FOND_FISS / FISSURE


Operator CALC_G: local vs global values

Global values: options CALC_G_GLOBand G_MAX_GLOB, …

Γ o

Local values: options CALC_G, CALC_K_G, G_MAX

The value G(s) printed in the result table corresponds to local value in J/m²

To induce a mean local value (J/m²), we need to divide by the crack length l :

( ) ( ) absc. curviligne dem 1 , os s s⋅ = ∀ Γθθθθ

The global G (J/m) printed in the result table corresponds to an uniform crack propagation.

G

Particular case: in 2D-axisymetric, the ‘local’ G (option CALC_G) corresponds to the energy by unit of radian. In order to obtain a local value of G, we need to divide by its radius R.

( ) ( ) ( ) ( )∫Γ

Θ∈∀=Γ0

,. θθθ dssmssG

( ) ( )θΓ== ∫Γ

ldssG

lG

11

0

( )θΓ=R

G1

( )θΓ


Operator CALC_G: advice

Remarks and advice:Theoretically, results do not depend on the contour of theta. Advise to choose

- Rinf different from 0 (imprecise computational results at crack front)

- Rsup ‘not too large’ (for example 5 or 6 elements)

- if possible, verify the independency for different contours

- compare G and G_Irwin in result table

Remark: in practice, we use mesh with a tore around the crack front …No obligation to use a tore in the mesh around the crack front

If a tore is meshed around the crack front, results will be more regular if the radii of the theta field correspond to the radius of the tore


Operator CALC_G: advice for 3D

Choice of smoothing in 3D : need to use different smoothing methods and compare the obtained results !

Energy release rate for an elliptical crack (relative quite coarse mesh)

0,0E+00

5,0E+03

1,0E+04

1,5E+04

2,0E+04

2,5E+04

3,0E+04

0 0,005 0,01 0,015 0,02 0,025 0,03 0,035

Curvilinear co-ordinate

G

LAGRANGE: no smoothing oscillations can occur

LAGRANGE_REGU: decrease of oscillations if nodes along crack

front are dense and regularly spaced

LEGENDRE: smooth results, but results from the nodes at the

extremities of the crack front should be used with care


G-theta method: computation of stress intensity factors

Definition: g(u,v) symmetric bilinear form associated with G(u)

( )1( , ) ( ) ( )

4g u v G u v G u v= + − − ( , ) ( )g u u G u=

Properties: singular displacements are orthogonal two to two referring to the scalar product defined by g(u,v)

( ) ( ) ( ) 0,,, === SIII

SII

SIII

SI

SII

SI uuguuguug

( ) ( ) ( ) 0,,, === SIII

RSII

RSI

R uuguuguug

( ) ( ), ,R S S S R S S SI I II II III III I I II II III IIIG g u u g u K u K u K u u K u K u K u= = + + + + + +

2 2 2( , ) ( , ) ( , ) ( , )S S S S S SI I I II II II III III IIIG g u u K g u u K g u u K g u u= = + +

( ) ( )2,

1S

I I

EK g u u

ν=

−

( ) ( )2,

1S

II II

EK g u u

ν=

−

( )2 , SIII IIIK g u uµ=

(so )

Demonstration: Rice’s integral (=G in elasticity) and symmetric properties of singular functions

R S S SI I II II III IIIu u K u K u K u= + + +


Additional physical zone: Between free surface (crack) and sound zone, interaction zone.

Particular behaviour law: Assuming the existence of a surface energy depending on displacement jump Cohesive law = relation between and force vector of separation (stress)

Parameters of cohesive law: critical surface energy and critical stress

Particular finite elements: Classical joint elements, linear, with regularisation: PLAN_JOINT, AXIS_JOINT , 3D_JOINT ; an additional parameter PENA_ADHERENCE

Discontinuous elements including, linear, without regularisation: PLAN_ELDI, AXIS_ELDI

Interface elements, quadratic, mixed formulation: PLAN_INTERFACE, AXIS_INTERFACE, 3D_INTERFACE; (an additional parameter PENA_LAGR)

Cohesive zone model (CZM): generalities

δ

r

δ

σ rr

∂Ψ∂=

Ψ δ

r

cG cσ


Cohesive zone model (CZM): possible laws

CZM_LIN_REG

+δr

σ

Cσ

C

CG

σ2

+δr

σ

Cσ

Keyword RUPT_FRAGof DEFI_MATERIAU:

MAT = DEFI_MATERIAU (RUPT_FRAG =_F( GC=…

SIGM_C =…PENA_ADHERENCE =…PENA_LAGR =…

),);

Principal parameters

For joint elements

For interface elements

CZM_EXP_REG

0κ κ 0κ κ

+δr

σ

Cσ

C

CG

σ2

C

CG

σ2κ nδ

nσ

Cσ CZM_OUV_MIX CZM_TAC_MIX


Summary for brittle behaviour:

Specificities for some laws: CZM_OUV_MIXpure mode ICZM_TAC_MIXboth sides of crack must be meshed

Orientation: Cohesive force direction: all element; MODI_MAILLAGE (ORIE_FISSURE =_F(GROUP_MA= ))

Local basis of crack: interface elements;AFFE_CARA_ELEM(MASSIF=_F(ANGL_REP= ))

STAT_NON_LINE=(CARA_ELEM= )

Cohesive zone model (CZM): some rules

Element JOINT DISCONTINUITE INTERFACE

Type Linear Linear Quadratic

Thickness Null or non null Non null Null or non null

Material Linear or non Linear Linear or non

Possible lawsCZM_LIN_REG

CZM_EXP_REGCZM_EXP

CZM_OUV_MIX

CZM_TAC_MIX

Regularisation PENA_ADHERENCE NonePENA_LAGR

(optional)

xr

zry

r

Xr

Yr

Zr


Accounting for confined plasticity by plastic correctionwe replace a crack of length a by a virtual crack of length a + ry, where ry is the plastic zone size (Irwin’s approach)

we compute stress intensity factors by analytical formula (RCC-M approach)

Simple accounting for plasticityIf the loading is radial and monotone, we can compute G for a crack with non-linear elastic behaviour (ELAS_VMIS_TRACor ELAS_VMIS_LINE under COMP_ELAS)

Advanced models for complex situations (see doc Aster): Research at EDF R&D

GTP approach for extended plasticity (addition of a plastic term in computation of G)Gp approach (extended energetic approach to account for plasticity for brittle fracture)Damage law ENDO_FRAGILEfor brittle fracture and ROUSSELIERductile fractureSpecific cohesive law CZM_TRA_MIX for ductile fracture

Non Linear fracture mechanics

21

6I

ys

Kr

π σ

=

ycp I

a rK K

aα

+=

α coefficient dependent on ratio crack length / pipe thickness


Code_Aster references

General user documentationApplication domains of operators in fracture mechanics of Code_Aster and advices for users [U2.05.01] Notice for utilisation of cohesive zone models [U2.05.07]Realisation for a computation of prediction for cleavage fracture [U2.05.08]

Documentation of operatorsOperators DEFI_FOND_FISS [U4.82.01], DEFI_FISS_XFEM [U4.82.08], CALC_G[U4.82.03] et

POST_K1_K2_K3[U4.82.05]

Reference documentation Computation of stress intensity factors by Displacement Jump Extrapolation Method [R7.02.08]Computation of coefficients of stress intensity in plane linear thermoelasticity [R7.02.05]Energy release rate in linear thermo-elasticity [R7.02.01]Energy release rate in non-linear thermo-elasticity [R7.02.03]Energy release rate in non-linear thermo-elasticity-plasticity: GTP approach [R7.02.07]

Elastic energy release rate en thermo-elasticity-plasticity by Gp approach [R7.02.16]


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14 Fracture Mechanics