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Fracture mechanics

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Why fracture mechanics ?


Fracture mechanics: objectives and generalities

Initiation Propagation FailureSane structure

Where ?When ?

Propagate or not ?

Speed ?Path ?

Does it fail ?What shape ?

What shape ?


Fracture mechanics: objectives and generalities

Initiation Propagation FailureSane structure

Where ?When ?

Propagate or not ?

Speed ?Path ?

Does it fail ?

Fracture mechanicsDamage mechanics


Applications of fracture mechanics

DesignLifetime assessment

Maintenance

OperationJustification

Repair

Brittle fracture Ductile fracture Fatigue propagation


Applications of fracture mechanics

Cracking of a dam

Cracks in UK

AGRs

Cracks in EDF turbines


Outline

Main criteria in fracture mechanics

Linear fracture mechanics in code_aster

Non linear fracture mechanics

References


Outline




References


Basis on LEFM: Vocabulary

FR EN code_aster 2D 3D

Front – Fond Front - Tip FOND_FISS Point Edge

Lèvres Lips LEVRE_SUP

Edge Face_INF

Crack : mater discontinuity


Basis on LEFM: Cracking modes

𝑢𝑥 = 0

𝑢𝑦 ≠ 0

𝑢𝑧 = 0

𝑢𝑥 ≠ 0

𝑢𝑦 = 0

𝑢𝑧 = 0

𝑢𝑥 = 0

𝑢𝑦 = 0

𝑢𝑧 ≠ 0

x

y

z


Basis on LEFM: local axis

Singular stress

θ

rMx

y

z

Global axis Crack local axis

a

M

𝑡

𝑛

𝜎 ~𝑟→0

𝐾𝑖 𝜎∞, 𝑎

𝑟𝑓 𝜃 𝑢 ~

𝑟→0𝐾𝑖 𝜎∞, 𝑎 𝑟 𝑔 𝜃


General cases:

2a

2a b

K depends on:crack geometry

structure geometry

loading conditions

2 examples:

Based on analytical solution,

approximated solution or FEM

calculations

Stress intensity factor K

𝐾𝐼 = 𝜎∞ 𝜋𝑎 cos2𝛼

𝐾𝐼𝐼 = 𝜎∞ 𝜋𝑎 cos𝛼 sin𝛼

𝐾𝐼 = 𝜎∞ 𝜋𝑎 cos𝜋𝑎

𝑏

−12


Codified approaches

Stress intensity factor K

Influence coefficients

t

R

a 𝜎

𝑥

0 𝑎

𝜎 𝑥 = 𝜎0 + 𝜎1𝑥

𝑡+ 𝜎2

𝑥

𝑡

2

+ 𝜎3𝑥

𝑡

3

+ 𝜎4𝑥

𝑡

4

𝐾𝐼 = 𝜋𝑎 𝜎0𝑖0 + 𝜎1𝑖1𝑎

𝑡+ 𝜎2𝑖2

𝑎

𝑡

2

+ 𝜎3𝑖3𝑎

𝑡

3

+ 𝜎4𝑖4𝑎

𝑡

4

Calcul

Polynomial fit

Mode ~𝑟→0

𝑓 𝜎𝑖𝑗 ~𝑟→0

𝑓 𝑢𝑖

𝜎𝜃𝜃 𝑟, 0 2𝜋 𝑟𝐸 2𝜋

8 1 − 𝜈2∙𝑢2𝑟

𝜎𝑟𝜃 𝑟, 0 2𝜋 𝑟𝐸 2𝜋

8 1 − 𝜈2∙𝑢1𝑟

𝜎𝜃𝑧 𝑟, 0 2𝜋 𝑟𝐸 2𝜋

8 1 + 𝜈∙𝑢3𝑟


Stress intensity factors K

IK

IIK

IIIK


Contour integral: Rice

Characterization of stress singularity

Induced from energy conservation

Independent of the considered contour

For a plane cracked solid subjected to

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

With the elastic energy density.

𝐽 =

𝐶1

𝑤𝑒𝑛1 − 𝜎𝑖𝑗𝑛𝑗𝜕𝑢𝑖𝜕𝑥𝑖

𝑑𝑠

𝑤𝑒 = 𝝈: 𝜺

𝑥2𝐶1

𝒏

𝒅𝒔

𝑥1


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 + Δl

Minimum total energy principle:

𝐸𝑡𝑜𝑡 𝑙 = 𝑊 𝑙 + 2𝛾𝑙 𝐸𝑡𝑜𝑡 𝑙 + ∆𝑙 = 𝑊 𝑙 + ∆𝑙 + 2𝛾 𝑙 + ∆𝑙

𝐸𝑡𝑜𝑡 𝑙 + ∆𝑙 < 𝐸𝑡𝑜𝑡 𝑙

𝑊 𝑙 + ∆𝑙 −𝑊 𝑙 < −2𝛾∆𝑙



F

U

G

l

l+Δl

F

U

G

l

l+Δl

𝑊 𝑙 + ∆𝑙 −𝑊 𝑙

𝑊 𝑙 + ∆𝑙 −𝑊 𝑙



Definition of G :

“variation of potential energy per virtual crack advance”

l l + dl

Potential energy

Cracking energy2D 3D

𝐺 = −𝑑𝑊

𝑑𝑙𝐺 = −

𝑑𝑊

𝑑𝐴

𝐺 = −𝑑𝑊

𝑑𝐴


G-theta method

G-θ method

Derivative difficult to compute directly

Γ 𝜃 :𝑥 → 𝑥 + ℎ 𝜃 𝑥 F

F’

𝜃 𝐹 = ℎ 𝑡

Γ0

𝐺𝜃 ∙ 𝑡 𝑑𝑠 = 𝐺 𝜃 = −𝑑𝑊

𝜕𝐴Solution of variational equation:


G-theta method: implementation

In 2D:

In 3D:

𝑅𝑖𝑛𝑓 𝑅𝑠𝑢𝑝0

𝑅𝑖𝑛𝑓𝑅𝑠𝑢𝑝

0

𝑟

𝜃

𝜃0

Discretisation of θ and G along front:

𝛾0 𝑠

𝑠 = 0 𝑠 = 𝑙

Γ0

𝛾1 𝑠

𝑠 = 0

𝑠 = 𝑙

Γ0

𝛾2 𝑠

𝑠 = 0

𝑠 = 𝑙

Γ0

𝜑1 𝑠

𝑠 = 0 𝑠 = 𝑙

Γ0

𝜑2 𝑠𝜑3 𝑠

θ0


Relation between parameters (Irwin)

Linear elasticity

Plane strain, 3D

Plane stress

Plane elasticity (plane strain + plane stress)

𝐺 =1

𝐸𝐾𝐼

2 + 𝐾𝐼𝐼2 +

1 + 𝜈

𝐸𝐾𝐼𝐼𝐼

2

𝐺 =1 − 𝜈

𝐸𝐾𝐼

2+ 𝐾𝐼𝐼2 +

1+ 𝜈

𝐸𝐾𝐼𝐼𝐼

2

𝐺 = 𝐽

𝐾𝐼𝑐 ≈ 1𝑀𝑃𝑎 𝑚






Material criterion

K ’s dimension: 𝑴𝑷𝒂 𝒎

Be careful with FEM units !

!• Concrete

• Hardened Steel

• Aluminium alloy

• Titanium Alloy

• Polymer

Some material criterionPropagation if

• Glass, ceramics

• Aluminium

• Steel

• Pure metals

G’s dimension: J/m² or N/m

𝐾𝐼 ≥ 𝐾𝐼𝑐

𝐺 ≥ 𝐺𝑐 𝑜𝑟 𝐽𝐼𝑐

𝐺𝑐 ≈ 2𝐽.𝑚−2

𝐺𝑐 ≈ 100𝑘𝐽.𝑚−2

𝐺𝑐 ≈ 1𝑀𝐽. 𝑚−2

𝐺𝑐 ≈ 10𝑘𝐽. 𝑚−2


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 : DK weak, slow or non propagation

– Stage B : DK moderate, propagation with a constant velocity

– Stage C : DK high, sudden failure See POST_RUPTURE operator

𝑑𝑎

𝑑𝑁= 𝑐∆𝐾𝑚


Outline




References


Fracture mechanics problem in code_aster

Step 1

Meshing cracked structures

Step 2

Thermo-mechanical computation

Step 3

Crack definition

Step 4

Computation of fracture mechanics parameters

Type of calculation:

Thermo-Elastic (linear or non linear)

Thermo-elastoplastic : See the end of the

presentation

Residual stresses (linear or non linear

elasticity)


Crack definition in code_aster



In 2D

In 3D

OUVERT

FERME



DECOLLE (α<5°)

crack crack

CONFIG_INIT

COLLEE

SYME

No Yes

LEVRE_SUP/INF

LEVRE_SUP

LEVRE_INF

NORMALE


Displacement Jump Extrapolation Method (1)

0,0E+00

5,0E-08

1,0E-07

1,5E-07

2,0E-07

2,5E-07

3,0E-07

3,5E-07

4,0E-07

4,5E-07

5,0E-07

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

Curvilinear co-ordinate

Dis

pla

ce

me

nt ju

mp

Computed displacement jump

function K.sqrt(r)

Extraction of node

displacements along the crack

front (normal direction)

ABSC_CURV_MAXI

Analytical model (𝑟 → 0):

Operator POST_K1_K2_K3

ABSC_CURV_MAXI

n

u2

N

𝐾1 =𝐸 2𝜋

8 1− 𝜈2∙𝑢2

𝑟

𝑢2 ~ 𝑟



3 methods to extrapolate the displacement:

Method 1

One value of K for each

consecutive node couple

With quarter-node

elements

Without quarter-

node elements

𝑢22

𝑟=

64 1− 𝜈2 2

2𝜋𝐸2 𝐾12

Maximal value



Maximal slope

Method 2:

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

Minimisation by least

square error of J(k):

One value of K

Without quarter-node elements

With quarter-node elements

𝑢22 = 𝑟

64 1 − 𝜈2 2

2𝜋𝐸2 𝐾12

𝐽 𝑘 =1

2

0

max _𝑎𝑏𝑠𝑐

𝑈 𝑟 − 𝑘 𝑟 2𝑑𝑟


Usage of the POST_K1_K2_K3 operator

FEM

X-FEM

From mechanical calculation

Use different material from the one in Result !

Maximal distance of calculation

Type of mesh of the crack

REGLE LIBRE


Limited to plane or quasi-planar cracks (possibility to define only one normal)

Choice of ABSC_CURV_MAXI: between 3 to 5 elements

Precision of computation: error < 10 % for validation tests

Precision is better if crack mesh is REGLE

POST_K1_K2_K3: advices

Verifications:

Compare with different ABSC_CURV_MAXI

Check errK1, errK2 and errK3 < 1%


Operator CALC_G


Operator CALC_G

Example: NB_POINT_FOND=7

R_INF

R_SUP

FEM

X-FEM

If crack has several fronts

several CALC_G

LAGRANGE only


Operator CALC_G

Result from mechanical calculation


Operator CALC_GOption = CALC_G

•In 3D/2D plane the local value G(s) is in J/m²

•in 2D-axisymetric, G is the energy by unit of

radian. In order to obtain a local value of G, we

need to divide by its radius R.

Option = CALC_K_G

•Also compute stress intensity factors

•Use of mathematical properties of G to separate

contributions of K1, K2 and K3

Option = CALC_G_GLOB

DO NOT USE !

𝐺 =1

𝑅Γ 𝜃


Operator CALC_G

Smoothing options in 3D

LISSAGE_THETA=LEGENDRE

LISSAGE_G=LEGENDRE

DEGRE=NLISSAGE_THETA=LAGRANGE

LISSAGE_G=LAGRANGE

THETA: [NB_POINT_FOND=N]

ORDefault


Operator CALC_G

Choice of smoothing in 3D : need to use different smoothing methods and

compare the obtained results !

Energy release rate for an elliptical crack (test case sslv154a)

Quadratic mesh with Barsoum elements

LAGRANGE: no smoothing,

oscillations can occur

LAGRANGE with

NB_POINT_FOND=20 (33):

decrease of oscillations

LEGENDRE: smooth results

•Results at the extremities of the crack

front should be used with care

•Good if G is polynomial

Reference (analytical solution)


Operator CALC_G

To calculate only at given steps


Operator CALC_G: general advice

R_INF > 0 (imprecise computational results at crack front) ~ 2 elements

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

Use OPTION=‘CALC_K_G’

If LISSAGE ‘LAGRANGE’:

Verifications:

Compare with different R_INF and R_SUP

Compare between LISSAGE ‘LEGENDRE’ and ‘LAGRANGE’

Compare different values of NB_POINT_FOND

If N front nodes > 25 use NB_POINT_FOND

NB_POINT_FOND = N/5 or N/10

5 ≤ NB_POINT_FOND ≤ 50


General advice

Computation on an unstructured meshComputation on a structured mesh

Use a mesh with a tore around the crack front

Not mandatory

Results will be more regular if the R_SUP ≤ radius of the tore

Use BlocFissure plugin in salome_meca to insert a crack in a mesh with a tore

POST_K1_K2_K3

CALC_Gadvice

General advices for meshed cracks:

Element type : prefer quadratic elements with Barsoum


General advice

crack crack crack

Linear Quadratic Quadratic + Quarter nodes

<< <<

See MODI_MAILLAGE with

OPTION=‘NŒUD_QUART’

3D if free or structured mesh

3D if structured meshadvice


Outline




References

𝛼 = 1 𝑟𝑦 ≤ 0,05 𝑡 − 𝑎

𝛼 = 1+ 0,15𝑟𝑦 − 0,05 𝑡 − 𝑎

0,035 𝑡 − 𝑎

2 0,05 𝑡 − 𝑎 < 𝑟𝑦 ≤ 0,12 𝑡 − 𝑎

𝛼 = 1,6 𝑟𝑦 > 0,12 𝑡 − 𝑎


Accounting for confined plasticity by plastic correction (RCC-M ZG5110 appendix)

Replace crack length a by a virtual crack a + ry (Irwin’s approach) with:

Compute corrected stress intensity factors

Non Linear fracture mechanics

𝑟𝑦 =1

6𝜋

𝐾𝐼

𝜎𝑠

2

𝐾𝑐𝑝 = 𝛼𝐾𝐼

𝑎 + 𝑟𝑦𝑎

𝜎𝑠: yield stress

𝑟𝑦: plastic zone size

With:


Accounting for confined plasticity by 3D approach:

Non Linear fracture mechanics

STAT_NON_LINE CALC_G ‘RELATION’

ELAS_VMIS_LINE ELAS_VMIS_LINE

ELAS_VMIS_PUIS ELAS_VMIS_PUIS

ELAS_VMIS_TRAC ELAS_VMIS_TRAC

VMIS_ISOT_LINE ELAS_VMIS_LINE

VMIS_ISOT_PUIS ELAS_VMIS_PUIS

VMIS_ISOT_TRAC ELAS_VMIS_TRAC!

Loading must be radial and monotonous

!See DERA_ELGA in CALC_CHAMP

Compare VMIS_ISOT_ and

ELAS_VMIS_ in STAT_NON_LINE

Compare by activating or not CALCUL_CONTRAINTE=‘NON’ in CALC_G


Outline




References


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], 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] and non-linear thermo-elasticity

[R7.02.03]

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

Other references :Plasticité et Rupture - Jean-Jacques Marigo course

Formation ITECH – Mécanique de la rupture


End of presentation

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Or feeling happy to have read such a clear tutorial?

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dedicated thread.

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Appendix for TP forma05b

CALC_G :

Rinf ≈ 2 elements

Rsup ≈ 5 elements

POST_K1_K2_K3 :

ABSC_CURV_MAXI ≈ 5 elements

ABSC_CURV_MAXI

R_INF

R_SUP

Best results with

quadratic and Barsoum

Quadratic

crack crack

Barsoum

K and G independent of parameters

CALC_G or POST_K1_K2_K3 in 2D ?

Similar results

Both very accurate


Appendix for TP forma07a

LISSAGE in CALC_GDEFI_FOND_FISS in 3D

CALC_G or POST_K1_K2_K3 ?

Some advice

POST_K1_K2_K3 :

3D if structured mesh

CALC_G :

3D if free mesh

Use OPTION=‘CALC_K_G’

Define crack front with GROUP_MA

FOND_FISS=_F( GROUP_MA='LFF',

GROUP_NO_ORIG='NFF1',

GROUP_NO_EXTR='NFF2'),

Define LEVRE_SUP and LEVRE_INF

(optional with CALC_G)

Accuracy :

LAGRANGE+NB_POINT_FONDBetter than

LEGENDREBetter than

LAGRANGE

Computational time :

LEGENDREFaster than

LAGRANGE+NB_POINT_FONDFaster than

LAGRANGE

Oscillations of G and K :

LEGENDREBetter than

LAGRANGE+NB_POINT_FONDBetter than

LAGRANGE

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