Are damage gradient Are damage gradient models applicable up models applicable up to ultimate fracture ?to ultimate fracture ?

Eric LorentzV. Godard

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Motivation : safety related to electricity generationMotivation : safety related to electricity generation

FOCUS

Crack propagationin concrete structures

(characteristic size : several meters)

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OutlineOutline

1. Continuum damage and crack propagation

2. A constitutive law designed for robustness and efficiency

3. Numerical applications

Continuum damage

and crack propagation

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The damage law : assumptions and commitmentsThe damage law : assumptions and commitments

Modelling assumptions for sake of simplicity

• No crack closure• Damage isotropy • No distinction between traction and compression• No irreversible strain

Complying with macroscopic quantities of interest

• Elastic properties• Fracture energy• Critical stress

Adjustable parameters

• « crack thickness » compatible with the structure size

• Softening part of the law so as to gain desirable properties

3 MPac

2100 J/mfG 30 GPa 0 2; .E

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Localisation control through gradient damage laws Localisation control through gradient damage laws

Local constitutive laws lead to spurious damage localisation

Introduction of the damage gradient into the constitutive law

• Usual meaning of the stress field• Thermodynamical framework

• Limited intrusion in a finite element code• Compatibility with usual solution algorithms

• 2 additional unknowns / element vertex• Availability of a symmetrical tangent matrix

Dependenceon mesh size

Dependence onmesh orientation

A non local law is necessary because of the high gradients of macroscopic fields

Bourdin et al., 2000Dimitrijevic & Hackl, 2008

Liebe et al., 2001Lorentz & Andrieux, 1999

Fremond & Nedjar, 1996

Pijaudier-Cabot & Burlion, 1996

Benallal & Marigo, 2007

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Description of the gradient constitutive lawDescription of the gradient constitutive law

State variables

Strain field

Damage field

ε

a

Parameters

Hooke’s tensor Stiffness function

Dissipated energy Nonlocal coefficient

E A a

a c

21, w

2Aa a a k c aa ε ε

Thermodynamical potentials Generalised Standard Material

0 1a

a k a 0a

1w

2ε ε : E : εElastic strain energy

Helmholtz’ free energy

Dissipation potential

Pointwise interpretation

A aσ E : ε

2f , A wa a ca a ε

f 0 ; 0 ; f 0a a

Stress

Yield function

Consistency

Boundary conditions

0a

0a n

0a ν

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QuestionsQuestions

How far can I push the model ?

• Is it able to describe a total loss of stiffness ?• Is it able to go up to ultimate fracture ?• Is it necessary to introduce a transition to a real crack ?

Related expected qualities

• Robustness Does the model always provide a result ?• Reliability What confidence can I grant to the result ?• Efficiency How long have I been staring at my

computer ?

A constitutive law designed

for robustness and efficiency

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Application in case of rectilinear and stable propagationApplication in case of rectilinear and stable propagation

• The damage model is consistent with coarser formulations : Griffith, cohesive zone model

BUT

• The computation exhibits a lot of snap-backs which slow down the convergence

• The number of iterations increase dramatically as soon as points are broken C

ZM

= 5

mn

CD

M =

5 h

displacement Propagation length

forc

e

Ext

rern

al w

ork

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Possible explanations for the drawbacksPossible explanations for the drawbacks

Spurious snap-backs because the strain remains bounded ?

Excessive iterations because of loading / unloading issues ?

Only a single point reaches a = 1 Wherever else, the strain is zero

A ( ) 0a x x

In order to avoid a snap-backthe strain should be at least a Dirach

displ.

stress.

Without enforcing damage increaseWould the band width reduce ?

x

damage

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Tuning the constitutive law to remedy the drawbacksTuning the constitutive law to remedy the drawbacks

Assumption

• The shape of the local softening response is not significant• But the peak stress and the fracture energy are prescribed

Exploration among many types of constitutive laws

• Design the stiffness function A(a) and the dissipated energy (a)• Submitted to monotonicity and convexity constraints

Validation

• Closed-form solution on a 1D problem• The constitutive law is retained if :

(1) Increase of the band width(2) No snap-back at the scale of the localisation band

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(1) Increase of the band width(1) Increase of the band width

Some constitutive laws used in the litterature

• Liebe, Steinmann, Benallal (2001)• Bourdin, Francfort, Marigo (2008),• Lorentz, Benallal (2005)

Power-law constitutive relations

1A 0 ; 1

1

ma

a ma

a k a

A aσ E : ε

2f , A wa a ca a ε

Non

e O

KO

K f

or s

ome

par

amet

ers

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(2) No snap-back at the scale of the localisation band(2) No snap-back at the scale of the localisation band

Observations

• Snap-back for m < 2• Finite opening for m = 2• Asymptotic fracture for m > 2

Average strain

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The selected constitutive lawThe selected constitutive law

Quadratic constitutive laws

Closed-form expression for the identification process

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1A 2.8

1

aa

a

ka a

A aσ E : ε

2 elasticity parameters and 3 damage parameters

3

4fk

G

D 21

2c k D 2

31

4f

c

G E

D

Peak stress

Fracture energy

Final band width

c

fG

2D

2f , A wa a ca a ε

Numerical applications

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Perforated platePerforated plate

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L-shaped panelL-shaped panel

400 000 dof25 h CPU

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ConclusionConclusion

Continuum damage is adapted to predict crack propagation

• Nonlocal formulation : damage gradient• Numerical demonstration on concrete structures

Robustness, reliability and reasonable efficiency are achieved

• Appropriate choice of the constitutive law• Complying with a critical stress and a fracture energy• Up to ultimate fracture• Consistent with coarser models (Griffith, cohesive zone models)

A future transition to discontinuous models what for ?

• Easier access to crack opening (leakage)• Modelling of large relative motions between crack lips

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Preview : mesh adaptationPreview : mesh adaptation

rigid inclusion( u = v = 0 )

prescribed v

Final damaged zone Final mesh

Salomé_Méca (Code_Aster, Homard)

Thank you

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Why using continuum damage for crack path prediction ?Why using continuum damage for crack path prediction ?

• Following the element edges ? Initial stiffness CPU cost Approximating a curve with fixed segments

• Ensuring crack path continuity In order to compute the correct dissipation Difficult to ensure step by step 3D continuity

• Crack orientation criterion Possibly inacurate in 2D (fixed crack) Theoretically questioned in 3D

Mesh dependency, Feyel (2005)

Jirasek & Zimmermann (2005)

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Solution algorithmSolution algorithm

Energetic formulation

non linearnon convexconstraints

2E , ,

2

, A w I

ca a a d

a a a a

u ε

ε ε 1na a

Decomposition – coordination

2 2,

2 2, ,,

cb

rb a b aa d bL a

u

u,a

b min min max min , , ,

baL ba

uu

Newton Closed-form

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Trapezium specimenTrapezium specimen