A shell fracture framework based on a full discontinuous Galerkin formulation combined with an extrinsic cohesive law

(1)

University of Liège

Aerospace & Mechanical Engineering

A shell fracture framework based on a full discontinuous

Galerkin formulation combined with an extrinsic cohesive law

G. Becker & L. Noels

Computational & Multiscale Mechanics of Materials, ULg

Chemin des Chevreuils 1, B4000 Liège, Belgium

Gauthier.Becker@ulg.ac.be

L.Noels@ulg.ac.be

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Context of cohesive approach

• Very appealing with finite element method

• Ideally

– Consistent

– Easy implementation (//)

• Two classical methods

– Intrinsic

– Extrinsic

(3)

Context of cohesive approach

• Intrinsic methods

– Cohesive law

+ Easy implementation: cohesive element inserted at beginning

– Not consistent: initial slope modifies stiffness structure

(4)

Context of cohesive approach

• Extrinsic methods

– Cohesive law

+ Consistent

(5)

Consistent without mesh modification ?

• Discontinuous Galerkin (DG) formulation

– Discontinuous tests functions at interfaces

– Consistency at interface

(a-1)

(a)

x

(a+1)

Field

(a-1)

+

_(a)

+

x

(a+1)

-Field

(a+1)

+

(a)

-(a-1)

-Continuous Galerkin

Discontinuous Galerkin

No fracture

Fracture

DG terms ensure continuity

Extrinsic cohesive element

Switch when

fracture

crtiterion is

reached

(6)

DG / Extrinsic cohesive law combination

• Application to 3D cases

[Radovitzky, Seagraves, Tupek &

Noels CMAME 2010]

(7)

DG / Extrinsic cohesive law combination

• Application to 3D cases

– Scalable approach

• Extension to thin bodies (beam, plate & shell) ?

[Radovitzky, Seagraves, Tupek &

Noels CMAME 2010]

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Topic

• DG Kirchhoff-Love shell formulation

• Coupling with extrinsic cohesive law (ECL)

• DG/ECL applications

(9)

DG Kirchhoff-Love shell formulation

• Conservation equations

– Linear momentum

– Angular momentum

• Integration on thickness

– Linear momentum

– Angular momentum

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• Expression in term of resultant stresses

– Stress Vector

– Torque Vector

(11)

DG Kirchoff-Love shell formulation

• Thin bodies C

1 _{continuity required}

– Test functions

• New DG interface terms

– Obtained as in literature (consistency, compatibility and

stability)

(a-1)

(a)

x

(a+1)

Field

(a-1)

+

_(a)

+

x

(a+1)

-Field

(a+1)

+

(a)

-(a-1)

-Extension

C

0 /DG formulation

[Noels & Radovitzky, CMAME 2008]

DG formulation

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DG Kirchhoff-Love shell formulation

• Pinched open hemisphere

P= 0 [N]

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• Pinched open hemisphere

DG Kirchhoff-Love shell formulation

P= 400 [N]

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Extrinsic cohesive law for shell

• Main problem: through the thickness crack propagation

– No element on thickness

– No fracture in compression

– Pure Bending

– How to move the neutral axis ?

Tension

Compression

Discontinuity

(15)

Extrinsic cohesive law for shell

• Shell equations are integrated on thickness

• Same for cohesive tensions

– How to combine reduced stress (

N

) and reduced torque (

M

) ?

• Examine beam case first

– Only mode I

t

D

*

N

,

M

N

M

Integration

on thickness

(16)

• Energy released = hG

_c

• Pure tension

• Pure bending

Extrinsic cohesive law for beam

x

z

_n

_xx

_n

xx

D

*

₌

D

x

z

D

*

N

₀

= h

s

_c

D

_c

D

*

M

₀

= h

2 /6

s

_c

D

_c

(17)

• Energy released = hG

_c

• Coupled case

– Obtained by superposition principle

– With the coupling parameter

h

_I

Extrinsic cohesive law for beam

D

*

_{= (1-}

h

I

)

D

x

+

h

I

h/6

D

r

D

*

N

,

M

N

₀

M

₀

x

z

D

_c

(18)

Extrinsic cohesive law for beam

• Coupled case

– Increase u

_z

until fracture

No fracture

Full broken

Transition

?

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Extrinsic cohesive law for beam

• Coupled case

– Geometry effect (no pre-strain)

Stable transition

ΔE

_int

(u

_z

) < hG

_c

Unstable transition

(20)

• Coupled case

– Pre-strain effect

hG

_c

= 22.00

Pre

-st

ra

in

Extrinsic cohesive law for beam

-2e

-5

0. 2e

-5

4e

-5

6e

-5

8e

-5

10e

-5

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• No mode III

– Kirchhoff-Love assumption: out-of-plate shearing is negligible

• Applied beam model on 2 fracture modes

– Mode I : tension and bending

– Mode II: shearing and torsion

• Combination: Camacho & Ortiz model

– Fracture criterion

– Resultant opening

Extrinsic cohesive law for shell

[Camacho & Ortiz Int. J Solid

Structures 1996]

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DG/ ECL Applications

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• Mode I dynamic crack propagation

Time: 0 [μs]

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• Mode I dynamic crack propagation

Time: 12.50 [μs]

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• Mode I dynamic crack propagation

Time: 25.00 [μs]

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• Mode I dynamic crack propagation

Time: 37.50 [μs]

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• Mode I dynamic crack propagation

Time: 50 [μs]

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• Mode I dynamic crack propagation

Time: 62.50 [μs]

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• Mode I dynamic crack propagation

Time: 75.00 [μs]

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• Mode I dynamic crack propagation

Time: 87.5 [μs]

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• Mode I dynamic crack propagation

Time: 100 [μs]

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• Mode I dynamic crack propagation

Time: 112.50 [μs]

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• Mode I dynamic crack propagation

Time: 125.00 [μs]

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• Mode I dynamic crack propagation

Time: 137.50 [μs]

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• Mode I dynamic crack propagation

Time: 150.00 [μs]

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• Mode I dynamic crack propagation

Time: 162.50 [μs]

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• Mode I dynamic crack propagation

Time: 175.00 [μs]

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• Mode I dynamic crack propagation

Time: 187.50 [μs]

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• Mode I dynamic crack propagation

Time: 200.00 [μs]

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• Mode I dynamic crack propagation

– Results

G =

Δ(W

ext

– W

int

)

bhΔa

DG/ ECL Applications

G

_c

(41)

• Blast of pressurized cylinder with an initial crack

Initial

crack

(42)

• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

(43)

• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

(44)

• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

(45)

• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

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• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

(47)

• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

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• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

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• Blast of pressurized cylinder with an initial crack

DG/ ECL Applications

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Conclusions

• DG methods

– Easy combined with extrinsic cohesive method

– Extended to shell

– Appealing for parallelization (ongoing work)

• Cohesive with shell

– Complex modelization of through the thickness crack

– Application on resultant stresses

• Extension to large deformations

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• Mode I dynamic crack propagation

– Result unstructured meshes

– Independent of mesh

• Max relative error (thin mesh worst crack

path)

– 2.7% on maximal force

No dissipation

Spectral radius 0.9

DG/ ECL Applications