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COMSOL CONFERENCE MILAN 2009 D i C kP ti i OCTOBER 14-16 Dynamic Crack Propagation in Fiber Reinforced Composites P. Lonetti, C. Caruso , A. Manna Presented at the COMSOL Conference 2009 Milan

LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

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Page 1: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

COMSOL CONFERENCE MILAN 2009

D i C k P ti i

OCTOBER 14-16

Dynamic Crack Propagation in Fiber Reinforced Compositesp

P. Lonetti, C. Caruso, A. Manna

Presented at the COMSOL Conference 2009 Milan

Page 2: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DYNAMIC FRACTURE MECHANICSINDEX:

INTRODUCTION

MOTIVATIONS

ALE MODEL

MONOLITIC MATERIALS

FORMULATION

FE MODEL

RESULTS

Crack Branching phenomena

Crack speeds are limited

U k th f th k

COMPOSITE STRUCTURES

CONCLUSIONS

Ravi-Chandar and Knauss, Int J Fract, 1984

Unknown path of the crack

High crack speedWeak plane

(Rosakis, A.J., “Intersonic shear cracks and fault ruptures propagation”, Advances in Physics, 2002)

Crack constrained along the interfaces

Page 3: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DYNAMIC CRACK GROWTH MODELING

Cohesive modeling

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODELInterface elements are introduced at the crack region

Damaged constitutive relationship is requiredw

FORMULATION

FE MODEL

RESULTS

Fracture Mechanics approaches

Static analyses:

CONCLUSIONS

Static analyses: (the time dependence is neglected “a priori”)

Steady state crack growth approaches:(Mo ing reference s stem ith the tip crack tip speed is constant)(Moving reference system with the tip, crack tip speed is constant)

Unsteady models :Full Time dependence , inertial forces, loading rate,….

Page 4: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DYNAMIC CRACK GROWTH MODELING

Node release technique

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

Virtual crack closure methods

Gradual release of the nodal forces behind the crack tip FORMULATION

FE MODEL

RESULTSV u c c c osu e e ods

The ERR is evaluated by the mutual work at the crack tip and behind the crack tip

CONCLUSIONS

Moving mesh methodology

and behind the crack tip

The nodes are moved to predict changes of the geometry produced by the crack motion

Page 5: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

MOTIVATION OF THE WORK AND SUMMARY

AIM OF THE WORK

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODELPropose a generalized modeling based on Fracture mechanics and moving mesh methodology to predict the dynamic behavior of composite laminated structures

SUMMARY

FORMULATION

FE MODEL

RESULTS

Review the main equations of the ALE formulation in view of the Dynamic Fracture Mechanics approach

SUMMARY CONCLUSIONS

Evaluate the specialized expressions of the ERR by the use of the decomposition methodology of the J-integral and propose a proper mixed mode crack toughness criterionp p g

Develop the finite element implementation. Propose validation by means of comparisons with experimental data and a parametric study to analyze dynamic crack behavior (i.e. crack arrest phenomena, allowable tip speedsdynamic crack behavior (i.e. crack arrest phenomena, allowable tip speeds and rate dependence of the interfacial crack growth)

Page 6: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

BASICS OF MOVING MESH STRATEGY: ARBITRARY-LAGRANGIAN EULERIAN FORMULATION

“Lagrangian Approach”“Eulerian Approach”

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODELppFORMULATION

FE MODEL

RESULTSBL

tat,at.x X,t

BE

t

: L EB B

CONCLUSIONSX ta,tat

x r,t

. xt

,

L E

x r t

:

,

R LB BX r t

a

X r,tx r,t

r

“Arbitrary Lagrangian Eulerian”

The Reference configuration is fixed and independent of any placement of the material body

BR

Page 7: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

BASICS OF MOVING MESH STRATEGY: ARBITRARY-LAGRANGIAN EULERIAN FORMULATION

a,tat. at ,at. ++ t t BL

t+ t

BL

t

time “t” time “t+dt” Physical quantities:

d d

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

X2

X3

X t X t+ t

X r,t+tt+ tX r,tt

X

dv X, t ,dt

“Material” “Referential”

r

dX r, tdt

FORMULATION

FE MODEL

RESULTS

X1

a

BR

t

r2r3

Time derivative rule df f X f X, tdX

CONCLUSIONS

r1

r“Referential configuration”

,dX

Physical fields in ALE formulationPhysical fields in ALE formulation

“Material accel.” x x x x x xu u 2 u X uX u X X u X X

11x ru u J

det J 0

“one-to-one relationship” “Grad. transform.”

Page 8: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DESCRIPTION OF THE DELAMINATION MODEL

F, F, Multi-layer Modeling

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

2D Kinematic formulation

The laminate is divided into

FORMULATION

FE MODEL

RESULTS

a,an mathematical layer representing the staking sequence

C tibilit ti LMMa,ta t. L-a t

CONCLUSIONS

Compatibility equations LMM:

hj Layer j-th

hn Layer j+1-th

u=0,v=0 1 10, 0, i i i i i iu u u v v v

v>0

h1 Layer j-2

hj-1 Layer j-1-th

u=0,v=0

1 0, i i iv v v

“undelaminated interfaces”

“delaminated interfaces”“delaminated interfaces”

Page 9: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DESCRIPTION OF THE DELAMINATION MODEL IN THE REFERENTIAL CONFIGURATION

Governing Equations: “Principle of d’Alembert”INDEX:

INTRODUCTION

MOTIVATIONS

ALE MODEL n n n n

dV dV t dA f dV

Internal work External work

FORMULATION

FE MODEL

RESULTS

1 1 1 1

i i i ii i i iV V V

udV u udV t udA f udV

a,tat.BL

t

: Jacobian

CONCLUSIONS

1 1

1 1det

i ri

n n

r r ri iV V

udV C uJ uJ J dV

X

X2

X3 X t

X r,tt

1 1

1 1

1 1 1 1

[ 2

] det

i ri

n n

r ri iV V

u udV u u J X u J X

u J J X X u J X J X u J dV

X1a

BL

t

r2r3

] det r r r r ru J J X X u J X J X u J dV

det( ) det( ) n n n n

r rt udA f udV t u J d f u J dV

r1

r1 1 1 1

( ) ( ) i i ri ri

r ri i i iV V

f f

Page 10: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

ERR RATE EVALUATION : J-INTEGRAL APPROACHRevision of the J-integral Dec. procedure (Rigby & Aliabady, 1998, Greco & Lonetti, 2009)

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

nP

Expressions of the ERR

FORMULATION

FE MODEL

RESULTS

X2

X

a,tat.

P'

10lim

uJ W K n t dsX

“Path independent”

pCONCLUSIONSX1

1

uJ W K n t ds u f u u u dAX

p

Decomposition of the ERR into symmetric and antisymmetric fields

(Nishioka,T, 2001)\

1 ,

SS S S S S S S S

I I ij j

ASAS AS AS AS AS AS AS AS

uJ G W K n n ds u f u u u dAx

uJ G W K n n ds u f u u u dA

1 ,

II II ij jJ G W K n n ds u f u u u dAx

Page 11: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DYNAMIC CRACK PROPAGATION ANALYSIS:GROWTH CRITERION

Crack growth criterion

dx 1“Critical value of the ERR”

Material parameter

0GG

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODELcR

V G

C c v ue o e(Freund, 1990; Ravi-Chandar, 2004)

m

1

D m

t

R

GcV

FORMULATION

FE MODEL

RESULTS

G G0

0

0 0

R D t

D t

c V G c

c G c G“Rayleigh wave speed”

“initiation value”

1) i i i

CONCLUSIONS

1) Mixed mode crack growth criterion

1.700

M i l

1 0 I IIf

ID t IID t

G GgG c G c

3,02,52003,0 2,0

2,51,52,0

700

GI

1.200Material parameter

0 0 ,

1 1

I IIID t IID tm m

t t

G GG c G cc c

GI1,01,5

GII 1,0 0,50,5

0,00,0

1 1 R RV V

Page 12: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

MOVING MESH METHOD: FOUNDAMENTAL EQUATIONS

ALE formulation to describe mesh motion

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

Mesh regularization technique

“Winslow Smoothing method”1 1 1 X X r 2 2 2 X X r

21 0,

X X 2

2 0. X X FORMULATION

FE MODEL

RESULTS

“Mesh displacements of nodesShould be regular” Minimize the mesh warping

B d diti C

CONCLUSIONS

Boundary conditions Example: DCB scheme

h2

1 3 1 2 1 2

2 3 4

0, 0 on , 0 on

X XX

h1

h2

tip

2

4

2 3 4

1

1

2

0 0 on ,

0 on ,

0 on

f

t f

X if g

X c if g

X

1 2 1 20 0, 0 0, 0 0, 0 0 X X X X

Page 13: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

VARIATIONAL FORMULATION AND FE IMPLEMENTATION

Weak forms: coupled equations for the ALE and PS formulations:n n

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

PS

1 1 1 1

1 1

1 1 1 1

det [ 2

] det

ri ri

n n

r r r r ri iV V

r r r r r

n n

C uJ uJ J dV u u J X u J X

u J J X X u J X J X u J dV

FORMULATION

FE MODEL

RESULTS

ALE

1 1det( ) det( )

ri ri

r ri i V

t u J d f u J dV

1 1 det 0,

r r r tV

XJ wJ J dV X c i Xi J ds

CONCLUSIONS

h2 2

1 3

Explicit equations for PS+ALE

r rV

h1

tip 4

Implicit equation the crack growth

Crack growth criterionCrack growth criterion

Page 14: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

VARIATIONAL FORMULATION AND FE IMPLEMENTATION

Quadratic Lagrangian interpolation functions

FE approximation by “Comsol Multiphysics”:

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODELQuadratic Lagrangian interpolation functions for displacements, velocity and acceleration fieldsQuadratic Lagrangian interpolation functions for mesh points displacements

Non Linear Equations System

FORMULATION

FE MODEL

RESULTSfor mesh points displacements

0 1 2 0

n n n n n

i i i i i i i i i i iM U CU K K K K U T P

FE equationsCONCLUSIONS

CHECK1 1 1 1 1

0,

i i i i i

W X Q X L

Solution Procedure

MESH ELEMENTSQUALITY

Implicit time integration scheme based on variable-step-size backward differentiation formula

Iterative-incremental Solving procedure

Page 15: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

RESULTS: VALIDATION OF THE STRUCTURAL MODEL

0 5

0.6

0 10

0.15

0.20

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODELa

hh

0.3

0.4

0.5

a/L=0.367, h/B=0.1, m=0.5, Experimental dataP d d la)

/L

0.0 0.3 0.6 0.9 1.2 1.50.00

0.05

0.10

FORMULATION

FE MODEL

RESULTSL

B hh

u

0.1

0.2 

hh

a

L

B

u

u

a/L 0.367, h/B 0.1, m 0.5, G

0/G

st=0.3, mm/s Proposed model

X

(a

25E

d

1

 

aBu

CONCLUSIONSL

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

L

tL15

20

d

E c)/(G

0BL)

0.1

a/L=0.367, h/B=0.1, m=0.5, G0/Gst=0.3, mm/s

hh

L

u

ct/csh

c t/csh

0

5

10

Ec

(Ed,

0.01

c

Comparisons with experimental data

DCB mode I loading scheme

AS 3501 6 Graphite/Epoxy 0.0 0.5 1.0 1.5 2.0 2.5 3.0

tLAS 3501-6 Graphite/Epoxy

Page 16: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

RESULTS: EFFECT OF THE LOADING RATE

0 035

0.040

0.045  

hh

aB

u

u

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

a

B hhh

0.020

0.025

0.030

0.035 hL

u

a/L=0.1, h/B=0.1, m=0.5, G0/Gst=50/150

c t/csh

FORMULATION

FE MODEL

RESULTSL

B hh

u

0.000

0.005

0.010

0.015 t

0.040

10 8 6 4 2 0

0.000

CONCLUSIONS

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

t/T1

0 020

0.025

0.030

0.035

c sh/csh

0 020

0.015

0.010

0.005

DCB mode I loading scheme

0.005

0.010

0.015

0.020

a/L=0.1, h/B=0.1, m=0.5, G0/Gst=50/150

c t/cc t/

0.035

0.030

0.025

0.020Influence of the loading rate

Evolution of the crack tip speed

0.0 0.1 0.2 0.3 0.4 0.50.000

Xt/L0.040

Page 17: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DEFORMED SHAPE OF THE BEAM UNDER

MODE I LOADING CONDITIONSMODE I LOADING CONDITIONS

t = 0.065 s

Horizontal displacement of the crack-tip front

t 0.065 s

t = 0.12 s

t 0 18t = 0.18 s

di tix direction

Page 18: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

DEFORMED SHAPES OF THE BEAM UNDER

MIXED MODE LOADING CONDITIONS

TRIANGULAR MESH ELEMENTS

C TCRACK - TIP

t = 0.10 s

Page 19: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

RESULTS : MODE II ENF SCHEMEINDEX:

INTRODUCTION

MOTIVATIONS

ALE MODELB(F,u)0.24

0.27

0 10

0.15

0.20

FORMULATION

FE MODEL

RESULTS

aB

hh

0 12

0.15

0.18

0.21

a/L=0.216, h/B=0.213, m=1,

Experimental data Proposed model

X(a

)/L

0.0 3.0x10-9 6.0x10-9 9.0x10-90.00

0.05

0.10

0.8

ct/csh  

aB

u

u

hh

CONCLUSIONSL

0.03

0.06

0.09

0.12

 

a

L

B(F,u)

hh

G0/Gst=0.52, mm/s,

0.6 a/L=0.367, h/B=0.1, m=1, G0/Gst=0.52, mm/s

Lh

0.0 3.0x10-9 6.0x10-9 9.0x10-9 1.2x10-8 1.5x10-80.00

L

tLtL

ENF mode II loading scheme

0.2

0.4

tL

Comparisons with experimental data

ENF mode II loading scheme

S2/8553 Glass/Epoxy0.0 2.0x10-9 4.0x10-9 6.0x10-9 8.0x10-9 1.0x10-8 1.2x10-8

S2/8553 Glass/Epoxy

Page 20: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

RESULTS : MODE II ENF SCHEME

5.0 /L 0 25 h/B 0 213 1 P

INDEX:INTRODUCTION

MOTIVATIONS

ALE MODEL

High amplifications in the ERR prediction

3.5

4.0

4.5a/L=0.25, h/B=0.213, m=1, G

0/G

st=0.52,

t0

t

P

t0 / T1= 0.1

FORMULATION

FE MODEL

RESULTS

2 0

2.5

3.0

3.5

t0 / T1= 0.5 t0 / T1= 0.4

G /

Gst

CONCLUSIONS

0 5

1.0

1.5

2.0t0 / T1= 1

t0 / T

1= 2

t0 / T1= 0.7

G

aB(F,u)

h

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.00.0

0.5

t / T

L

h

t / T1

Page 21: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

RESULTS : MIXED MODE ANALYSISINDEX:

INTRODUCTION

MOTIVATIONS

ALE MODEL

FORMULATION

FE MODEL

RESULTS

 

aB(F,u)

hh

12

0.30

0.35

0.40

0.05

0.10

0.15

0.20

CONCLUSIONSL

0 15

0.20

0.25 

aB(F,u)

h2

Proposed model Experimental data0.0 2.0x10-9 4.0x10-9 6.0x10-9

0.00

X(a

)/L

Mesh tip discretization

0.05

0.10

0.15

L

h12

a/L=0.220, H/B=0.485, h1/h2=0.66, m=0.5, G0/Gst=0.526, mm/s

0.0 5.0x10-9 1.0x10-8 1.5x10-8 2.0x10-8 2.5x10-80.00

tL

AS 3501 6 Graphite/EpoxyAS 3501-6 Graphite/Epoxy

Page 22: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

RESULTS : MIXED MODE ANALYSISINDEX:

INTRODUCTION

MOTIVATIONS

ALE MODEL0.0400.0035 0.0030 0.0025 0.0020 0.0015 0.0010

1

FORMULATION

FE MODEL

RESULTS

a

L

B(F,u)

hh

12

0.025

0.030

0.035

sh u/u

u/ust=[1.2; 1.42; 1.74; 2.36; 3.98;5.08; 7.12]Eq. (22), g

f=[1.5; 3.0; 5.0; 10; 30;50; 100]

0.3

0.4

)/L)

CONCLUSIONSL

0 005

0.010

0.015

0.020

c t/cs u/ust

0.1

0.2

max

X(a

)

0 27

0.30

/u/ust=[1.2; 1.42; 1.74; 2.36; 3.98;5.08; 7.12]

0.0035 0.0030 0.0025 0.0020 0.0015 0.00100.25

1

u,gf

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.000

0.005

t/T1

0.0

0.15

0.18

0.21

0.24

0.27

u/ust

u/ust

X

(a)/L

)

Eq. (22), gf=[1.5; 3.0; 5.0; 10; 30;50; 100]

X

(a)/L 0.15

0.20

Crack arrestg

f=100

gf=1.5

.....

.....

.....

f

u/u=7.12st

u/u=1.20st

0 0 0 5 1 0 1 5 2 0 2 5 3 00.00

0.03

0.06

0.09

0.12

max

0.00

0.05

0.10

Loading curves

Crack arrest phenomenon

t

T1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

t/T1

Page 23: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

RESULTS : MIXED MODE ANALYSISINDEX:

INTRODUCTION

MOTIVATIONS

ALE MODEL

FORMULATION

FE MODEL

RESULTS

CONCLUSIONS

Time iincrem

entaation

Page 24: LONETTI P. - Dynamic Crack Propagation in Fiber Reinforced Composites (COMSOL 2009)

CONCLUDING REMARKSINDEX:

INTRODUCTION

MOTIVATIONS

ALE MODELA delamination model for general loading conditions based onmoving mesh methodology and fracture mechanics is proposed.

New expressions of the ERR mode components based on the

FORMULATION

FE MODEL

RESULTS

Comparisons with experimental data are proposed to validate the

New expressions of the ERR mode components based on theJ-integral decomposition procedure.

CONCLUSIONS

p p p pdelamination modelling

The analyzed parametric study shows that delamination phenomenay p y pare quite influenced by the loading rate, inertial effects leading to highamplifications in the ERR prediction and the crack growth.