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
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
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,….
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
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)
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
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.”
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”
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
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
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
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
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
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
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
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
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
DEFORMED SHAPES OF THE BEAM UNDER
MIXED MODE LOADING CONDITIONS
TRIANGULAR MESH ELEMENTS
C TCRACK - TIP
t = 0.10 s
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
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
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
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
RESULTS : MIXED MODE ANALYSISINDEX:
INTRODUCTION
MOTIVATIONS
ALE MODEL
FORMULATION
FE MODEL
RESULTS
CONCLUSIONS
Time iincrem
entaation
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.