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COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING
A. Airoldi*, C. Davila** *Dipartimento Ingegneria Aerospaziale, Politecnico di Milano **NASA Langley Research Center
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
CONTENTS
Introduction & Motivation
o Cohesive zone models and fibre bridging
experiments and numerical model
o DCB tests on fiberglass specimens and numerical model
Superposed cohesive laws approach for bridging
o Superposition of cohesive elements and analytical identification of material parameters
Numerical identification
o Response surface and optimization approaches to material parameter identification
Conclusions
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
INTRODUCTION AND MOTIVATION
Bi-linear cohesive laws
can be successfully in FE models of delaminations They are adequate when toughness is constant with crack length.
Characterisation Material model Application Verification Analysis of crack growth in curved fabric laminates
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
INTRODUCTION AND MOTIVATION
The crack growth resistance can significantly increase in the presence of fibre bridging In large scale fibre bridging a very long process zone develops before toughness reaches a steady level G C
Cohesive laws with linear softening are inadequate
curve effect. to model the G-a
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
INTRODUCTION AND MOTIVATION
The measurement of bridging tractions in the wake of crack confirms that they do not have a linear softening (Sorensen et al. 2008). Other shapes must be employed for the softening law The
superposition of two linear softening laws
has been proposed for intralaminar fracture (Davila et. Al 2009).
It can be considered an appealing practical approach (conventional cohesive elements can be used)
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
INTRODUCTION AND MOTIVATION
Objectives:
o Apply the
superposed element approach
to model the R-a curve effects in interlaminar fracture in glass fiber reinforced laminates o Develop an
analytical approach for the calibration
of material parameters from the experimental R-a curve o Apply numerical techniques for the
automatic identification
of such parameters based on the force vs. displacement response of DCB tests
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
EXPERIMENTS AND NUMERICAL MODEL
DCB tests have been performed on [0] 48 laminates of
S2 Glass fibre
reinforced tape with an Epoxy Cycom SP250 matrix (5 Tests) Pre-crack has been obtained by means of a PTFE insert Pre-opening test were performed Subsequent opening tests Crack advance monitored by dye penetrant inspection.
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
EXPERIMENTS AND NUMERICAL MODEL
Four data reduction techniques: Beam Theory (
BT
), Compliance Calibration (
CC
), Modified Beam Theory (
MBT
), Modified Compliance Calibration (
MCC
) Large scale fibre bridging and a marked G-a curve effect. The length of the process zone (LPZ) is approximately 80 mm
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
EXPERIMENTS AND NUMERICAL MODEL
A 2 mm wide strip of the specimen has been analysed in Abaqus Standard
Incompatible modes C3D8I elements Imposed displacement 0.5 mm equispaced grid COH3D8 cohesive elements
Material stiffness from previous characterisation and transverse isotropy assumptions
E a (MPa) 45670 G ta (MPa) 5900
v
ta 0.257
E t (MPa) 13600 G t (MPa) 5230
v
t 0.3
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
EXPERIMENTS AND NUMERICAL MODEL
Preliminary numerical evaluation: o cohesive law with linear softening o G IC = 1.0 KJ/m 2 o 0 =20 MPa and 0 =50 MPa Bi-linear cohesive law largely overestimates the force in DCB tests Peel strength has a little influence on DCB response as expected
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
SUPERPOSED COHESIVE LAWS APPROACH
In the presence of bridging, the softening law is non-linear
G I
J I
G tip
* 0
d
by means of two superimposed cohesive laws
c
1
n
c
c
2 ( 1
n
)
c G
1
m G c G
2 ( 1
m
)
G c
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
SUPERPOSED COHESIVE LAWS APPROACH
l c
E G c
/
c
2 reference length of the process zone
G R
G
1 ( 1
n
) 3 2
a l c
G C
Linearised expression of the G-a curve by Davila et al. 2009 G 1
a
ss
G c Parameter
m
is G 1 /G c
n
is obtained by imposing G R experimental
a
ss
= G C in correspondance of the
n
1 2 3 1
m
a
ss
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
E G c
c
2
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
SUPERPOSED COHESIVE LAWS APPROACH
The previous formulation has been applied and verified for a compact tension specimen (Davila et al. 2009) In DCB test adherends are thin and LPZ becomes much shorter than
l c
Turon et al. (2008) suggested a correction of reference process zone based on an undetermined factor H
l c
t t
H
A refined model using a single cohesive (linear softening law) has been used to asses an appropriate expression of reference LPZ
l c
Symmetry damage = 0 damage = 1 Process Zone 0 < damage < 1
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
SUPERPOSED COHESIVE LAWS APPROACH
Two corrections are considered: LPZ 1 LPZ 2
l c
t
t l
c l c
l c
t
t l c
l c
FEM 2D
l c
LPZ 1 LPZ 2 The errors in the uncorrected l c are very large when LPZ is long For large LPZ a correction factor with the additional parameter provides the best results is set to 0.48 for best correlation
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
SUPERPOSED COHESIVE LAWS APPROACH
Using
l c
t
t l
c
0 .
48
l c
and
m
=2 Sigma (MPa) n 15 0.9800
25 0.9928
35 0.9963
superposed cohesive elements model:
Numerical G( a)
G IC
P
2 2
B dC da
LPZ and Force vs. Displacement curves captured for Sigma = 15 and 25 MPa
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
The presented model proved effective to accurately capture the forces and the process zone lenght for moderate values of peel strength Analytical calibration of material parameters requires the knowledge of the G-a curve An alternative strategy is explored, based on a numerical identification technique The objective is the identification of material parameters considering the Force vs. Displacement curve A cost function is defined response surfaces techniques is applied to explore the feasibility of the approach Optimization procedures is applied to minimize the error
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Cost Functions
d 1 d 2 d 3 d 4
Mean Square Error between numerical and average test
MSE
F num
F test
2 Average MSE values in 4 selected zones
E i
1
d i
1
d i
d d i i
1
MSE
d
Global error index
E
E
1 2
E
2 2
E
2 3
E
2 4
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Implementation Ichrome/NEXUS Optimisation Suite Abaqus runs E i Matlab post processing variables Error zones
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
Total error
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Response surface techniques Response surfaces have been built by means of a Kriging approximation (second order polynomial + local gauss functions) The surface has been created by allocating 300 points within the domain Steady state toughness has been set at 1.0 kJ/m 2 Sigma(MPa) m n min 15 0.000
0.500
max 50 0.500
0.999
The database allows the creation of different surfaces of the cost function in the space m-n at a given value of peel strength (Sigma)
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Response surface for Sigma = 15 MPa Minimum of cost function is found along a valley for high values of n An interval 0.05
< m
< 0. 2 can be identified along the valley
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Response surface for Sigma = 25 MPa As Sigma is increased optimal slightly moves towards 1.0
n optimal m seems to be lower than m=0.2, but derivatives are small in such direction
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Response surface for Sigma = 35 MPa For Sigma = 35 MPa qualitative tendencies are confirmed. Overall minimum values of cost function are about 20 N.
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Following the meta-model indications three solutions have been selected Sigma (MPa) 15 25 35 m 0.19
0.14
0.11
n 0.985
0.985
0.990
Cost (N) 16.40
17.12
21.14
LPZ (mm) 74 76 73 Meta-model allows identifying acceptable approximations
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
Optimization : Gradient-based method Sigma = 15 Mpa , Gc = 1 kJ/m 2 Initial guess
m
=0.3,
n
=0.7
(meta-model indications ignored) m n 0.169
0.977
Cost (N) 15.61
Optimized Solution LPZ (mm) 67 Evolution of m,n, Objective
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
For Sigma =25 and 35 MPa meta-model indication have been used as initial guess for a gradient based method The application of different weights to error indices in the different zones of the curve has been investigated Initial Guess
E k
kE
1 2 2 .
0
kE
2 2
E
2 3
E
2 4 Interesting results have been found by increasing the weights in the first 2 zones of the domain
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
NUMERICAL IDENTIFICATION
minimization of cost function lead to increase
m
m Initial 0.140
optim
0.15
2
n 0.985
0.986
Sigma = 25 Mpa m Initial 0.110
optim
0.146
n 0.990
0.991
Sigma = 35 Mpa Improvement of Force-displacement and G-a correlation in the initial part of the response Final G C is almost unchanged (imposed value of 1 kJ/m 2 )
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011
IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi, C. Davila
CONCLUSIONS
Bi-linear softening laws can model delamination processes in the presence of fibre bridging An analytical calibration procedure of the model has been assessed for moderate values of peel strength (more refined models could be required for higher values) Numerical identification ( response surface / optimization ) can obtain approximate solutions without requiring the knowledge of the G-a curve Numerical procedures can be extended to multi-linear softening laws which could be more flexible for capturing both force response, G-a curve and process zone lengths
COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14 th 16 th , 2011