Introduction

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(No Transcript)
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CONTENTS

• Introduction Motivation
• experiments and numerical model
• Superposed cohesive laws approach for bridging
• Numerical identification
• Conclusions

• Cohesive zone models and fibre bridging

• DCB tests on fiberglass specimens and numerical
  model

• Superposition of cohesive elements and analytical
  identification of material parameters

• Response surface and optimization approaches to
  material parameter identification

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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
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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 GC
Cohesive laws with linear softening are
inadequate to model the G-a curve effect.
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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)
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INTRODUCTION AND MOTIVATION
Objectives

• Apply the superposed element approach to model
  the R-a curve effects in interlaminar fracture in
  glass fiber reinforced laminates

• Develop an analytical approach for the
  calibration of material parameters from the
  experimental R-a curve

• Apply numerical techniques for the automatic
  identification of such parameters based on the
  force vs. displacement response of DCB tests

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EXPERIMENTS AND NUMERICAL MODEL
DCB tests have been performed on 048 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.

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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
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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
Ea (MPa) 45670 Gta (MPa) 5900 vta 0.257
Et (MPa) 13600 Gt (MPa) 5230 vt 0.3
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EXPERIMENTS AND NUMERICAL MODEL
Preliminary numerical evaluation

• cohesive law with linear softening
• GIC 1.0 KJ/m2
• ?0 20 MPa and ?050 MPa

Bi-linear cohesive law largely overestimates the
force in DCB tests Peel strength has a little
influence on DCB response as expected
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SUPERPOSED COHESIVE LAWS APPROACH
In the presence of bridging, the softening law
is non-linear
?
the complete cohesive law is approximated by
means of two superimposed cohesive laws
.
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SUPERPOSED COHESIVE LAWS APPROACH
reference length of the process zone
Linearised expression of the G-a curve by Davila
et al. 2009
Parameter m is G1/Gc
n is obtained by imposing GR GC in
correspondance of the experimental
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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
Turon et al. (2008) suggested a correction of
reference process zone based on an undetermined
factor H
A refined model using a single cohesive (linear
softening law) has been used to asses an
appropriate expression of reference LPZ
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SUPERPOSED COHESIVE LAWS APPROACH
Two corrections are considered
FEM 2D
LPZ 1
LPZ 1
LPZ 2
LPZ 2

• The errors in the uncorrected lc 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

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SUPERPOSED COHESIVE LAWS APPROACH
Sigma (MPa) 15 25 35
n 0.9800 0.9928 0.9963
Using
and m2
superposed cohesive elements model
Numerical G(?a)
LPZ and Force vs. Displacement curves captured
for Sigma 15 and 25 MPa
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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

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NUMERICAL IDENTIFICATION
Cost Functions
Mean Square Error between numerical and average
test
Average MSE values in 4 selected zones
d4
d3
d1
d2
Global error index
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NUMERICAL IDENTIFICATION
Implementation
Ichrome/NEXUS Optimisation Suite
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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
min max
Sigma(MPa) 15 50
m 0.000 0.500
n 0.500 0.999
Steady state toughness has been set at 1.0 kJ/m2
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)
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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 lt m lt 0. 2 can be identified
along the valley
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NUMERICAL IDENTIFICATION
Response surface for Sigma 25 MPa
optimal m seems to be lower than m0.2, but
derivatives are small in such direction
As Sigma is increased optimal n slightly moves
towards 1.0
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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.
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NUMERICAL IDENTIFICATION
Sigma (MPa) m n Cost (N) LPZ (mm)
15 0.19 0.985 16.40 74
25 0.14 0.985 17.12 76
35 0.11 0.990 21.14 73
Following the meta-model indications three
solutions have been selected
Meta-model allows identifying acceptable
approximations
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NUMERICAL IDENTIFICATION
Optimization Gradient-based method
Sigma 15 Mpa, Gc 1 kJ/m2 Initial guess
m0.3, n0.7 (meta-model indications ignored)
m n Cost (N) LPZ (mm)
0.169 0.977 15.61 67
Optimized Solution
Evolution of m,n, Objective
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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
Interesting results have been found by increasing
the weights in the first 2 zones of the domain
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NUMERICAL IDENTIFICATION
m n
Initial 0.140 0.985
optim 0.152 0.986
m n
Initial 0.110 0.990
optim 0.146 0.991
minimization of cost function lead to increase m
Sigma 35 Mpa
Sigma 25 Mpa
Improvement of Force-displacement and G-a
correlation in the initial part of the response
Final GC is almost unchanged (imposed value of 1
kJ/m2)
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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