IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi*, C. Davila** *Dipartimento Ingegneria Aerospaziale,

IDENTIFICATION OF MATERIAL PARAMETERS

FOR MODELLING DELAMINATION IN THE

PRESENCE OF FIBRE BRIDGINGA. Airoldi*, C. Davila**

*Dipartimento Ingegneria Aerospaziale, Politecnico di Milano

**NASA Langley Research Center

COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL) Switzerland – February 14th 16th, 2011

COMPTEST 2011 – Ecole Polytechnique Fédérale de Lausanne (EPFL)

Switzerland – February 14th 16th, 2011

IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGINGA. Airoldi, C. Davila

Introduction & Motivation

experiments and numerical model

Superposed cohesive laws approach for

bridging

Numerical identification

Conclusions

CONTENTS

o Cohesive zone models and fibre bridging

o DCB tests on fiberglass specimens and numerical model

o Superposition of cohesive elements and analytical identification of material parameters

o Response surface and optimization approaches to material parameter identification




INTRODUCTION AND MOTIVATIONBi-linear cohesive laws can be successfully in FE models of delaminations

They are adequate when toughness is constant with crack length.

Characterisation Material model

Analysis of crack growth in curved fabric laminates

Application

Verification




INTRODUCTION AND MOTIVATIONThe 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.




INTRODUCTION AND MOTIVATIONThe measurement of bridging tractions in the wake of crack confirms that they do not have a linear softening (Sorensen et al. 2008).

The superposition of two linear softening laws has been proposed for intralaminar fracture (Davila et. Al 2009).

Other shapes must be employed for the softening law

It can be considered an appealing practical approach (conventional cohesive elements can be used)




INTRODUCTION AND MOTIVATIONObjectives:

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




EXPERIMENTS AND NUMERICAL

MODELDCB tests have been performed on [0]48 laminates of S2 Glass fibre reinforced tape with an Epoxy Cycom SP250 matrix (5 Tests)

Crack advance monitored by dye penetrant inspection.

Pre-crack has been obtained by means of a PTFE insert

Pre-opening test were performed

Subsequent opening tests





MODELFour 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





MODELA 2 mm wide strip of the specimen has been analysed in Abaqus Standard

Incompatible modes C3D8I elements

0.5 mm equispaced gridCOH3D8 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

Imposed displacement





MODELo cohesive law with

linear softening

o GIC = 1.0 KJ/m2

0 =20 MPa and 0=50 MPa

Preliminary numerical evaluation:

Bi-linear cohesive law largely overestimates the force in DCB tests

Peel strength has a little influence on DCB response as expected




SUPERPOSED COHESIVE LAWS

APPROACH

*

0

dGJG tipII

the complete cohesive law is approximated by means of two superimposed cohesive laws

.

cc n 1 cc n )1(2

cGmG 1 cGmG )1(2

In the presence of bridging, the softening law is non-linear





APPROACH C

cR G

l

anGaG

2

3)1(1

reference length of the process zone 2/ ccc GEl

Linearised expression of the G-a curve by Davila et al. 2009

G1

ssaGc

Parameter m is G1/Gc

232 1

1c

c

ss

GE

a

mn

n is obtained by imposing GR = GC in correspondance of the experimental ssa





APPROACHThe previous formulation has been applied and verified for a compact tension specimen (Davila et al. 2009)

Turon et al. (2008) suggested a correction of reference process zone based on an undetermined factor H

cc l

Ht

tl

cl

dam age = 0 dam age = 1

0 < dam age < 1

Process Zone

S ym m etry

A refined model using a single cohesive (linear softening law) has been used to asses an appropriate expression of reference LPZ

In DCB test adherends are thin and LPZ becomes much shorter than





APPROACHcl

FEM 2D

c

cc l

lt

tl

Two corrections are considered:

c

cc l

lt

tl

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

LPZ 1

LPZ 2

LPZ 1

LPZ 2




Using

c

cc l

lt

tl

48.0

and m=2Sigma

(MPa)15 25 35

n 0.9800 0.9928 0.9963

LPZ and Force vs. Displacement curves captured for Sigma = 15 and 25 MPa

superposed cohesive elements model:

Numerical G(a)


APPROACH

da

dC

B

PGIC 2

2




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




NUMERICAL IDENTIFICATION Cost Functions

2 testnum FFMSE

dMSEdd

Ei

i

d

dii

i

1

1

1

d1 d2d3 d4

Mean Square Error between numerical and average test

Average MSE values in 4 selected zones

24

23

22

21 EEEEE

Global error index




NUMERICAL IDENTIFICATION Implementation Ichrome/NEXUS Optimisation Suite

variables

Abaqus runs

Matlab post-processing

Error zones

Ei

Total error




NUMERICAL IDENTIFICATION Response surface techniquesResponse 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 0.999Steady 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)




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





As Sigma is increased optimal n slightly moves towards 1.0

optimal m seems to be lower than m=0.2, but derivatives are small in

such direction





For Sigma = 35 MPa qualitative tendencies are confirmed. Overall minimum values of cost function are about 20 N.




NUMERICAL IDENTIFICATION Following the meta-model indications three solutions have been selected

Meta-model allows identifying acceptable approximations

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




NUMERICAL IDENTIFICATION Optimization: Gradient-based method

Sigma = 15 Mpa, Gc = 1 kJ/m2

Initial guess m=0.3, n=0.7

(meta-model indications ignored)

m n Cost (N) LPZ (mm)

0.169 0.977 15.61 67

Optimized Solution

Evolution of m,n, Objective




NUMERICAL IDENTIFICATION For Sigma =25 and 35 MPa meta-model indication have been used as initial guess for a gradient based methodThe application of different weights to error indices in the different zones of the curve has been investigated

0.2

24

23

22

21

k

EEkEkEEInitial Guess

Interesting results have been found by increasing the weights in the first 2 zones of the domain




NUMERICAL IDENTIFICATION

Sigma = 25 Mpa

m nInitial 0.140 0.985optim 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

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)




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

Documents

IDENTIFICATION OF MATERIAL PARAMETERS FOR MODELLING DELAMINATION IN THE PRESENCE OF FIBRE BRIDGING A. Airoldi*, C. Davila** *Dipartimento Ingegneria Aerospaziale,