USNCCM-XI11th US National Congress on Computational Mechanics

MinneapolisJuly 25-29, 2011

M. Alfano, G. Lubineau, A. MoussawiComposite and Heterogeneous Materials Analysis and Simulations,

King Abdullah University of Science and TechnologyKingdom of Saudi Arabia

G. H. PaulinoDepartment of Civil and Environmental Engineering,

University of Illinois at Urbana-Champaign, USA

Minneapolis, July 25th 2011

Identification of fracture properties for a cohesive model using digital

image correlation

M. Alfano et al. (http://cohmas.kaust.edu.sa) 2

1. Introduction and motivation

2. Objective

3. Proposed approach

4. Algorithm and implementation of the inverse procedure

5. Target applications and experimental set-up (current status)

6. Follow-up work

Outline-

3

Introduction and motivations (1/4)Mechanical testing

as produced

θ ≈ 83o

laser treated

20o

δ

griparea

1.5 mm

25 mm

45 mm

40 mm

100 mm

100 mm

P, P

laser treated

grit-blasted

Alfano M, Furgiuele F, Lubineau G, Paulino GH. Role of laser surface preparation on damage and decohesion of Al/epoxy joints. Submitted for Journal publication.

4

Introduction and motivations (2/4)PPR based cohesive model

Park K, Paulino GH, Roesler JR. A unified potential-based cohesive model of mixed-mode fracture. Journal of the Mechanics and Physics of Solids. 2009;57(6):891-908.

Γn = −φn

α

m

m

m =α (α− 1)λ2

n

1− αλ2n

δn =φn

σmaxαλn (1− λn)

α−1 α

m+ 1

·

α

mλn + 1

m−1

non-dimensional exponent;

energy constant;

final crack opening width;

T (∆n) =∂Ψ∂∆n

=

= Γnδn

m

1− ∆n

δn

α mα + ∆n

δn

m−1− α

1− ∆n

δn

α−1 mα + ∆n

δn

m

Ψ(∆n) = φn + Γn

1− ∆n

δn

α m

α+

∆n

δn

m

unknown properties to be identifiedX = φn,σmax,λn,α : unknown cohesive properties;

5

Introduction and motivations (3/4)PPR for mode I fracture

Park K, Paulino GH, Roesler JR. A unified potential-based cohesive model of mixed-mode fracture. Journal of the Mechanics and Physics of Solids. 2009;57(6):891-908.

Cohesive energy (N/mm)

Coh

esiv

e st

reng

th (M

Pa)

(b)

2.8 3 3.2 3.4 3.6 3.8 4 4.2

10

20

30

40

50

60

70

80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Φ

6

Introduction and motivations (4/4)Identification of bond toughness using the CZM

Φ = U(δ)EXP − U(δ)FE 2

U(δ) =

δi

δi−1

P (δ)dδ = (δi − δi−1)×P (δi) + P (δi−1)

2

Alfano M, Furgiuele F, Lubineau G, Paulino GH. Identification of mode-I cohesive zone parameters of Al / epoxy T-peel joints with laser treated substrates. Submitted for Journal publication.

φn = 0.7÷ 0.9 N/mm

σmax = 0.5÷ 5 MPa

φn = 3.2÷ 3.8 N/mm

σmax = 20÷ 85 MPa

The response function is not affected by shape factor and

slope indicator

P

δ

Exp

FEA

7

Remarks and objective of the work-

A global response is often obtained from experiments, however, it may have low sensitivity to certain cohesive properties.

The uniqueness of the obtained cohesive zone model is not guaranteed.

Although the cohesive models obtained using global data can yield satisfactory predictive capabilities in FEA simulations of fracture, the development of an alternative procedure is needed, e.g. to determine cohesive strength.

A solution may be provided by the original combination of experimental full-field measurements techniques and inverse problems.

Gain AL, Carroll J, Paulino GH, Lambros J. A hybrid experimental / numerical technique to extract cohesive fracture properties for mode-I fracture of quasi-brittle materials. International Journal of Fracture. 2011;169:113-131.Shen B, Paulino GH. Direct Extraction of Cohesive Fracture Properties from Digital Image Correlation: A Hybrid Inverse Technique. Experimental Mechanics. 2011;51(2):143-161.

8

Forward versus inverse problemForward problem

Ωσ : δ dΩ−

Γext

Text · δ∆u dΓext +

−

Γcoh

Tcoh · δ∆u dΓcoh = 0

Principle of virtual work Ω : specimen domainΓext : external boundaryΓcoh : cohesive surfaces∆u : cohesive surfaces opening displacementσ : stress tensor : strain tensor

Forward problem P

CMOD

Step 1

1

u

Kb =

ΩBTEB dΩ

Kcoh =

Γcoh

NT ∂T

∂∆uN dΓcoh

Fext =

Γext

TextdΓext

(Kb +Kc(u,X))u = Fext

P-CMOD

+T

∆u

9

P

CMOD

Step 2

Step 3

Step 1

crack growth

ROI

P-CMOD

Inverse problem(optimization)

T

∆u

X = φn ,σmax ,α ,λn

ωi (X) =1

(Umax,i)2

nn

j=1

[uexp − u (X)]2j

X = argminX∈RM

Π =m

i=1

ωi (X)

m : available measurement instants (load levels)nn : nodal displacements in the ROI

Forward versus inverse problemInverse problem (objective of the work)

u(X) = K−1b F

ext(uexp ;X)

Fext

(uexp ;X) = Fext −Kc(uexp ;X)uexp

Optimization algorithm?

10

Exploration algorithm based on the mechanism of natural selection and genetics: the strongest individuals (chromosomes) in a population survive and generate offsprings.

Genetic algorithmFundamentals concepts

A chromosome represents a generic solution of the problem, in our context a set of cohesive fracture parameters (X):

X Φn σmax λn α

1. Random generation of the initial population (individuals X) satisfying suitable restraint conditions (e.g. fracture energy must not be negative);

Basic steps of the GA

11

2. The chromosomes are evaluated, using some measures of fitness. We defined the following objective function (or cost function):

Genetic algorithmBasic steps of the algorithm

ωi (X) =1

(Umax,i)2

nn

j=1

[uexp − u (X)]2j

X = argminX∈RM

Π =m

i=1

ωi (X)

m : available measurement instants (load levels)nn : nodal displacements in the ROI

3. Individuals for reproduction are firstly chosen based on their fitness

4. and some of them are processed by means of genetic operators (crossover and mutation) to create a new populations

12

Crossover (type 1)

X1 Φn σmax λn α X2 Φn σmax λn α

X3 Φn σmax λn α X4 Φn σmax λn α

Crossover (type 2)

X3 = a ·X1 + (1− a) ·X2

X4 = (1− a) ·X1 + a ·X2a ∈ [0, 1]

5. New chromosomes, called offspring, are formed by merging two chromosomes from current generation

Genetic algorithmBasic steps of the algorithm

13

X1 Φn σmax λn α

X2 Φn σ’max λn α

σmax = σmax + r ·∆σmax

r ∈ [−1, 1] ,∆σmax = cost.

5. New chromosomes are also formed by modifying a chromosome using a mutation operator

Genetic algorithmBasic steps of the algorithm

6. The newly created population replace the old one and the process restarts.

14

Genetic algorithmFundamentals concepts

Initialsolutions

Start Population P(t)chromosomes

(φn ,σmax ,λn ,α)1(φn ,σmax ,λn ,α)2(φn ,σmax ,λn ,α)3

(φn ,σmax ,λn ,α)n

.

.

.

chromosomesOffsprings C(t)

crossover

mutation

CC(t)

CM(t)

Solution candidates

(fitness computation)

Newpopulation

Terminating condition?

Yes

Bestsolution

Stop

No

t0

t0+1

(φn ,σmax ,λn ,α)1(φn ,σmax ,λn ,α)2

(φn ,σmax ,λn ,α)12(φn ,σmax ,λn ,α)21

(φn ,σmax ,λn ,α)j

(φn ,σmax ,λn ,α)j

selection evaluation

Abaqus/Matlab interaction by means of Linux shell scripts

readoutput file

solve non-linearfracture problem

+

writeinput file

15

Target applications and experimental set-upcurrent status

Hot press 15T/300°C

Specialized cutting devices

Ultrasonic Dispersion

propagation in CNT (carbon nanaotube) and fiberreinforced hybrid composites. Wichmann et al. [8]have reported a decrease in the Mode I interlaminarfracture toughness of DWCNT and glass fiber rein-forced epoxy matrix composites, and pointed somedifficulties during the tests caused by the obstructedtracking of the crack tip in the opaque resin usingthe conventional visual crack tracking during DCBtesting. In our previous studies we have reportedsome indirect delamination resistance improvementin a nanotube/fiber reinforced system [9]. The aimof this research is to directly characterize the effectof carbon nanotube filling on the interlaminarmechanical properties of fiber reinforced compos-ites through standard DCB tests.

2. Experimental2.1. Materials

FM-20 epoxy laminating resin was used (P+MPolimerkémia Kft., Hungary) with T-16 curingagent (P+M Polimerkémia Kft., Hungary) asmatrix. The recommended mixing weight ratio was100:20, the resin had a curing treatment of 4 h at60°C.Baytubes® BT150 HP (Bayer, Germany) multi-walled carbon nanotubes (MWCNTs) were used asfiller in one portion of the matrix (Figure 1). Thenanotubes have been produced in a CVD based cat-alytic process resulting in an average outer diame-ter between 13–16 nm, length above 1 µm andcarbon purity above 99% according to the manu-facturer. The carbon nanotubes have been mixed tothe epoxy component of the resin using a three rollmill, four pass-throughs have been carried out to

achieve uniform dispersion and appropriate particlesize (<10 µm).Zoltek PX35FBUD0300 unidirectional carbon fab-ric (Zoltek Ltd., Hungary) has been used as fiberreinforcement in the composites. The fabric con-sisted of 50k rovings, and had a surface weight of300 g/m2.

2.2. Composite preparation

To characterize the effect of the nanotube filling ofthe matrix on the properties of carbon fiber rein-forced composites one carbon fiber/epoxy laminateand four carbon nanotube/carbon fiber/epoxy lami-nates with 0.1, 0.3, 0.5, and 1 weight% nanotubefilling have been produced under the same circum-stances.The laminates have been produced by hand lamina-tion of 10 plies of carbon fabric impregnated withthe resin, the fiber orientation of all laminae hasbeen 0°. A 50 µm thick PET film has been used as adelamination initiator insert in the center plane(between the 5th and 6th lamina) of the laminates.Both sides of the film have been coated with mouldrelease agent to minimize adhesion between thefilm and the matrix of the composite. To avoidtrapped in air bubbles, the laminate has been rolledafter every two plies.To achieve uniform thickness and fiber content thelaminate has been pressed for 12 hours under30 kN at room temperature. The uniform thicknessof all of the laminates has been achieved by using a4 mm thick steel plate placed as a spreader next tothe laminates in the press. The fiber contents were49.2±1.1, 51.9±2.8, 51.7±3.2, 51.9±2.7, and

146

Romhány and Szebényi – eXPRESS Polymer Letters Vol.3, No.3 (2009) 145–151

Figure 1. SEM micrographs of the MWCNT aggregates (a) and MWCNTs (b) used (raw materials)

MWCNT Dispersion

Material processing - DCB with metal and composites substrates

Composite laminates

High resolution

CCD camera(PixelFly)

QM100 long distance microscope(QUESTAR)

INSTRON Testing machine

High intensityLed spotlight In-house developed DIC algorithm

Full-field experimental measurements

(http://cohmas.kaust.edu.sa)