HOMOGENIZATION METHOD APPLIED TO THE DEVELOPMENT OF COMPOSITE MATERIALS

HOMOGENIZATION METHOD APPLIED TO THE DEVELOPMENT OF COMPOSITE MATERIALS

Emílio Carlos Nelli SilvaAssociate Professor

Department of Mechatronics and Mechanical System Engineering

Escola Politécnica da Universidade de São PauloBrazil

Emílio Carlos Nelli SilvaAssociate Professor

Department of Mechatronics and Mechanical System Engineering

Escola Politécnica da Universidade de São PauloBrazil

US-South America Workshop: Mechanics and AdvancedMaterials – Research and Education

Rio de Janeiro, August 2004

Outline

h Introduction to Homogenization MethodhHomogenization of FGM MaterialshTopology Optimization MethodhMaterial Design ConcepthConclusions and Future Trends

h Introduction to Homogenization MethodhHomogenization of FGM MaterialshTopology Optimization MethodhMaterial Design ConcepthConclusions and Future Trends

F F

unit cell

unit cell

homogenizedmaterial

a)

b)

brick wall

perforated beam

homogenizedmaterial

Concept of Homogenization Method

Homogenized Material

Homogenized Material

Example of application:

Homogenization method allows the calculation of composite effective properties knowing the topology of

the composite unit cell.

Homogenization method allows the calculation of composite effective properties knowing the topology of

the composite unit cell.

Complex unit cell topologies implementation using FEM

Concept of Homogenization Method

It allows the replacement of the composite medium by an “equivalent” homogeneous medium to solve the global problem.

It allows the replacement of the composite medium by an “equivalent” homogeneous medium to solve the global problem.

• it needs only the information about the unit cell • the unit cell can have any complex shape• it needs only the information about the unit cell • the unit cell can have any complex shape

Analytical methods

Advantage in relation to other methods:

• Mixture rule models - no interaction between phases• Self-consistent methods - some interaction, limited to

simple geometries

• Periodic composites; hAsymptotic analysis, mathematically correct;h Scale of microstructure must be very small compared to

the size of the part;• Acoustic wavelength larger than unit cell dimensions.

(Dispersive behavior can also be modeled)

Component EnlargedPeriodic Microstructure

x y EnlargedUnit Cell (Microscale)

Assumptions

Concept of Homogenization Method

Literature Review

Theory development (elastic medium):hSanchez-Palencia (1980) - FrancehDe Giorgi and Spagnolo (1973) (G-convergence) - ItalyhDuvaut (1976) and Lions (1981) - FrancehBakhvalov and Panasenko (1989) - Soviet Union

Numerical Implementation using FEM:hLéné (1984) - FrancehGuedes and Kikuchi (1990) - USA

Dispersive behavior:hTurbé (1982) - France

Theory development (elastic medium):hSanchez-Palencia (1980) - FrancehDe Giorgi and Spagnolo (1973) (G-convergence) - ItalyhDuvaut (1976) and Lions (1981) - FrancehBakhvalov and Panasenko (1989) - Soviet Union

Numerical Implementation using FEM:hLéné (1984) - FrancehGuedes and Kikuchi (1990) - USA

Dispersive behavior:hTurbé (1982) - France

hflow in porous media - Sanchez-Palencia (1980)hconductivity (heat transfer) - Sanchez-Palencia (1980)h viscoelasticity - Turbé (1982)hbiological materials (bones) - Hollister and Kikuchi (1994)helectromagnetism - Turbé and Maugin (1991)hpiezoelectricity - Telega (1990), Galka et al. (1992),

Turbé and Maugin (1991), Otero et al. (1997)

etc …

hflow in porous media - Sanchez-Palencia (1980)hconductivity (heat transfer) - Sanchez-Palencia (1980)h viscoelasticity - Turbé (1982)hbiological materials (bones) - Hollister and Kikuchi (1994)helectromagnetism - Turbé and Maugin (1991)hpiezoelectricity - Telega (1990), Galka et al. (1992),

Turbé and Maugin (1991), Otero et al. (1997)

etc …

Extension to Other Fields

• Properties cijkl are Y-periodic functions (Y - unit cell domain).

• Asymptotic expansion:- displacements:

where y=x/ε and ε>0 is the composite microstructuremicroscale, and u1 is Y-periodic first order variation term.

Theoretical Formulation

Component EnlargedPeriodic Microstructure

x y EnlargedUnit Cell (Microscale)

ε

u u x u x yε ε= +0 1( ) ( , )

uε u x y1( , )

microscopic equations (δu1(x,y) terms)

FEM solution of microscopic equations

for χu x y u x1 0= χ ε( , ) ( ( ))

u u x u x yε ε= +0 1( ) ( , )

Energy Functional for the Medium

Theory of Asymptotic Analysis macroscopic equations (δu0(x) terms)

where χ is Y-periodic characteristic functions of the unit cell

Due to linearity:

Theoretical Formulation

; y=x/ε

Substitute in the system of microscopic equations (χ)

FEM Solution

χ χi I iIN≅ I=1,NN NN =→→

48

nodes 2D case nodes 3D case

[ ] K Fmn mnχ ( ) ( )=

Bilinear (2D) and trilinear (3D) interpolation functions

FEM system of equations: 6 for 3D3 for 2D

loadcasesmn

periodicity conditionsenforced in the unit cell

Homogenization Implementation

cH,

Unit Cell

6 for 3D3 for 2D

Number ofload cases:

FEM model and Data Input

Assembly of Stiffness Matrix

Solver

Last?

Calculation of Homogenized Coefficients

Y

N

12

periodicity conditionsenforced in the unit cell

Physical Concept of Homogenization

Calculation of effective properties (cH)

Unit Cell

Load Cases (2D model)

Solutions using FEM

Unit Cell

Solid phase Fluid phase

ExampleHomogenization of composite material with solid and

fluid phases

Discretized Unit Cell

Homogenization of woven fabric composites

Example

Discretized Unit Cell

230.000 brick elements230.000 brick elements

Homogenization of bone microstructure

Solid phase Fluid phase

Example

(Hollister and Kikuchi - 1997)

“Representative Volume Element (RVE)” concept

Micrograph of Metal Matrix Composites (MMC)

There must be “statistic” periodicity !!!

Example

Cr (fiber) - NiAl (matrix)RVE unit cell RVE unit cell

Force

Displacement

Electric potential

Electric charge

Mechanical Energy

Mechanical Energy

Electrical Energy

Electrical Energy

Piezoelectric Material

Piezoelectric Material

Examples: Quartz (natural)Ceramic (PZT5A, PMN, etc…)Polymer (PVDF)

Examples: Quartz (natural)Ceramic (PZT5A, PMN, etc…)Polymer (PVDF)

Applications: Pressure sensors, accelerometers, actuators,acoustic wave generation (ultrasonic transducers, sonars,and hydrophones), etc...

Applications: Pressure sensors, accelerometers, actuators,acoustic wave generation (ultrasonic transducers, sonars,and hydrophones), etc...

Homogenization for Coupled Field MaterialsExample: Piezoelectric Material

cEijkl - stiffness property

eikl - piezoelectric strain property

εSik - dielectric property

Tij - stress

Skl - strain

Ek - electric field

Di - electric displacement

Constitutive Equations of Piezoelectric Medium

T c S e ED E e S

ij ijklE

kl kij k

i ikS

k ikl kl

= −= +

ε

Elasticity equation

Electrostatic equation

hProperties cEijkl, eijk, and εS

ij are Y-periodic functions (Y - unit cell domain).

hAsymptotic expansion:- displacements:

- electric potential:

where y=x/ε and ε>0 is the composite microstructuremicroscale, and u1 and φ1 are Y-periodic first order variation terms.

Homogenization for Piezoelectricity

u u x u x yε ε= +0 1( ) ( , )

φ φ εφε = +0 1( ) ( , )x x y

Telega (1990), Galka et al. (1992), and Turbé and Maugin (1991)

20

periodicity conditionsenforced in the unit cellperiodicity conditions

enforced in the unit cell

Homogenization Implementation

Calculation of effective properties (cE

H, eH, and εSH)

Unit Cell

9 for 3D model5 for 2D model

Number ofload cases

Load Cases (2D model)Load Cases (2D model)

Solutions using FEM

Unit Cell

21

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

Ceramic Volume Fraction (%)

d h (p

C/N

)

Rectangular inclusionCircular inclusionRectangular inclusion (hex.)

polymer piezoceramic

3D Piezocomposite Unit Cell

“staggeredformation”

“squareinclusion”

dh

“circularinclusion”

Example

Performance quantity

Concept of FGM materials

FGM materials possess continuously graded properties with gradual change in microstructure which avoids interface problems, such as, stress concentrations.

FGM materials possess continuously graded properties with gradual change in microstructure which avoids interface problems, such as, stress concentrations.

1-D

2-D

3-D

THot

Ceramic matrix with metallic inclusions

Metallic matrix

with ceramic inclusions

Transition region

Metallic PhaseTCold

Ceramic Phase

MicrostructureMicrostructure Types of gradationTypes of gradation

Collaboration USP/UIUC

• Emilio visited UIUC in December 2003• Glaucio visited USP and LNLS (Syncroton Light NationalLaboratory – Campinas, SP) in April 2004• Conference papers presented at FGM2004 and ICTAM2004• The following journal paper is at final stage: “Topology Optimization Applied to the Design of Functionally Graded Material (FGM) Structures”

• Emilio visited UIUC in December 2003• Glaucio visited USP and LNLS (Syncroton Light NationalLaboratory – Campinas, SP) in April 2004• Conference papers presented at FGM2004 and ICTAM2004• The following journal paper is at final stage: “Topology Optimization Applied to the Design of Functionally Graded Material (FGM) Structures”

University ofSão Paulo

University of Illinoisat Urbana-Champaign

“Inter-Americas Collaboration in Materials Research and Education”

NSF project CMS 0303492 P.I.: Professor Wole Soboyejo (Princeton University)

“Inter-Americas Collaboration in Materials Research and Education”

NSF project CMS 0303492 P.I.: Professor Wole Soboyejo (Princeton University)

Acknowledgment

Homogenization for FGM Materials

FGM Composite material example

These materials possess continuously graded properties with gradual change in microstructure;

These materials possess continuously graded properties with gradual change in microstructure;

Calculation of effective properties is very difficult using analytical methods

Calculation of effective properties is very difficult using analytical methods

Homogenization method can be applied

Homogenization method can be applied

FGM unit cell

Homogenization for FGM Materials

( ) ( )∑=

=nnodes

III NEE

1xx

To solve homogenization equations the graded finiteelement (Kim and Paulino 2002) is used which considers a

continuous distribution of material inside unit cell

E: material propertyEI: material property evaluated at FEM nodesx=(x, y): position Cartesian coordinates

E: material propertyEI: material property evaluated at FEM nodesx=(x, y): position Cartesian coordinates

I J

KL

EI EJ

EK

EL

Example

Material 1Material 2

Plane Strain assumption

3.0.2 .;1 .;1E ;8

21

2121

αβαα

E=======

vvE

Material properties:x

y

.0

72.673.5

;29.1.0.0.083.41.1.01.168.3

H

=

= βHEHomogenized

properties

Elasticproperties

Thermoelasticproperties

Unit Cell

( )( ) 2/'2cos)(

2/'2cos)(

2121

2121

ββπβββπ

++−=++−=

xx EEEEE

FGM law:

ExampleAxyssimetric composite

(Application to bamboo andnatural fiber composites)

Material 1

Material 2

Unit Cell

3.0 .;2E ;10

21

21

====

vvEElastic properties:

89.1.0.0.0.049.61.6562.1.01.655.8501.2.062.101.239.5

=HEHomogenized

properties

( ) 2/'2cos)( 2121 EEEEE ++−= rπFGM law:

Material Design - Introduction

Specify the desiredMaterial propertiesSpecify the desiredMaterial properties

Design the CompositeMaterial

Design the CompositeMaterial

Inverse Problem (Synthesis)Inverse Problem (Synthesis)

How to implement it??

Homogenization method can be combined with optimization algorithms to design composite materials with

desired performance

Homogenization method can be combined with optimization algorithms to design composite materials with

desired performance

Consider the periodic composite:

Unit Cell

Effective Properties(“equivalent” homogeneous

medium)

Effective Properties(“equivalent” homogeneous

medium)Depend on

unit cell topologyDepend on

unit cell topology

Change effective

properties !!

Change effective

properties !!Changing unitcell topology

Changing unitcell topology

Material Design - Introduction

Material Design Method

Calculation of Effective properties

Calculation of Effective properties

Homogenization Method

Homogenization Method

Change of Unit Cell Topology

Change of Unit Cell Topology

Topology OptimizationTopology

Optimization

+

Design in a mesoscopic scale rather than amicroscopic scale (Physics approach)

• Design of negative Poisson’s ratio materials (Bendsoe 1989, Sigmund 1994, Fonseca 1997)

• Design of thermoelastic materials (Sigmund and Torquato 1996, Chen and Kikuchi 2001)

• Design of Piezocomposite materials(Silva and Kikuchi 1998)

• Design of Band-Gap materials(Sigmund and Jensen 2002)

How to build them?• Rapid Prototyping Techniques• Microfabrication technique (described ahead)

Material Design Method

Topology Optimization Concept

It combines FEM with optimization algorithms to find the optimum material distribution inside of a fixed design

domain

It combines FEM with optimization algorithms to find the optimum material distribution inside of a fixed design

domain

It turns the design process more generic and systematic,and independent of engineer previous knowledgment.Largely applied to automotive and aeronautic industries to design optimized parts. In addition, has been applied to:• Design of compliant mechanisms;• Design of piezoelectric actuators;• Design of “MEMS”;• Design of electromagnetic devices;• Design of composite materials

It turns the design process more generic and systematic,and independent of engineer previous knowledgment.Largely applied to automotive and aeronautic industries to design optimized parts. In addition, has been applied to:• Design of compliant mechanisms;• Design of piezoelectric actuators;• Design of “MEMS”;• Design of electromagnetic devices;• Design of composite materials

?

Topology Optimization Concept

Optimum topology

Topology Optimization Concept

Based on two main concepts:

•Extended Fixed Domain

•Relaxation of the Design Variable

Based on two main concepts:

•Extended Fixed Domain

•Relaxation of the Design Variable

Old approach: Find the boundaries of the unknown structure (Zienkiewicz and Campbell 1973)

Old approach: Find the boundaries of the unknown structure (Zienkiewicz and Campbell 1973)

New approach: Find thematerial distribution in the extended fixed domain (Bendsφe and

Kikuchi 1988)

New approach: Find thematerial distribution in the extended fixed domain (Bendsφe and

Kikuchi 1988)

Extended Fixed Domain

t

Ωd - Unknown DomainΩ − Extended Domain

t

The material model formulation for intermediate materials defines the level of problem relaxation.

?

0 1

The use of discrete values will cause numerical instabilities due to multiple local minimum.Thus, the material must assume intermediate property values during the optimization mixture law or material model.

How to change the material from zero to one?

Relaxation of the Design Problem

x2

x1

Ω

t

Structure Design Domain

1

1y2

y1

θa

b

Relaxation of the Design Problem

A point with no material

A point with material

Material Model: Density MethodMaterial Model: Density Method

E x Eijklp

ijkl= 0

property

fraction of material in each point

In each finite element n property cn is given by:In each finite element n property cn is given by:

Example of DiscretizedUnit Cell Domain

Example of DiscretizedUnit Cell Domain

Material Design

;0cxc pnn =

0c : property of basic material

End of Optimization: xn=xlow element is “air”xn=1 element is full of material

Maximize: F(x), where x=[x1,x2,…,xn,…,xNDV]x

subject to: cijkl clow, i, j, k, l are specified values0<xlow xn 1

symmetry conditions

Maximize: F(x), where x=[x1,x2,…,xn,…,xNDV]x

subject to: cijkl clow, i, j, k, l are specified values0<xlow xn 1

symmetry conditions

F(x) - function of effective propertiesx - design variablesW - constraint to reduce intermediate densities

(Vn - volume of each element)

Optimization Problem

≤ ≤

W x V Wnp

n

NDV

n low= >=

∑1

≥

Converged?

Flow Chart of the Optimization Procedure

Initial Guess

Final Topology

Y

N

Plotting ResultsPlotting Results

Optimizing (SLP)with respect to x

Optimizing (SLP)with respect to x

Initializing andData Input

Initializing andData Input

ObtainingHomogenized

Properties

ObtainingHomogenized

Properties

Updating Material

Distribution

Updating Material

Distribution

Calculating Sensitivity

Calculating Sensitivity

(Fonseca 1997)•Plane Stress•Isotropic•Poisson’s ratio = -0.5

Example

Composite MaterialUnit Cell

Example

• Orthotropic • Two negative and one positive Poisson’s ratio

(Fonseca 1997)

Unit Cell

Example

(Sigmund&Torquato 1996)

Thermoelastic Composites

(zero thermal expansion)

(negative thermal expansion)

(high positive thermal expansion)

(negative thermal expansion)

45

PZT5A

air

2D Piezocomposite Unit Cell (hydrophone)

Improvement in relation to the 2-2 piezocomposite unit cell:|dh|: 3. times

dhgh: 9.22 timeskh: 3.6 times stiffness constraint: cE

33>1.1010N/m2

Improvement in relation to the 2-2 piezocomposite unit cell:|dh|: 3. times

dhgh: 9.22 timeskh: 3.6 times stiffness constraint: cE

33>1.1010N/m2

PZT5A

1

3

Initially Optimized Microstructure Piezocomposite

“optimized porous ceramic”“optimized porous ceramic”

Example

46

Composite Manufacturing

Theoretical unit cell

Fugitive

Ceramic

Feedrod

Reduction Zone

ExtrudateSEM Image Crumm and Halloran (1997)

Microfabrication by coextrusion techniqueMicrofabrication by coextrusion technique

1

380 µm

Experimental Verification

Measured Performancesdh(pC/N) dhgh (fPa-1)

Solid PZT 68. 220.Optimized 308. 18400. (Simulation) (257.) (19000.)

TheoreticalTheoretical PrototypePrototype

48

z y

x

piezoceramic

Improvement in relation to the reference unit cells:|dh|: 5 times

dhgh: 45 timeskh : 3.71 times stiffness constraint: cE

zz>4.109N/m2

Improvement in relation to the reference unit cells:|dh|: 5 times

dhgh: 45 timeskh : 3.71 times stiffness constraint: cE

zz>4.109N/m2

3D Piezocomposite Unit Cell (hydrophone)

Poled in the z direction

Example

Composite Manufacturing

Rapid Prototyping: Stereolithography

Technique3D prototypes

Rapid Prototyping: Stereolithography

Technique3D prototypes

Recent works in the Field

•Fujii et al. 2001 - Design of 2D thermoelasticmicrostructures;

•Torquato et al. 2003 - Design of 3D composite with multifunctional characteristics;

• Guedes et al. 2003 - Energy bounds for two-phase composites

• Diaz and Bénard 2003 - Material Design using polygonal cells

•Fujii et al. 2001 - Design of 2D thermoelasticmicrostructures;

•Torquato et al. 2003 - Design of 3D composite with multifunctional characteristics;

• Guedes et al. 2003 - Energy bounds for two-phase composites

• Diaz and Bénard 2003 - Material Design using polygonal cells

• The results presented give us an idea about the potentiality of applying homogenization and optimization methods to model and design composite materials. However, synthesis methods for designing these materials are still in the beginning, and the performance limits of advanced composite materials can be improved more;

• Design of FGM materials using topology optimization will allow us to explore the potential of FGM concept;

• As a future trend, the design of composite materials considering nanoscale unit cells started been studied by some scientists.

• The results presented give us an idea about the potentiality of applying homogenization and optimization methods to model and design composite materials. However, synthesis methods for designing these materials are still in the beginning, and the performance limits of advanced composite materials can be improved more;

• Design of FGM materials using topology optimization will allow us to explore the potential of FGM concept;

• As a future trend, the design of composite materials considering nanoscale unit cells started been studied by some scientists.

Conclusion

Theoretical Formulation

Homogenizaton Procedure

• Strains are expanded as a global function plus a local oscillation proportional to the global strain;• The unit cell response (microscopic strain) is obtained considering independent load cases (unit strains) under periodic boundary conditions;• The microscopic strains are integrated to obtain composite “average” properties;• After a global analysis, the strains inside the cell can be obtained by using the localization functions;

• Strains are expanded as a global function plus a local oscillation proportional to the global strain;• The unit cell response (microscopic strain) is obtained considering independent load cases (unit strains) under periodic boundary conditions;• The microscopic strains are integrated to obtain composite “average” properties;• After a global analysis, the strains inside the cell can be obtained by using the localization functions;