HOMOGENIZATION METHOD APPLIED TO THE HOMOGENIZATION METHOD APPLIED - - PowerPoint PPT Presentation

HOMOGENIZATION METHOD APPLIED TO THE DEVELOPMENT OF COMPOSITE MATERIALS HOMOGENIZATION METHOD APPLIED TO THE DEVELOPMENT OF COMPOSITE MATERIALS Emílio Carlos Nelli Silva

Associate Professor Department of Mechatronics and Mechanical System Engineering Escola Politécnica da Universidade de São Paulo Brazil

Emílio Carlos Nelli Silva

Associate Professor Department of Mechatronics and Mechanical System Engineering Escola Politécnica da Universidade de São Paulo Brazil US-South America Workshop: Mechanics and Advanced Materials – Research and Education

Rio de Janeiro, August 2004

Outline

h Introduction to Homogenization Method h Homogenization of FGM Materials h Topology Optimization Method h Material Design Concept h Conclusions and Future Trends h Introduction to Homogenization Method h Homogenization of FGM Materials h Topology Optimization Method h Material Design Concept h Conclusions and Future Trends

F F unit cell unit cell homogenized material a) b) brick wall perforated beam homogenized material

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

Enlarged

Periodic Microstructure x y Enlarged Unit Cell (Microscale)

Assumptions

Concept of Homogenization Method

Literature Review

Theory development (elastic medium):

hSanchez-Palencia (1980) - France hDe Giorgi and Spagnolo (1973) (G-convergence) - Italy hDuvaut (1976) and Lions (1981) - France hBakhvalov and Panasenko (1989) - Soviet Union

Numerical Implementation using FEM:

hLéné (1984) - France hGuedes and Kikuchi (1990) - USA

Dispersive behavior:

hTurbé (1982) - France

Theory development (elastic medium):

hSanchez-Palencia (1980) - France hDe Giorgi and Spagnolo (1973) (G-convergence) - Italy hDuvaut (1976) and Lions (1981) - France hBakhvalov and Panasenko (1989) - Soviet Union

Numerical Implementation using FEM:

hLéné (1984) - France hGuedes 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 microstructure microscale, and u1 is Y-periodic first order variation term.

Theoretical Formulation

Component

Enlarged

Periodic Microstructure x y Enlarged Unit Cell (Microscale)

ε

u u x u x y

ε

ε = +

1

( ) ( , ) uε u x y

1( , )

microscopic equations (δu1(x,y) terms) FEM solution of microscopic equations for χ

u x y u x

1

= χ ε ( , ) ( ( ))

u u x u x y

ε

ε = +

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 χ

i I iI

N ≅

I=1,NN NN = → →    4 8 nodes 2D case nodes 3D case

[ ]{

} { }

K F

mn mn

χ

( ) ( )

=

Bilinear (2D) and trilinear (3D) interpolation functions FEM system of equations:

6 for 3D 3 for 2D   

load cases mn

periodicity conditions enforced in the unit cell

Homogenization Implementation

cH,

Unit Cell

6 for 3D 3 for 2D   

Number of load cases: FEM model and Data Input Assembly of Stiffness Matrix Solver Last? Calculation of Homogenized Coefficients Y N

12

periodicity conditions enforced 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

Example

Homogenization of composite material with solid and fluid phases

Discretized Unit Cell

Homogenization of woven fabric composites

Example

Discretized Unit Cell 230.000 brick elements 230.000 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 Materials

Example: Piezoelectric Material

cE

ijkl - stiffness property

eikl - piezoelectric strain

property

εS

ik - dielectric property

Tij - stress Skl - strain Ek - electric field Di - electric displacement

Constitutive Equations of Piezoelectric Medium

T c S e E D E e S

ij ijkl E kl kij k i ik S k ikl kl

= − = +    ε

Elasticity equation Electrostatic equation

hProperties cEijkl, eijk, and εSij are Y-periodic functions

(Y - unit cell domain).

hAsymptotic expansion:

• displacements:
• electric potential:

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

Homogenization for Piezoelectricity

u u x u x y

ε

ε = +

1

( ) ( , )

φ φ εφ

ε =

+

1

( ) ( , ) x x y

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

20

periodicity conditions enforced in the unit cell periodicity conditions enforced in the unit cell

Homogenization Implementation

Calculation of effective properties (cEH, eH, and εSH)

Unit Cell

9 for 3D model 5 for 2D model   

Number of load cases Load Cases (2D model) Load Cases (2D model)

Solutions using FEM

Unit Cell

21

10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1

Ceramic Volume Fraction (%) dh (pC/N)

Rectangular inclusion Circular inclusion Rectangular inclusion (hex.)

polymer piezoceramic

3D Piezocomposite Unit Cell

“staggered formation” “square inclusion”

dh

“circular inclusion”

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 Phase

TCold

Ceramic Phase

Microstructure Microstructure Types of gradation Types of gradation

Collaboration USP/UIUC

• Emilio visited UIUC in December 2003
• Glaucio visited USP and LNLS (Syncroton Light National

Laboratory – 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 National

Laboratory – 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 of São Paulo University of Illinois at 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 I I I N

E E

1

x x

To solve homogenization equations the graded finite element (Kim and Paulino 2002) is used which considers a continuous distribution of material inside unit cell E: material property EI: material property evaluated at FEM nodes x=(x, y): position Cartesian coordinates E: material property EI: material property evaluated at FEM nodes x=(x, y): position Cartesian coordinates

I J K L EI EJ EK EL

Example

Material 1 Material 2

Plane Strain assumption

3 . . 2 .; 1 .; 1 E ; 8

2 1 2 1 2 1

α β α α E = = = = = = = v v E

Material properties: x y

. 72 . 6 73 . 5 ; 29 . 1 . . . 83 . 4 1 . 1 . 1 . 1 68 . 3

H

          =           = β

H

E

Homogenized properties

Elastic properties Thermoelastic properties Unit Cell

( ) ( ) 2

/ ' 2 cos ) ( 2 / ' 2 cos ) (

2 1 2 1 2 1 2 1

β β π β β β π + + − = + + − = x x E E E E E FGM law:

Example

Axyssimetric composite (Application to bamboo and natural fiber composites)

Material 1 Material 2 Unit Cell

3 . .; 2 E ; 10

2 1 2 1

= = = = v v E

Elastic properties:

89 . 1 . . . . 49 . 6 1.65 62 . 1 . 1.65 5.85 01 . 2 . 62 . 1 01 . 2 39 . 5             =

H

E

Homogenized properties

( ) 2

/ ' 2 cos ) (

2 1 2 1

E E E E E + + − = r π FGM law:

Material Design - Introduction

Specify the desired Material properties Specify the desired Material properties Design the Composite Material Design the Composite Material Inverse Problem (Synthesis) Inverse Problem (Synthesis)

How to implement it??

Homogenization method can be combined with

• ptimization algorithms to design composite materials with

desired performance Homogenization method can be combined with

• ptimization 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 topology Depend on unit cell topology Change effective properties !! Change effective properties !! Changing unit cell topology Changing unit cell 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 Optimization Topology Optimization

+

Design in a mesoscopic scale rather than a microscopic 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 the material distribution in the extended fixed domain (Bendsφe and Kikuchi 1988) New approach: Find the material distribution in the extended fixed domain (Bendsφe and Kikuchi 1988)

Extended Fixed Domain

t Ωd - Unknown Domain Ω − Extended Domain t

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

?

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 1

y2 y1

θ a b

Relaxation of the Design Problem

A point with no material A point with material

Material Model: Density Method Material Model: Density Method

E x E

ijkl p ijkl

=

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 Discretized Unit Cell Domain Example of Discretized Unit Cell Domain

Material Design

; c x c

p n n =

c : 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 values 0<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 values 0<xlow xn 1 symmetry conditions F(x) - function of effective properties x - design variables W - constraint to reduce intermediate densities (Vn - volume of each element)

Optimization Problem

≤ ≤

W x V W

n p n NDV n low

= >

=

∑

1

≥

Converged?

Flow Chart of the Optimization Procedure

Initial Guess Final Topology

Y N Plotting Results Plotting Results Optimizing (SLP) with respect to x Optimizing (SLP) with respect to x Initializing and Data Input Initializing and Data Input Obtaining Homogenized Properties Obtaining Homogenized Properties Updating Material Distribution Updating Material Distribution Calculating Sensitivity Calculating Sensitivity

(Fonseca 1997)

• Plane Stress
• Isotropic
• Poisson’s ratio = -0.5

Example

Composite Material Unit 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 times kh: 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 times kh: 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 Extrudate SEM Image Crumm and Halloran (1997)

Microfabrication by coextrusion technique Microfabrication by coextrusion technique

1 3

80 µm

Experimental Verification

Measured Performances dh(pC/N) dhgh (fPa-1)

Solid PZT 68. 220. Optimized

• 308. 18400.

(Simulation)

(257.) (19000.) Theoretical Theoretical Prototype Prototype

48

z y x

piezoceramic Improvement in relation to the reference unit cells: |dh|: 5 times dhgh: 45 times kh : 3.71 times stiffness constraint: cE

zz>4.109N/m2

Improvement in relation to the reference unit cells: |dh|: 5 times dhgh: 45 times kh : 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 Technique 3D prototypes Rapid Prototyping: Stereolithography Technique 3D prototypes

Recent works in the Field

• Fujii et al. 2001 - Design of 2D thermoelastic

microstructures;

• 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 thermoelastic

microstructures;

• 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
• scillation 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
• scillation 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;