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ABMTENC - www.abmtenc.civ.puc-rio.br

MODELLING BAMBOO AS A FUNCTIONALLY

GRADED MATERIAL

MODELLING BAMBOO AS A FUNCTIONALLY

GRADED MATERIAL

Emílio Carlos Nelli Silva

Associate Professor

Department of Mechatronics and Mechanical Systems Engineering

Escola Politécnica da Universidade de São Paulo, São Paulo, SP,Brazil

Glaucio H. Paulino and Mattew C. Walters

Department of Civil and Environmental Engineering

University of Illinois at Urbana-Champaign, Urbana, IL, USA

Emílio Carlos Nelli Silva

Associate Professor

Department of Mechatronics and Mechanical Systems Engineering

Escola Politécnica da Universidade de São Paulo, São Paulo, SP,Brazil

Glaucio H. Paulino and Mattew C. Walters

Department of Civil and Environmental Engineering

University of Illinois at Urbana-Champaign, Urbana, IL, USA

Acknowledgments:

NSF (USA)NSF


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Outline

hIntroduction to Natural Fibers and FGM

hObjective and Motivation

hGraded Finite Element

hHomogenization Applied to AxisymmetricComposites

h

ResultshConclusions

hIntroduction to Natural Fibers and FGM

hObjective and Motivation

hGraded Finite Element

hHomogenization Applied to AxisymmetricComposites

hResults

hConclusions


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Introduction Natural Fibers in Engineering

• Optimized to the loading conditions they are subjected;• Multifunctional and adaptable;

• Optimized to the loading conditions they are subjected;• Multifunctional and adaptable;

• Low cost production;• Available mainly in tropical and

subtropical regions of the world;

• Examples of Natural Fibers: bamboo,

coconut fibers, sisal, etc...• Promising material in housing

construction at underdeveloped or

developed countries (also composites such

as, bamboo + concrete);

• Low cost production;

• Available mainly in tropical andsubtropical regions of the world;

• Examples of Natural Fibers: bamboo,

coconut fibers, sisal, etc...

• Promising material in housing

construction at underdeveloped or

developed countries (also composites such

as, bamboo + concrete);

Natural Fibers

Biological Structures

Bamboo (Prof. Ghavami)


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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

}THotCeramic matrix

with metallic

inclusions}}}}

}}}}}}}}Metallic matrixwith ceramicinclusions

Transition region

}}}} Metallic PhaseTCold

Ceramic Phase

MicrostructureMicrostructure Types of gradationTypes of gradation


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• Superheat-resistance

Thermal barrier coatings, aero-spacestructures

• Biomedical

Dental and bone implants, Artificialskin

• MilitaryMilitary vehicles and personal body

armor

• Electro-magnetic and MEMS

Piezoelectric and thermoelectricdevices, sensors

• Optical

Graded refractive index material

Applications of FGM Materials


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Natural FGM Materials

Bamboo is a FGM materialBamboo is a FGM material

Cross section of culm(70 % is made of natural fibers)

(Prof. Ghavami)

Distribution of

fibers in the

thickness

FGM sections

along length


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Motivation

• Most part of works done with bamboo are experimental work

(to find bamboo strength and stiffness properties). Very few

works about Bamboo modeling (usually analytical work);

• Due to complicated shapes and material distribution, the use of

numerical methods such as (finite element method) FEM can be a

great tool to help us to understand the mechanical behavior ofthese structures;

• Bamboo is a composite material where a microstructure can be

identified, thus, multiscale methods, such as homogenization can

be applied;• Traditional FEM gives a wrong stress distribution for the FGM

materials (layer approximation) graded finite element concept;

• Most part of works done with bamboo are experimental work

(to find bamboo strength and stiffness properties). Very few

works about Bamboo modeling (usually analytical work);• Due to complicated shapes and material distribution, the use of

numerical methods such as (finite element method) FEM can be a

great tool to help us to understand the mechanical behavior of

these structures;

• Bamboo is a composite material where a microstructure can be

identified, thus, multiscale methods, such as homogenization can

be applied;• Traditional FEM gives a wrong stress distribution for the FGM

materials (layer approximation) graded finite element concept;


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Objective

• To apply computational techniques such as FEM and

a multiscale method (based on homogenization) to

characterize the bamboo tree behavior.

• FEM formulation will be based on the so-calledgraded finite element concept continuous material

distribution inside of the domain;

• Homogenization theory is extended for axissymetriccomposites;

• To apply computational techniques such as FEM anda multiscale method (based on homogenization) to

characterize the bamboo tree behavior.

• FEM formulation will be based on the so-calledgraded finite element concept continuous material

distribution inside of the domain;

• Homogenization theory is extended for axissymetriccomposites;


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Graded Finite Element

• Traditional FEM layered

approximation (highly inaccurate)

• Graded finite element [Kim and

Paulino 2002] continuous

material distribution inside unit cell

• Traditional FEM layered

approximation (highly inaccurate)

• Graded finite element [Kim andPaulino 2002] continuous

material distribution inside unit cell

I

J

KL

EIEJ EK

EL

x

EContinuous

distribution

Layered

approximation

E : material property E I : material property

evaluated at FEM nodes

E : material property E I : material property

evaluated at FEM nodes


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Homogenization - Multiscale Method

F F

unit cell

unit cell

homogenizedmaterial

a)

b)

brick wall

perforated beam

homogenized

material

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.


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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


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• Periodic composites;

hAsymptotic analysis, mathematically correct;

h Scale of microstructure must be very small compared tothe size of the part;

• Acoustic wavelength larger than unit cell dimensions.

(Dispersive behavior can also be modeled)

Component EnlargedPeriodic Microstructure

x y

Enlarged

Unit Cell (Microscale)

Assumptions

Concept of Homogenization Method


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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


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Axisymmetric Composites

Bamboo is a FGM

axisymmetric

composite

Bamboo is a FGM

axisymmetric

composite

Unit cell

The unit cell has a

plane strain behavior!!

The unit cell has a

plane strain behavior!!


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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


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Thickness of

walls

lacunaDiaphragm

Internodal region

Bamboo StructureBamboo Structure

(Prof. Ghavami)

Bamboo Modelling


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Properties considered for bamboo [Nogata &

Takahashi, 1995]:

• Young’s modulus fiber: 55 GPa• Young’s modulus matrix: 2 GPa

• Poisson’s ratio: 0.35

• FGM Law:Dimensions:

• External diameter: 80mm

• Internal diameter: 56mm• Thickness: 12mm

• Internodal distance: 350mm

Properties considered for bamboo [Nogata &

Takahashi, 1995]:

• Young’s modulus fiber: 55 GPa• Young’s modulus matrix: 2 GPa

• Poisson’s ratio: 0.35

• FGM Law:Dimensions:

• External diameter: 80mm

• Internal diameter: 56mm

• Thickness: 12mm

• Internodal distance: 350mm

2.2 /

3.75 r t

E e=

Bamboo Modelling


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Homogenization Results

Axisymmetric

tensor properties

Axisymmetric

tensor properties

Unit cell mesh: 20 x 20

isoparametric 4-node finite

elements

Unit cell mesh: 20 x 20

isoparametric 4-node finite

elements

Obtained homogenized properties for bamboo:Obtained homogenized properties for bamboo:

12.54 5.37 5.37 0

5.37 18.41 6.81 0 GPa

5.37 6.81 17.33 0

0 0 0 3.58

H

=

E

Unit cell

These properties allow us to model bamboo as

an orthotropic homogeneous medium

These properties allow us to model bamboo as

an orthotropic homogeneous medium


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Macro Behavior Modelling

Two different material distributions are considered:• Homogeneous isotropic with properties averaged

along bamboo thickness: E=13.68 Gpa, ν=0.35;

• Isotropic FGM considering the described FGM law;

Two different material distributions are considered:• Homogeneous isotropic with properties averaged

along bamboo thickness: E=13.68 Gpa, ν=0.35;

• Isotropic FGM considering the described FGM law;

Three load cases: tension, torsion, bendingThree load cases: tension, torsion, bending


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FEM Models

mesh: 7,380

20-node brick

finite elements

(33,794 nodes)

mesh: 7,380

20-node brick

finite elements

(33,794 nodes)

One Cell

Two Cells

mesh: 14,760

20-node brickfinite elements

(66,417 nodes)

mesh: 14,760

20-node brickfinite elements

(66,417 nodes)


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Applied loads and Boundary Conditions

TensionTension TorsionTorsion

BendingBending


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Tension Results

Deformed ShapeDeformed Shape

Homogeneous IsotropicHomogeneous Isotropic FGMFGM


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Tension Results

Von Mises Stress DistributionVon Mises Stress Distribution



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Torsion Results


FGMFGMHomogeneous IsotropicHomogeneous Isotropic


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Torsion Results




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Bending Results




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Bending Results




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Comparison

Displacements

23.5727.61Bending

0.1210.143Torsion

22.8023.22Tension

FGMHomogeneousLoad Case


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Continuation of the Work

How optimal is bamboo?

Structural optimization techniques such as

topology optimization can be applied to answer

this question!

Structural optimization techniques such as

topology optimization can be applied to answer

this question!


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?

Topology Optimization Concept

Optimum topology

F l i f O i i i P bl


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Formulation of Optimization Problem

Max

such that [ ]{ } { }

10

0

1

≤≤

≤

=

∑=

I

N

I

I fV

ρ

ρ

FUK

ρ I (for each node){ } { }FU

t

meanC =

ρ i

ρ 1 ρ 2 ρ 3• Layered structure

• Plane stress behavior

• Layered structure

• Plane stress behavior

Material Model:Material Model: 21 )1( EEE ii ρ ρ −+=

E l


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Example

Design of horizontal layered FGM structures20 % volume constraint

Two materials are considered E1=1, E2=10, ν1= ν2=0.3

Design of horizontal layered FGM structures20 % volume constraint

Two materials are considered E1=1, E2=10, ν1= ν2=0.3

Obtained property

distribution in y-

directionOptimal topologies

Boundary

conditions

x

y

C l i


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Conclusions

• Numerical simulations of bamboo structure using finite element

method and a multi-scale method were performed;• By using the graded finite element concept the continuous

change of bamboo properties along the thickness could be taken

into account, and its influence in the bamboo mechanical

behavior was shown;• By using homogenization method, the effective properties of

bamboo, were calculated allowing us to model bamboo as a

homogeneous medium;

• Numerical simulation is a powerful tool to model natural fibercomposites helping us to understand their behavior.

• Numerical simulations of bamboo structure using finite element

method and a multi-scale method were performed;

• By using the graded finite element concept the continuous

change of bamboo properties along the thickness could be taken

into account, and its influence in the bamboo mechanical

behavior was shown;• By using homogenization method, the effective properties of

bamboo, were calculated allowing us to model bamboo as a

homogeneous medium;

• Numerical simulation is a powerful tool to model natural fibercomposites helping us to understand their behavior.

Fl Ch f h O i i i P d


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Flow Chart of the Optimization Procedure

Initializing and

data input

Initializing and

data input

Calculating (FEM)

Mean Compliance

Calculating (FEM)

Mean Compliance

Calculating

objective function

and constraints

Calculating

objective function

and constraints

Initially

Converged?Plotting resultsPlotting results

Calculating

sensitivity

Calculating

sensitivity

Optimizing

(Optimality Criteria)

with respect to ( ρ )

Optimizing

(Optimality Criteria)

with respect to ( ρ )

Updating material

distribution (design

variables)

Updating material

distribution (design

variables)

Final Topology

N

Y

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