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    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 Axisymmetric Composites

    h

    Results hConclusions

    hIntroduction to Natural Fibers and FGM

    hObjective and Motivation

    hGraded Finite Element

    hHomogenization Applied to Axisymmetric Composites

    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 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);

    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

    }THot Ceramic matrix

    with metallic

    inclusions }}}}

    }}}}}}}}Metallic matrixwith ceramic inclusions

    Transition region

    }}}} Metallic PhaseTCold

    Ceramic Phase

    MicrostructureMicrostructure Types of gradationTypes of gradation

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    • Superheat-resistance

    Thermal barrier coatings, aero-space structures

    • Biomedical

    Dental and bone implants, Artificial skin

    • Military Military vehicles and personal body

    armor

    • Electro-magnetic and MEMS

    Piezoelectric and thermoelectric devices, 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-called graded finite element concept continuous material

    distribution inside of the domain;

    • Homogenization theory is extended for axissymetric composites;

    • 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-called graded finite element concept continuous material

    distribution inside of the domain;

    • Homogenization theory is extended for axissymetric composites;

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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 and Paulino 2002] continuous

    material distribution inside unit cell

    I

    J

    KL

    EI EJ EK

    EL

    x

    E Continuous

    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

    homogenized material

    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 to the size