MODELLING BAMBOO AS A FUNCTIONALLY GRADED MATERIAL Emílio Carlos Nelli Silva Associate Professor Department of ofmechatronics and Mechanical Systems Engineering Escola Politécnica da dauniversidade de de São Paulo, São Paulo, SP, Brazil Glaucio H. H. Paulino and Mattew C. C. Walters Department of of Civil and Environmental Engineering University of of Illinois at at Urbana-Champaign, Urbana, IL, IL, USA Acknowledgments: NSF NSF (USA)
Outline hintroduction to Natural Fibers and FGM hobjective and Motivation hgraded Finite Element hhomogenization Applied to Axisymmetric Composites hresults hconclusions
Introduction Natural Fibers in Engineering Biological Structures Optimized to to the the loading conditions they are are subjected; Multifunctional and adaptable; Natural Fibers Low cost production; Available mainly in in tropical and subtropical regions of of the the world; Examples of of Natural Fibers: bamboo, coconut fibers, sisal, etc... Promising material in in housing construction at at underdeveloped or or developed countries (also composites such as, as, bamboo + concrete); Bamboo (Prof. Ghavami)
Concept of FGM Materials FGM materials possess continuously graded properties with gradual change in in microstructure which avoids interface problems, such as, stress concentrations. T Hot Microstructure } Ceramic Phase }} Ceramic matrix with metallic inclusions } Metallic matrix Types of of gradation Transition region 1-D T Cold with ceramic inclusions } Metallic Phase 2-D 3-D
Applications of FGM Materials 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
Natural FGM Materials Bamboo is is a FGM material Distribution of fibers in the thickness Cross section of culm (70 % is made of natural fibers) FGM sections along length (Prof. Ghavami)
Motivation Most part of of works done with bamboo are are experimental work (to (to find bamboo strength and stiffness properties). Very few works about Bamboo modeling (usually analytical work); Due to to complicated shapes and material distribution, the the use of of numerical methods such as as (finite element method) FEM can be be a great tool to to help us us to to understand the the mechanical behavior of of these structures; Bamboo is is a composite material where a microstructure can be be identified, thus, multiscale methods, such as as homogenization can be be applied; Traditional FEM gives a wrong stress distribution for for the the FGM materials (layer approximation) graded finite element concept;
Objective To apply computational techniques such as as FEM and a multiscale method (based on homogenization) to to characterize the bamboo tree behavior. FEM formulation will be based on the so-called graded finite element concept continuous material distribution inside of of the domain; Homogenization theory is is extended for axissymetric composites;
Graded Finite Element E Continuous distribution Layered approximation x Traditional FEM layered approximation (highly inaccurate) Graded finite element [Kim and Paulino 2002] continuous material distribution inside unit cell E E L E I J E K I L J K E: E: material property E I : I : material property evaluated at at FEM nodes
Homogenization - Multiscale Method Homogenization method allows the calculation of of composite effective properties knowing the topology of of the composite unit cell. Example of application: F F a) perforated beam unit cell Homogenized homogenized Material material b) Homogenized homogenized Material material brick wall unit cell
Concept of Homogenization Method It It allows the replacement of of the composite medium by an equivalent homogeneous medium to to solve the global problem. Advantage in relation to other methods: it it needs only the information about the unit cell the unit cell can have any complex shape Complex unit cell topologies implementation using FEM Analytical methods Mixture rule models - no interaction between phases Self-consistent methods - some interaction, limited to simple geometries
Concept of Homogenization Method Assumptions x Component Enlarged Periodic Microstructure 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) y Enlarged Unit Cell (Microscale)
Extension to Other Fields hflow in 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 et al. (1992), Turbé and Maugin (1991), Otero et et al. (1997) etc
Axisymmetric Composites Unit cell The unit cell has a plane strain behavior!! Bamboo is is a FGM axisymmetric composite
Physical Concept of Homogenization Unit Cell Load Cases (2D model) Unit Cell periodicity conditions enforced in the unit cell Solutions using FEM Calculation of effective properties (c H )
Bamboo Modelling Bamboo Structure Internodal region Thickness of walls Diaphragm lacuna (Prof. Ghavami)
Bamboo Modelling Properties considered for bamboo [Nogata & Takahashi, 1995]: Young s modulus fiber: 55 GPa Young s modulus matrix: 2 GPa Poisson s ratio: 0.35 2.2 r / t FGM Law: E = 3.75e Dimensions: External diameter: 80mm Internal diameter: 56mm Thickness: 12mm Internodal distance: 350mm
Homogenization Results Unit cell Unit cell mesh: 20 20 x 20 20 isoparametric 4-node finite elements Obtained homogenized properties for for bamboo: E H 12.54 5.37 5.37 0 5.37 18.41 6.81 0 = 5.37 6.81 17.33 0 0 0 0 3.58 GPa Axisymmetric tensor properties These properties allow us us to to model bamboo as as an an orthotropic homogeneous medium
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; Three load cases: tension, torsion, bending
FEM Models One Cell mesh: 7,380 20-node brick finite elements (33,794 nodes) Two Cells mesh: 14,760 20-node brick finite elements (66,417 nodes)
Applied loads and Boundary Conditions Tension Torsion Bending
Tension Results Deformed Shape Homogeneous Isotropic FGM
Tension Results Von Mises Stress Distribution Homogeneous Isotropic FGM
Torsion Results Deformed Shape Homogeneous Isotropic FGM
Torsion Results Von Mises Stress Distribution Homogeneous Isotropic FGM
Bending Results Deformed Shape Homogeneous Isotropic FGM
Bending Results Von Mises Stress Distribution Homogeneous Isotropic FGM
Comparison Displacements Load Case Homogeneous FGM Tension 23.22 22.80 Torsion 0.143 0.121 Bending 27.61 23.57
Continuation of the Work How optimal is bamboo? Structural optimization techniques such as topology optimization can be applied to answer this question!
Topology Optimization Concept? Optimum topology
Formulation of Optimization Problem Layered structure Plane stress behavior ρ 1 ρ 2 ρ 3 ρ i Material Model: E = ρie1 + ( 1 ρi ) E2 Max C = mean ρ I (for each node) such that [ K]{ U} = { F} I = 1 0 N { U} t { F} ρ 1 I I ρ fv 0
Example Design of of horizontal layered FGM structures 20 20 % volume constraint Two materials are are considered E 11 =1, E 22 =10, ν 11 =ν =ν 22 =0.3 Boundary conditions Optimal topologies Obtained property distribution in y- direction y x
Conclusions Numerical simulations of of bamboo structure using finite element method and a multi-scale method were performed; By By using the the graded finite element concept the the continuous change of of bamboo properties along the the thickness could be be taken into account, and its its influence in in the the bamboo mechanical behavior was shown; By By using homogenization method, the the effective properties of of bamboo, were calculated allowing us us to to model bamboo as as a homogeneous medium; Numerical simulation is is a powerful tool to to model natural fiber composites helping us us to to understand their behavior.
Flow Chart of the Optimization Procedure Initializing and data input Calculating (FEM) Mean Compliance Initially Calculating objective function and constraints Updating material distribution (design variables) Final Topology Plotting results Y Converged? Optimizing (Optimality Criteria) with respect to to (ρ) (ρ) Calculating sensitivity N