HOMOGENIZATION METHOD APPLIED TO THE HOMOGENIZATION METHOD APPLIED TO THE DEVELOPMENT OF COMPOSITE MATERIALS DEVELOPMENT OF COMPOSITE MATERIALS Emílio Carlos Nelli Silva Emílio Carlos Nelli Silva Associate Professor Associate Professor Department of Mechatronics and Mechanical System Department of Mechatronics and Mechanical System Engineering Engineering Escola Politécnica da Universidade de São Paulo Escola Politécnica da Universidade de São Paulo Brazil Brazil US-South America Workshop: Mechanics and Advanced Materials – Research and Education Rio de Janeiro, August 2004
Outline h Introduction to Homogenization Method h Introduction to Homogenization Method h Homogenization of FGM Materials h Homogenization of FGM Materials h Topology Optimization Method h Topology Optimization Method h Material Design Concept h Material Design Concept h Conclusions and Future Trends h Conclusions and Future Trends
Concept of Homogenization Method Homogenization method allows the calculation of Homogenization method allows the calculation of composite effective properties knowing the topology of composite effective properties knowing the topology of the composite unit cell. the composite unit cell. Example of application: F F Homogenized homogenized a) Material material unit cell perforated beam Homogenized homogenized b) Material material unit cell brick wall
Concept of Homogenization Method It allows the replacement of the composite medium by an It allows the replacement of the composite medium by an “equivalent” homogeneous medium to solve the global “equivalent” homogeneous medium to solve the global problem. problem. Advantage in relation to other methods: • it needs only the information about the unit cell • it needs only the information about the unit cell • the unit cell can have any complex shape • 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 y x Enlarged Enlarged Component Periodic Microstructure Unit Cell (Microscale) • Periodic composites ; h Asymptotic 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)
Literature Review Theory development (elastic medium): Theory development (elastic medium): h Sanchez-Palencia (1980) - France h Sanchez-Palencia (1980) - France h De Giorgi and Spagnolo (1973) (G-convergence) - Italy h De Giorgi and Spagnolo (1973) (G-convergence) - Italy h Duvaut (1976) and Lions (1981) - France h Duvaut (1976) and Lions (1981) - France h Bakhvalov and Panasenko (1989) - Soviet Union h Bakhvalov and Panasenko (1989) - Soviet Union Numerical Implementation using FEM: Numerical Implementation using FEM: h Léné (1984) - France h Léné (1984) - France h Guedes and Kikuchi (1990) - USA h Guedes and Kikuchi (1990) - USA Dispersive behavior: Dispersive behavior: h Turbé (1982) - France h Turbé (1982) - France
Extension to Other Fields h flow in porous media - Sanchez-Palencia (1980) h flow in porous media - Sanchez-Palencia (1980) h conductivity (heat transfer) - Sanchez-Palencia (1980) h conductivity (heat transfer) - Sanchez-Palencia (1980) h viscoelasticity - Turbé (1982) h viscoelasticity - Turbé (1982) h biological materials (bones) - Hollister and Kikuchi h biological materials (bones) - Hollister and Kikuchi (1994) (1994) h electromagnetism - Turbé and Maugin (1991) h electromagnetism - Turbé and Maugin (1991) h piezoelectricity - Telega (1990), Galka et al. (1992), h piezoelectricity - Telega (1990), Galka et al. (1992), Turbé and Maugin (1991), Otero et al. (1997) Turbé and Maugin (1991), Otero et al. (1997) etc … etc …
Theoretical Formulation ε u ε u x y 1 ( , ) y x Enlarged Enlarged Component Periodic Microstructure Unit Cell (Microscale) • Properties c ijkl are Y-periodic functions (Y - unit cell domain). • Asymptotic expansion: ε = + ε u u ( ) x u x y ( , ) - displacements: 0 1 where y=x/ ε and ε >0 is the composite microstructure microscale, and u 1 is Y-periodic first order variation term.
Theoretical Formulation ; y=x/ ε ε = + ε u u ( ) x u ( , ) x y 0 1 Energy Functional for the Medium macroscopic equations Theory of Asymptotic Analysis ( δ u 0 (x) terms) microscopic equations FEM solution of ( δ u 1 (x,y) terms) microscopic equations for χ Due to linearity: = χ ε u ( , ) ( x y u x ( )) 1 0 where χ is Y-periodic characteristic functions of the unit cell
FEM Solution → 4 nodes 2D case χ ≅ χ N NN = I=1,NN i I iI → 8 nodes 3D case Bilinear (2D) and trilinear (3D) interpolation functions Substitute in the system of microscopic equations ( χ ) FEM system of equations: 6 for 3D load [ ] { } { } cases χ = ( mn ) ( mn ) K F 3 for 2D mn
Homogenization Implementation FEM model and Data Input Unit Cell Assembly of Stiffness Matrix Number of load cases: Solver 6 for 3D periodicity conditions 3 for 2D enforced in the unit cell N Last? Y Calculation of Homogenized c H , Coefficients
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 ) 12
Example Homogenization of composite material with solid and fluid phases Discretized Unit Cell Solid phase Fluid phase
Example Homogenization of woven fabric composites Discretized Unit Cell 230.000 brick elements 230.000 brick elements
Example Homogenization of bone microstructure Solid phase Fluid phase (Hollister and Kikuchi - 1997)
Example “Representative Volume Element (RVE)” concept Micrograph of Metal Matrix Composites (MMC) Cr (fiber) - NiAl (matrix) RVE unit cell RVE unit cell There must be “statistic” periodicity !!!
Homogenization for Coupled Field Materials Example: Piezoelectric Material Force Electric potential Mechanical Mechanical Piezoelectric Electrical Piezoelectric Electrical Energy Energy Material Energy Material Energy Displacement Electric charge Examples: Quartz (natural) Examples: Quartz (natural) Ceramic (PZT5A, PMN, etc…) Ceramic (PZT5A, PMN, etc…) Polymer (PVDF) Polymer (PVDF) Applications: Pressure sensors, accelerometers, actuators, Applications: Pressure sensors, accelerometers, actuators, acoustic wave generation (ultrasonic transducers, sonars, acoustic wave generation (ultrasonic transducers, sonars, and hydrophones), etc... and hydrophones), etc...
Constitutive Equations of Piezoelectric Medium Elasticity equation = − E T c S e E ij ijkl kl kij k = ε + S D E e S i ik k ikl kl Electrostatic equation T ij - stress c E ijkl - stiffness property S kl - strain e ikl - piezoelectric strain E k - electric field property ε S ik - dielectric property D i - electric displacement
Homogenization for Piezoelectricity h Properties c Eijkl , e ijk , and ε Sij are Y-periodic functions (Y - unit cell domain). h Asymptotic expansion: ε = + ε u u ( ) x u x y ( , ) - displacements: 0 1 ε = φ φ + εφ ( ) x ( , ) x y - electric potential: 0 1 where y=x/ ε and ε >0 is the composite microstructure microscale, and u 1 and φ 1 are Y-periodic first order variation terms. Telega (1990), Galka et al. (1992), and Turbé and Maugin (1991)
Homogenization Implementation Load Cases (2D model) Load Cases (2D model) Unit Cell Unit Cell periodicity conditions periodicity conditions enforced in the unit cell enforced in the unit cell Solutions using FEM 9 for 3D model Number of load cases 5 for 2D model Calculation of effective properties (c EH , e H , and ε SH ) 20
Example 3D Piezocomposite Unit Cell polymer piezoceramic Performance quantity 80 “circular 70 inclusion” 60 50 d h ( p C/N) Rectangular inclusion Circular inclusion 40 “square Rectangular inclusion (hex.) d h 30 inclusion” 20 10 0 “staggered 0 0.2 0.4 0.6 0.8 1 Ceramic Volume Fraction (%) formation” 21
Concept of FGM materials FGM materials possess continuously graded properties FGM materials possess continuously graded properties with gradual change in microstructure which avoids with gradual change in microstructure which avoids interface problems, such as, stress concentrations. interface problems, such as, stress concentrations. Types of gradation Microstructure Types of gradation Microstructure } T Hot Ceramic Phase } Ceramic matrix with metallic } inclusions Transition region 1-D } Metallic matrix 2-D with ceramic inclusions } Metallic Phase 3-D T Cold
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