HOMOGENIZATION METHOD APPLIED TO THE DEVELOPMENT OF COMPOSITE MATERIALS Emílio Carlos Nelli Silva Associate Professor Department of of Mechatronics and Mechanical System Engineering Escola Politécnica

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HOMOGENIZATION METHOD APPLIED TO THE DEVELOPMENT OF COMPOSITE MATERIALS Emílio Carlos Nelli Silva Associate Professor Department of of Mechatronics and Mechanical System Engineering Escola Politécnica da da Universidade de 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 Concept of Homogenization Method brick wall 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 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 y Enlarged Unit Cell (Microscale) 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) 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 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 Theoretical Formulation ε x Component u ε Enlarged Periodic Microstructure u ( 1 x, y ) y Enlarged Unit Cell (Microscale) Properties c ijkl are Y-periodic functions (Y - unit cell domain). Asymptotic expansion: ε - displacements: u = u0( x) + εu1( x, y) where y=x/ε and ε 0 is the composite microstructure microscale, and u 1 is Y-periodic first order variation term. Theoretical Formulation ε u = u ( x) + εu ( x, y) 0 1 ; y=x/ε Energy Functional for the Medium Theory of Asymptotic Analysis Due to linearity: microscopic equations (δu 1 (x,y) terms) u = χ( x, y) ε( u ( x)) 1 0 where χ is Y-periodic characteristic functions of the unit cell macroscopic equations (δu 0 (x) terms) FEM solution of microscopic equations for χ FEM Solution χ N χ I=1,NN NN = i I ii 4 nodes 8 nodes 2D case 3D case Bilinear (2D) and trilinear (3D) interpolation functions Substitute in the system of microscopic equations (χ) FEM system of equations: [ ]{ mn K } { mn χ ( ) = F ( ) } load cases mn 6 for 3D 3 for 2D Homogenization Implementation FEM model and Data Input Unit Cell Number of load cases: 6 for 3D 3 for 2D Assembly of Stiffness Matrix N Solver Last? periodicity conditions enforced in the unit cell Y Calculation of Homogenized Coefficients c H, 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 brick elements brick elements Example Homogenization of bone microstructure Solid phase Fluid phase (Hollister and Kikuchi ) Example Representative Volume Element (RVE) concept Micrograph of Metal Matrix Composites (MMC) RVE unit cell Cr (fiber) - NiAl (matrix) RVE unit cell There must be statistic periodicity!!! Homogenization for Coupled Field Materials Force Mechanical Energy Displacement Example: Piezoelectric Material Electric potential Piezoelectric Electrical Material Energy Electric charge Examples: Quartz (natural) Ceramic (PZT5A, PMN, etc ) Polymer (PVDF) Applications: Pressure sensors, accelerometers, actuators, acoustic wave generation (ultrasonic transducers, sonars, and hydrophones), etc... Constitutive Equations of Piezoelectric Medium T ij -stress S kl -strain E k - electric field E T = c S e E ij ijkl D S = ε E + e S i ik D i - electric displacement c E ijkl e ikl ε S ik Elasticity equation kl kij k k ikl kl Electrostatic equation - stiffness property - piezoelectric strain property - dielectric property Homogenization for Piezoelectricity hproperties c E ijkl, e ijk, and εs ij (Y - unit cell domain). are Y-periodic functions hasymptotic expansion: - displacements: - electric potential: ε u = u ( x) + εu ( x, y) 0 1 φ ε = φ ( x) + εφ ( x, y) 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 Unit Cell Load Cases (2D model) Unit Cell Solutions using FEM periodicity conditions enforced in in the the unit cell Calculation of effective properties (c E H,e H, and εs H ) Number of load cases 9 for 3D model 5 for 2D model 20 Example 3D Piezocomposite Unit Cell polymer piezoceramic Performance quantity circular inclusion square inclusion dh (pc/n) d h Rectangular inclusion Circular inclusion Rectangular inclusion (hex.) staggered formation Ceramic Volume Fraction (%) 21 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 Types of of gradation T Cold } } Metallic matrix with ceramic inclusions Transition region } Metallic Phase 1-D 2-D 3-D Collaboration USP/UIUC University of São Paulo University of Illinois at Urbana-Champaign Emilio visited UIUC in in December 2003 Glaucio visited USP and LNLS (Syncroton Light National Laboratory Campinas, SP) in in April 2004 Conference papers presented at atfgm2004 and ICTAM2004 The following journal paper is is at atfinal stage: Topology Optimization Applied to to the the Design of of Functionally Graded Material (FGM) Structures Acknowledgment Inter-Americas Collaboration in in Materials Research and Education NSF project CMS P.I.: Professor Wole Soboyejo (Princeton University) Homogenization for FGM Materials These materials possess continuously graded properties with gradual change in in microstructure; Calculation of of effective properties is is very difficult using analytical methods Homogenization method can be applied FGM unit cell FGM Composite material example Homogenization for FGM Materials 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 nnodes ( x) = E ( ) I N I x E L I = 1 E K E: E: material property E I I :: material property evaluated at at FEM nodes x=(x, y): position Cartesian coordinates E I L E J K I J Example y Plane Strain assumption x Material properties: Elastic properties Homogenized properties Material 2 E v 1 1 FGM law: Material 1 = 8; E2 = 1.; α1 = 1.; α 2 = 2. = v2 = 0.3 β = Eα E = (( E1 E2)cos 2πx' + E1 + E2 )/ β = (( β β )cos 2πx' + β + β )/ 2 H E 1 2 Unit Cell = ; β H = Thermoelastic properties Example Unit Cell Axyssimetric composite (Application to bamboo and natural fiber composites) Material 2 Material 1 Elastic properties: E v 1 1 = 10; = v 2 E 2 = 0.3 = 2.; FGM law: ( E E )cos 2π ' + E )/ 2 E = r + ( E2 Homogenized properties H E = Material Design - Introduction Homogenization method can be combined with optimization algorithms to to design composite materials with desired performance Specify the desired Material properties Design the Composite Material Inverse Problem (Synthesis) How to implement it?? Material Design - Introduction Consider the periodic composite: Unit Cell Effective Properties ( equivalent homogeneous medium) Changing unit cell topology Depend on unit cell topology Change effective properties!!!! Material Design Method Calculation of of Homogenization Effective properties Method + Change of of Unit Cell Topology Topology Optimization Design in a mesoscopic scale rather than a microscopic scale (Physics approach) Material Design Method 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) Topology Optimization Concept It It combines FEM with optimization algorithms to to find the optimum material distribution inside of of a fixed design domain It Itturns the design process more generic and systematic, and independent of ofengineer previous knowledgment. Largely applied to toautomotive and aeronautic industries to to design optimized parts. In Inaddition, has been applied to: Design of ofcompliant mechanisms; Design of ofpiezoelectric actuators; Design of of MEMS ; Design of ofelectromagnetic devices; Design of ofcomposite materials Topology Optimization Concept? Optimum topology Topology Optimization Concept Based on two main concepts: Extended Fixed Domain Relaxation of the Design Variable Extended Fixed Domain t t Ω d - Unknown Domain Old approach: Find the boundaries of of the unknown structure (Zienkiewicz and Campbell 1973) Ω Extended Domain New approach: Find the material distribution in in the extended fixed domain (Bendsφeand Kikuchi 1988) Relaxation of the Design Problem How to change the material from zero to one?? 0 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. The material model formulation for intermediate materials defines the level of problem relaxation. Relaxation of the Design Problem Material Model: Density Method t property E ijkl = p x E 0 ijkl a fraction of material in each point 1 b θ Structure Design Domain y 2 y 1 1 Ω x 2 A point with material x 1 A point with no material Material Design Example of of Discretized Unit Cell Domain In In each finite element n property c n is is given by: 0 c n = x n p c 0 ; c : property of basic material End of Optimization: x n =x low element is air x n =1 element is full of material Optimization Problem Maximize: F(x), where x=[x 1,x,x 2,,x n,,x NDV NDV ] x subject to: c ijkl ijkl c low low,, i, i, j, j, k, k, l l are specified values 0 x low low x n 1 NDV p W = xv W n= 1 symmetry conditions n n F(x) - function of effective properties x - design variables W - constraint to reduce intermediate densities (V n - volume of each element) low Flow Chart of the Optimization Procedure Initializing and Data Input Updating Obtaining Material Homogenized Distribution Properties Initial Guess Converged? N Y Plotting Results Optimizing (SLP) with respect to to x Calculating Sensitivity Final Topology Example Plane Stress Isotropic Poisson s ratio = -0.5 (Fonseca 1997) Unit Cell Composite Material Example Orthotropic Two negative and one positive Poisson s ratio (Fonseca 1997) Unit Cell Example Thermoelastic Composites (negative thermal expansion) (negative thermal expansion) (high positive thermal expansion) (Sigmund&Torquato 1996) (zero thermal expansion) Example 2D Piezocomposite Unit Cell (hydrophone) Initially Optimized Microstructure Piezocomposite air PZT5A PZT5A Improvement in in relation to to the the 2-2 piezocomposite unit cell: d d h h : : times d hh g h h :: 9.22 times k h h :: times stiffness constraint: c E 33 N/m optimized porous ceramic 45 Composite Manufacturing Theoretical unit cell Microfabrication by coextrusion technique Fugitive Ceramic Feedrod Reduction Zone SEM Image Crumm and Halloran (1997) Extrudate 46 Experimental Verification Theoretical Prototype 3 1 Measured Performances d h (pc/n) d h g h (fpa -1 ) Solid PZT Optimized (Simulation) (257.) (19000.) 80 µm Example 3D Piezocomposite Unit Cell (hydrophone) piezoceramic Poled in the z direction z y x Improvement in in relation to to the the reference unit cells: d d h h : : 5 times d hh g h h :: times k h h :: 3.71 times stiffness constraint: c E zz N/m 22 48 Composite Manufacturing Rapid Prototyping: Stereolithography Technique 3D prototypes Recent works in the Field Fujii et et al Design of of 2D thermoelastic microstructures; Torquato et et al Design of of 3D composite with multifunctional characteristics; Guedes et et al Energy bounds for two-phase composites Diaz and Bénard Material Design using polygonal cells Conclusion The results presented give us us an idea about the potentiality of of applying homogenization and optimization methods to to model and design composite materials. However, synthesis methods for designing these materials are still in in the beginning, and the performance limits of of advanced composite materials can be improved more; Design of of FGM materials using topology optimization will allow us us to to explore the potential of of FGM concept; As a future trend, the design of of composite materials considering nanoscale unit cells started been studied by some scientists. Theoretical Formulation Homogenizaton Procedure Strains are expanded as as a global function plus a local oscillation proportional to to the global strain; The unit cell response (microscopic strain) is is obtained considering independent load cases (unit strains) under periodic boundary conditions; The microscopic strains are integrated to to obtain composite average properties; After a global analysis, the strains inside the cell can be obtained by using the localization functions;
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