FINITE ELEMENT APPROACHES TO MESOSCOPIC MATERIALS MODELING Andrei A. Gusev Institute of Polymers, Department of Materials, ETH-Zürich, Switzerland Outlook Generic finite element approach (PALMYRA) Random composites Periodic boundary conditions Morphology adaptive unstructured meshes Back-field solution for effective properties Modeling viscoelastic responses Frequency domain vs. time domain methods Application: High Impact Polystyrene (HIPS) Perspectives
Random composites MC AAG, J. Mech. Phys. Solids 1997, 45, 1449 AAG, Macromolecules 2001, 34, 3081
Morphology-adaptive meshes Surface triangulation Volume mesh Delaunay tessellation spacing between nodes local curvature distance-based refinement ill-shaped tetrahedrons: slivers, caps, needles & wedges AAG, Macromolecules 2001, 34, 3081
Quality refinement Ne w Coating layers General purpose periodic mesh generator - sequential Boyer-Watson algorithm -10 4 nodes a second on a PC, independently of the mesh size - adaptive-precision floating-point arithmetic AAG, Macromolecules 2001, 34, 3081
Asymptotic back-field approach Divergence-less condition divj = 0 n = 4 p =1 J is a vector (e.g., electric or magnetic induction, mass or heat flux) or a tensor (e.g., the stress tensor) h n = 10 p =2 Linear constitutive equation J=P(Sv - Z) v is a suitable field variable (e.g., temperature T or displacement u) P is a local property tensor (e.g., heat conductivity χ or elastic constants C) S is a linear geometric operator, e.g. 1 T S= T or S= ( u+ ( u) ) 2 Z is a back-field (e.g., strain tensor) n = 20 p =3 Effective property π from J = -πz The error in π decreases as p δπ h log( δπ) p AAG, Adv. Eng. Mater. 2007, 9, 117; Adv. Eng. Mater. 2007, 9, 1009
Rapid exponential convergence Effective stiffness (B & G are the bulk & shear moduli, respectively) Rigid E-glass inclusions (E = 70 GPa, ν = 0.2) Glassy polymer matrix (E = 3 GPa, ν = 0.35) AAG, Adv. Eng. Mater. 2007, 9, 1009
Rapid exponential convergence Effective diffusivity D Assuming D inc /D mat = 1000 AAG, Adv. Eng. Mater. 2007, 9, 1009
Dental composites, Ivoclar Adv. Eng. Mater. 2003, 5, 113 Applications Foams, BASF Gold nanoparticles in polymers Dow Chemical Adv. Eng. Mater. 2003, 5, 713 E Rubber/Talcum/PP, DSM Adv. Eng. Mater. 2001, 3, 427 CNT/polymer membranes EC 6 th network program MULTIMATDESIGN Adv. Mater. 2007, 19, 2672 Clay/Polymer nanocomposites Nestlé, Alcan & KTI Adv. Mater. 2001, 13, 1641
Viscoelastic responses Linear constitutive equation at a given frequency ω ( ) ω (& & ) σ = D( ω) ε ε + D ( ) ε ε 0 d 0 for systems with isotropic constituents: λ+ 2μ λ λ 0 0 0 λ λ+ 2μ λ 0 0 0 λ λ λ+ 2μ 0 0 0 D = 0 0 0 μ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ D d 4 2 2 0 0 0 2 4 2 0 0 0 η 2 2 4 0 0 0 = 3 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 ε 0 is a prescribed function of time (e.g., a harmonic function) Principle of virtual work (weak form of the equilibrium equations) δ ε T σ dv = 0 V
Numerical Resulting discrete equations for the nodal displacement vector a with T K = B DBdV Ka + Ca& = f T T C B D d B dv f = B ( Dε + Ddε& ) = and B standing for the strain-displacement matrix dv Steady-state solution a= ( a + ia ) e iωt i t a& = ( a + ia ) ωe ω f = f 0e iωt substituting and rearranging terms gives a saddle-point optimization problem K -ωc a f0 ω = 0 C K a Uzawa s iterations, cg-solvers on normal equations, GMRES, etc. but neither is suitable for large-scale (10 6 & larger) problems
Numerical Imposing dynamic equilibrium at time t n ( n) ( n) ( n) Ka + Ca& = f and using the central difference formula ( n) 1 ( ( n+ 1) ( n) a& = a a ) 2Δt one obtains an implicit scheme for stepping from a (n) at time t n to a (n+1) at t n +Δt 1 1 Ca = f Ka + Ca 2Δt 2Δt ( n+ 1) ( n) ( n) ( n 1) at each time step, it requires solution of the above linear-equation system despite the fact that the new estimates for a (n+1) were derived entirely from historical information about velocity It thus makes sense to directly impose dynamic equilibrium at time t n+1
Numerical Imposing dynamic equilibrium at time t n+1 ( n+ 1) ( n+ 1) ( n+ 1) Ka + Ca& = f and using the trapezoidal ( average velocity ) difference formula ( n+ 1) 1 ( ( n+ 1) ( n) ) ( n) a& = a a a& 2Δt one obtains a Crank-Nicolson type, unconditionally stable algorithm for stepping from a (n) at time t n to a (n+1) at t n +Δt Δt ( n+ 1) Δt ( n+ 1) ( n) C+ K a = f + g 2 2 ( n) ( n) Δt ( n) g = Ca + a& 2 at each time step, one should solve the above linear-equation system iterative, preconditioned conjugate gradient solver as both K and C are symmetric positive definite matrices
Extracting effective viscoelastic properties Under external harmonic strain ε = ε 0 sinωt the steady-state system stress is also harmonic the effective complex modulus μ * = σ 0 ε 0 the effective loss factor tan δ = μ / μ 1.0 0.5 ( ) σ = σ0 sin ω t + δ ω = 1 rad/s ε 66 0.0-0.5-1.0 0.0 0.2 0.4 0.6 0.8 1.0 μ M = 1.1 + i 0.022 [GPa] μ L = 0.01 + i 0.005 [GPa] K M = K L = 3 [GPa] σ 66 [GPa] 0.2 0.0-0.2 0.0 0.2 0.4 0.6 0.8 1.0 t [s]
Core-Shell PS/Rubber/PS Materials for Vibration & Noise Damping (e.g. HIPS) 0.10 0.08 f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc μ M = 1.1 + i 0.022 [GPa] μ " / μ ' 0.06 μ L = 0.01 + i 0.005 [GPa] 0.04 K M = K L = 3 [GPa] 0.02 1E-4 1E-3 0.01 0.1 1 μ ' [GPa] 1.0 0.8 0.6 0.4 f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc μ " [GPa] 0.06 0.05 0.04 0.03 δ f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc 0.2 1E-4 1E-3 0.01 0.1 1 δ = Δ/R 0.02 0.01 1E-4 1E-3 0.01 0.1 1 δ
Perspectives Advanced materials for vibration and noise damping e.g., fan blades of airplane s turbines (Rolls Royce & Airbus) epoxy matrixes filled with glass microballones + interfacial layers CA screening of the parameter space Interfacial delamination, gradient materials, etc. Multiscale (atomistic + continuum) modeling