FINITE ELEMENT APPROACHES TO MESOSCOPIC MATERIALS MODELING

Transcription

1 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

2 Random composites MC AAG, J. Mech. Phys. Solids 1997, 45, 1449 AAG, Macromolecules 2001, 34, 3081

3 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

4 Quality refinement Ne w Coating layers General purpose periodic mesh generator - sequential Boyer-Watson algorithm nodes a second on a PC, independently of the mesh size - adaptive-precision floating-point arithmetic AAG, Macromolecules 2001, 34, 3081

5 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

6 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

7 Rapid exponential convergence Effective diffusivity D Assuming D inc /D mat = 1000 AAG, Adv. Eng. Mater. 2007, 9, 1009

8 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

9 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 λ λ+ 2μ λ λ λ λ+ 2μ D = μ μ μ D d η = 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

10 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

11 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

12 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

13 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 ( ) σ = σ0 sin ω t + δ ω = 1 rad/s ε μ M = i [GPa] μ L = i [GPa] K M = K L = 3 [GPa] σ 66 [GPa] t [s]

14 Core-Shell PS/Rubber/PS Materials for Vibration & Noise Damping (e.g. HIPS) f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc μ M = i [GPa] μ " / μ ' 0.06 μ L = i [GPa] 0.04 K M = K L = 3 [GPa] E-4 1E μ ' [GPa] f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc μ " [GPa] δ f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc 0.2 1E-4 1E δ = Δ/R E-4 1E δ

15 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