Loss Amplification Effect in Multiphase Materials with Viscoelastic Interfaces Andrei A. Gusev Institute of Polymers, Department of Materials, ETH-Zürich, Switzerland Outlook Lamellar morphology systems Predicting effective viscoelastic responses of particulate morphology systems Time domain finite element method Four-phase composite sphere model High Impact Polystyrene (HIPS) Silica reinforced polymers Perspectives γ = γ i t 0e ω μ = μ + iμ 2 E = μ γ 0
Lamellar Morphology Phase 1: Phase 2: μ = 1 b μ ( iη ) = a + 2 1 b = 30 GPa, a = 0.02 GPa, η = 1 Effective shear modulus 1 f 1 f = + μ μ μ eff 2 1 Asymptotic behavior for ab 0 The maximum occurs at f = 1+ η 2 ( a b) With a maximum value of 2 ( ) μ eff = b 1+ η 1 2η Free & constrained layer damping treatments widely used in practice 1 f 1 f = + C C C eff 2 1 C 1 = 80 GPa ( i ) C2 = 3.5 1+ 0.1 GPa
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 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
Viscoelastic Responses Linear constitutive equation at a given frequency ω for systems with isotropic constituents ( 0) d ω ( 0) ( K = λ + 2μ 3) σ = D( ω) ε ε + D ( ) ε ε 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
Steady-State Approach Resulting discrete equations for the nodal displacement vector a with T K = B DBdV and B standing for the strain-displacement matrix Steady-state solution Ka + Ca = f T C B D d B dv f = B T ( Dε0 + Ddε 0) dv = i t a = ( a + ia ) e ω i t a = ( a + ia ) ωe ω i t f = ( f + if ) e ω substituting and rearranging terms gives a saddle-point optimization problem K -ω C a f = ω C K a f Uzawa s iterations, cg-solvers on normal equations, GMRES, etc. but neither is suitable for large-scale (10 6 & larger) problems
Time-Domain Approach 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) 2 ( ( n+ 1) ( n) ) ( n) a = a a a Δt 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 2 a = f + g 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
Extraction of Effective Properties Under external harmonic strain ε ε 0 sinωt in the steady-state, the system stress is also harmonic the effective complex modulus the effective loss factor tan δ = μ / μ = σ = σ ( ω + δ) μ * = σ ε 0 0 0 sin t μ M =0.01+i 0.01 [GPa] μ I =1+i 0.02 [GPa] K M = K I =3 [GPa] 3 CPU days on a 1.6 GHz Itanium processor
Four-Phase Composite Sphere Models Pure dilation θ 0 at infinity Displacement vector u = (u r, 0, 0) And is a function of r alone In spherical coordinates (in phase i) 2 () i ( rur ) 1 d div u = = const = 3a 2 i r dr Or () i bi ur = ar i + r 2 () i 4μi The radial stress σ rr = 3Ka i i 3 bi r Continuity at the boundaries () ( ( ) ) ( ) () ( ) ( ) ( ) i i σ = σ + 1 i i = + 1 R R u R u R rr i rr i r i r i Self-consistent condition: the average dilation in the composite assembly is θ 0 6 equations, 6 unknown coefficients K eff
Four-Phase Composite Sphere Models Pure shear γ 0 at infinity Displacement vector u =,, ( ur uθ uφ ) In spherical coordinates (in phase i) () i 6ν ν i 3 Ci 5 4 i Di 2 ur = Ar i Br i + 3 + sin θ cos2φ 4 2 1 2νi r 1 2νi r () i 7 4ν i 3 Ci Di uθ = Ar i Br i 2 + 2 sinθ cosθ cos 2φ 4 2 1 2ν i r r () i ν 7 4 i 3 Ci Di uφ = Ar i Br i 2 + 2 sinθsin2φ 4 2 1 2ν i r r Stress and displacement continuity at the boundaries Self-consistent condition: the average shear in the composite assembly is γ 0 12 second-order equations, 12 unknown coefficients μ eff Three-phase model: formulated by Kernel (1947), solved for μ eff by Christensen & Lo (1979) Four-phase model was then solved by Herve and Zaoui (1991)
Spherical Interfaces Random Composite Sphere Model Bcc R inclusion radius Δ interface thickness R = R 2 Δ = R 2 R 1
High Impact Polystyrene (HIPS) L M μ M = μ I = 1 + i 0 [GPa] μ L = 0.01 (1+ i 0.5) [GPa] (T g ) K M = K I = K L = 3 [GPa] I PS PIB PS
Spherical Interfaces Shearing in the xy-plane 0.09 0.06 f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc tanδ 0.03 0.00 1E-4 1E-3 0.01 0.1 1 Δ / R z y x μ M = μ I = 1 + i 0 [GPa] μ L = 0.01 + i 0.005 [GPa] (T g ) K M = K I = K L = 3 [GPa] μ " [GPa] 0.04 0.03 0.02 0.01 0.00 1E-4 1E-3 0.01 0.1 1 Δ / R
Two Modes of Energy Dissipation Energy-dissipation rate per unit volume E εσ Δ/R = 0.2 Δ/R = 0.007 y y z z x x ε 0 5 i 0 0 = + 0 1.3 i0.3 0 + 0 0 3.3 i0.7 Shearing in the xy-plane 1 0 0 iωt ε = ε = 0 e 0 1 0 e 0 0 0 iωt ε 0 1 0 0 = ( 45 i10) 0 1 0 0 0 0
Silica Reinforced Polymers K [GPa] μ [GPa] Silica 40 30 Solid polymer below T g 4 1 + i 0.02 Polymer at T g 3.5 (1 + i 0.1) 0.01 (1 + i) Viscoelastic polymer above T g 3 0.001 (1 + i 0.1) 0.4 0.3 at T g above T g 0.20 0.15 μ " [GPa] 0.2 tanδ 0.10 0.1 0.05 0.0 1E-5 1E-4 1E-3 0.01 0.1 1 0.00 1E-5 1E-4 1E-3 0.01 0.1 1 Δ / R Δ / R
Lossy Syntactic Foams Void Epoxy matrix Rigid SiO 2 shell Lossy coating ( T g )
Perspectives Advanced materials for noise reduction and vibration damping Epoxy matrixes filled with core-shell particles (syntactic foams) such as glass micro-balloons coated with a lossy polymer layer Short fiber and platelet reinforced polymers Numerical screening of the parameter space