Loss Amplification Effect in Multiphase Materials with Viscoelastic Interfaces

Transcription

1 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

2 Lamellar Morphology Phase 1: Phase 2: μ = 1 b μ ( iη ) = a 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 = GPa

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 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 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 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

6 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

7 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

8 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 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

9 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

10 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 sin θ cos2φ νi r 1 2νi r () i 7 4ν i 3 Ci Di uθ = Ar i Br i sinθ cosθ cos 2φ ν i r r () i ν 7 4 i 3 Ci Di uφ = Ar i Br i sinθsin2φ ν 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)

11 Spherical Interfaces Random Composite Sphere Model Bcc R inclusion radius Δ interface thickness R = R 2 Δ = R 2 R 1

12 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

13 Spherical Interfaces Shearing in the xy-plane f = 0.2 bcc f = 0.4 bcc f = 0.4 random f = 0.6 bcc tanδ E-4 1E Δ / R z y x μ M = μ I = 1 + i 0 [GPa] μ L = i [GPa] (T g ) K M = K I = K L = 3 [GPa] μ " [GPa] E-4 1E Δ / R

14 Two Modes of Energy Dissipation Energy-dissipation rate per unit volume E εσ Δ/R = 0.2 Δ/R = y y z z x x ε 0 5 i 0 0 = i i0.7 Shearing in the xy-plane iωt ε = ε = 0 e e iωt ε = ( 45 i10)

15 Silica Reinforced Polymers K [GPa] μ [GPa] Silica Solid polymer below T g i 0.02 Polymer at T g 3.5 (1 + i 0.1) 0.01 (1 + i) Viscoelastic polymer above T g (1 + i 0.1) at T g above T g μ " [GPa] 0.2 tanδ E-5 1E-4 1E E-5 1E-4 1E Δ / R Δ / R

16 Lossy Syntactic Foams Void Epoxy matrix Rigid SiO 2 shell Lossy coating ( T g )

17 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