finite element approaches to mesoscopic materials …
TRANSCRIPT
FINITE ELEMENT APPROACHES TOMESOSCOPIC MATERIALS MODELING
Andrei A. Gusev
Institute of Polymers, Department of Materials, ETH-Zürich, Switzerland
OutlookGeneric 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
AAG, Macromolecules 2001, 34, 3081
MC
AAG, J. Mech. Phys. Solids 1997, 45, 1449
Morphology-adaptive meshes
Surface triangulation Volume mesh
Delaunay
tessellation
AAG, Macromolecules 2001, 34, 3081
spacing between nodes ↔ local curvature
distance-based refinement
ill-shaped tetrahedrons:
slivers, caps, needles & wedges
Quality refinement
AAG, Macromolecules 2001, 34, 3081
■ General purpose periodic mesh generator- sequential Boyer-Watson algorithm- 104 nodes a second on a PC, independently of the mesh size- adaptive-precision floating-point arithmetic
New
Coating layers
h
divJ = 0
J = P
Asymptotic back-field approachDivergence-less condition
– J is a vector (e.g., electric or magnetic induction, mass or heat flux)
– or a tensor (e.g., the stress tensor)
Linear constitutive equation
– 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.
– Z is a back-field (e.g., strain tensor)
(Sv - Z)
T1or ( ( ) )2
T= −∇ = ∇ + ∇S S u u
AAG, Adv. Eng. Mater. 2007, 9, 117; Adv. Eng. Mater. 2007, 9, 1009
Effective property π from
The error in π decreases as
=J -πZ
log( )ph pδ δ∝ ⇒ ∝π π
n = 4 ↔ p =1
n = 10 ↔ p =2
n = 20 ↔ p =3
Rapid exponential convergenceEffective 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 convergenceEffective diffusivity D
– Assuming Dinc/Dmat = 1000
AAG, Adv. Eng. Mater. 2007, 9, 1009
ApplicationsDental composites, IvoclarAdv. Eng. Mater. 2003, 5, 113
Gold nanoparticles in polymersDow ChemicalAdv. Eng. Mater. 2003, 5, 713
Rubber/Talcum/PP, DSMAdv. Eng. Mater. 2001, 3, 427
Foams, BASF
E
Clay/Polymer nanocompositesNestlé, Alcan & KTIAdv. Mater. 2001, 13, 1641
CNT/polymer membranesEC 6th network programMULTIMATDESIGNAdv. Mater. 2007, 19, 2672
Viscoelastic responsesLinear constitutive equation at a given frequency ω
– for systems with isotropic constituents:
– ε0 is a prescribed function of time (e.g., a harmonic function)
Principle of virtual work (weak form of the equilibrium equations)
( ) ( )0 d 0( ) ( )ω ω= − + −σ D ε ε D ε ε& &
2 0 0 02 0 0 0
2 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0
λ μ λ λλ λ μ λλ λ λ μ
μμ
μ
+⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟+
= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
D d
4 2 2 0 0 02 4 2 0 0 02 2 4 0 0 0
0 0 0 3 0 030 0 0 0 3 00 0 0 0 0 3
η
− −⎛ ⎞⎜ ⎟− −⎜ ⎟⎜ ⎟− −
= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
D
T d 0V
Vδ =∫ ε σ
NumericalResulting discrete equations for the nodal displacement vector a
– with
– and B standing for the strain-displacement matrix
Steady-state solution
– substituting and rearranging terms gives a saddle-point optimization problem
– Uzawa’s iterations, cg-solvers on normal equations, GMRES, etc.
– but neither is suitable for large-scale (106 & larger) problems
+ =Ka Ca f&
T dV= ∫K B DB Td dV= ∫C B D B ( )T dd V= +∫f B Dε D ε&
( ) i ti e ω′ ′′= +a a a ( ) i ti e ωω′′ ′= − +a a a& 0i te ω=f f
0-0
ωω
′⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟′′⎝ ⎠⎝ ⎠ ⎝ ⎠
K C a fC K a
NumericalImposing dynamic equilibrium at time tn
– and using the central difference formula
– one obtains an implicit scheme for stepping from a(n) at time tn to a(n+1) at tn+Δt
– 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 tn+1
( ) ( ) ( )n n n+ =Ka Ca f&
( ) ( 1) ( )1 ( )2
n n nt
+= −Δ
a a a&
( 1) ( ) ( ) ( 1)1 12 2
n n n nt t
+ −= − +Δ ΔCa f Ka Ca
NumericalImposing dynamic equilibrium at time tn+1
– and using the trapezoidal (“average velocity”) difference formula
– one obtains a Crank-Nicolson type, unconditionally stable algorithm• for stepping from a(n) at time tn to a(n+1) at tn+Δt
– 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
( ) ( ) ( )
( ) ( ) ( )
1 1
2 2
2
n n n
n n n
t t
t
+ +Δ Δ⎛ ⎞+ = +⎜ ⎟⎝ ⎠
Δ⎛ ⎞= +⎜ ⎟⎝ ⎠
C K a f g
g C a a&
( 1) ( 1) ( 1)n n n+ + ++ =Ka Ca f&
( 1) ( 1) ( ) ( )1 ( )2
n n n nt
+ += − −Δ
a a a a& &
Under external harmonic strain– the steady-state system stress is also harmonic– the effective complex modulus– the effective loss factor
Extracting effective viscoelastic properties
ε ε ω= 0 sin t( )σ σ ω δ= +0 sin t
μ σ ε= 0 0*δ μ μ′′ ′=tan /
μM = 1.1 + i 0.022 [GPa]
μL = 0.01 + i 0.005 [GPa]
KM = KL = 3 [GPa]
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
0.0
0.2
σ 66 [G
Pa]
t [s]
0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.5
0.0
0.5
1.0
ε 66
ω = 1 rad/s
Core-Shell PS/Rubber/PS Materials for Vibration & Noise Damping (e.g. HIPS)
μM = 1.1 + i 0.022 [GPa]
μL = 0.01 + i 0.005 [GPa]
KM = KL = 3 [GPa]
1E-4 1E-3 0.01 0.1 10.2
0.4
0.6
0.8
1.0
μ ' [G
Pa]
δ = Δ/R
f = 0.2 bccf = 0.4 bccf = 0.4 randomf = 0.6 bcc
1E-4 1E-3 0.01 0.1 10.02
0.04
0.06
0.08
0.10f = 0.2 bccf = 0.4 bccf = 0.4 randomf = 0.6 bcc
μ " / μ
'
δ
1E-4 1E-3 0.01 0.1 10.01
0.02
0.03
0.04
0.05
0.06
μ " [G
Pa]
δ
f = 0.2 bccf = 0.4 bccf = 0.4 randomf = 0.6 bcc
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
Perspectives