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