HOMOGENIZATION OF VIBRATING PERIODIC STRUCTURES

Stefano Gonella , Massimo Ruzzene

Georgia Institute of Technology,School of Aerospace Engineering

Atlanta, GA 30332

ASME 2005 International Design Engineering Technical Conferences& Computers and Information in Engineering Conference

Long Beach, CA September 27 2005

Objectives and motivations

OBJECTIVES:Derivation of continuum equations for periodic lattice structures

Approximation of dynamic behavior and wave propagation characteristics (Harmonic response and dispersion relations)

Formulation of macro-elements for the numerical solution of continuum equations

MOTIVATIONS:Full description of the dynamic behavior may be computationally intensive (large number of elements)

Continuum equations may provide insight in the “equivalent”mechanical properties of periodic media

FOAMS featuringstochastic architecture

CHIRAL honeycomb withNEGATIVE Poisson’s ratio

Material-likeWOVEN composites

Periodic assemblies of engineering interest*

* = Fish, J. Chen, W. RVE Based Multilevel Method for Periodic Heterogeneous Media With Strong Scale Mixing Journal of Applied Mathematics, Vol. 46, pp. 87-106, (2003).

Outline

Definition of periodic assemblies

Fourier Analysis of discrete periodic media

Homogenization:Multiscale formulationDerivation of continuum equations

Example 1 : one dimensional system

Example 2 : Two-dimensional system

Conclusions

Periodic structures

PERIODIC STRUCTURE : STRUCTURE FEATURING A REPETITIVE ELEMENT KNOWN AS ELEMENTARY UNIT CELL

EXAMPLES OF PERIODIC STRUCTURES

One-dimensional bi-material bar E1 , ρ1

Unit Cell

One-dimensional truss structure

Fourier analysis for discrete media

Elasto-dynamic equationsCell’s degrees of freedom

Discrete spatial Fourier Transform

Applying Transformationto dynamic equations

“SYMBOL” OFTHE SYSTEM

Solving the linear system and evaluating the Inverse Fourier Transform yields

Locations of nodeswithin the cell

Homogenization

HOMOGENIZATION : technique to derive equivalent continuum equations that approximate the behavior of a given periodic structure

CONCEPT OF DIFFERENT LENGTH SCALES between the macroscopic structure and the repetitive element is captured through the introduction of parameter ε such that

x = characteristic dimension of the repetitive elementy = characteristic dimension of the macroscopic system

xy

Homogenization of lattice equationsScaled latticeequations become

It’s shown in the literature that the symbol can be scaled like:

With the scaled symbol the solution in the Fourier domain becomes

A Taylor expansion of the symbol can be performed such that

Recalling the use of the Inverse Fourier Transform, a continuous variable

One dimensional spring-mass system… …

EXACT DISPERSION RELATIONS

APPROXIMATE DISPERSION RELATIONS

I ORDER

II ORDER

III ORDER

MONO-ATOMIC STRUCTURE

PROGRESSIVETAYLOR SERIESOF TRIGONOMETRICQUANTITIES

SYMBOL

One dimensional spring-mass system

Low order approximations work only in the low-wave-number limitIncreasing the order yield increasing agreement with the exact curve

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

[rad/

s]

ExactI order approximationII order approximationIII order approximation

One-dimensional bi-material bar

Homogenized equation

TRUNCATION AT FIRST ORDER (p=1)

E2 , ρ2E1 , ρ1

whereSame resultsUsing the Ruleof Mixtures

O.D.E.

BOUNDARY CONDITIONS

One-dimensional bi-material bar

DEFINITION OF PROPERBOUNDARY CONDITIONSMAJOR LIMITATION TOHIGHER ORDER PDEs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-3

10-2

10-1

100

101

102

103

104

Two-dimensional structure

Simply supported 2DSpring-mass system

REPETITIVE ELEMENT

Assumption : OUT- OF- PLANE DISPLACEMENTS

Simply supported along the contour

Derivation of continuum equationsCONTINUUM EQUIVALENT PDE TRUNCATED AT FIRST ORDER

Special case :

DYNAMIC EQUATION OF A VIBRATING MEMBRANE

Approximate solution found though a Finite Element Formulation for the weak statement of the continuum PDEFE analysis conducted discretizing the bi-dimensional domain into rectangular “super” elements

Analysis of dispersion relations

Considered structure

PHASE CONSTANTSURFACES

III order

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

I order

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

II order

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

IV order

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Response to harmonic load

Considered structure

FRF at the mid point

Discrete structure - FEMContinuum equivalent - FEM

Refinement of the approximationORIGINAL APPROXIMATE SYMBOL

IMPOSED “FITTING” SYMBOL

a1 a2 a3 are determined through least square method in order to FIT the EXACT dispersion relations

Refinement of the approximation

Discrete structure Continuum equivalent

Refined approximationthrough least squares fitStandard homogenization

Discrete structure Continuum equivalent

Refined approximationthrough least squares fitStandard homogenization

Refinement of the approximationBI-MATERIAL BAR

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-3

10-2

10-1

100

101

102

103

104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-3

10-2

10-1

100

101

102

103

104

Multi-field analysis

SINGLE-FIELD APPROACHMULTI-FIELD APPROACH

TWO-FIELD

NEW SETS OF DEGREES OF FREEDOMObtained through a change of variables

Average and Semi-difference of theOriginal degrees of freedom

The Average is related to average propertiesof the whole structureIt approximates the behavior of the structurein the long wavelength limit

The semi-difference is related to the small differences/changes within the unit cellIt approximates the behavior of the structure in the short wavelength limit

Two-field analysis - ResultsTRUSS STRUCTURE

0.5 1 1.5 2 2.5 3

0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 30

2000

4000

6000

8000

10000

ξ [1/m]

ω[ra

d/s]

ξ [1/m]

ξ [1/m]

ω[ra

d/s]

ω[ra

d/s]

0

2000

4000

6000

8000

10000

0

2000

4000

6000

8000

10000

0 0.5 1 1.5 2 2.5 30

500

1000

1500

2000

ξ [1/m]

ω[ra

d/s]

ExactBranch IBranch II

Conclusions and future work

THE HOMOGENIZATION PROCEDURE PROVIDES A WAY TO OBTAIN APPROXIMATE DISPERSION RELATIONS AND HARMONIC RESPONSE FOR A LATTICE

REFINEMENT OF THE APPROXIMATION CAN BE ACHIEVED THROUGH INTERPOLATIONOF THE COEFFICIENTS OF THE PDE TO FIT THE EXACT DISPERSION RELATIONS

Current and future work

Conclusive Remarks

ATTEMPT TO MAP A CORRESPONDENCE BETWEEN A GIVEN LATTICE STRUCTUREAND THE CORRESPONDENT FAMILY OF EQUIVALENT PDEs