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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
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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
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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).
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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
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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
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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
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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
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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
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One dimensional spring-mass system… …
EXACT DISPERSION RELATIONS
APPROXIMATE DISPERSION RELATIONS
I ORDER
II ORDER
III ORDER
MONO-ATOMIC STRUCTURE
PROGRESSIVETAYLOR SERIESOF TRIGONOMETRICQUANTITIES
SYMBOL
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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
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One-dimensional bi-material bar
Homogenized equation
TRUNCATION AT FIRST ORDER (p=1)
E2 , ρ2E1 , ρ1
whereSame resultsUsing the Ruleof Mixtures
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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
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Two-dimensional structure
Simply supported 2DSpring-mass system
REPETITIVE ELEMENT
Assumption : OUT- OF- PLANE DISPLACEMENTS
Simply supported along the contour
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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
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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
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Response to harmonic load
Considered structure
FRF at the mid point
Discrete structure - FEMContinuum equivalent - FEM
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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
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Refinement of the approximation
Discrete structure Continuum equivalent
Refined approximationthrough least squares fitStandard homogenization
Discrete structure Continuum equivalent
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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
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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
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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
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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