dynamicsdynamics- ---fe formulation of sandwich fe ... · cnam paris – 17th november, 2006 2...

$: dynamicsdynamics- ---FE formulation of sandwich FE ... · CNAM Paris – 17th November, 2006 2 Objectives •To implement a fractional derivative model into a finite element code$
Ana Cristina Ana Cristina Ana Cristina Ana Cristina GalucioGalucioGalucioGalucio****JeanJeanJeanJean----FranFranFranFranççççois Deois Deois Deois Deüüüü**, Fran**, Fran**, Fran**, Franççççois Dubois***ois Dubois***ois Dubois***ois Dubois***

*EADS Corporate Research Center France**Structural Mechanics and Coupled Systems Lab., CNAM

***Mathematics Dept., CNAM & Univ. Orsay

DDDDéééérivation fractionnaire en mrivation fractionnaire en mrivation fractionnaire en mrivation fractionnaire en méééécanique canique canique canique –––– EtatEtatEtatEtat----dededede----llll’’’’artartartart et applicationset applicationset applicationset applications

CNAM Paris CNAM Paris CNAM Paris CNAM Paris –––– 17th November, 2006 17th November, 2006 17th November, 2006 17th November, 2006

On the use of On the use of On the use of On the use of fractionalfractionalfractionalfractional derivativederivativederivativederivative operatorsoperatorsoperatorsoperators to to to to describedescribedescribedescribe viscoelasticviscoelasticviscoelasticviscoelastic dampingdampingdampingdamping in structural in structural in structural in structural

dynamicsdynamicsdynamicsdynamics ---- FE formulation of sandwich FE formulation of sandwich FE formulation of sandwich FE formulation of sandwich beamsbeamsbeamsbeamsand approximation of and approximation of and approximation of and approximation of fractionalfractionalfractionalfractional derivativesderivativesderivativesderivatives by by by by

usingusingusingusing the Gthe Gthe Gthe Gαααα schemeschemeschemescheme

2CNAM Paris – 17th November, 2006

ObjectivesObjectivesObjectivesObjectives

• To implement a fractional derivative model into a finite element code in structural dynamics

• To test a finite difference based on scheme to approximate fractional derivative operators in viscoelastic constitutive equations

viscoelastic core

elastic face

elastic face

Damped oscillator

Sandwich beam

Viscoelastic beam


OutlineOutlineOutlineOutlineI. I. I. I. Implementation of the fractional Implementation of the fractional Implementation of the fractional Implementation of the fractional ZenerZenerZenerZener model into model into model into model into a FE code a FE code a FE code a FE code –––– application to sandwich beamsapplication to sandwich beamsapplication to sandwich beamsapplication to sandwich beams

• Constitutive equations – fractional Zener model

• Grünwald-Letnikov approximation

• FE formulation & algorithm

• Examples: validation & results

II.TestingII.TestingII.TestingII.Testing the Gthe Gthe Gthe Gαααα schemeschemeschemescheme

• Outline

• Elementary tests

• Error estimates

• The damped oscillator problem

• Viscoelastic cantilever beam


I. I. I. I. ImplementationImplementationImplementationImplementation of the of the of the of the fractionalfractionalfractionalfractional ZenerZenerZenerZenermodel model model model intointointointo a FE code a FE code a FE code a FE code –––– application to application to application to application to

sandwich sandwich sandwich sandwich beamsbeamsbeamsbeams


Fractional derivative viscoelastic modelFractional derivative viscoelastic modelFractional derivative viscoelastic modelFractional derivative viscoelastic model

R.L. Bagley and P.J. Torvik, AIAA (1983)

OneOneOneOne----dimensional constitutive equationdimensional constitutive equationdimensional constitutive equationdimensional constitutive equation

In time domain

In frequency domain

⇒⇒⇒⇒ Fractional derivative Zener Model (4 parameters)

with


Identified model parameters:

Storage modulus Loss factor

Master curves of the ISD112 (3MMaster curves of the ISD112 (3MMaster curves of the ISD112 (3MMaster curves of the ISD112 (3M™™™™) at 27) at 27) at 27) at 27°°°°Fractional model versus classical Zener model


with

GrGrGrGrüüüünwald approximation & time discretizationnwald approximation & time discretizationnwald approximation & time discretizationnwald approximation & time discretization

with

Discretizing…

J. Padovan, Comp. Mech. (1987)

A. Schmidt and L. Gaul, Proc. 2nd ECCMR (2001)

Variable changing Only one fractional derivative term

Approximation for fractional derivatives


The structure consists on a three-layer beam ⇒ constrained layer damping treatmentconstrained layer damping treatmentconstrained layer damping treatmentconstrained layer damping treatment

membranebendingshear

membranebending

membranebending

Finite element formulationFinite element formulationFinite element formulationFinite element formulation

a

b

cviscoelastic

elastic

elastic

• Sandwich beam element

• Fractional model implementation

• Dynamic time integration


SandwichSandwichSandwichSandwich----beam kinematicsbeam kinematicsbeam kinematicsbeam kinematicsMead and Markus, JSV (1969), Trindade, Benjeddou and Ohayon, IJNME (2001)

• Laminated elastic faces: Euler-Bernoulli beam• Viscoelastic core: Timoshenko beam

Constrained layer damping treatmentConstrained layer damping treatmentConstrained layer damping treatmentConstrained layer damping treatment


Viscoelastic coreViscoelastic coreViscoelastic coreViscoelastic core

“Anelastic displacements” are evaluated as functions of their own time histories

modified stiffness matrixmodified stiffness matrixmodified stiffness matrixmodified stiffness matrix modified loadingmodified loadingmodified loadingmodified loading

Elementary dof vector


Equation of motionEquation of motionEquation of motionEquation of motion

� For an elastic material (τ = 0), classical elastic stiffness matrix

is obtained

� For the Zener model (α = 1), previous equations correspond to

a backward Euler discretization of the constitutive equations

RemarkRemarkRemarkRemarkssss+ initial conditions

0

0


Dynamic time integrationDynamic time integrationDynamic time integrationDynamic time integration

Newmark scheme

storage of “anelasticdisplacements”

evaluatedonce


M. Enelund et B.L. Josefson, AIAA Journal (1997)

Classical algorithmTaylor et al., IJNME (1970)

Fractional derivatives model

Viscoelastic fixed-free bar subjected to unit step load

Validation Validation Validation Validation –––– Example 1Example 1Example 1Example 1


Validation Validation Validation Validation –––– Example 2Example 2Example 2Example 2

T.M. Chen, IJNME (1995)

Viscoelastic simply-supported Timoshenko beam subjected to a uniform step function load

Validation for α = 1

good agreement with the “hybrid Laplace transform/FE method”


ResultsResultsResultsResultsSandwich beam subjected to a triangular pulse

loadlength = 280 mm width = 25 mm thickness:• core = 0.2 mm• skin = 1.5 mm

viscoelastic core

elastic faces


ResultsResultsResultsResults

Calculations performedwith whole history

Remark: Energy balance is strictly zero.

Influence of the time step and truncation


Existence of a truncation time where the recent history must be stored.

Very accurate solution with a fine time discretization:Nstep = 2500 Nt = 2500

Influence of the time step and truncation


AA

BC B C


II. II. II. II. TestingTestingTestingTesting the Gthe Gthe Gthe Gαααα schemeschemeschemescheme


How to How to How to How to obtainobtainobtainobtain a a a a fractionalfractionalfractionalfractional derivativederivativederivativederivative operatoroperatoroperatoroperator


The GThe GThe GThe Gαααα schemeschemeschemescheme


Elementary testsElementary testsElementary testsElementary tests


Error estimates in L normError estimates in L normError estimates in L normError estimates in L norm∞


The damped oscillator problemThe damped oscillator problemThe damped oscillator problemThe damped oscillator problem


AlgorithmAlgorithmAlgorithmAlgorithm


Extension to Extension to Extension to Extension to viscoelasticviscoelasticviscoelasticviscoelastic beamsbeamsbeamsbeamsConstitutive Constitutive Constitutive Constitutive equationsequationsequationsequations


ViscoelasticViscoelasticViscoelasticViscoelastic cantilever cantilever cantilever cantilever beambeambeambeam


AlgorithmAlgorithmAlgorithmAlgorithm


Quelques rQuelques rQuelques rQuelques rééééfffféééérencesrencesrencesrences1. A.C. GALUCIO, J.-F. DEÜ AND R. OHAYON. Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Computational Mechanics, 33:282–291, 2004.

2. A.C. GALUCIO, J.-F. DEÜ AND R. OHAYON. Atténuation des vibrations de structures par traitement piézoélectrique/viscoélastique en utilisant un mode`le a` de´rive´es fractionnaires. Revue Europeénne des Eléments Finis, 13/5-6-7:509–521, 2004.

3. A.C. GALUCIO, J.-F. DEÜ AND R. OHAYON. A fractional derivative viscoelastic model for hybrid active/passive damping treatments in time domain – Application to sandwich beams. Journal of Intelligent Material Systems and Structures, 16:33–45, 2005.

4. A.C. GALUCIO, J.-F. DEÜ , S. MENGUE AND F. DUBOIS. An adaptation of the Gear scheme for fractional derivatives. Computer Methods in Applied Mechanics and Engineering, 195:6073–6085, 2006.

5. J.-F. DEÜ, A.C. GALUCIO, R. OHAYON. Dynamic responses of flexible-link mechanisms with passive/active damping treatment, Accepted for publication on the Special Issue of the International Journal of Computer and Structures, dedicated to the II ECCOMAS Thematic Conference on Smart Structures and Materials, 18-21 July, 2005, Lisbon.

6. A.C. GALUCIO, J.-F. DEÜ , R. OHAYON. Hybrid active-passive damping treatment of sandwich beams in nonlinear dynamics, Journal of Vibration and Control, Accepted for publication.

7. A.C. GALUCIO. Atténuation des réponses transitoires par traitement hybride piézoélectrique/viscoélastique en utilisant un modèle à dérivées fractionnaires (in French), PhD thesis, CNAM Paris, 2004.


Conclusions and PerspectivesConclusions and PerspectivesConclusions and PerspectivesConclusions and Perspectives

� Sandwich beam with viscoelastic core and elastic faces

� FE implementation of a viscoelastic fractional derivative model

� The Gα scheme is no more costly than the Grünwald-Letnikov one and it appears to be (α+2)-order accurate for power functions

� The combination Newmark- Gα yields a two-order accuracy for the damped oscillator problem

Coming work: to investigate the order of convergence obtained by using the Gα scheme in the case of sandwich beams

dynamicsdynamics- ---fe formulation of sandwich fe ... · cnam paris – 17th november, 2006 2...

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