dynamicsdynamics- ---FE formulation of sandwich FE ... · CNAM Paris – 17th November, 2006 2...

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

3CNAM Paris – 17th November, 2006

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

4CNAM Paris – 17th November, 2006

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

5CNAM Paris – 17th November, 2006

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

6CNAM Paris – 17th November, 2006

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

7CNAM Paris – 17th November, 2006

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

8CNAM Paris – 17th November, 2006

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

9CNAM Paris – 17th November, 2006

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

10CNAM Paris – 17th November, 2006

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

11CNAM Paris – 17th November, 2006

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

12CNAM Paris – 17th November, 2006

Dynamic time integrationDynamic time integrationDynamic time integrationDynamic time integration

Newmark scheme

storage of “anelasticdisplacements”

evaluatedonce

13CNAM Paris – 17th November, 2006

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

14CNAM Paris – 17th November, 2006

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”

15CNAM Paris – 17th November, 2006

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

16CNAM Paris – 17th November, 2006

ResultsResultsResultsResults

Calculations performedwith whole history

Remark: Energy balance is strictly zero.

Influence of the time step and truncation

17CNAM Paris – 17th November, 2006

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

ResultsResultsResultsResults

AA

BC B C

18CNAM Paris – 17th November, 2006

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

19CNAM Paris – 17th November, 2006

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

20CNAM Paris – 17th November, 2006

The GThe GThe GThe Gαααα schemeschemeschemescheme

21CNAM Paris – 17th November, 2006

Elementary testsElementary testsElementary testsElementary tests

22CNAM Paris – 17th November, 2006

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

23CNAM Paris – 17th November, 2006

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

24CNAM Paris – 17th November, 2006

AlgorithmAlgorithmAlgorithmAlgorithm

25CNAM Paris – 17th November, 2006

ResultsResultsResultsResults

26CNAM Paris – 17th November, 2006

ResultsResultsResultsResults

27CNAM Paris – 17th November, 2006

ResultsResultsResultsResults

28CNAM Paris – 17th November, 2006

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

29CNAM Paris – 17th November, 2006

ViscoelasticViscoelasticViscoelasticViscoelastic cantilever cantilever cantilever cantilever beambeambeambeam

30CNAM Paris – 17th November, 2006

AlgorithmAlgorithmAlgorithmAlgorithm

31CNAM Paris – 17th November, 2006

ResultsResultsResultsResults

32CNAM Paris – 17th November, 2006

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.

33CNAM Paris – 17th November, 2006

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