Topics on Mathematics for Smart Systems: Proceedings of the European Conference Rome, Italy, 26-28 October,2006

Topics on

Mathematics for Smart

Systems

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Topics on

Mathematics for Smart

Systems

Proceedings o f t h e European Conference 26 - 28 October 2006

Rome, Italy

Editors

Bernadette Miara ESIEE, Paris, France

Georgios Stavroulakis Technical University of Crete, Greece

Vanda Valente IAC-CNR, Rome, ltaly

r p World Scientific N E W JERSEY

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v

PREFACE The European Conference on Smart Systems was held in Rome during the period 26-28 October 2006. The present volume contains the papers of several lectures delivered at the Conference and provides an overview of the activities carried out within the Research and Training Network "New Materials, Adaptive Systems and their Nonlinearities: Modeling, Control and Numerical Simulation" supported by the European Commission within the 5th Framework Programme. The aim of the book is to present a multidisciplinary approach to smart systems from both the engineering and the mathematical perspectives. The originality relies on the necessity of a cross-disciplineapproach. Actually interaction between such disciplines as applied mathematics, physics, structural mechanics, fluid mechanics, computer sciences is necessary in order to deal with the field of smart systems. "New materials" is used here as a generic term for functional materials such as piezoelectric, magnetostrictive, electrostrictive materials, piezoceramics, electrorheological fluids, shape memory alloys, whose physical or chemical properties are used in the design of intelligent systems. Their use as control elements, for example, is widespread: sensors or actuators bonded to a surface or embedded within the structure yielding smart structures or adaptive structures that can be monitored and controlled. Due to recent developments in miniaturization, active transducers are present in almost all domains of industry from telecommunications to optics, biomechanics and environment and they are used even in extreme environments: high pressure, high temperature, presence of radiations. They are now very popular in various applications in particular through MOEMS (Micro Optical-Electrical-Mechanical Systems): adaptive micro-mirrors, microaccelerometers, capacitive microphones, pressure sensors, micro-positioners, ultrasonics transducers, noise reduction, vibrations suppressor, damage detection. Another challenge, rarely addressed, is the question of nonlinearity, which is an important aspect inherent to adaptive structures. Nonlinear

vi problems appear, among others, when hysteresis has to be dealt with, or when phase transitions occur in shape memory alloys. The application-oriented approach followed by the research team aims at reducing the gap and reinforcing the ties between engineers and applied mathematicians. It has been designated to provide mathematical tools dedicated to a better understanding of the behavior of complex systems and of the associated control strategies. The results will hopefully help improve the performances of technical devices and applications, or even invent new ones. We wish to thank the other members of the Scientific Committee of the Conference, namely C.C. Baniotopoulos, M. Bernadou, J. Holnicki-Szulc, C. Lovadina, 3. Martins, A. Mielke, R. Stenberg, J. Viaiio, E. Zuazua, and the Referees of the papers who helped us in the editing procedure of this volume. Special thanks are due to all the authors who have contributed to the success of the book and to World Scientific Publishing Co. for allowing us to publish the proceedings volume.

The Editors Rome, 31 October 2006

vii vii

CONTENTS

Preface A Phenomenological 3D Model Describing Stress-Induced Solid Phase Transformations with Permanent Inelasticity F. Auricchio, A. Reali and U. Stefanelli Numerical Analysis of a Frictionless Piezoelectric Contact Problem Arising in Viscoelasticity M. Barboteu, J. R. Ferncindez and Y. Ouafik

A Stabilized MITC6 Triangular Shell Element L. Beiriio da Vezga, D. Chapelle and I. Paris A New Family of C0 Finite Elements for the Kirchhoff Plate Model L. Beiriio da Veiga, J. Niiranen and R. Stenberg Modeling and Simulation of Piezoelectric-Active Control of Wind-Induced Vibrations on Beams M. Betti, C. C. Baniotopoulos and G. E. Stavroulakis A Numerical Library for Shells Described by the Intrinsic Geometric Modeling via the Oriented Distance Function J. Cagnol and V. Sansalone

A Contact Problem for Viscoelastic Materials with Long Memory Involving Damage M. Carnpo, J. R. Ferncindez and A. Rodm"guez-Arbs Memory Effects Arising in the Homogenization of Composites with Inclusions L. Faella and S. Monsurrd

viii Numerical Experiments on the Controllability of the Ginzburg-Landau Equation R. Garz6n and V. Valente Homogenization of Thin Piezoelectric Perforated Shells M. Ghergu, G. Griso and B. Miara Damaged Support Identification in Aluminium Curtain-Walls Using Neural Networks P. Nazarko, L. Ziemianski, Ch. Efstathiades, C. C. Baniotopoulos and G. E. Stavroulakis Mathematical Results on the Stability of Quasi-Static Paths of Elastic-Plastic Systems with Hardening A. Petrov, J. A. C. Martins and M. D. P. Monteiro Marques

167

Mathematical Results on the Stability of Quasi-Static Paths of Smooth Systems N. V. Rebrova, J. A. C. Martins and V. A. Sobolev

183

Sensitivity Analysis of Acoustic Wave Propagation in Strongly Heterogeneous Piezoelectric Composite E. Rohan and B. Miara

193

New Results on the Stability of Quasi-Static Paths of a Single Particle System with Coulomb Friction and Persistent Contact F. Schmid, J. A. C. Martins and N. Rebrova

208

218 Numerical Experiments on Smart Beams and Plates G. E. Stavroulakis, D. G. Marinova, G. A. Foutsitzi, E. P. Hadjigeorgiou, E. C. Zacharenakis and C. C. Baniotopoulos On Modeling, Analytical Study and Homogenization for Smart Materials A. Timofte The Cardiovascular System as a Smart System M. Tringelova', P. Nardinocchi, L. Teresi and A. Di Carlo

253

Author Index

271

Topics on

Mathematics for Smart

Systems

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1

A PHENOMENOLOGICAL 3D MODEL DESCRIBING STRESS-INDUCED SOLID PHASE TRANSFORMATIONS WITH PERMANENT INELASTICITY F. A U R I C C H I O ~ A. ~ ~ REALIa* , and U. STEFANELLI~ a

Dipartimento di Meccanica Struttumle, Universitci deglz Studi di Pavia, via Fewata 1, 27100, Pavia, Italy *E-mail: [email protected] Istituto di Matematica Applicata e Tecnologie Informatiche via Fewata 1, 27100, Pavia, Italy

The diffusion and the use of shape memory alloys in many aeronautical, biomedical and structural engineering applications is resulting in an increasing research effort toward a reliable and complete modeling of their macroscopic behaviour. As many models for SMA available in the literature consider fully reversible phase transformations (i.e. no permanent inelastic strains), which are proved by experiments to be sometimes a not fully realistic approximation, we present here a 3D model capable of including permanent inelastic effects combined with a good description of pseudo-elastic and shape-memory behaviours. We also show numerical results from both uniaxial and non-proportional biaxial tests, which aim at assessing model features and performance. Keywords: Shape memory alloys; Permanent inelasticity; Phase transformation; Pseudo-elasticity; Shape-memory effect; Non-proportional biaxial tests.

1. Introduction The great and always increasing interest in SMA materials (cf. Refs. 1,2.) and their industrial applications is deeply stimulating the research on constitutive laws. As a consequence, many models able to reproduce one or both of the well-known SMA macroscopic behaviours, referred to as pseudoelasticity and shape-memory effect, have been recently proposed in the literature (see for instance Refs. 3-11). In particular, the constitutive law proposed in Ref. 12 and improved in Ref. 13 seems to be attractive. Developed within the theory of irreversible thermodynamics, this model is in fact able to describe both pseudoelasticity and shape-memory effect and the corresponding solution algo-

2 rithm, based on a plasticity-like return map procedure, is simple and robust. However, we have to stress that most of the SMA models present in the literature are not able to reproduce other experimentally observed SMA behaviours such as permanent inelasticity and degradation effects, as shown by Fig. 1 (Ref. 14). In fact, such a figure presents pseudo-elastic loops showing an increasing level of permanent inelasticity that saturates on a stable value after a certain number of cycles. The same figure highlights that degradation effects should be taken into account as well. For a description of these behaviours from a physical point of view the interested reader is referred to classical SMA textbooks such as Refs. 15,16. Moving from these experimental evidences, some models accounting for permanent inelastic effects have been recently proposed in the literature (see e.g. Refs. 17-20).

Fig. 1. Experimental results on a SMA Ni-Ti wire. Cyclic tension test: stress versus strain up to 6% strain.

In particular, in this work we discuss a phenomenological constitutive model, able to reproduce pseudo-elastic and shape-memory behaviours as well as to include permanent inelasticity and degradation effects, which has been introduced in Ref. 21,22. The model consists of an extension of the model discussed in Ref. 13, by means of the introduction of a new internal variable describing permanent inelastic strains. In the following, an analytic description of the constitutive equations is presented together with numerical experiments which show main features and performance of the model.

3 2. 3D phenomenological model for stress-induced solid

phase transformation with permanent inelasticity 2.1. Time-continuous frame

The model assumes the total strain E and the absolute temperature T as control variables, the transformation strain etTand the permanent inelastic strain q as internal ones. As in Ref. 13, the second-order tensor etTdescribes the strain associated to the transformation between the two solid phases referred to as martensite and austenite. Here, this quantity has no fully reversible evolution and the permanent inelastic strain q gives a measure of the part of etTthat cannot be recovered when unloading to a zero stress state. Moreover, we require that

lletTll I EL,

(1)

where 11 . 11 is the usual Euclidean norm and EL is a material parameter corresponding to the maximum transformation strain reached at the end of the transformation during an uniaxial test. Assuming a small strain regime, justified by the fact that the approximation of large displacements and small strains is valid for several applications, the following standard additive decomposition can be considered

where 8 = tr(e) and e are respectively the volumetric and the deviatoric part of the total strain e, while 1 is the second-order identity tensor. The free energy density function Q for a polycrystalline SMA material is then expressed as the convex potential

where K and G are respectively the bulk and the shear modulus, P is a material parameter related to the dependence of the critical stress on the temperature, Mf is the temperature below which only martensite phase is stable, h defines the hardening of the phase transformation, H controls the saturation of the permanent inelastic strain evolution, and A controls the degradation of the model. Moreover, we make use of the indicator function

0 if lletTll I EL otherwise,

+CCI

4 in order to satisfy the transformation strain constraint (1); we also introduce the positive part function (.), defined as aifa>O 0 otherwise. We remark that in the expression of the free energy we neglect the contributions due to thermal expansion and change in temperature with respect to the reference state, since we are not interested here in a complete description of the thermomechanical coupled problem. However, the interested reader may find in Refs. 13,23 how it is possible to take into account these aspects in the formulation. We also stress that, due to the fact that we do not consider a fully thermomechanical coupled model, 9 should be more properly referred t o as a temperature-parameterized free energy density function. Moreover, since we use only a single internal variable second-order tensor to describe phase transformations, at most it is possible to distinguish between a generic parent phase (not associated to any macroscopic strain) and a generic product phase (associated to a macroscopic strain), as in Ref. 13. Accordingly, the model does not distinguish between the austenite and the twinned martensite, as both these phases do not produce macre scopic strain. We furthermore highlight that, for the sake of simplicity, the present model does not reflect the difference existing between the austenite and the martensite elastic properties. Starting from the free energy function Q and following standard arguments, we can derive the constitutive equations /

<

,

dQ

p =

--- KO,

S =

d9 --- 2G(e - etr), de

'I=

a9

( T - Mf ) IT- Mfl' etr - Q etr - P(T - Mf) Iletr - qll - hetr + A q - y-lletrll'

m - -Plletr

39 x=--= 3.t~

a9

- qII

etr

-

+

(3)

Q = --=P(TMf) - H q Aetr, \ a4 lletr - qII where p = tr(u)/3 and s are respectively the volumetric and the deviatoric part of the stress a, X is a thermodynamic stress-like quantity associated to the transformation strain etr, Q is a thermodynamic stress-like quantity

5 associated to the permanent inelastic strain q, and 7 is the entropy. The variable y results from the indicator function subdifferential 8Z,, (etr) and it is defined as y = 0 if lletTll< E L , y 2 0 if IletTll= EL,

etr so that 8z,,(etr)= yIletTll' To describe phase transformation and inelasticity evolution, we choose (following a plasticity-like terminology) a limit function F defined as

where K. is a material parameter defining a scaling modulus between the inelastic effect and the phase transformation, while R is the radius of the elastic domain. We stress that, in order to reproduce the asymmetric behaviour in tension and compression shown by SMA in many experiments, different and more complicate choices for F should be introduced in (4), as it is done in Ref. 13 where a Prager-Lode type limit function is employed. Considering an associative framework, the flow rules for the internal variables take the form

The model is finally completed by the classical Kuhn-Tucker conditions

3. Time-discrete frame

Let us now focus on the crucial issue of computing the stress and internal variable evolution of a SMA sample in a strain-driven situation. We shall directly concentrate ourselves on the solution of the time-incremental problem. Namely, we discretize the time-interval of interest [0,t f ] by means of the partition I = (0 = to < tl < ... < tlvV1 < t, = tf), assume to be given the state of the system (p,, s,, qn, e;, q,) at time t,, the actual total strain (8, e) and temperature T at time tn+l (note that for notation simplicity here and in the following we drop the subindex n 1 for all the

+

6 variables computed at time tn+l), and solve for (p, s,v, etr,q). For the sake of numerical convenience, instead of solving (3) we prefer to perform some regularization. Indeed, we let be defined as

-

IIaII = d

m - 4,

(6 is a user-defined parameter controlling the smoothness of the norm regularization) and introduce the regularized free energy density and limit function as

Finally, the updated values (p, s,7, etr,q) for regularized constitutive model can be computed from the following relations

along with the requirements

where A( = - & = fffl(dt is the time-integrated consistency parameter. We shall clearly state that our numerical experiments are not performed on the model of Sec. 2 but rather on its above-introduced 6-regularized version. This choice turns out to be quite convenient from the numerical

7 viewpoint and preserves most of the characteristic features of the model. Moreover, it can be proved that the ®ularized model converges t o the original one as the regularization parameter 6 goes to 0. This fact along with additional mathematical analysis of the model are the subject of the forthcoming contribution Ref. 24 (see also Ref. 25 for similar problems). 3.1. Solution algorithm

The solution of the discrete model is performed by means of an elasticpredictor inelastic-corrector return map procedure as in classical plasticity problems (cf. Ref. 26). An elastic trial state is evaluated keeping frozen the internal variables, then a trial value of the limit function is computed to verify the admissibility of the trial state. If this is not verified, the step is inelastic and the evolution equations have to be integrated. We remark that, as in Ref. 13, we distinguish two inelastic phases in our model: a non-saturated phase (IJetrJJ < EL, y = 0) and a saturated one (JJetrll= EL, 2 0). In our solution procedure we start assuming to be in a non-saturated phase, and when convergence is attained we check if our assumption is violated. If the non-saturated solution is not admissible, we search for a new solution considering saturated conditions. For each inelastic step, we have to solve the nonlinear system constituted by equations (9). As the aim of this paper is to show the model behaviour without focusing on algorithmic problems, we find a solution to the nonlinear system by means of the function fsolve implemented in the optimization toolbox of the program MATLAB@. 4. Numerical results

To show the model capability of reproducing the macroscopic behaviour of SMA materials, we perform a number of stress-driven numerical experiments, in particular uniaxial and biaxial tests. In all tests we consider the material properties specified in Table 1 and compatible with Cu-based alloys (see e.g. Refs. 27,28), where E and v are respectively the Young's modulus and the Poisson's ratio, while all the other material constants have already been introduced in Sec. 2. Uniaxial tests represent the simplest setting on which it is possible to show the main features of the model as well as to appreciate the role played by the single material parameters; while biaxial tests allow to assess the model behaviour under complex non-proportional multi-axial loading conditions. All the numerical experiments have been performed in both the

8 Table 1. Material parameters. --

parameter

value

unit

E v

5 . lo4 0.35 2 223 1000 50 4 lo-8

MPa

B Mf h

R EL

6

M P ~ K - ~ K MPa MPa

%

pseudo-elastic and the shape-memory regimes, but for brevity we report here only the most significative examples.

4.1. Uniaxial tests To begin with, we consider the following uniaxial tests in the pseudo-elastic regime single and multiple tension cycles with permanent inelasticity, multiple tension cycles followed by multiple compression cycles with saturating permanent inelasticity, multiple tension cycles with saturating permanent inelasticity, including degradation effect. On the other hand, in the shape-memory regime we consider multiple tension cycles at T = Mf, each one followed by heating strain recovery. For each experiment, we plot the output axial stress-axial strain curve. Single and multiple tension cycles with permanent inelasticity. The first considered uniaxial test consists in studying the response of the model under tension cycles reaching a maximum axial stress of urn,, = 300 MPa. The numerical experiments are performed at a temperature T = 298 K and using the following model parameters: H = 0 MPa, A = 0 MPa and K = 2%. The choice of a non-zero parameter K gives rise to a permanent inelasticity phenomenon, as shown by the axial stress-axial strain curves of Fig. 2, referring, respectively, to one and ten tension cycles. Multiple tension cycles followed by multiple compression cycles with saturating permanent inelasticity.

9 The goal of this test is to show the saturation of the permanent inelasticity The experiment is performed at a temperature T = 298 K and using the following model parameters: H = 1.5 lo4 MPa, A = 0 MPa and rc = 2%. The left part of Fig. 3 shows the response to ten tension cycles. We note that, since H is different from zero, the permanent strain saturates and does not exceed the threshold (cf. Ref. 22)

The right part of Fig. 3 reports the results when fifteen compression cycles follow the tension ones. Again, we can observe that permanent inelasticity is accumulated and saturates when reaching the same threshold as in the case of tension. Multiple tension cycles with saturating permanent inelasticity, including degradation effect. We now want to investigate the effect induced on the model by the parameter A coupling the two internal variables. The experiment consists of fifteen tension loops performed at a temperature T = 298 K and using the following model parameters: H = 1.5 . lo4 MPa, A = 2 . lo3 MPa and rc = 2%. As shown in Fig. 4, the choice of a non-zero value for A results in shifting down the loops. This sort of degradation effect is an important feature of the model as an analogous phenomenon is observed in experimental tests (see Fig. 1). Multiple tension cycles at T = Mf, each one followed by heating strain recovery. The aim of this last uniaxial experiment is to study the behaviour of the model when reproducing the shape-memory effect. The input consists of ten cycles, each one constructed as a tension loop with a maximum stress a,, = 150 MPa at a temperature T = Mf followed by a heating process at a constant zero stress up to a temperature of 298 K. The left part of Fig. 5 refers to a test with H = 0 MPa, A = 0 MPa and rc = 2%, while the right part refers to a test with H = 1.5 lo4 MPa, A = 2 lo3 MPa and rc = 2%. Both of them show that an inelastic effect is activated, so that we observe only a partial shape recovery. We'finally stress that in the first case, since A = 0 MPa, inelasticity is activated only during the heating process.

10 4.2. Biaxial tests

The goal of biaxial tests is to verify the behaviour of the model and its capability of reproducing permanent inelasticity when subjected to nonproportional multi-axial loading. Accordingly, we study the model response under the two following loading conditions non-proportional hourglass-shaped test, combined uniaxial tests. For both of these numerical experiments, we report the stress input and the corresponding strain output plots. Non-proportional hourglass-shaped test. The first considered biaxial test consists of a non-proportional test where ( ~ 1 1and (Tla are led to , , ,T( = 300 MPa in the hourglass shaped loading history of Fig. 6 (left), which is repeated five times. The experiment is performed at a temperature T = 298 K and using the following model parameters: H = 1.5. lo4 MPa, A = 4 . lo3 MPa and K = 10%. The numerical results, reported in terms of first and fifth cycle in Fig. 6 (right), show that the new formulation proposed is capable of introducing and controlling permanent inelasticity effects even in non-proportional multi-axial tests. Ten tension cycles in direction 1 followed by twenty tension cycles in direction 2. This last numerical experiment aims at showing the model response under loading conditions changing in their direction of application. It consists of uniaxial tension cycles whose direction is suddenly rotated of 7r/2 and is performed at a temperature T = 298 K using the following model parameters: H = 1.5-lo4MPa, A = 0 MPa and K, = 2%. Figure 7 shows the axial stress-axial strain curves for the two loading directions. The numerical results prove the capability of the model of reproducing the features shown in uniaxial tests even under multi-axial loading conditions. 5. Conclusions

The present work discusses the 3D constitutive model for describing the macroscopic behaviour of SMA proposed in Refs. 21,22. With respect to the previous model presented in Ref. 13, this new one is able to describe SMA macroscopic behaviours taking into account also permanent inelasticity effects. Such effects can be introduced both with a saturating or a non-

11 saturating evolution. Moreover, also degradation can be included. Many numerical experiments have been presented in order to show and assess the model performance both in uniaxial and non-proportional multi-axial problems. Acknowledgements This work has been partially supported by the European Project HPRNCT-2002-00284 "New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation".

Fig. 2. Uniaxial tests: tension cycles with permanent inelasticity ( H = 0 MPa, A = 0 MPa, tc = 2%,T=298K). Axial stress-axial strain output for single (left) and multiple (right) tension loops.

Fig. 3. Uniaxial tests: ten tension cycles (left) and ten tension followed by fifteen cornpression cycles (right) with saturating permanent inelasticity ( H = 1.5.lo4 MPa, A = 0 MPa, n = 2%,T=298K). Axial stress-axial strain output.

12

Fig. 4. Uniaxial tests: fifteen tension cycles with saturating permanent inelasticity, including degradation effect ( H = 1.5. lo4 MPa, A = 2 . lo3 MPa, n = 2%, T = 298 K). Axial stress-axial strain output.

Fig. 5. Uniaxial tests: multiple (ten) tension cycles a t T = M f ,each one followed by heating strain recovery, with H = 0 MPa, A = 0 MPa (left) and H = 1 . 5 . lo4 MPa, A = 2 . lo3 MPa (right) and n = 2%. Axial stress-axial strain output.

Fig. 6. Biaxial tests: non-proportional hourglass-shaped test ( H = 1 . 5 . lo4 MPa, A = 4 . lo3 MPa, n = lo%,T = 298 K). ~ ~ 1 1o12 - input (left) and lStand 5th cycle €11 - ylz output (right).

13

Fig. 7. Combined uniaxial tests: ten tension cycles in direction 1 followed by twenty in direction 2 ( H = 1.5. lo4 MPa, A = 0 MPa, n = 2%, T = 298 K). ull - E I I output (left)and u 2 z - €22 output (right).

References 1. T . Duerig and A. Pelton (eds.), SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology Conference (ASM International, 2003). 2. T . W . Duerig, K . N. Melton, D. Stoekel and C . M . Wayman, Engineering aspects of shape memory alloys (Butterworth-Heinemann, London, 1990). 3. C . Bouvet, S . Calloch and C. Lexcellent, European Journal of Mechanics A/Solids 23, 37-61 (2004). 4. S . Govindjee and C. Miehe, Computer Methods i n Applied Mechanics and Engineering 1 9 1 , 215-238 (2001). 5. D. Helm and P. Haupt, International Journal of Solids and Structures 40, 827-849 (2003). 6. S . Leclercq and C. Lexcellent, Journal of Mechanics and Physics of Solids 44, 953-980 (1996). 7. V . I . Levitas, International Journal of Solids and Structures 3 5 , 889-940 (1998). 8. V . I . Levitas, and D. L. Preston, Physical Review B 6 6 , 134206:l-9 (2002). 9. V . I . Levitas, and D. L. Preston, Physical Review B 6 6 , 134207:l-15 (2002). 10. B. Peultier, T . Benzineb and E. Patoor, Journal de Physique IV France 115, 351-359 (2004). 11. B. Raniecki and C. Lexcellent, European Journal of Mechanics, A: Solids 13, 21-50 (1994). 12. A. C. Souza, E. N . Mamiya and N., Zouain, European Journal of Mechanics, A: Solids 17, 789-806 (1998). 13. F . Auricchio and L. Petrini, International Journal for Numerical Methods i n Engineering 6 1 , 807-836 (2004). 14. M . Arrigoni, F . Auricchio, V . Cacciafesta, L. Petrini and R. Pietrabissa, Journal de Physique IV France 11, 577-582 (2001). 15. H . Funakubo (ed.), Shape Memory Alloys (Gordon and Breach Science Publishers, New York, 1987).

14 16. K. Otsuka and C. M. Wayman (eds.), Shape Memory Materials (Cambridge University Press, 1998). 17. Z. Bo and D. C. Lagoudas, D. C., International Journal of Engineering Science 37, 1175-1203 (1999). 18. S. Govindjee and E. P. Kasper, Journal for Intelligent Material Systems and Structures 8, 815-823 (1997). 19. D. C . Lagoudas and P. Entchev, Mechanics of Materials 36, 865-892 (2004). 20. A. Paiva, M. A. Savi, A. M. B. Braga and P. M. C. L. Pacheco, International Journal of Solids and Structures 42, 3439-3457 (2005). 21. F. Auricchio and A. Reali, Mechanics of Advanced Materials and Structures 14, 43-55 (2007). 22. F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, International Journal of Plasticity, available online (2007). 23. F. Auricchio and L. Petrini, International Journal for Numerical Methods i n Engineering 61, 716-737 (2004). 24. F. Auricchio, A. Reali and U. Stefanelli, Analysis of a model describing stressinduced solid phase transformation with permanent inelasticity, in preparation (2006). 25. F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the evoultion of shape memory materials, in preparation (2006). 26. J. C. Simo and T. J. R. Hughes, Computational Inelasticity (Springer-Verlag, New York, 1998). 27. P. Sittner, Y. Hara and M. Tokuda, Metallurgical and Materials Transactions 26A,2923-2935 (1995). 28. P. Sittner and V. N o v a , International Journal of Plasticity 16, 1243-1268 (2000).

15

NUMERICAL ANALYSIS OF A FRICTIONLESS PIEZOELECTRIC CONTACT PROBLEM ARISING IN VISCOELASTICITY . M . BARBOTEU and Y . OUAFIK Laboratoire de Mathdmatiques et Physique pour les Systtmes (MEPS), Universitd de Perpignan, Bhtiment B3, case courrier 12, 52 Avenue Paul Alduy, 66860 Perpignan, France E-mail: (barboteu,yowsef.ouafik) @perpignan.fr

Departamento de Matemdtica Aplicada, Universidade de Santiago de Compostela Facultade de Matemdticas, Campus Sur s/n, 15782 Santiago de Compostela, Spain 'E-mail: jramon@usc. es http://web.usc. es/-jmrnon A quasistatic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle is numerically studied. T h e linear electroviscoelastic constitutive law is employed t o model the piezoelectric material and the normal compliance condition is used t o model the contact. T h e variational formulation is derived in a form o f a coupled system for the displacement and electric potential fields. A n existence and uniqueness result is recalled. T h e n , a fully discrete scheme is introduced based on the finite element method t o approximate t h e spatial variable and an Euler scheme t o discretize the time derivatives. A main error estimates result is stated from which, under suitable regularity conditions, the linear convergence o f the algorithm is deduced. Finally, a two-dimensional example is presented t o demonstrate the performance o f the algorithm. Keywords: Piezoelectricity, Viscoelasticity, Normal compliance, Error estimates, Numerical simulations.

1. Introduction

In this work, we study, from the numerical point of view, a frictionless contact problem between a viscoelastic piezoeletric body and a deformable obstacle. Piezoelectricity is the ability of certain cristals, like the quartz (also ceramics (BaTi03, KNb03, LiNb03, etc) and even the human mandible or

16 the human bone), to produce a voltage when they are subjected to mechanical stress. The piezoeletric effect is characterized by the coupling between the mechanical and the electrical properties of the material: it was observed that the appearance of electric charges on some cristals was due to the action of body forces and surface tractions and, conversely, the action of the electric field generated strain or stress in the body. This kind of materials appears usually in the industry as switches in radiotronics, electroacoustics or measuring equipments. Different models have been developed early to describe the interaction between the electric and mechanical fields (see, e.g., [I-41 and the references therein). Recently, contact problems involving elastic-piezoelectric materials ( [5-91) or viscoelastic piezoelectric materials ( [lo]) have been studied. Moreover, in [ll]the well-known thermistor problem was numerically studied. In this paper, we consider a viscoelastic piezoelectric body which may become in contact with a deformable obstacle, the so-called foundation. The contact is assumed frictionless and a normal compliance condition is employed to model it (see [12]).

2. Mechanical and variational formulations

We begin with the description of the model, following [13], where more details can be found. Denote by Sd the space of second order symmetric tensors on IRd and by "." and 11 . 11 the inner product and the Euclidean norms on IRd and Sd. Let R c IRd, d = 1,2,3, denote a domain occupied by a viscoelastic piezoelectric body with a smooth boundary I' = do. We denote by v the unit outer normal vector to and we assume that this boundary is decomposed into three measurable parts rD,rF,rc, on one hand, and on two measurable parts I'A and I'B, on the other hand, such that meas (I'D) > 0, meas (FA) > 0, and rc r B . Finally, let [0,TI, T > 0, be the time interval of interest (see Fig. 1). Let x E R and t E [0,TI be the spatial and time variables, respectively, and, in order to simplify the writing, we do not indicate the dependence of the functions on x and t. Moreover, a dot above a variable represents the derivative with respect to the time variable. Let us denote by u the displacement field, a the stress tensor, E(U)=

c

17

Fig. 1. A viscoelastic piezoelectric body in contact with a foundation.

(cij (u))$=~ the linearized strain tensor given by

and cp the electric potential. The body is assumed viscoelastic piezoelectric and satisfying the following consitutive law (see [10,14]),

where A and B are the fourth-order viscosity and elastic tensors, respectively, E(cp) = ( ~ ~ ( c p ) )represents $~ the electric field defined by

and E* = (e:jk)f,3k=l denotes the transpose of the third-order piezoelectric tensor E = (eijk)ij,k=l. We recall that e:jk = ekij,

.

for all i, j, Ic = 1,. . ,d.

Following [I] the following constitutive law is satisfied for the electric potential,

18 where D is the electric displacement field and ,B is the electric permittivity tensor. Since the process is assumed quasistatic, the inertia effects are negligible and therefore,

where f o is the density of the body forces acting in R and qo is the volume density of free electric charges. Moreover, Div and div represent the divergence operators for tensor and vector functions, respectively. We turn now to describe the boundary conditions. On the boundary part FD we assume that the body is clamped and thus the displacement field neglects there, that is u = 0 on F D x (0,T). Moreover, we assume that a density of traction forces, denoted by f F, acts on the boundary part rFli.e.,

On the part rc the body can become in contact with a deformable insulator obstacle, the so-called foundation. According to [12] the following normal compliance contact condition is employed,

where (T, = uv . Y is the normal stress, u, = u . v denotes the normal displacement, g represents the gap between the body and the obstacle measured along the normal direction Y and p is a given function whose properties will be described below. Finally, we assume that the contact is frictionless and therefore, a, = a - cr,v = 0. Let R be subject to a prescribed electric potential c p on ~ rA and to a density of surface electric q~ on rg, that is,

We assume that q~ = 0 on rC,that is, the foundation is supposed to be insulator. We note that it is straightforward to extend the results presented below to more general situations by decomposing I' in a different way. The mechanical problem of the quasistatic contact of a viscoelastic piezoelectric body with a deformable obstacle is then written as follows. Problem P. Find a displacement field u : R x (0,T) -, Rd, a stress field a : R x (0, T ) -+B*, a n electric potential field cp : R x (0, T) 4 R and

19 an electric displacement field D : R x (0,T ) -+ IRd such that,

u=d~(u +B ) E ( u ) -E*E((p) in R x (O,T), D = E e ( u ) + P E ( ( p ) in R x (O,T), Diva+ f o = O in R x (O,T), divD=qo in R x ( O , T ) , u = O on ~ D X ( O , T ) , uv = f on I'F x (0,T ) , uT= 0 , -CTV = p(uV - g ) on Fc x (0,T ) , on F A X (O,T), D . Y = ~ F on I ' g x ( O , T ) , V=VA

u ( 0 ) = uo in R. Here, uo represents an initial condition for the displacement field. In order t o obtain the variational formulation o f Problem P, let us introduce the variational spaces V , Q and W , and the convex set W A as follows, V = {v E [ H 1 ( R ) l dv; = O on I'D), Q = {T = ( ~ i j ) t E~ [=L~~ ( R ) ] ;~ ~X i~=j ~ W = { $ J E H ~ ( R ) ; + = on O FA), W A = {$J E H 1 ( R ); $ = V A on I'A),

= 1,. . . , d),

j i , i ,j

and denote by H = [ L ' ( R ) ] ~ . The viscosity tensor A(x)= ( a i j k l ( ~ ) ) t ~ :, T~ ,E~Sd = ~ A(x)(T) E Sd satisfies: -+

( a ) aijkl = aklij = ajikl for i ,j, k , 1 = 1,. . . ,d. ( b ) aijkl E LoO(Q) for i ,j, k , 1 = 1,. . . ,d. (c) There exists mA > 0 such that A ( x ) T .T mA 1 V T E S d , a.e. x E R.

>

The elastic tensor B ( x ) = (bijkl(~))t,j,k,l,l : T E Sd verifies:

-+

(a) eijk ( b ) eijk

E

= eircj for i,j, k = 1,.. . ,d. E L w ( R ) for i ,j,k = I , . . .,d.

(11)

~(x)(T E )Sd

(a) bijkl = bklij = bjikl for i ,j, k , 1 = 1, . . . ,d. ( b ) bijkl E L w ( R ) for i ,j, k,1 = 1, . . . ,d. The piezoelectric tensor E ( x ) = (eijk(x))f,j,k,l : T Itd satisfies:

1~11~

(12)

Sd -+ E ( x ) ( T )

E

(13)

20 The permittivity tensor P(x) = (@ij(x))f,j,k,l,l : w E Rd -+ P(x)(w) E Rd verifies: (")Pij=& for i , j = l ,..., d. (b) Pij E Lm(R) for i , j = 1,..., d. (c) There exists mp > 0 such that P ( x ) w . w 2 mp llw112 Q w E Rd, a.e. x E R.

(14)

The normal compliance function p(x) : r E IW --t p(x, r) E 10, rn) satisfies: (a) There exists m, > 0 such that IP(x, TI)- P(X,r2)I I mpIrl - r ~ i Q r l , r z E R, a.e. x E r c . (b) (p(x,r i ) - p(x, rz))(ri - rz) 2 0 Q r l ,rz E R, a.e. x E r c . (15) (c) The mapping x E rc +-+ p(x, r) is measurable on rc, for all r E R. (d) p(x, r ) = 0 for all r 5 0. The following regularity is assumed on the density of volume forces, tractions, volume electric charges and surface electric changes:

Finally, we assume that the gap function, the initial displacements and the boundary condition ( P A satisfy

Using the Riesz' Theorem, we define the linear mappings f : [0,TI and q : [0,TI -t W as follows,

--t

V

We notice that the regularity assumptions (16) imply that f E C([O,TI;V) and q E C([O,TI;W). Let us denote by j : V x V -t R the normal compliance functional given by

where, for all v E V, we let v, = v v.

21 Plugging (1) into (3) and (2) into (4), keeping in mind that D = -Vcp and using the boundary conditions (5)-(9), applying a Green's formula we derive the following variational formulation of Problem P. P r o b l e m V P . Find a displacement field u : [0,TI -t V and an electric potential field cp : [0,T] -+ WA such that u(0) = uo and for all t E [0,TI,

Using analogous ideas to those employed in [15] for a normal compliance elastic problem or in [lo] for the case of viscoelastic materials, the following theorem, which states the existence of a unique weak solution to Problem VP, is obtained. T h e o r e m 2.1. Assume that (11)-(17) hold. Then there exists a unique solution to Problem VP with the following regularity

3. Fully discrete approximations

In order to simplify the writing we assume, in this section, that c p = ~ 0 (and then WA = W). It is straightforward to extend the results presented below to a more general case. The discretization of (18)-(19) is as follows. First, we consider two finite dimensional spaces vh c V and wh C W approximating the spaces V and W , respectively. h > 0 denotes the spatial discretization parameter. To discretize the time derivatives, we use a uniform partition of LO, TI, denoted by 0 = to < tl < . . . < tN = T , and let k be the time step size, k = T I N . For a continuous function f (t) let f, = f (t,), and for a sequence { ~ , ) z = we ~ let 6wn = (w, - wn-1)/k denote the divided differences. In this section, no summation is assumed over a repeated index, and c denotes a positive constant which depends on the problem data, but it is independent of the discretization parameters h and k. Thus, using an Euler scheme, the fully discrete approximation of Problem VP is the following. P r o b l e m vphk. Find a discrete displacement field uhk= { u ~ ~ c) ~ = ~ vh and a discrete electric potential field cphk = { c p 2 k ) L c wh such that

22 ukk = uk and for all n = 1,.. . , N ,

where U! is an appropriate approximation of the initial condition uo. We notice that the fully discrete problem Whkcan be seen as a coupled system of variational equations. Using classical results of nonlinear variational equations (see [16]) we obtain that Problem VPhk admits a unique solution u h k c vh and cphk c w h . Now, we have the following main error estimates result (see [13] for details). Theorem 3.1. Assume that (11)-(17) hold. Let (u, cp) and ( u h k ,cphk)denote the solutions to problems V P and vphk,respectively. Then, the following error estimates hold for all w h = {W?)~N_~ c vh and $h = c Wh,

{$:)cl

We notice that the above error estimates are the basis for the analysis of the convergence rate of the algorithm. Thus, let 0 be a polyhedral domain and denote by Th a triangulation of R compatible with the partition of the boundary 'I = dR into I'D, I'F, I'c on one hand, and on I'A and I'B, on the other hand. Let Vh and wh be defined in the following form,

where P l ( T r ) represents the space of polynomials of global degree less or equal to one in T r . Assume that the discrete initial condition u; is obtained by

23 where IIh = (.rrh)f=l: [C(n)ld-+ vh,and .rrh : C ( n ) -+ B h is the standard finite element interpolation operator (see, e.g., [17]). Then, we have the following corollary which states the linear convergence of the algorithm under suitable regularity conditions.

Corollary 3.1. Assume that (11)-(17) hold. Let ( u , cp) and ( u h k ,cphk) denote the solutions to problems V P and v p h k , respectively, and let the discrete initial condition be given by (25). Under the following regularity conditions

the linear convergence of the algorithm is achieved, that is, there exists a positive constant c > 0 , independent of the discretization parameters h and k, such that

4. Numerical results

In order to show the behaviour of the fully discrete method presented in the previous section, some experiments have been done in the study of twodimensional problems. First, in Section 4.1 we describe the algorithm used to solve Problem vphkand, secondly, in Section 4.2, we consider a twodimensional example in order to describe some mechanical aspects of the frictionless viscoelastic piezoelectric contact behaviour. 4.1. Numerical algorithm

The algorithm, used in solving the fully discrete frictionless contact problem v p h k , is based on a backward Euler difference for the time derivatives and on a penalty approach (see 1181 for more details) to simulate the normal compliance law. In order to give the solution algorithm, we have to introduce the expressions of the functions w h , uh and 6uh(resp. cph and $Jh)by considering theirs values a t the ith nodes of Th and the basis functions ai (resp. yi) of the space vh (resp. w h ) for i = 1 , . . . , NtOt(Nt,t is the total number of nodes), Ntot

.luh

=

=C w i a i ,

Ntot

uh =

i=l

Ntot

u'ai,

i=l Ntot

and

$Jh

6uh =

=

Ntot $

J

~

~

p~ h, =

6uiai, i=l

piyi.

(27)

24 The penalty approach shows us that the Problem by the following system of nonlinear equations

+

vphkcan be governed

+

A(8un) G(un, ~ n ).T(un) = 0 . (29) The vectors Sun, u, and cp, represent respectively the generalized vectors defined as follows = {uin )Ntot Z=I 7 Sun = {S~k)Zv=t,.~ and cp, = {cpk)zv=t.,

(30) The penalized contact operator .T(u,) = cpdist(uv(t,) - g, lR+)vt denotes the gradient of the penalized contact functional ,Chk = +cpdist2(uv(tn)g, lR+) in the direction u ; cp is the surface stiffness coefficient and so l/cp is the deformability coefficient. In addition, the terms A(Su,) and G(u,, cp,) represent respectively the viscous term and the elastic-piezoelectric term given by U,

(A(6un) . w ) ~ =N( d~&~( ~~u n ) , & ( w ~Qwh ) ) ~E (G(un, 9,)

'

vh,

+

(w,$))!RNtot = ( B & ( ~ ~ ) , & ( w ~ ()t)&Q( w h ) , v v n ) H

-(fn,w h ) v - (E&(un),v $ ~ ) H+ ( P v ( ~ nv, $ ~ ) H- (q, $ h ) ~ , for all wh E vh and $h E wh,where w (resp. $) represents the generalized vector constitued by the values wi (resp. q i ) for i = l , . . . ,Ntot. We can remark that the volume and surface efforts are contained in the term G(un, 9,). The solution algorithm consists in a combination between the finite differences (backward Euler difference) and the linear iterations methods (Newton method). To solve (29), at each time increment the variables (u,, cp,) are treated simultaneously through a Newton method and therefore in what follows we use x, to denote the pair (u,, cp,). The algorithm that we used in the viscoelastic piezoelectric case can be developed in three steps which are the following: A prediction step This step gives the initial displacement and the velocity by the following formula 0

0

q n + l = cpn+l, 4 + 1 = un+l and A Newton linearization step At an iteration i of the Newton method, we have

= Sun.

25 with K;+, = D ~ , , G ( u h + ~ , c p ~~ + "~ ) 1, = ~ u ~ ( b u h + ~l )h , + 1 = , + ~P ,i+l , + ~ ) ;i and n reD u . F ( ~ ~ + and l ) , where xhyl denotes the pair ( ui+l present the Newton iteration index and the time index, respectively. Here, Du,,G, D u A and DuF denote the differentials of the functions G , A and .F with respect to the variables u and p. This leads us to solve the resulting linear system

cpkyl

uhY1

where A x = ( A u i , A P i ) with Aui = - u:+, and Acpi = - cp;+, . We solve the linear system of equations (31) by using a Conjugate Gradient Method with efficient preconditioners to overcome the poor conditioning of the matrix due to the penalized contact terms. We can remark that in the case where the operator A and G are linear the matrices Q ; + ~and K;+l do not change during the Newton iterations. a A correction step Once the system (31) is resolved, we update xi% and 6 u h 3 by

xi+' n+l = x : + ~+ n x i

Aui + k.

and d ~ i f :=, dt&+l

4.2. Numerical results i n a two-dimensional example

As a two-dimensional example of problem P, we consider the body R with the boundary r which can come into contact with a deformable foundation (see the setting shown on the left-hand side of Fig. 2). To fix the geometry we set the points PI = ( 0 , 1 ) , P:! = ( 1 , 0 ) ,P3 = ( 3 , 1 ) , P4 = (1.5,1),Ps = (1,1.5)and P6 = ( 1 , 4 ) .We define FB = rc = [ P I P2], , FD = [P4,P5]and rF= r\(l?cUl?~);(X2, X 3 ) denotes the canonical orthonormal basis. Here, we use as material the viscoelastic piezoelectric body whose constants are given in Tables 1 and 2. We suppose that the body is clamped on rDand Table 1. Material elastic and viscoelastic constants of the considered piezoelectric body. Elastic b22

210

I 1

b23

105

I 1

(GPa) b33

211

I b44 1 42.5

viscoelastic a22

21

1 1

a23

10.5

1 1

(GPa a33

21.1

. s)

I a44 1 4.25

26 Table 2. Material electric constants of the considered piezoelectric body. Piezoelectric (C/m2) e32 e33 1 e24 -0.61 1 1.14 1 -0.59

1

Permittivity ( C 2 / ~ m 2 )

PZZ/EOI -8.3

1

P33/€0

-8.8

we employ the following data:

This physical setting permits to show the inverse piezoelectric effect that corresponds to the appearance of strain or stress in the body due to the action of the eletric field. This example represents a contactor stimulated by an electric field.

Fig. 2. Problem setting (left) and the discretization of the body (right).

We suppose that the time interval [O, 11 is discretized with a uniform partition. The picture on the right-hand side of Fig. 4 shows a uniform triangulation of the domain C2. According to Fig. 3, it can be seen that

27

Fig. 3. Initial and amplified deformed mesh with contact interface forces in the viscoelastic piezoelectric case.

the action of the difference of the electric field on [P3,P4]and [P5, Ps]induces a deformation of the body. That results in to make come into contact the body with the foundation on Fc. Indeed, we remark that some contact nodes are in slip case and the contact forces are following the exterior normal on rc.This is due to the fact that the problem is frictionless. Moreover, Fig. 4 shows the distribution of the electric displacement field and the electric potential in the body. We can notice a certain correspondence between the distribution of the electric displacement field and the viscoelastic constraints presented in Fig. 5. This happens since the higher values of the electric field are located at the zones where the viscoelastic constraints are stronger.

Acknowledgement The work of J.R. Fernbndez was partially supported by MCYT-Spain (Project BFM2003-05357) and by the project "New materials, Adaptive Systems and their Nonlinearities; Modelling, Control and Numerical Simulation" carried out in the framework of the european community program "Improving the Human Research Potential and the Socio-Economic Knowledge Base" (Contract No. HPRN-CT-2002-00284).

28

Fig. 4. The electric displacement field (arrows) and the electric potential in the deformed configuration.

Fig. 5. Initial boundary and the viscoelastic constraints in the deformed configuration.

References 1. R.C. Batra and J.S. Yang, Saint-Venant's principle in linear piezoelectricity, J. Elasticity 38 (1995), 209-218. 2. T. Ideka, Fundamentals of piezoelectricity, Oxford University Press, Oxford, 1990. 3. R.D. Mindlin, Polarisation gradient in elastic dielectrics, Internat. J. Solids Structures 4 (1968), 637-663. 4. R.A. Toupin, The elastic dielectrics, J. Rational Mech. Anal. 5 (1956), 84% 915.

29 5. M. Barboteu, J.R. FernBndez and Y. Ouafik, Numerical analysis of two frictionless elastic-piezoelectric contact problems, Preprint (2006). 6. S. Hiieber, A. Matei and B.I. Wohlmuth, A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity, Bull. Math. Soc. Sci. Math. Roumanie 48(96) (2005), 209-232. 7. F. Maceri and B. Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support, Math. Comput. Modelling 28 (1998), 19-28. 8. M. Sofonea and E.-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction, Math. Model. Anal. 9(3) (2004), 229-242. 9. M. Sofonea and Y. Ouafik, A piezoelectric contact problem with normal compliance, Appl. Math. 32 (2005), 425-442. 10. M. Sofonea and E.-H. Essoufi, Quasistatic frictional contact of a viscoelastic piezoelectric body, Adv. Math. Sci. Appl. 14(1) (2004), 25-40. 11. J.R. FernBndez, Numerical analysis of the quasistatic thermoviscoelastic thermistor problem, M2AN Math. Model. Numer. Anal. 40(2) (2006), 353-366. 12. A. Klarbring, A. Mikelib and M. Shillor, Frictional contact problems with normal compliance, Internat. J. Engrg. Sci. 26 (1988), 811-832. 13. M. Barboteu, J.R. FernBndez and Y. Ouafik, Numerical analysis of a frictionless viscoelastic piezoelectric contact problem, Preprint (2006). 14. G. Duvaut and J.L. Lions, Inequalities in mechanics and physics. Springer Verlag, Berlin, 1976. 15. Y. Ouafik, A piezoelectric body in frictional contact, Bull. Math. Soc. Sci. Math. Roumanie 48(96) (2005), 233-242. 16. R. Glowinski, Numerical methods for nonlinear variational problems, Springer, New York, 1984. 17. P. G. Ciarlet, The finite element method for elliptic problems, In: Handbook of Numerical Analysis, eds. P.G. Ciarlet and J.L. Lions, Vol. 11, North Holland, 1991, 17-352. 18. P. Wriggers, Computational contact mechanics, John Wiley & Sons, 2002.

30

A STABILIZED MITC6 TRIANGULAR SHELL ELEMENT

Dipartimento di Matematica F.Enriques, Universitd di Milano, Milano 20133, Italy E-mail: beiraoOmat.unimi.it

D. CHAPELLE and I. PARIS MACS team, ZNRIA, Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, Z h n c e E-mail: fisLname. 1asLnameOinria.fr T h e phenomenon o f spurious membrane modes in t h e triangular MITCGa shell element is discussed, and a remedy based on a stabilized bilinear form is proposed. Numerical tests are included in order t o show the good performance o f t h e method both in membrane and bending dominated problems. Keywords: Shells; Triangular MITC elements; Spurious modes, Stabilization.

1. Introduction

In engineering analysis shell structures are frequently encountered, hence shell finite elements are of utmost interest. As is now well established,CB03 one of the great challenges in the design of shell elements is to ensure reliability of the numerical procedures for all types of asymptotic behaviours. The main difficulty in this respect - which can be summarized as the "asymptotic dilemma" - consists in circumventing the locking phenomena arising in bending-dominated asymptotic behaviours without destroying the ability of the procedure to accurately represent membrane-dominated and mixed behaviours. Some shell finite element procedures have been shown to be quite reliable for general asymptotic behaviours, and in particular the quadrilateral elements MITC4, MITC9 and MITC16, see.B1C00However, in practice complex geometries must be handled, and triangular elements are often more adequate - or even necessary - in this respect. Some triangular MITC elements have also been proposed,PSL04but they have been found to feature

31 some limitations, particularly in membrane-dominated situations. Namely, rotations convergence is very poor and in some test problems there exist non-physical displacement modes with zero membrane energy. The purpose of this paper is to illustrate and summarize these difficulties for the MITCGa element, and to propose some remedies carefully designed and assessed in order to improve the membrane-dominated behaviour without deteriorating the reliability in bending-dominated and mixed situations. The outline of the paper is as follows. In Section 2, we introduce a test problem for which the MITCGa exhibits spurious membrane energy modes and we provide detailed convergence results which show the shortcomings of the element. In Section 3, based on these numerical results we propose a stabilization strategy to improve the convergence. Then, in Section 4 we numerically assess the stabilized element. Finally we give some conclusions. 2. The hyperbolic triangle test problem

As demonstrated iqcP for certain geometries there exist some discrete displacement fields for which the MITCG membrane energy (shear energy not included) vanishes, although pure bending is inhibited in theory. We call these particular displacement fields "spurious membrane modes", and the aim of this section is to analyse the impact of such modes in actual solutions. Noting that spurious membranes modes per se are not present in the clamped hyperboloid test problem used in,CP3PSL04 we now introduce a new test problem specifically designed to investigate this issue. As in,cP we consider a shell of uniform thickness t given by the midsurface defined as the image of the triangular 2 0 domain of vertices (-1, O), (1,O) and (0,112) by the following mapping

c2

The structure is clamped along the boundary corresponding to = 0, hence pure bending is inhibited (seecP). The loading applied is given by (2) where "tr" denotes tensor traces. This choice of loading clearly ensures the admissibility condition F E which is crucial to obtain a well-posed membrane-dominated problem, see.CB03In addition, this loading is distributed over the whole domain, hence it does not induce internal boundary layers that would entail meshing difficulties.

v;,

32

Fig. 1. Hyperbolic triangle test problem - Left: undeformed midsurface mesh with boundary refinement ( E = Right: deformed and undeformed meshes

Figure 1 shows the geometry of the structure and the deformed configuration computed with displacement-based P2 elements. Note that we resort to mesh refinement to adequately capture the boundary layer corresponding to the clamped boundary conditions, see.cP Based on preliminary computations, the width of the refined band is taken as 3&L, where L denotes a characteristic dimension of the structure - here, L = 1 - and E = t/L. 2.1. MITC6a spurious membrane modes

We henceforth need to clearly distinguish between membrane and shear energies. Therefore, we now denote by A, and A, the scaled membrane and shear bilinear form, respectively. Likewise, when referring to an MITC formulation we will denote by A&, and A! the bilinear forms obtained by the corresponding mixed interpolation of strain components. We will call "spurious membrane mode" a non-zero discrete displacement field C such that

A ~ , ( G ,C) = 0.

(3)

We recall that the existence of such modes for the geometry corresponding t o the hyperbolic triangle test problem has been demonstrated in.CP~BadVCP In addition, we point out that these modes cannot be well detected by an error indicator using reduced strains such as the s-norm - although the s-norm is a meaningful indicator in many cases, seeCBo3- since they have near-zero membrane reduced strains. Therefore, in the sequel we also consider the error measures provided by the unreduced membrane and shear energies to assess the solutions.

33 2.2. Displacement-based and MITCGa solutions

For the above-described hyperbolic triangle test problem, we computed the P2 displacement-based and MITCGa solutions corresponding to various thickness values and meshes. The reference solution was given by the P2 solution for a very fine mesh. We show the corresponding convergence curves in Figures 2 and 3. -

P2 Am norm mM near

o

,

.

,

.

-06.

-1.2

-

-1 8

-

-2.4 ' -3 -3.8

-

.

-42-4.8

:'

.

5 4 -

-8 -1.8

;'

/ "

-1.5

"

-12 4 . 8 -0.8 IW$U

-0.3

Fig. 2. Convergence curves associated with the s-norm and A, norm (with or without shear terms) for the hyperbolic triangle problem and P2 displacement-based shell finite elements. The dotted line shows the optimal convergence rate which is 4.

As expected, P2 displacement-based elements provide accurate numerical solutions (see Fig. 2), and uniform quadratic convergence is fairly well achieved for strains (as seen through the s-norm) and displacements and rotations (by means of the A, norm). Total and partial energy values associated with the numerical solutions are summarized in Table 1. The membrane-dominated behavior is well illustrated: shear and bending energies vanish as the thickness decreases, whereas the membrane energy becomes dominant and it can be scaled by a factor of E . Also, given a thickness value total and partial energy values converge as the mesh is being refined. Considering the MITCGa element (see Fig. 3), quadratic convergence is observed for the strains as seen through the s-norm, albeit with some significant sensitivity with respect to the thickness. However, displacements and rotations are not well predicted for small thicknesses as measured by

34

Fig. 3. Convergence curves associated with the s-norm and membrane energy norm (with or without shear terms) for the hyperbolic triangle problem and MITC6a shell finite element. The dotted line shows the optimal convergence rate which is 4.

Table 1. Energy values associated with the hyperbolic triangle problem and P2

E

= lo-2

Total Bending Membrane Shear =

Total Bending Membrane Shear =

Total Bending Membrane Shear

h = 0.25 1.625614138e-10 2.899686587e-12 1.592970032e-10 4.32304665Oe-13

h = 0.125 1.636227424e-10 3.443922227e-12 1.600679730e-10 1.761754264e-13

h = 0.0625 1.637919918e-10 3.583705711e-12 1.601860052e-10 8.726474189-14

Ref sol 1.638092595e-10 3.602163776e-12 1.601980158e-10 7.403760514e-14

h = 0.25 1.745541193e-09 1.318061523e-11 1.731816291e-09 5.598601652e-13

h = 0.125 1.75305636Oe-09 1.598817421e-11 1.736784713e-09 3.001356466e-13

h = 0.0625 1.754706364e-09 1.617578273e-11 1.738305088e-09 2.422948594e-13

Ref sol 1.755258783e-09 1.624272790e-11 1.738917129e-09 1.156832852e-13

h = 0.25 1.791391461e-08 2.889606850e-11 1.788109973e-08 3.922139514e-12

h = 0.125 1.801289895e-08 5.421977711e-11 1.795776910e-08 9.142441537e-13

h = 0.0625 1.802635849e-08 5.999243985e-11 1.796613404e-08 2.363593308e-13

Ref sol 1.802860067e-08 5.937903190e-11 1.796901784e-08 2.082007634e-13

means of the Am-norm. Even when discarding the shear terms from the This poor Am-norm, displacements per se remain inadequate for e = convergence is also apparent when plotting the displacements and rotations

35 for the Pz and MITCGa solutions, see Figs. 4 and 5.

Fig. 4. Deformed midsurface plots for the hyperbolic triangle problem: Pz (left) and MITC6a (right) elements for E = lod4 and a mesh of 96 elements -h = 0.25- and 217 nodes, boundary layer of width 3&L (scale = 1 x lo6).

Fig. 5. Magnitude of the rotation field for the hyperbolic triangle problem: P2 (left) and MITC6a (right) elements for E = and a mesh of 96 elements -h = 0.25- and 217 nodes, boundary layer of width 3&L.

We also give in Table 2 the total and partial energy values computed for the MITCGa solutions, and both with MITC-reduced and unreduced strains. Like for the displacement-based solutions, membrane energy is dominant when considering reduced strains for the MITCGa solutions. However,

36 energy convergence is much slower than for the P2 solution. Furthermore, we note a discrepancy between unreduced and reduced membrane energy values for small thicknesses, a likely indication of the presence of membrane spurious modes. Another noteworthy observation is that shear energy is dominant in the unreduced energy for small thicknesses, in fact by several orders of magnitude for e = This will be further discussed in the next section.

Table 2. Reduced and unreduced energy values associated with the hyperbolic triangle problem and MITC6a triangular shell finite element for different values of E and h.

E

Reduced Unreduced Reduced Unreduced Reduced Unreduced Reduced Unreduced

h = 0.25 1.635943071e-10 1.782743337e-10 3.707275380e-12 3.704225021e-12 1.597151945e-10 1.594303696e-10 2.239399956e-13 1.519089629e-11

h = 0.125 1.637427893e-10 1.649619303e-10 3.602355501e-12 3.601609453e-12 1.600937939e-10 1.598377553e-10 1.079039190e-13 1.583109201e-12

h = 0.0625 1.638004556e-10 1.639188685e-10 3.606538670e-12 3.606281368e-12 1.601762770e-10 1.601017737e-10 8.162652771e-14 2.745765666e-13

Reduced Unreduced Reduced Unreduced Reduced Unred~Ced Reduced Unreduced

h = 0.25 1.824418394e-09 2.199030724e-08 5.165339248e-11 5.147666241e-11 1.766887584e-09 2.35258 1643e-09 5.879353460e-12 1.958620655e-08

h = 0.125 1.761419957e-09 2.843743147e-09 2.140567412e-11 2.14056073Oe-11 1.739098512e-09 1.772663452e-09 9.253863961e-13 1.049680093e-09

h = 0.0625 1.754729267e-09 1.781053800e-09 1.698796935e-11 1,698723834-11 1.737541751e-09 1.738327613e-09 2.143617913e-13 2.575325829e-11

Reduced Unreduced Reduced Unreduced Reduced Unreduced Reduced Unredzlced

h = 0.25 1.981605327e-08 4.812321121e-06 6.515344241e-10 6.505662494e-10 1.895572441e-08 7.029725398e-08 2.087915606e-10 4.741373215e-06

h = 0.125 1.837589483e-08 2.282277524e-06 1.417090352e-10 1.416300378e-10 1.820591533e-08 2.51982605Oe-08 2.827106350e-11 2.256937611e-06

h = 0.0625 1.811792133e-08 7.336077997e-07 1.001703188e-10 1.001495601e-10 1.801124243e-08 1.894450248e-08 6.510695501e-12 7.145631429e-07

=

Total Bending Membmne Shear

=

Total Bendinq Membmne Shear

=

Total Bendinq Membrane Shear

37 3. Improving the MITCGa element

As discussed in the previous section, the presence of spurious membrane modes in the MITCGa solution of some membrane-dominated problems is the likely cause of unsatisfactory error curves in the membrane norm. Moreover, the amplitude of these parasitic modes may dominate that of the underlying "correct" solution, giving erroneous displacement graphs. The main difficulty in the attempt of curing this phenomenon, is that the property that membrane energy vanishes for the membrane spurious modes is essentially shared by all the discrete inextensional modes in bending dominated problems, see for example.CB03As a consequence, any cure for membrane spurious modes will be at a strong risk of deteriorating the method's performance in bending dominated situations. The aim of this section therefore becomes to develop a cure for the MITCGa element in membrane dominated problems, which does not strongly hinder the method in bending dominated cases. In the energy Table 2 for the MITCGa element, a particular phenomenon can be immediately noticed. For the solution Ui, particularly when E << 1, we have that

In other words, there is an amplification of several orders of magnitude when comparing the reduced and unreduced shear energies, while the amplification is limited in the membrane energy. Furthermore, whenever the relative thickness is sufficiently small, the unreduced shear energy A,(Ui, U,E) is dominant over the total energy. This phenomenon is in complete contrast with the benchmark results in Table 1, where it is shown that the shear part of the energy in the "correct" solution should be negligible. These observations, which hold for all the membrane shell problems considered, seem to indicate that the nature of the spurious part of the solution is mainly reflected in the shear energy. Moreover, it is well known that in typical bending dominated situations the main source of locking is the membrane energy, and not the shear energy. The combination of these two arguments suggests t o stabilize/modify only the shear part of the MITCGa energy, in order to reach our aforementioned goal. The most natural modification is to propose a MITCGa scheme in which the parasitic modes are handled by adding an unreduced weighted shear part to the discrete method.

38 Namely, we modify the original bilinear form of the problem as follows & , ( ~ h ,vh) = E3Ab(uh,vh)

+ & A k ( ~ vh) h, +

where T represents a general triangle of the mesh T h , h~ the diameter of T, L the characterstic size of the shell, and As,T,At,T the local bilinear forms integrated over the element. The chosen h~-scalingfor the stabilization term can be justified by observing carefully at the energy tables of the previous section; for the related study see.BadVCPNote that we include no dependence of C on E in order to have a method which, in principle, provides an improved stability also in the E --+ 0 membrane limit problem. 4. Numerical results for t h e stabilized M I T C 6 a element

In the present section we show a set of numerical tests in order to assess the performance of the newly introduced stabilized method. In addition to the hyperbolic triangle, we consider also a bending dominated problem. Additional numerical tests and results can be found in.BadVCP 4.1. Hyperbolic triangle

In Figs. 6-7 we plot the convergence graphs for the stabilized MITCGa, with the choices of the parameter C = 1, 115. The norms considered are the same as in the previous section, namely the s-norm and the membrane norm with and without the shear part (i.e. Am and A,, norm, respectively). Comparing the results with those of the original MITC6a element in Fig. 3, we observe the following. As expected, the convergence curves in the s-norm, which where already satisfactory, are essentially the same. Furthermore, the plots for the A, norm are now strongly improved with respect to the original case; the highly t dependent and unsatisfactory results of Fig. 3 have been substituted by the fairly good results of Figs 6-7. In addition there is also an even larger improvement for the convergence in the full Am norm, i.e. in the approximation of the rotations, although the approximation properties for very small values of the thickness are still rather poor. Finally, note that as expected the results improve when increasing C, although the difference between the two choices is mostly visible when considering rotation convergence for very small thicknesses.

39 In Fig. 8 we plot the deformed midsurface of the shell as computed by the MITCGa stabilized method. As it can be clearly appreciated by comparing with Fig. 4, the unphysical modes which completely dominate the displacement plot of the original MITCGa element are now efficiently filtered. Moreover, the same positive effect can be seen on the rotation plots of Fig. 9, which do not exhibit the oscillatory behavior of Fig. 5. -

mHaBs A norm ulhanhear (

-0.8

-

norm

--A

1 by ~ $A,$ 4

~

(atabllked bym*&

.

0

.'

.

0

-

-0.8 .

-

-1.2

.

1

.

.

.

-1 2

-24 -

4 . 2-

;

-

4 4 .

-

..

-4.8

.

;

4.8 4.4

-

; '

:

'

.

"

-1.8 -1.5 -1.2 4.0 -01 4 3 @!o@')

4

Ublllrn

A~111Ca

.';

--e--

:

4

:

4 2 -

"

Ub1110m0

"

-18 -1.5 -1.2 -0.8 -0.0 1 ~ , 0 ~ 0

-0.3

Fig. 6. Convergence curves associated with the s-norm and membrane energy norm (with shear terms discarded or not) for the hyperbolic triangle problem and stabilized MITC6a shell finite element with C = 1. The dotted line shows the optimal convergence rate which is 4.

4.2. h

e a x i s y m m e t r i c hyperboloid

In order to assess the behavior of the stabilized method in a bending dominated situation, we consider the following classical benchmark. The analyzed shell is an axisymmetric hyperboloid of uniform thickness t and length 2L, as displayed in Fig. 10. The hyperboloid is free at both ends and subjected to the smooth periodic pressure load (6) P(V) = pows(2cp), where cp represents the usual angular coordinate on the shell surface. Due to the free boundary conditions, it is well known that this benchmark constitutes a bending dominated shell problem. We solve this problem numerically with the original MITCGa element and with the stabilized

40

Fig. 7. Convergence curves associated with the s-norm and membrane energy norm (with shear terms discarded or not) for the hyperbolic triangle problem and stabilized MITC6a shell finite element with C = 115. The dotted line shows the optimal convergence rate which is 4.

stabilized MITCGa C = 1

stabilized MITCGa C = 115

Fig. 8. Deformed midsurfaces for the hyperbolic triangle problem and stabilized 217 MITC6a element: C = 1 (on the left) and C = 115 (on the right) ( E = nodes, 96 elements, scale = 1 x lo6).

MITCGa for the parameter choices C = 115 and C = 1. The convergence graphs for the s-norm and for the unreduced energy norm are depicted in Figs. 11-13; note that, due to its periodic structure, the discrete problem is

41

Fig. 9. Magnitude of the rotation field for the hyperbolic triangle problem and stabilized 217 nodes, MITC6a element: C = 1 (on the left) and C = 1/5 (on the right) ( E = 96 elements).

Fig. 10. Axisymmetric hyperboloid shell problem loaded by smoothly varying periodic pressure normal to the surface (L = 1.0, E = 2.0 x lo1', u = 1/3 and po = 1.0).

actually solved on the subdomain depicted in Fig. 10. We recall that our added stabilization term consitutes an unreduced contribution in the shear energy, and is therefore a possible additional source of locking in a bending dominated situation. Therefore, as already mentioned, our stabilization is actually a potential hindrance for the method in bending dominated problems. As we can appreciate in Fig. 11, the MITCGa gives fairly satisfactory results. Regarding the stabilized MITCGa, the convergence curves of Figs. 1213 clearly show that the element behaves almost as well as the original MITCGa element, for both choices of the stabilization parameter C.

42

Fig. 11. Convergence curves associated with the s-norm and Ab + A m norm for the free hyperboloid shell problem and MITCGa shell finite element. The dotted line shows the optimal convergence rate which is 4.

+

Fig. 12. Convergence curves associated with the s-norm and Ab Am norm for the free hyperboloid shell problem and stabilized MITCGa shell finite element with C = 1. The dotted line shows the optimal convergence rate which is 4.

43

Fig. 13. Convergence curves associated with the s-norm and Ab+ A , norm for the free hyperboloid shell problem and stabilized MITCGa shell finite element with C = 115.The dotted line shows the optimal convergence rate which is 4.

5 . Concluding remarks

We have introduced a test problem named the hyperbolic triangle for which the MITCGa element displays spurious membrane energy norm and poor rotation convergence. Based on the detailed energy values obtained for this test problem we proposed a stabilization strategy designed to improve the MITC6a convergence in membrane-dominated problems. The stabilized element displays a much better convergence behaviour in the hyperbolic triangle test problem, and the bending-dominated behaviour - assessed with the free axisyrnmetric hyperboloid test problem - is not significantly deteriorated. Hence, the stabilized MITC6a element is a valuable triangular element that behaves well in the main two categories of asymptotic behaviours. References

BadVCP. L. Beirh da Veiga, D. Chapelle, and I. Paris. Towards improving the MITCGa element. In preparation. BICOO. K.J. Bathe, A. Iosilevich, and D. Chapelle. An evaluation of the MITC shell elements. Comput. €4 Structures, 75(1):1-30, 2000. CB03. D. Chapelle and K.J. Bathe. The Finite Element Analysis of Shells Fundamentals. Springer-Verlag, 2003.

44 CP. PSL04.

D. Chapelle and I. Paris. Detailed assessment of MITC6 elements. In preparation. K.3. Bathe Phill-Seung Lee. Development of MITC isotropic triangular shell finite elements. Computers €4 Structures, 82:945-962, 2004.

45

A NEW FAMILY OF C0 FINITE ELEMENTS FOR THE KIRCHHOFF PLATE MODEL L.

BEIMO DA VEIGA

Dipartimento di Matematica nFederigo Enriques", Universitci di Milano, via Saldini 50 Milan, 20133, Italy E-mail: beimoOmat.unimi.it

J. NIIRANEN and R. STENBERG Institute of Mathematics, Helsinki University of Technology, P. 0. Box 1100 Espoo, 02015 T K K , Finland, E-mail: jarkko.niimnenOtkk.fi, rolf.stenbergQtkk.fi We present a finite element family for the Kirchhoff problem by using C0continuous basis functions for the deflection and the rotation, i.e., the same approach as typically used for the Reissner-Mindlin model. A theoretical error analysis shows that optimal +priori error estimates are valid for the method, in the presence of free boundary conditions as well. Furthermore, a reliable and efficient a-posteriori error estimator is introduced for adaptive mesh refinements. Benchmark computations verify the optimal rate of convergence and illustrate the robustness of the a-posteriori error estimator.

Keywords: Finite elements; Kirchhoff plate model; Free boundary; A-priori error analysis; A-posteriori error analysis.

1. Introduction

The fundamental difficulty in designing a finite element method for the Kirchhoff plate bending problem originates from the corresponding variational formulation: the natural variational space for the biharmonic problem is the second-order Sobolev space H2. For a conforming finite element approximation this requires globally C1-continuous elements. As an alternative, we present here a method originating from a C0continuous formulation of the Reissner-Mindlin problem. In the presence of free boundary conditions, we introduce additional terms in the bilinear form guaranteeing the consistency, symmetry and the stability of the meth0d.l In the next two Sections, we motivate and derive the new finite element

46 family. In Sections 4 and 5, respectively, we state the main results of the theoretical a-priori and a-posteriori error analysis. In Section 6, we present some numerical results, confirming the theory and the applicability of the method. 2. T h e Kirchhoff plate bending problem

We consider the bending problem of an isotropic linearly elastic plate under the transverse loading g. The midsurface of the undeformed plate is described by a convex polygonal domain 52 c R2. The plate is considered to be clamped on the part rc of its boundary 1352, simply supported on the part I's c 65'2 and free on rFc 6'52. With V we indicate the collection of all the corner points in rFcorresponding to an angle of the free boundary. 2.1. The biharmonic formulation

First, we define the material constants for the model: The bending modulus and the shear modulus, respectively, are denoted by

D=

Et3

12(1 - v2)

and

E 2(1+ v) '

G =-

with the Young modulus E and the Poisson ratio v for the material. The thickness of the plate is denoted by t. In the sequel, we need the following differential operators: The strain tensor E is defined as the symmetric tensor gradient

The vector gradient and the vector divergence are defined as usual, while the tensor divergence is defined as diva

=

+ +

aaXx/dx dax,/dy anyx/ax aayy/ay

Next, we define the physical quantities for the problem: The bending moment is defined as

which implies that the shear stress Q satisfies the equilibrium equation: Q=-divM

and

-divQ=g.

(2.5)

47 With these notation, and assuming that the load is sufficiently regular, the Kirchhoff plate bending problem can be written as the well known biharmonic problem: Find the deflection w such that

where n and s are, respectively, the unit outward normal and the unit counterclockwise tangent to the boundary. By the indices 1 and 2 we denote the sides of the boundary angle at a corner point c. 2.2. The scaled mixed formulation

In order to analyse the differences between the Kirchhoff and ReissnerMindlin formulations, we introduce the mixed formulation.' For the analysis, it is first assumed, as usual, that the loading is scaled as g = G t 3 f with f fixed. Then the problem (2.6) becomes independent of the plate thickness:

The corresponding scaled moment and shear stress are, respectively,

Gt3 ~ ( V W=) M(Vw)

and q = -.Q

Gt3

In the mixed formulation, the rotation and the shear stress, respectively, are taken as new unknowns:

P = Vw

and q = -divm(P) = - L P ,

(2.9)

where we have introduced a partial differential operator L. Now, the scaled mixed problem reads:

48 Find the deflection w, rotation -divq= f , L P + q = O ,

and the shear stress q such that

Vw-P=O

in R ,

2.3. The variational formulation

For deriving the variational formulation of the problem (2.10), we denote by and respectively, the L2 inner products in the domain R and on the boundary r. Then, by taking inner products between Eqs. (2.10)1 and the corresponding test functions, we first obtain (a,

0

)

(a,

a),

-(div 9, v) = ( f , v) , ( L P + 9777) = 0 , (Vw - p , r ) = 0 . Integration by parts implies for the first equation above that

By the definitions (2.2) and (2.9), integration by parts and the symmetry properties of the moment tensor, the second equation above takes the form

Altogether, we obtain the variational formulation

for all (v, 77, r ) E

M

4,P 7

W x e x S , with the bilinear forms

7r) =4

,)

)

+(

V-4

1 +P

Vv - 77,

(2.17)

R(z, 4 , p ; v, 77, r ) = -(m(4)n, TI) - (P - n, v) = -(mnn ( 4 ) 77 . n ) - (mns (4) 77 . s ) - ( p n v) , (2.18)

where m,, = mn . s, m,,

-

= mn n and p, = p n .

49 We now search for the correct boundary conditions to be enforced in the variational spaces, in order to obtain a formulation equivalent to Eqs. (2.10): For the clamped boundary rc,there is no natural boundary conditions in Eq. (2.10)2, which requires the space conditions These together make the boundary part R vanish on rc. For the simply supported boundary rs,the natural boundary condition mnn = 0 in Eq. (2.10)3 gives the corresponding space conditions v=0, q.s=O

onrs.

(2.20)

For the free boundary FF, the natural boundary conditions mnn = 0 and ~a m , ,- qn = 0 in Eq. (2.10)* imply that

Integration by parts with the natural corner conditions (2.10),~jimplies

which gives the space condition

In addition, the bilinear form M has to be well defined. Thus, the variational spaces for the problem (2.10) are defined as

W X ~ = { ( V , ~ ) ) E H ~ ( R ) X [ H ~ ( R ) ] ~ ) ~ = O O I I I ' ~ (2.24) UI'~, q=Oonrc, q.s=OonrS, (Vv-q).s=OonrF), S = [ L ~ ( R ).] ~

(2.25)

We note that directly imposing the mixed space condition (Vv - q ) ' I ~ F = 0 in the finite element level is complicated in practice and it can cause a locking phenomena. Thus, a weakening of the constraint is necessary. This leads us to consider the variational spaces of the Reissner-Mindlin formulation. 2.4. Inconsistency between the Kirchhoff problem a n d the Reissner-Mindlin limit problem

For a deeper analysis of the t-dependence in the Reissner-Mindlin problem, we refer to Refs. 2-6.

50 Formally, the difference between the Reissner-Mindlin and Kirchhoff problems is the condition for the shear deformation and the boundary conditions on the free boundary. Namely, instead of the last equation in (2.10)l and Eqs. (2.10)4, for the Reissner-Mindlin problem it holds

and the corresponding bilinear form of the problem is defined as Mt(z, 4 , ~v,; rl, r ) = M ( z , 4 , ~v,; rl, r )

+ t2(p,r ) ,

(2.27)

with the Kirchhoff bilinear form M defined in Eq. (2.17). Now, for the free boundary r F , the natural boundary conditions mnn = 0, m,, = 0 and qn = 0 in Eq. (2.26)2 imply that the boundary part R of Eq. (2.18) vanishes on rFwithout any essential boundary conditions. This means that the variational spaces for the Reissner-Mindlin problem can be written as

In order to compare the Reissner-Mindlin and the Kirchhoff formulations, we first take the limit t --+ 0 in the bilinear form (2.27) of the Reissner-Mindlin problem, which gives the bilinear form (2.17) of the Kirchhoff problem. However, when comparing the variational spaces for these two problems, we see that x # W x V , because of the different space conditions on the free boundary. More precisely, with the space W x V the limit problem formulation includes the inconsistency term of Eq. (2.22):

W

Finally, we conclude that if the space W x V is used for the Kirchhoff problem, the space condition (Vv-q).s,rF = 0 has to be taken into account in another way, e.g. in a weaker sense with Lagrange multipliers. 2.5. The limit problem with Lagmnge multipliers

Here we formulate the Kirchhoff problem as the limit of the ReissnerMindlin problem by taking into account the essential free boundary condition with Lagrange multipliers: Find (w, f3,, q, A) E W x V x S x A

51 for all (v, q, r, p ) E W x form

V

x S x A, with the additional symmetric bilinear

corresponding to the free boundary FF. In order to see the physical meaning for the Lagrange multiplier, we integrate by parts the bilinear form M above and obtain

Now, the inner products (.,.) above give the Kirchhoff equations (2.10)1. Respectively, the boundary inner products (., .) above corresponding to the clamped and simply supported boundaries give the boundary conditions (2.10)2-(2.10)3. For the free boundary, we combine the boundary terms in M above with the ones in C and obtain

This implies the following connections: First, by testing with qes, and then with v, we obtain, respectively, mns(P)-X=O

and q n - -

ax = O . ds

(2.36)

These two identities together, and testing with p and q . n, respectively, imply the correct boundary conditions of Eq. (2.10)4. 3. The finite element formulation

Our formulation for the Kirchhoff problem is a displacement method based on the limit of the Reissner-Mindlin problem described in the previous Section. The starting point is the stabilized displacement formulation introduced in Ref. 7 for the Reissner-Mindlin problem. In that formulation, the shear stress is already locally eliminated, which suits well for our approach, since the shear stress does not appear in the additional free boundary condition which is our main concern.

52 In order to write a formulation for the Kirchhoff problem, the first step is to set the thickness parameter equal to zero in the stabilized ReissnerMindlin formulation. Second, we show how to take into account the additional free boundary condition, either with Lagrange multipliers or with the Nitsche technique.8 We emphasize that with certain choices, these two approaches finally lead us to the same finite element formulation. 3.1. Elimination of the Lagrange multipliers

The discrete displacement formulation with Lagrange multipliers is now of the following form: Find (wh, Ph,Ah) E Wh x V h x Ah such that

with the bilinear form ) Ch(z, 476'; v, q , P) Wh(z, 4, P; v, 7 7 7 P) = Bh(z, 4 ; ~ 7 7 +

(3.2)

where Bh stands for the displacement Kirchhoff formulation t o be specified later and Ch represents the stabilized Lagrange multiplier form

where 6 > 0 is a stabilizing parameter and h is the mesh size. For simplicity, the mesh is assumed to be quasiuniform in this context. We note that the stabilizing term is consistent due to the relation (2.36). Next, we locally eliminate the Lagrange multiplier Ah by the L~ projection nh onto Ah. By testing with p E Ah in Eq. (3.1) (and v = 0, q = 0) we get (nh(Vwh - Ph)

S, p)rF - dh(Ah

- nhmns(Ph), P)II, = 0

(3.4)

for all p E Ah. For the free boundary edges E of rF,this implies the relation

Now, with this identity, testing with (v, q ) (and p = 0) gives

53 F'urthermore, the projection identity ITh = IIi = II;l implies the symmetric form 1 Lh(wh,Ph,Xh;v,q,p) = %(nh(Vwh - Ph) s,nh(Vv - 9 ) . s)rF

+ (nhmns(Ph),nh(Vv - V) f

'

S)~F

(3.7)

(nh(vwh - oh) ' 9 , n h m n s ( q ) ) r ~

- bh((nh - I)mns

(P) (nh - I)mns (q))rF

Finally, the choice Ah = L2(rF) implies that llh= I and thus

1 - P h ) S, (Vv - q ) s ) r F 6h (mns(Ph), (Vv - V) . s ) r F +

= -((Vwh

(3.8)

+

+

- Ph)

. s,mns(q))rF .

This is a symmetric stabilized displacement form for the Lagrange terms. 3.2. The Nitsche technique

The Nitsche method analysed in Ref. 8, is a straightforward way to end up with the formulation derived above without explicitly using Lagrange multipliers. We start from the original formulation which is inconsistent, a s discussed in Sec. 2.4. Combining the variational formulation (2.16) with the boundary part (2.22) gives the corresponding discrete formulation B h ( ~ hPh; , v , ~= ) (ftv) - (mns(Ph), (Vv - q) ' S ) ~ F

(3.9)

for all (v,q) E Wh x Vh. This means that we have to add the inconsistency term on both sides and symmetrize the formulation with a consistent counterpart:

+

Bh(wh,Ph; v , ~ ) (mns(Ph), ( v v

+

V) s ) ~ F ((Vwh - Ph) S, mns(q))rF = (f,V) V(V,71) E Wh x Vh

(3.10)

Finally, we stabilize the method with another consistent term: Bh(wh,Ph; 21, V)

+ (mns(Ph),(Vv - 11)

'

S)~F

+ ((Vwh - Ph) S, mns(v))rF + j1j $ ( V ~ h - Ph) . S, (VV- q ) - s)rF = (f,V) *

(3.11)

for all (v, q ) E Wh x Vh. Now, we have ended up with the formulation (3.1) combined with the eliminated Lagrange form (3.8).

54 3.3. The stabilized displacement formulation

In this Section, we formulate our displacement finite element method motivated and derived in the previous Sections. First, we assume that the boundary conditions of the problem prevent the possible rigid body movements of the plate, in order to neglect the related technicalities. For simplicity, we consider triangular elements, but all the results hold for rectangular elements as well. Hence, let a regular family of triangular meshes on R be given.21gFor the integer k >. 1, we then define the discrete spaces

for the approximations of the deflection and the rotation, respectively. Here Ch represents the collection of all the triangles K of the mesh and Pk(K) is the space of polynomials of degree k on K . In the sequel, we will indicate with h~ the diameter of each element K, while by h we denote the maximum size of all the elements in the mesh. Moreover, we will indicate with E a general edge of the triangulation and with h E the length of E. Finally, let the positive stability constants a and y be assigned. Now, the finite element method for the problem (2.7) with (2.9) reads: Method 3.1 Find (wh,Ph) E Wh x V h such that dh(wh, Ph; v, 1)) = (f, v)

V(v,q) E Wh

X

Vh

,

(3.14)

where the bilinear form is defined as

where

for all (z, 4 ) , (v,q) E Wh X Vh, where 3 h represents the collection of all the boundary edges i n I?F and mn, = mn . s.

55 The bilinear form Bh constitutes essentially of the original method of Ref. 7 with the thickness t set equal to zero, while the additional bilinear form Dh is introduced in order to avoid the convergence deterioration in the presence of free b o u n d a r i e ~ . ~ ~ ~ ~ 4. A-priori error estimates

Here we show that our method is optimally convergent. For the deflection and the rotation ( v , q ) E Wh x V h ,we introduce the following mesh dependent norms:

where I[.] represents the jump operator which is assumed to be equal to the function value on boundary edges. Eh represents the collection of all the edges of the triangulation. For the analysis of the shear stress, we introduce the following notation: Given the space

V , = (77 E [ H ~ ( o ) (]7 7~= 0 on Fc, 77.s = 0 on FF U r s ) , we introduce the dual norm

which bounds from above the classical norm 11 . II-l. following a-priori error estimate, as proved in Ref. 1:

(4.3)

We then have the

Theorem 4.1. Let ( w ,P ) be the solution of the problem (2.7) with (2.9) and (wh, Ph) the solution of the problem (3.14). Then it holds - ~ h Pr - Ph)lIlh

for all 1

+ 119 - qhII-l,*

5 ChSIIwlls+2

(4.5)

< s 5 k, where the exact and discrete shear stresses are

We note that if the exact shear stress q belongs locally to the space L 2 , then the abstract norm 11 . 11-1,, in the estimate (4.5) can be replaced by the practical mesh-dependent norm defined as I I ~ I I - 1 .= h

(

x~%II~IIH,~)~

KECh

56 For the error estimation of the deflection and the rotation, with the norm

we note as well that the following useful norm equivalence holds:

Lemma 4.1. There is a positive constant C such that

5. A-posteriori error estimates We now briefly recall the reliability and the efficiency results for the aposteriori error estimator presented in Ref. 1. To this end, we introduce

where fh is some approximation of the load f. Then, for any element K E Ch, the local error indicator is defined as

where Zh represents the collection of all the internal edges, while Sh and Fh represent the collections of all the boundary edges in rs and r ~re-, spectively. Finally, the global error indicator is defined as

For the error analysis, we have assumed that a classical saturation assumption holds. We then have the following reliability and efficiency results for the error estimator:'

Theorem 5.1. There exists a positive constant C such that

57 Theorem 5.2. It holds

represent, respectively, the norms where 11 I 11 lh,WK, 11 . I ~ - I , * , ~ ~ and 11 and 1) JJo restricted to the domain WK,the set of all the 1)) . \\Ih, 1) . triangles sharing an edge with K.

-

6. Numerical results

In this Section, we first show the convergence results concerning the apriori error estimates for the new and the original method, i.e., respectively, with and without the additional bilinear form Dh in Eq. (3.17) for the free boundary edges. Second, we present some results for the a-posteriori error estimator of the new method. 6.1. A-priori test - semi-infinite free boundary plate

We consider the following Kirchhoff bending problem of a semi-infinite plate: The midsurface and the boundary of the plate, respectively, are described by the sets

a = {(x,y) E El2 I

y

> 0) and I' = {(x,y) E I W ~I y = 0).

(6.1)

The plate is assumed to be free on the boundary r and subjected to the transverse loading f(x) = cosx/G. The exact x-periodic solution of this problem is given in Ref. 4: w

=

(1/A

+ be-" + dye-")

cos x ,

b, = ( - 1/A - be-" - dye-") sin x , by = ( - be-" + d(1- y)e-") cosx ,

(6.2)

where A = G/(6(1 - v ) ) . The coefficients b and d, depending on G and v , are given in Ref. 4 for different types of boundary conditions on I?. Due to the smoothness of the solution, from Th. 4.1 and Lemma 4.1 it immediately follows the convergence rate

On the contrary, according to the observations in Refs. 1 and 10, the convergence rate for the original method, without the additional bilinear form Dh, should be of order 0(h1/'). We discretize the domain D = (0,7r/2) x (0,37r/4) and set the symmetry conditions on the vertical boundaries {x = 0,Q 5 y 5 3n/4) and {x =

58 7r/2,O 5 y 5 37~/4),while on the upper horizontal boundary {y = 37r/4,O 5 x 5 7r/2) we use the non-homogeneus Dirichlet conditions adopting the exact solution as a reference. Finally, let Db represent the boundary domain [o,~ 1 2 x1 10,~ 1 4 1 . First, in Fig. 1 (left), we show the error convergence for the moment component m,, in the norm L~( ~ b ) ,for the polynomial degrees k = 1,2. The dashed line represents the convergence graph for the original method, i.e., without the correction Dh in Eq. (3.17), while the solid line refers to the new one, Meth. 3.1. As predicted by the theory, the convergence rate for the original method is ~ ( h ' / ~while ) , the modified method follows the rate O(hk). In Fig. 1 (right), the error is measured in the norm of Eq. (6.3) which is equivalent to the norm II I . 11 lh (cf. Lemma 4.1). In this norm as well, the convergence is of the correct order O(hk). 6.2. A-posteriori test - simply supported L-domain

As a nonconvex plate problem, we consider the L-shaped domain with the corners (0, O), (2, O), (2, I), (1,I), (1,2) and (0,2). The plate is uniformly loaded, f = 1, and all the boundaries are simply supported. According t o Refs. 11 and 12, the regularity in the critical L-corner is now w E ~ ~This1implies, ~ by . Th. 4.1 and Lemma 4.1, the theoretical convergence rate O(hu), a = min{l/3, k) = 113. The convergence graphs for the uniformly (circles) and adaptively (triangles) refined meshes are shown in Fig. 2 (right). The two upper graphs (solid lines) represent the global error estimator, while the lower ones (dashed lines) indicate the maximum local estimator. Moreover, we show in the same figure the theoretical convergence rates O(h) and ~ ( h l / (dashed ~) lines). Finally, two example meshes from the adaptive process are depicted in Fig. 3 and the deflection is illustrated in Fig. 2 (left). With the uniform refinements (circles), the convergence rate of the error estimator clearly follows the theoretical value 0(h1l3). This holds for both the global error estimator and the maximum local estimator. Differently, after the first adaptive steps, the method shows its robustness in finding the corner singularity of the solution and refining locally near the L-corner. This is seen in both the convergence graphs (triangles), and it is clear as well when looking at the meshes in Fig. 3. References 1. L. Beirk da Veiga, J. Niiranen and R. Stenberg, A family of CO finite elements for Kzrchhoff plates, Research Reports A 483, Helsinki

59 1o"

10"

0

e

i

w

i

m

i

i

Io4 1o"

1o1 Number of elements in x-direction

10'

18

, , ,

10' Number of elements in x-direction

lo'

Fig. 1. Free boundary: Convergence of the error for the moment component m,,(P) ~ ) k = 1 , 2 (left); Convergence of the error in the norm IIP in the norm L ~ ( Dwith Phll~,DbI(w - whrP - Ph)lh,Dbwith k = 1 , 2 (right); Dashed lines for the original, solid lines for the new method.

+

University of Technology, Institute of Mathematics (January 2006), http://wwu.math.tkk.fi/reports.

2. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer Verlag, New York, 1991). 3. D. N. Arnold and R. S. Falk, S I A M J. Math. Anal. 27, 486 (1996). 4. D. N. Arnold and R. S. Falk, Edge effects in the Reissner-Mindlin plate theory, in Analytic and Computational Models of Shells, eds. A. K . Noor, T. Belytschko and J. C. Simo (ASME, New York, 1989). 5. D. N. Arnold, A. L. Madureira and S. Zhang, J. Elasticity 67, 171 (2002). 6. P. Destuynder and M. Salaun, Mathematical Analysis of Thin Plate Models (Springer-Verlag, Berlin, 1996). 7. R. Stenberg, A new finite element formulation for the plate bending problem, in Asymptotic Methods for Elastic Structures, eds. P. Ciarlet, L. Trabucho and J. M. Viano (Walter de Gruyter & Co., Berlin - New York, 1995) pp. 209-221. 8. R. Stenberg, Journal of Computational and Applied Mathematics 63, 139 (1995). 9. P. G. Ciarlet, The Finite Element Method for Elliptic Problems (NorthHolland, Amsterdam, 1987). 10. L. B e i r k da Veiga, S I A M J. Num. Anal. 42, 1572 (2004). 11. H. Melzer and R. Rannacher, Bauzngenieur 5 5 , 181 (1980). 12. H. Blurn and R. Rannacher, Math. Meth. Apll. Sci. 2, 556 (1980).

60

0.031

. . . . .

,

lo2

Rmsolullon.3

loo

lo'

+

.,*,

..--.A lo3

,

10'

. 105

Number of Elements

Fig. 2. Simply supported L-corner: Deflection distribution for the first mesh (left); Convergence of the global estimator (solid lines) and the maximum local estimator (dashed lines) (right); Circles for the uniform refinements, triangles for the adaptive refinements.

Fig. 3. Simply supported L-corner: Distribution of the error estimator for two different refinement steps.

61

MODELING AND SIMULATION OF PIEZOELECTRIC-ACTIVE CONTROL OF WIND-INDUCED VIBRATIONS ON BEAMS MICHELE BETTI CRIACIVDIC, University of Florence, Via di Santa Marta, 3 Florence, 1-50134, Italy CHARALAMPOS C. BANIOTOPOULOS Dept. of Civil Engineering, Aristotle University of Thessaloniki (AUTH) Thessaloniki,GR-54124, Greece GEORGIOS E. STAVROULAKIS Dept. of Production Engineering and Management, Technical University of Crete Chania, GR-73100, Greece Dept. of Civil Engineering, Carolo Wilhelmina Technical University Braunschweig, 0-38106, Germany Strong wind could induce severe vibration on tall buildings and then discomfort to the occupants (and maybe also damage of sensitive equipments on the buildings). Insofar active vibrations control, due to the increasing requirements of high structures in modem building, is an actual engineering concern. As a matter of fact, finite element methods are able to take into account the integration of smart components on classical structural parts. Special fmite elements have been developed for the consideration of the piezoelectric effect. For the design of smart piezoelectric structures for active vibration control both structural dynamics and control need to be considered. In this paper an active vibration control technique for a smart beam is presented. The structure is made of two layers of piezoelectric material (PZT-8)embed on the surface of an aluminium beam. A general purpose finite element code, ANSYS, is utilized in order to develop a computational model for the study of the dynamic response of the structures with the active piezoelectric elements. Herein a cantilever beam under dynamic loads is analyzed, and comparison between results for beam with and without control is made. Finally a typical stochastic wind-type load is considered.

1. Introduction

Due to the increasing requirements of high-rise and slender structures in modem buildings, active vibrations control is an active area of research and applications. After the intensive research efforts dedicated to the field of smart structures in the last 15 years, the use of piezoelectric materials in engineering applications is now widely diffused: piezoelectric laminated composites are

62 used in the field of active vibration, in sensors and actuators, in measuring instruments, in medical apparatus, in micro-electromechanical systems (Crawley et al. [I I]). It is well know that strong winds could induce severe vibration on tall buildings and then discomfort to the occupants (or even damage of sensitive equipments on the buildings). A smart or intelligent structure is a self-adaptive structure, whlch reacts to the environmental noise and minimize the effect of the applied disturbances (Abramovich [I], Tong et al. [20]). Typically a smart structure consists of a host structure equipped with sensors and actuators that are coordinated by a controller. Some of the most widely used distributed sensors and actuators are made of piezoelectric materials. The piezoelectric effect consists of the ability of certain crystalline materials to generate an electrical charge in proportion of an externally applied force and to induce an expansion on the piezoelectric material in proportion to an electric field parallel to the direction of polarization. Piezoelectric materials, by generating an electric chargelvoltage, respond to mechanical forceslpressures and this system has the ability to perform selfdiagnosis and to adapt itself to the environmental changes. In more details, the direct piezoelectric effect consists of the ability of certain crystalline materials (i.e. ceramics) to generate an electrical charge in proportion of an externally applied force. The electric chargelfield applied to the piezoelectric material induce mechanical stresses and strains: this phenomenon is the converse piezoelectric effect. Consequently, the direct effect is used for structural measurements (the sensor) and the converse effect is used for active vibration controls (the actuator). The most widely used examples of smart materials based on piezoelectricity, are usually surface bonded or embedded into the host structures as distributed sensors and actuators (Arvanitis et al. [3], Betti et al. [lo], Stavroulakis et al. [19]). There are two broad classes of piezoelectric materials used in vibration control: polymers and ceramics. The piezopolymers are used mostly as sensors because they require extremely high actuating voltages; the piezoceramics are extensively used both as sensors and actuators, since they require lower actuating voltages, and are used for a wide range of frequency. Under the assumption that piezoelectric sensors and actuators are perfectly bonded to the host structure, we deal with the Finite Element Model of the resulting composite beam, which constitutes the basis for the study of vibration control. As a matter of fact the application of the finite element modeling techniques on smart materials has been in continuous growth during the last

63 years and some piezoelectric elements have already become available in commercial finite element codes (C6t15 et al. [12]). A great variety of both theoretical and numerical applications have been proposed, such as for active vibration suppression (see Hwu et al. [16], Huang et al. [15], Deu et al. [13]). Analytical models of piezolaminated composites have been developed as well, (see Abramovich [I]), based on the linear piezoelectric theory and pin-force models for static loadings. Extension of this model in order to take into account a piezoelectric sensor layer and study of the direct effect has been done (see Abramovich et al. [2]). Since the 1970s, many FE models have been proposed for the analysis of smart piezoelectric structural systems. A survey on the advances in piezoelectric FE modeling of adaptive structural elements is presented by Benjeddou [6]. Herein the integration of control actions into the commercial, general purpose finite element code ANSYS is presented. This work could be considered as the first phase of a research devoted to develop a simplified finite element model for multilayered structures with sensors and actuators embedded on an homogenous number of layers. As a first step, the procedure is tested on the active vibration control problem of a cantilever beam. The final aim of this paper is to develop a computational tool in order to provide a simplified but accurate description of the dynamic response of beam structures incorporating active piezoelectric elements. This way allows us to study several effects on the effectiveness of the optimally controlled smart structures: the influence of the shape, position and thickness of actuators and sensors, the binding layer (glue), the delay introduced by the electric signal in the dynamical response of the structure etc. Wind-type loads (see Balamurugan et al. [4], Betti et al. [8], Bartoli et al. [5]) are taken into account for the numerical results.

2. Finite element modeling of the electromechanical system The behaviour of the electromechanical system (elastic beam embedded with sensors and actuators made of piezoelectric materials), is described by means of a linear model. Each point of the material is affiliated with both mechanical and electrical variables. The mechanical variables, in the usual engineering-oriented vector description of the underlying tensor quantities, are the 6x1 stress vector and the 6x1 strain vector denoted, respectively, with {o) and { E ) . The electrical variables, the 3x1 electric displacement (chargelarea along the corresponding directions) and the 3x1 electric field vector (voltagellength along the corresponding directions) are denoted, respectively, by {D) and {E). With the

64 classical linear approach the corresponding (for a non piezoelectric material) mechanical and electrical equations could be written as follow:

.I{[ (4

{D>=

(1)

Here [Q] is the 6x6 elastic stiffness coefficient matrix (in constant electric fields conditions) and [$] is the 3x3 permittivity matrix. In a piezoelectric material, eqs. (1) and (2) are coupled and in literature several approaches have been proposed, by taking into account that there are 18 state scalar variables and 9 linear equations. The well-known formulation is described by assuming in the independent state variables the stress vector {o)and the electric field vector {E). This is the so-called model with respect to the actuators. In this approach the constitutive relations, where two electrical and two mechanical variables are involved, can be written:

Here [Q]-' is evaluated with constant electric field conditions while the permittivity matrix [$I is evaluated with constant stress conditions. The matrix [dl, with elements dV representing the coupling between the electric field in direction i and the strain in the j direction, is a 6x3 matrix. If it is assumed that piezoelectric materials are polarized in the z-direction (thickness direction) and exhlbit transverse isotropic properties in the xy-plane, only few elements of this matrix are nonzero, and this matrix could be write as follow:

This formulation is useful when piezoelectric materials are used as actuators: by applying an electric field, deformations are induced on the material. The first equation in (3) gives an elastic contribution of the strain plus the piezoelectric one. Another formulation of the piezoelectric coupling between electrical and mechanical stare variables could de done by assuming as independent state variables the stress vector {o) and the electric displacement [Dl. This

65 formulation is useful when piezoelectric materials are used as sensors. In this case the following constitutive relations can be written:

With respect to this formulation [Q]-' is evaluated with constant electric displacement conditions, whilst the inverse of the permittivity matrix [c] is evaluated with constant stress conditions. The matrix [g], whose elements gg represent the coupling between the electric field in direction i and the stress in the j direction, is a 6x3 matrix. The following relation exists between matrix [g] and matrix [dl:

By using eq. (5) it is possible to obtain directly the electric field induced to a mechanical stress (under electric displacement zero). The finite element code ANSYS contains several 3D and 2D finite elements with piezoelectric capabilities. It uses a different formulation for the description of the coupling effect between mechanical and electrical variables. The program assumes as state variables the strain vector { E ) and the electric field {-El. In this case the constitutive relations for a piezoelectric material lead:

The connection between the elements of the constitutive matrix in eq. (7) and the usual formulation is obtained by the following equations:

The piezoelectric effect is taken into account in eq. (7) by the [el matrix. This matrix joins the elements of the electric field vector {E) with the elements of the stress vector {o).Furthermore, the permittivity matrix [(I in constant

66 strain conditions is obtained by the corresponding one in constant stress conditions. With ANSYS it is possible to use several elements in order to build a piezoelectric model. In this work Solids, an 8-noded brick element has been used. This element has 6 degrees of freedom (DOFs) for each node: three displacements (U,, U,, Ua, the electric potential (V), the temperature and the magnetic potential. With the use of eq. (7) the vectors { E ) and {E) are easily obtained by the program simply including the piezoelectric equations on the DOFs of the joints of the element. The strain vector is derived by the spatial derivative of the nodal displacements {Ui), and the electric vector field is obtained by the spatial derivative of the electric potential V. Take into account the ANSYS formulation for the coupling effect between mechanical and electrical behaviour it is important because both the commercial datasheets and the literature give the electric properties of piezoelectric material with respect to matrix [dl or [g].

I

2

Aluminium Beam

\

\

L

X

\Piezoelectric actuator Figure 1 . Laminated slender beam with piezoelectric sensors and actuators.

The finite element code ANSYS is used to simulate the time history behaviour of coupled electromechanical systems. Thls is a difficult step because multiple fields simulation is much more complicated then the usual structural analyses due to the fact that mechanical and electrical variable may interact with each other. The ANSYS Parametric Design Language (APDL) is used in order to develop closed loop feedback control laws (in which the drive voltages are computed using feedback control laws). APDL is a macro language that enables to run commands for an external file, it is used to perform the control law and has originally been created to allow the variation of model parameters in an

67 optimization design. It contain the basic control logic flow (f.i. the itfthen statements and loops) and it is used for the creation of a control law. Boundary conditions for each load step are generated by the controller based on the output of the previous simulation step results.

3. Case study

In this section case studies are presented in order to demonstrate the validity and the efficiency of the proposed scheme. Piezoelectric smart structure systems for active vibration control are designed using a commercial finite element code as stated before. Three dimensional coupled field solid elements Solid5 with piezoelectric option are used in order to model the piezoelectric patches, and three dimensional structural elements Solid45 are utilized in order to model the host aluminium structure. A cantilever beam (see Figure 1) is considered as a simple example. The structure consists of a host aluminium beam (80 x 2 x 2 cm) embedded with a bottom layer of PZT-8 sensor, and a top layer of PZT-8 actuator. Both the sensor and the actuator are assumed to be perfectly bounded to the beam surface. The material properties of the aluminium are reported in Appendix A. Appendix B reports the piezoelectric properties of the PZT-8 material utilized in the analysis. It is assumed that the poling direction for the piezoelectric layer is the z-direction. 3.1. Numerical model The finite element model consists of 2784 nodes, 690 Solid5 elements for the two piezoelectric layers and 1725 Solid45 elements for the host aluminum beam (see Figure 2). The Solid5 element has eight nodes: each node has up to six degrees of freedom (three displacements, voltage, temperature and magnetic displacements; rotations are not included). These elements, that are used for the modeling of the piezoelectric layer, are posed adjacent to the structural elements modeled by Solid45 elements (see Figure 2). A rigid region has been created in order to couple the DOFs of the piezoelectric element to the structural elements. In order to evaluate the benefits of the active control, two kinds of dynamic loadings are used as disturbances. The first one is a sinusoidal load, i.e. a force acting in the free end of the beam with an amplitude of 10 kg, and a frequency of 25 Hz. The second one simulates a turbulent wind load acting on the upper surface of the beam as it is described in the next paragraph. In this work we assume that the piezoelectric layer of sensor and actuator are perfectly bounded on the host structure, and that they cover the entire length of the smart beam. It is interesting to observe at this point that, by taking into account the modal

68 shape of the beam, it is possible to suggest a better position for this piezoelectric layer [9].

Figure 2. ANSYS numerical model.

3.2. Sinusoidal load The first load analysed is a sinusoidal load acting on the free end of the cantilever beam. A sinusoidal load with an amplitude of 10 kg, and a frequency of 25 Hz has been considered.

Figure 3. a) Longitudinal displacements on the free end of the beam (along x); b) Transversal displacements on the free end of the beam (along z)

Figure 3 reports the dynamic response of the controlled and uncontrolled beam. In particular, Figure 3a shows the diagram of the longitudinal displacements for the free end of the beam, and Figure 3b reports the diagram of the vertical (along z) displacements. Figure 4a+b reports the contour plot of the

69 electric voltage on the 3D finite element model of the beam at several instants of the time history.

(a)

(b)

Figure 4. Contour plot of the 3D FEM results for the electric voltage: a) t=1.5 sec; a) e 2 . 0 sec

3.3. Wind load The load induced by wind turbulence on the vertical beam was numerically generated as a random pressure field obtained by the multiplication of the squared velocity field by the pressure coefficient and the air density:

Pressure acts only on the negative z-ward face of the beam and is positive when it acts towards the positive z axis. Recent studies showed that the pressure probability density hnctions (PDF) can be well approximated by a log-normal distribution, but here for simplicity a Gaussian expression for the velocity will be considered, resulting in a chi-square distribution for the pressure field. In its turn, the chi-square is extremely close to the Gaussian bell, as the contribution of the squared fluctuation is very small, as usual. Such approximation leads to satisfactory results (see Bartoli et al., [ 5 ] ) and the extreme value distribution of the system response is not usually affected by deviations from Gaussianity of the loading process. The velocity field on the windward face was divided into its mean v,,, (varying according an exponential profile) and its fluctuation vo, modelled by means of a linear combination of radial basis functions with timedependent coefficients (see for details [8]). The spectral function of the velocity field was modelled by means of a Von Karman auto-spectral density with

70 exponentially decaying coherence and phase angle (see Simiu and Scanlan, [181):

With the described procedure a realization of the wind stochastic process has been generated and, for a node on the free end of the beam, Figure 5a shows a realization of the wind velocity history, while Figure 5b shows the corresponding load.

U

1

"

t

S

C

I

I

T

U

I

Lm

(b) Figure 5. Velocity field on the free end of the beam (a) and corresponding wind load acting @)

Figure 6. Contour plot of the 3D FEM results for the electric voltage: a) t=1.0 sec; a) t=2.0 sec

71 In Figure 6-10 contour plots of the electric voltage on the 3D finite element model of the beam at several instances of the wind time history are reported.

Figure 7. Contour plot of the 3D FEM results for the electric voltage: a) t3.0sec; a) t4.0 sec

Figure 8. Contour plot of the 3D FEM results for the electric voltage: a) t=5.0 sec; a) e6.0sec

Figure 9. Contour plot of the 3D FEM results for the electric voltage: a) t7.0sec; a) F8.0 sec

72 -.

....--.-. -

t*,. ,~.-. .". ! R*. %.*a?*

Figure 10. Contour plot of the 3D FEM results for the electric voltage: a) +9.0 sec; a) t=10.0 sec

Figure 11. a) Longitudinal displacements on the free end of the beam (along x); b) Transversal displacements on the free end of the beam (along z)

In Figure l l a and Figure l l b the structural displacements responses are reported, respectively, for the along-beam direction and for the along-wind direction, and for the free end of the beam. It is possible to see clearly the benefit induced by the control on the maximum value of these displacements. 4. Conclusion

In this paper the modelling of a smart cantilever beam was carried out. The paper also illustrates the process of converting typical manufacturer piezoelectric data into the format required for their ANSYS modeling. By this finite element code is possible to deal with the integration of smart component

73 and classical structural parts. In this paper an active vibration control technique for a smart beam is presented. The structure analysed is assumed to be made of two layers of piezoelectric material (PZT-8) embed on the surface of an aluminium beam. This cantilever beam under dynamic loads is analyzed, and comparison between results for beam with and without control is made. First a generic sinusoidal load has been taken into account and then a typical stochastic wind-type load has been considered. Acknowledgments The financial support of the European Commission to the present research activity is gratefully acknowledged ("Smart Systems1'-HPRN-CT-2002-00284). Appendix A. Material properties of the host beam structure Material properties of the aluminium are reported in Table 1 Table 1. Material properties of the host beam structure. Properties E1.

73 . lo9 Pa

Elastic modulus of host beam

Psm

2770 kg/m3

Density of host beam

Appendix B. Piezoelectric constitutive matrices Considering the standard notation introduced in chapter 2 for the piezoelectric constitutive equations, following numerical values for the constitutive matrices are assumed:

74 These value are used for the finite element simulations on the 3D model. They refer to the material properties of PZT-8.

References 1. H. Abramovich, Composite structure, Deflection control of laminated composite beams with piezoceramic layers - closed form solutions, 217 (1998). 2. H. Abramovich and A. Livshits, J. Int. Muter. Syst. Struct., Flexural vibrations of piezolaminated sheet actuator, 46 (2000). 3. K.G. Arvanitis, E.C. Zacharenakis, A.G. Soldatos, and G.E. Stavroulakis, Selected Topics in Structronic and Mechatronic Systems (Series on Stability, Vibration and Control of Systems), New trends in optimal structural control, 32 1 (2003). 4. V. Balarnurugan and S. Narayanan, J. Sound Vibr., Finite element formulation and active vibration control study on beams using smart constrained layer damping (SCLD) treatment, 227 (2002). 5. G. Bartoli, C. Borri and L. Facchini, Comput Struct, Simulation of nonGaussian wind pressures on a 3-D bluff body and estimation of design loads, 1061 (2002). 6. A. Benjeddou, Comput Struct, Advances in piezoelectric finite element modelling of adaptive structural elements: a survey, 347, (2000). 7. M. Betti, G.E. Stavroulakis and C.C. Baniotopoulos, Proceedings of the 7th National Greek Congress of Mechanics, Stochastic wind load on a smart beam: design for suppression of induced vibrations, 336 (2004). 8. M. Betti, P. Biagini and L. Facchini, Proceedings of the Fourth EuropeanAfiican Conference on Wind Engineering, Comparison among different techniques for reliability assessment of no-tensile structures under turbulent wind, 34 (2005). 9. M. Betti, C.C. Baniotopoulos, G.E. Stavroulakis, 77th Annual Meeting of the Gesellschaft f i r Angewandte Mathematik und Mechanik e. V., (GAMM 2006, Berlin, 27-31 Mayl, Active vibrations suppression on beams under stochastic loading by means of piezoelectric actuators and sensors, (2006) to appear in PAMM. 10. M. Betti, C. Borri, C.C. Baniotopoulos, G.E. Stavroulakis, Proceeding of IX Convegno Nazionale di Ingegneria del Vento (IN- VENT0 2006, Pescara, 18-21 June), Piezoelectric-active suppression of wind induced vibrations on beams: modelling and simulation, (2006), in press. 11. E.F. Crawley and J. De Luis, AIAA J , Use of piezoelectric actuators as elements of intelligent structures, 1373 (1987).

75 12. F. C8t6, P. Masson, N. Mrad and V. Cotoni, Composite structure, Dynamic and static modeling of piezoelectric composite structures using a thermal analogy with MSCNASTRAN, 47 1 (2004). 13. J.F. Deu and A. Benjeddou, Int. J. Sol. Struct., Free-vibration analysis of laminated plates with embedded shear-mode piezoceramic layers, 2059 (2005). 14. C. Fuller, S. Elliot and P. Nelson, US Ed. Sun Diego: Academic Press Inc., Active control of vibration (1996). 15. D. Huang, B.H. Sun, Comput Struct, Approximate solution on smart composite beams by using MATLAB, 197 (2001). 16. C. Hwu, W.C. Chang and H.S. Gai, Int. J. Sound, Vibr., Vibration suppression of composite sandwich beams, 1 (2004). 17. L.E. Mackriell, K.C.S. Kwok and B. Samali, Engineering Structures, Critical mode control of a wind-loaded tall building using an active tuned mass damper, 834 (1997). 18. E. Simiu and R.H. Scanlan, Wilq New York, Wind effects on structures, (1978). 19. G.E. Stavroulakis, G. Foutsitzi, E. Hadjigeorgiou, D. Marinova and C.C. Baniotopoulos, Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing, Design of Smart Beams for Suppression of Wind-Induced Vibrations, paper 114 (2003). 20. L. Tong, Q. Luo, Int. J. Sol. Struct., Exact dynamic solutions to piezoelectric smart beams including peel stresses I: Theory and application, 4789 (2003).

76

A NUMERICAL LIBRARY FOR SHELLS DESCRIBED BY THE INTRINSIC GEOMETRIC MODELING VIA THE ORIENTED DISTANCE FUNCTION

Pble Universitaire Ldonard de Vinci, ESILV, DER-CS, 92916 Paris La Ddfense Cedex, France. t E-mail: john. cagnolOdevinci.f r 5 E-mail: vittorio.sansaloneOdevinci.fr

A shell can be described via the oriented distance function, which provides a natural and simple tool t o describe the geometric properties of a set and of its boundary. The advantage of this approach is the total absence of Christoffel symbols. This description is well suited for the area of smart systems, where the adaptation of the existing techniques of control theory is made easier. In this presentation, we shall provide the numerical tools t o carry out the computa, tions within the framework and language of intrinsic modeling. The Geometry of the shell will be given by a Non-Uniform Rational B-Spline (NURBS). We shall present the implementation of the intrinsic geometric operators, and show their use in a test case: the vibration of a shell described by NURBS under a variety of mechanical constraints. Keywords: Shells; Numerical libraries; NURBS; FreeFEM++; Simulation; Intrinsic Modeling.

1. Introduction

A shell is a continuous three-dimensional object of which one dimension is small compared to the other two. The small dimension is called the thickness. While the object is three-dimensional, one usually takes advantage of its small thickness to obtain a twedimensional model based on the middle surface (or mid-surface). As a consequence, shell theory is linked to the representation of surfaces, which can be done in several ways. The most widespread way to describe a shell (or the mid-surface thereof) is by using one or several maps: one considers the usual Euclidean space * Supported by the European Commission under the 5th F.P. grant number HPRN-CT2002-00284.

77 referred to an orthonormal fixed system (0, el, e2, es) and an open bounded subset R in W2. Then, the mid-surface S = $(n)where $ is a function from W2 to W3. The geometrical information can be derived from $. The pioneers of this approach were Kirchhoffl and Love.2 We refer to 3 and references therein, for further details. Another way to describe a shell is via the oriented distance function (also called signed distance function) that provides a natural and simple tool t o describe the geometric properties of a set and of its boundary. The advantage of this approach is the total absence of Christophel symbols. The pioneers of this approach were Michel Delfour and Jean-Paul ZolBsio. An overview of this method is described in Section 3. The point of this article is certainly not to compare these models in view of establishing a hierarchy nor is it to make any claim about which model is better. However, in the area of control, the adaptation of the existing techniques of control theory turns out to be not so easy when the midsurface of the shell is described by several maps. This situation is mainly due to the presence of Christoffel symbols that make computations difficult. We acknowledge that work in control has been achieved with models describing the mid-surface with maps but we believe the intrinsic geometric modeling is more suited for this work, To our knowledge, there is no numerical library to easily deal with models expressed in terms of the oriented distance function. So far, whoever wanted to do some numerical computations with the intrinsic geometric models had to "translate" it first. The goal of this article is t o provide a numerical library so computations can be carried out within the framework and language of intrinsic modeling. The Geometry of the shell will be given by a Non-Uniform Rational BSpline (NURBS). We chose this representation because it provides a flexible way to design a variety of shapes and are widely used in the computer aided design industry. In section 2, we give a brief overview of NURBS, and in section 3 we recall the basics of intrinsic geometric modelling of surfaces. We provide some details on the implementation of the numerical library in Section 4 and give an example in section 5. 2. Overview of the Non-Uniform Rational B-Splines

Non-Uniform Rational B-Splines (NURBS) are a generalization of BBzier splines, which can be regarded as uniform, non-rational B-splines. In order to improve readability we include in this section a brief discussion of NURBS are and how to deal with them. Since by necessity this overview

78 will lack detail, the reader is referred to 4 for a definitive exposition on this topic. NURBS development began in the 1950's. They are now a mathematical model commonly used by engineers in computer aided design. Among their useful properties, NURBS provide a common mathematical setting for dealing with both analytical and free form shapes; they always maintain mathematical exactness and resolution independence; they offer great flexibility in representing arbitrary shapes with remarkable small amount of data; they are easy to control by means of a set of control points and knots, which guides the shape of the curve or surface and can be manipulated in order to achieve a desired smoothness or curvature; and they can be quickly evaluated by numerically stable and accurate algorithms. It is worth to point out that in spite of these great features, NURBS do not provide any geometrical interpretation of represented objects: detecting the geometrical features other than local minimum continuity degree is difficult.

2.1. NURBS Curves Spline functions take their name from the flexible tool used in past decades by car and ship designers in order to draw geometrically perfect objects. Therefore, this name hints to their capability to represent smooth curves. Splines are parametric functions of the type: C(t>= {X(t>,Y(t>)

(1)

In NURBS curves, the shape is defined by the position of a set of n, control points Cj, j = l..n,, and the value of the two functions {X(t), Y(t)) represent, at time t, a weighted average of the influence of the control points on the position of the particle p. An key feature is that this influence is only local: that is, at any time, only the control points "close" to the position of the particle at that time will influence its motion. B-splines are a special class of spline functions for which a basis is defined. Each of them is a function oft related to a control point: Nj(t),j = l..n,. Each basis function represents the weight of the associated control point. Therefore, eq. (1) reads:

In this way, the adjective "close" used at the end of previous paragraph can be better defined in terms of the mathematical properties of the basis:

79 the influence of each control point over the motion of p is restricted only to the support of the corresponding basis function. The support of each basis function being an interval of the time line, it follows that each control points Cjinfluences the motion of p only in that time interval. Moreover, a t any time, the amount of its influence is just the value attained by Njat that time. The computation of NURBS is always made starting from this functions, then accurate, effective algorithms have been developed in order to compute them and their derivatives. Usually, basis functions are smooth polynomial functions with compact support. In its basic formulation, all the shape functions have the same shape and cover equal intervals of time. In general, it is useful t o generalize this definition, letting different control points affect the curve in different ways. In order to obtain that, it is possible to modify the corresponding basis functions, namely their shape (i.e. sharpness and maximum value) and support (i.e., the amplitude of time interval on which they act). In practice, this is realized introducing a partition of the time interval by means of the knot vector, which defines the amplitude and the shape of the basis functions. Roughly speaking, the closer the knots in some time interval, the stronger and sharper the effect of the nearest control points. In this way, a non uniform knot vector allows to modify the relative influence of the control points on the shape of the curve, and then to obtain sharp corners or extended plateaux. It is also permitted to have multiple knots, that is knots with multiplicity greater than one. The knot vector affects the local continuity of the curve: the greater the multiplicity of a knot, the lower the continuity of the curve at that time. In general, given a basis of degree n, it defines a curve Cn-' continuous. Special care must be taken at the time values corresponding to the knot points: if a knot has multiplicity m, then at that time the curves has continuity Cn-m. Curves defined by eq. (2) cannot represent conic sections. So far, each control point has the same importance. In some sense, it as all the c.p. carry the same weight 1.0. It is possible to generalize this definition, letting control points to carry different weighs. In this way, increasing the weight of a point, it is possible to let it more strongly "attract" the curve. Vice versa, decreasing its weight, it will push the curve away. The introduction of a weight is equivalent to define the control points of the plane curve in a 3D space: the two Euclidean coordinates and the weight itself. Since curves are actually represented in a 2D space, it is then necessary to project on it

80 the control points. This is easily achieved by rewriting eq. (2) as:

where wj is the weight of control point Cj. From the computational point of view, the best way to deal with NURBS is to resort to the so called homogeneous coordinates. The outline of the procedure is then: (1) Add one dimension to the control points: Cj = (Xj,Yj) -+ C3. = (xj,q,l.o) (2) Represent the control points in homogeneous coordinates, by multiplying them by their weight: Cj = (Xj, Y ,,1.0) -+ Cj = (wjXj, wjK, wj), where wj is the weight of control point Cj. (3) Compute the NURBS in homogeneous coordinates:

(4) Map the NURBS back in the Euclidean space, which leads to eq. (3). 2.2. NURBS Surfaces

NURBS surfaces are the direct product of two NURBS curves, that is one of the curve is let glide along the other, so describing a surface. Then, given two NURBS curves Cl(tl) and Cz(tz),where tl and t2 are the curvilinear abscissas along the curves, they define a NURBS surface as:

The definition in terms of the basis functions of the two curves is straightforward. Of course, there is no need for the two curves being of the same order (the corresponding basis made of polynomials do not meet to be of the same degree). 3. Intrinsic Geometric Modeling

We here include a brief discussion of the oriented distance function (also known as signed distance function). We refer to 5,6 for more details.

81 3.1. The Oriented Distance Function

Consider a domain O c R3 whose nonempty boundary 80 is a C1 twodimensional sub-manifold of R3.Define the oriented (or signed) distance function to O as

where d is the Euclidean distance from the point x to the domain 0.In other words, b(x) is simply the positive or negative distance to the boundary d o , depending on if we are outside or inside the domain 0. It can be shown that for every x E dO, there exists a neighborhood where the function Vb = u, the unit outward external normal to Consider a subset I' dO which will eventually become the mid-surface of our shell. We define the projection p(x) of a point x onto I? as p(x) = x - b(x)Vb(x). Then, we define a shell Shof thickness h as

c

A natural curvilinear coordinate system (X, z) is thus'induced on the shell Sh,where the coordinate vector X gives the position of a point on the midsurface r, and z E (-$, $) gives the vertical (normal) distance from the mid-surface. Using this notation, we also define the "flow mapping" T,(X) as

for all X and z in Sh. This allows us to reconstruct the action at a given height z of the shell, once we know the action of the mid-surface r. Define as F Z the surface T,(I') at the 'altitude' z. Then, one can also describe the shell Shas

The curvatures of the shell will be denoted H and K. These can be reconstructed from the boundary distance function b(x) by noting that at any point (X, z), the matrix D2b has eigenvalues 0, XI, X2. The curvatures are then given by tr (D2b) = 2H = XI A 2 and K = X1X2.

+

3.2. Tangential Differential Calculw

Next, we mention briefly some useful aspects of the tangential differential calculus. Given f E C1(I'), we define the tangential gradient Vr of the

82 scalar function f by means of the projection as

This notion of the tangential gradient is equivalent to the classical definition using an extension F of f in the neighborhood of I?, i.e. Vr f = VFJ, Following the same idea we can define the tangential Jacobian matrix of a vector function v E C1(l?)3as

Ev.~

the tangential divergence as

the Hessian D; f of f E C2(I') as

the Laplace-Beltrami operator of f E C2(r)as

the tangential linear strain tensor of elasticity as

and the tangential vectorial divergence of a second-order tensor A as

Finally, f r ~ r nwe ~ . have ~ that ( V ~ WVb) , = 0,

DrvVb = 0

(10)

by definition for any scalar w and vector v. In addition, if we consider a purely tangent vector v = vr, i.e. (vr, Vb) = 0, we can take the tangential gradient of both sides of this expression and derive the following useful formula

Throughout this paper the conventions of7 concerning tensors are used. For instance, we will make no distinction between a second-order tensor or a matrix, nor will we make a distinction between a first-order tensor and a vector. Consequently we will not distinguish simple contraction and multiplication. Finally, the notation dtcp is used to denote the partial derivative of cp with respect to the time variable, and the notation A . .B denotes the double contraction of two matrices - i.e. A . .B = tr (AB),

83 3.3. More Tangential Operators and Spaces

Let e be a vector function, we note er and w the tangential component of e and its algebraic magnitude on the normal part respectively, i.e. w = (e, Vb) er = e - w V b Vector function e will physically represent the displacement of the midsurface, its components er and w will represent the in-place displacement and the normal deflection respectively. Following the definitions introduced in,8 we will use the subsequent notations. Definition 3.1. Let w be a scalar and u a vector. We define the following operators:

1 Frw = 5((Vb B Vrw)

G

+( V ~ W 8 Vb)) 1 ~ = w -((Vb B vrw)D2b + D2b(vrw B Vb)) 2

1 Vru = ,((D2b U) @ Vb 1 Srw = -(D:w 2

+ Vb @I (D2bu))

+ *D;w)

Cr, Ck, Fr and Gr are 1st-order tangential operators, Vr is a zero-order tangential operator, and Sr is the symmetrization of the Hessian (which is not symmetric in the tangential calculus6). 4. Implementation

In this section we will describe some details about the implementation of the differential operators previously introduced. We assume our model surface is described by a NURBS surface. In order to deal with this surface, we shall use NURBS++, a C++ library to manipulate and create NURBS curves and surfaces. It should be noted that the NURBS++ package classes and methods belong to the namespace PLib,it is therefore necessary to include it. We refer to 4 for details on this library.

84 The computer code for implementing the differential operators has been written from scratch in C++, and suitably linked to the NURBS++ library. There are two levels of implementation: first, a low-level set of routines which can be completely controlled in all the parameters they use for computations; then, a high-level reduced set of routines which have been thought for the end-user and provide an easy way to deal with the differential geometry of the shell surface. High-level routines do nothing but calling low-level functions with a set of parameters suitably optimized for most cases. Only few of these parameters can be set by calling the high-level routines. There are several functions, which can be grouped on the basis of what they do:

Points and Projection Handle Euclidean points, locate them on a NURBS surface by means of suitable projection algorithms and act as an easy-to-use interface with the NURBS++ internal variable structure. Derivatives Compute covariant derivatives on the NURBS surface. Normal fields Compute normal and unit normal field on the NURBS surface. Differential geometry Other useful functions dealing with differential geometry of the surface: 1" and 2"* fundamental forms, mean and Gauss curvature, etc. Oriented distance Oriented distance and its derivatives. For a complete list and description of such functions, as well as the most important C++ classes, see 9. In order to validate the numerical routines implemented, we considered the case of a semisphere and we tested the accuracy of the numerical representation of the differential operators involved in our shell model. The sphere is parametrized in terms of the parameters (u, v) E [0, l [ x [O,l[, which have the meaning of the spherical coordinates (8, p) E [O, 2 ~ [[O,x~ / 2 (the [ longitude and colatitude, respectively) scaled to the unit square. In Fig. 1 we show the logarithmic error of Vb and D2b, while in Fig. 2 we show the logarithmic error of the mean and Gauss curvature. It is possible t o notice that in each case the error is always very small, and the maximum value is always attained close to the point (u, v) = (0, O), where the parametrization of the sphere is singular. Not surprisingly, the error with Vb is always much smaller than the other cases, since the order of

85 differentiation is smaller. Finally, we point out that the peaks which appear at u = 1/4,1/2,3/4 are due to the NURBS representation of the sphere, since at those points two pieces of surfaces are joined together and a loss of accuracy, as larger as the differentiation order involved, should be expected.

U

U

Fig. 1. Logarithmic error in the numerical representation of Vb (left) and ~

' (right). b

5. Numerical Simulations We implemented the shell model in F ' ~ ~ ~ F E M + + ,an " open source software able to deal with a large class of mechanical problems governed by partial differential equations. FreeFern++ has an advanced automatic mesh generator and a general purpose elliptic solver. Moreover, it easily allows for dealing with hyperbolic and parabolic problems. FreeFEM++ provides a high level language and macro expansion capabilities, by which it is possible to describe even complex problems and a quick specification of any system of partial differential equations.

86

Fig. 2. Logarithmic error in the numerical representation of the mean curvature H (left) and of the Gauss curvature K (right).

5.1. Some Numerical Insights

A limitation of FreeFEM++ concerns the available types of finite element. In particular, only lagrangean FE are provided by default. This class of FE is only CO,while in the weak form of our shell model at the least C1 is required. Even if it is possible to implement new FE classes in FreeFEM++, this requires additional work and an in-depth knowledge of its data structures. We decided not to deal with this issue and to use the FE classes already available, modifying our formulation in order to reduce the continuity requirements. C1 continuity is required in the bilaplacian-like operators involving the normal deflection w, namely operator Srw and few other contributions in the energy involving A ~ w . We introduce a new independent vector field cp describing the rotation of the mid surface of the shell, related to the normal deflection w by: cp E V ~ W

(13)

With this position, we can rewrite some of the differential operators appearing in our model as:

87 Equation (13) is a constraint which must be suitably imposed. We tested few possibilities (penalty functions, least square method), and among them the most effective came out to be a Lagrangean multiplier technique. In this setting, the constraint above is imposed as:

which, in the weak form of the energy, reads:

The lagrangean multiplier X is the reaction responsible for maintaining the constraint and is related to the shear stress, so giving our model a mixed formulation. In order t o test the performances of different FE classes, we consider the bending problem of a square plate simply supported on its edges and subjected to a uniform vertical load. In a compatible setting, the weak form of the problem reads:

Using the lagrangean multipliers scheme, previous equation becomes:

Figure 3 shows the value of the vertical deflection w in the middle of the plate obtained by different classes of FE as a function of the total number of degrees of freedom (i.e. including w, cp, and A). In the same figure it is shown the contour map of w in a typical FreeFEM++ plot. The names in the plot refer to the type of interpolation for w, cp, and X respectively. Among the other, best performances are shown by the P2P2P2 family, i.e. assuming a quadratic interpolation for all the three fields (here, exact Gauss integration is used). We point out that we need to choose the interpolation for the different fields carefully to avoid either locking and spurious rigid modes. In the following, we will refer to the P2P2P2 family of FE for the bending part of the problem (w, cp, and A) and also P2 for the membrane part (er). This choice, which proved to be effective, means that we assume quadratic interpolation and exact numerical integration for all the fields involved in our FE model.

88

Fig. 3. Bilaplacian problem. Relative error in the vertical deflection w in the middle of the plate for different types of finite element, and FreeFEM++ contour map of w for the first point of P2P2P2 family (ndof % 2000).

5.2. Test Case: Circular Cylinder As a case study we considered a circular cylinder (radius R = 2.0, thickness t h = 0.01), subjected to an applied radial displacement to one end, the other being left free. In the following, we will refer to C as an abscissa running along the axis of the cylinder, and to 0 as an absissa running in the circumferential direction. In view of the symmetry conditions, we studied only a quarter of the cylinder, applying suitable boundary conditions in order to recover the periodicity. Numerical result are perfectly in agreement with the exact solution.'' On the top of fig. 5 the lateral radial displacement w of the wall of the cylinder a t 0 = 7r/4 is plotted along the abscissa: curves obtained using meshes labeled 2n and split follow exactly the exact solution, but also that obtained with mesh n is in quite good agreement. In the same picture the wave length of oscillations is also pointed out: again the numerical value is in very good agreement with the theoretical one (w.1. = 2 * 0.586). Finally, on the bottom of the figure the contour map of w related to mesh split is shown (continuous black lines indicate w = 0). The deformed configuration obtained from this analysis has been used as initial condition for studying the lateral oscillations of the cylinder (pure aluminum is again considered as target material). Figure 6 shows some

89

H

split

3,

Fig. 4. Left: sketch of geometry and boundary conditions. Right: Different meshes used in the simulations. 0.0012

1

Fig. 5. Radial displacement. Top: section at 0 = n/4. Bottom: contour map (continuous black lines indicate w = 0).

90 snapshots of the lateral displacement w (mesh 3n).

Fig. 6. Top: lateral oscillations of the cylinder at 8 = n/4. Bottom panels: time evolution of w a t the ends of the cylinder.

The time integration scheme used in our calculations is the Constant Average Acceleration (CAA) method, a variant of the Newmark's family based on the hypothesis that the acceleration in the time step is constant and equal to the average value between the initial and final time. In general, for a displacement based problem, Newmark's time integration scheme is based on the following time interpolation for the displacement u and the velocity u:

91 where ii is the a ~ c e l e r a t i o n .This ~ assumption leads t o the iterative procedure:

where M , D , and K are the mass, damping, and stiffness matrix, respectively. a and 6 are parameters which can be tuned in order t o obtain the desired convergence and accuracy of the scheme (for the CAA scheme: a = 0.25 and 6 = 0.5). Moreover, since we consider free oscillations, in our case D = O a n d p = O .

References 1. G. Kirchhoff, Vorlesungen uber Mathematische Physik (Mechanik, Leipzig, 1876). 2. A. E. H. Love, The Mathematical Theory of Elasticity (Cambridge University Press, 1934). 3. M. Bernadou and J.-M. Boisserie, The finite element method i n thin shell theory, Progress in Scientific Computing, Vol. 1 (Birkhauser Boston, Mass., 1982). 4. C. Hardy, P. Lavoie and K. Wetzel, The NURBS++ package http://libnurbs.sourceforge.net/. 5. M. C. Delfour and J.-P. Zolbsio, Differential equations for linear shells: comparison between intrinsic and classical models, in Advances i n mathematical sciences: CRM's 25 years (Montreal, PQ, 1994), , CRM Proc. Lecture Notes Vol. 11 (Amer. Math. Soc., Providence, RI, 1997) pp. 41-124. 6. M. C. Delfour and J.-P. Zolkio, Intrinsic differential geometry and theory of thin shells (to appear, 200x). 7. P. Germain, Mecanique (Ecole Polytechnique, 1986). 8. J. Cagnol, I. Lasiecka, C. Lebiedzik and J.-P. Zolkio, J. Differential Equations 186, 88 (2002). aIn our case, similar assumptions clearly hold for each field involved in the analysis: the formal expression above is unchanged if one think of u as the entire set of unknown fields: u {er ,w ,cp, A).

92 9. J. Cagnol, V. Sansalone and C. Lebiedzik, A numerical library for controlling shells within the framework of intrinsic geometry, tech. rep., P61e Universitaire Lkonard de Vinci, ESILV, DER-CS (2006). 10. F. Hecht, 0 . Pironneau, A. Le Hyaric and K. Ohtsuka, Reefem++ http://www.freefem.org/ff++/. 11. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, NY, 1959).

93

A CONTACT PROBLEM FOR VISCOELASTIC MATERIALS WITH LONG MEMORY INVOLVING DAMAGE M. CAMP0 and J.R. FERNANDEZ Departamento de Matemdtica Aplicada, Universidade de Santiago de Compostela, Facultade de Matemdticas, Campus Sur s/n, Santiago de Compostela, 1578.2 A Corufia, Spain E-mail: {macampo,jramon) Ousc. es

Departamento de MCtodos Matemdticos e de Representacidn, Universidade de A Corufia E.T.S.E. Camifios, Canais e Portos, Campus de ElwirTa s/n, 15071 A Corufia, Spain 'E-mail: [email protected] In this work, a model for the quasistatic frictionless contact between a viscoelastic body with long memory and a foundation is studied. The material constitutive relation is assumed t o be nonlinear and the contact is modelled with the normal compliance condition, i.e., the obstacle is assumed t o be deformable. The evolution of the mechanical damage of the material, caused by the opening and growth of microcracks under excessive stress or strain, is modelled by a nonlinear partial differential equation. We discuss the existence and uniqueness of weak solution for the mechanical problem and we introduce a fully discrete scheme based on the finite element method and finite differences to approximate this solution. The existence and uniqueness of the approximate solutions are presented and some estimates of the numerical error are provided. Finally, we show some results obtained in the simulation of a two-dimensional test problem.

Keywords: Viscoelastic material with long memory; Damage; Frictionless contact; Weak solutions; Finite elements; Error estimates; Numerical simulations.

1. Introduction Contact problems of deformable bodies which may undergo damage are present in everyday life and in many industrial proceses. Damage is caused by the opening and growth of microcracks under excessive stress or strain and leads to the decrease of the load carrying capacity of the body, thus affecting the effective functioning and safety of a mechanical system. Be-

94 cause of the importance of this topic, an increasing number of engineering publications dealing with damage models appeared in the last ten years. Here, we follow the early works of ~ r 6 m o n d "where ~ damage is modelled using an internal variable [ taking values between 0 and 1, in such a way that when [ = 1 the material is damage-free, when [ = 0 the material is completely damaged, and when 0 < [ < 1 there is partial damage. Mathematical problems including damage have been studied recently for elastic elastic-viscoplastic or viscoelastic material^.^^^^ Moreover, one-dimensional damage problems have been also considered.ll On the other hand, contact problems for viscoelastic materials with long memory which did not take damage into account have been recently studied, as well. 12-14 Here, a frictionless contact problem involving a viscoelastic material with long memory and damage is considered. The obstacle is assumed deformable and thus, a normal compliance contact condition is used. The paper is organized as follows. In Section 2 we state the mechanical problem and establish the existence of a unique weak solution. In Section 3, we analyze a fully discrete scheme for the problem. We use the finite element method to discretize the domain and a forward Euler scheme to discretize the time derivatives. We also provide related error estimates. Finally, in Section 4 we present numerical simulations in the study of two-dimensional test problems.

2. Problem statement and variational formulation

We consider a viscoelastic body which occupies a domain R C Itd for d = 1,2,3. The boundary r of the domain R is assumed to be Lipschitz, and it is divided into three disjoint measurable parts r ~rF , and rc such that x (0, T) and so the displacement meas ( r D ) > 0. The body is clamped on field vanishes there; a volume force of density f acts in R x (0, T), and surface tractions of density f act on FF x (0, T). Here, T > 0 and [0,TI is the time interval of interest. The body may come in contact with a reactive foundation over the part r c . A gap g exists between the potential contact surface rc and the foundation, and it is measured along the outward normal vector v (see Fig. 1). We denote by u the displacement field, a the stress tensor and E ( U ) the linearized strain tensor. Moreover, let [ be the damage field which measures the density of the microcracks in the material, and will be described below. The material is assumed viscoelastic with long memory, which is modelled

95

Fig. 1. Contact problem of a viscoelastic body with long memory.

by using the following constitutive law,

1 t

~ ( t =) d&(u(t))+

0

G(t - s, e(u(s)), I ( s ) )ds,

where d represents the elasticity tensor and Q is a prescribed nonlinear constitutive function. This law is a generalization of the linear equation used by other authors15-l7 to model the mechanical behavior of a variety of solid materials, as organic polyrneers or some kinds of wood and it has been used in the context of contact mechanics.12-l4 To simplify the notation, we do not indicate explicitly the dependence of various functions on the independent variables x E R U r and t E [0,TI. Also, dots above a variable represent time derivatives. We assume that the normal stress on rc,a, = uv . v , is given by the following normal compliance contact condition,

Here, u, = u . v denotes the normal displacement. When u, > g, the difference u, -g represents the interpenetration of the body's asperities into those of the foundation. The normal compliance function p is prescribed and satisfies p ( r ) = 0 for r 5 0, since then there is no contact. Moreover, we assume that the contact is frictionless, i.e. the tangential component of the stress o, = o - a,v vanishes on the contact surface. We turn to describe the damage process. As a result of the tensile or compressive stresses in the body, microcracks open and grow and this, in turn, causes the load bearing of the material to decrease. This reduction in the strength of the material is modelled by introducing the damage field C = [(x,t ) . Following the derivation in,'?' the evolution of the microscopic

96 cracks responsible for the damage is described by the partial differential equation (seel8?l9),

Here, K > 0 is a constant and 4 is a given constitutive function which describes damage sources in the system. F o l l ~ w i n ~ an , ' ~ homogeneous Neumann boundary condition for the damage field is used, so the normal derivative of C, denoted by &$, vanishes on I'. Let Sd be the space of second order symmetric tensors on Rd, or equivalently, the space of symmetric matrices of order d , and denote by the usual inner product defined in these spaces and by 11 . 11 their norm. The classical form of the mechanical problem of the quasistatic frictionless contact with damage of a viscoelastic body with long memory may be written as follows. Problem P. Find a displacement field u : R x [0,TI -, Rd, a stress field u : R x [O,T] + S d , and a damage field 5 : R x [0,T] -t R such that C(0) = Co zn R and,

a< -0 dv

u

=o

uu = f

uT = 0,

on

r x (O,T),

o n r x~ (O,T), on FF x (0, T), -a, = p(u, - g )

on rc x (0, T).

We now proceed to obtain a variational formulation of Problem P. For this purpose, we introduce additional notation and assumptions on the problem data. Here and in what follows the indices i and j run between 1 and d , the summation convention over repeated indices is adopted and the index that follows a comma indicates a partial derivative with respect to the corresponding component of the independent variable. Let Y = L2(R), E = H 1(R), H = [L2(R)ld and define the following variational spaces:

97

For every element v E V, we denote by v, and v, the normal and the tangential components of v on the boundary I? given by v, = v . v and v, = v -v,v, respectively. Similarly, for a regular (say C1) tensor field u : 52 -+ Sd we define its normal and tangential components by a, = ( u v ) . v and a, = a v - a,v. On V we consider the inner product given by (u,v ) =~ (&(a),E(v))Q, and the associated norm IIvllv = IIE(v)IIQ. Since m e a s ( I ' ~ )> 0, from Korn's inequality it follows that (V, 11 . [Iv) is a real Hilbert space. Moreover, by the Sobolev's trace theorem we have a constant Q , depending only on the domain 52, I'D and Fc,such that I l ~ l ( [ ~ z ( ~ , ,5~ d~ l l v l l Vv ~ E V. For every real Hilbert space X , let (1 . [Ix be the associated norm on X , we use the classical notation for the spaces LP(0, T; X) and wkJ"PO, T; X), 1 5 p 5 +oo, k 2 1 and we denote by C([O,TI; X) and C1([O,TI; X) the spaces of continuous and continuously differentiable functions from [0,TI to X , respectively. In the study of the mechanical problem (1)-(7), we assume that the elasticity tensor A(X) = (aijkl(~))~j,k,l,l : T E sd-+ A(x)T E sdsatisfies (a) aijkl = aklij = ajikl for i,j, k, 1 = 1,.. . ,d. (b) aijkl E Lm(R) for i , j , k , l = 1,...,d. (c) There exists r n >~ 0 such that A(x)T. T 2 r n 1 ~1 Q rcsd, a.e. x E R. The viscoelastic long memory operator G : R x [0,T] x Sd x R

~11~

-+

Sd satisfies

(a) There exists MG > 0 such that l l G ( ~ , t l t i , P l) G ( x , ~ , < ~ , P ~5) IMG I (ll
98 The damage source function

4 : R x Sd x R --+ R satisfies

(a) There exists L4 > 0 such that I4(.,El1P1) - 4(x,~2,P2)1I L4 (11~1-&all + IP1 - P2I) V E ~ , E Sd, ~ P1,P2 E W, a.e. x E R. (b) The mapping x H +(x, E, P) is Lebesgue measurable on R, V E E s d , 0 E R. (c) The mapping x I+ +(x, 0,0) E L2(R). (d) 4(x, E, p) is bounded VE E Sd, P E R, a.e. x E R. (e) 4 ( x , ~ , P I) O i f P 2 1, 4 ( x , ~ , P 2) O i f P I C * .

Remark 2.1. We notice that condition (10)(e) allows us to prove that the damage field satisfies the constraint c(x) L C,, Vx E R, where C, > 0 is assumed small enough. We restrict the damage to 5 since, when the effective modulus is small, the material is full of microcracks and it no longer makes sense to model it by a viscoelastic material law.

c

c* c

We suppose that the body forces and surface tractions are given as,

f o E C([O,T];H),

f 2 E C([O,TI; [ L 2 ( r ~ ) l d ) ,

(I1)

and the initial damage field satisfies The normal compliance function p : PC x R -+R+ verifies (a) There exists L, > 0 such that IP(x, rl) - P(X,r2)l I L, Irl - 7-21 V r l , 7-2 E R, a.e. x E rc. (b) The mapping x I+ p(x, r ) is Lebesgue measurable on rc, V r E R. (c) (P(x, r l ) - P(X,r2)) . (TI - 7-2) 0 V r l , 7-2 E R, a.e. x E rc. (d) The mapping x H p(x, r ) = 0 for all r 5 0.

>

Next, we define the function f : [0,TI -+ V, the functional j : V x V and the bilinear form a : E x E -+ R by the following equations:

-t

R

Notice that the integral (15) is well defined by (13), and conditions (11) imply the regularity

f

E C([O,TI; V).

99 By using standard arguments we derive the following variational formulation of the mechanical problem P . Problem V P . Find a displacement field u : [O,T] + V , a stress field a : [0,TI 4 Q, and a damage field C : [O,T] -+ E such that C(0) = Co, and for a.e. t E [0,TI,

The existence of a unique solution to Problem V P is proved using some results for parabolic variational equations and Banach fixed point arguments, which we summarize in the following (see2' for the details). Theorem 2.1. Assume that (8)-(13) hold. Then, there exists a unique solution (u, a , C) to Problem V P with the following regularity:

u E C([O,T];V), a E C([O,T];Q),

Moreover, the damage field

C E w ~ * ~ ( o ,Y)T ;n L ~ ( o , TE). ;

C satisfies [(t,x)

E

[C*, 11 a.e. t

E [O,T] and

x E R. 3. Fully discrete approximation: error estimates

In this section we consider a numerical scheme for solving Problem V P and derive some error estimates on the approximate solutions. We use two finite dimensional spaces Vh C V and E h C E to approximate the spaces V and E, respectively. Here, h > 0 denotes the spatial discretization parameter. To discretize time derivatives, let us consider a uniform partition of the time interval [0,TI, denoted by 0 = to < tl < . . . < t N = T , and let k = T I N be the time step size. For a continuous function f ( t ) ,we use the notation f, = f (t,). Moreover, for a sequence { w , ) ~ ~let, dw, = (w, - w,-l)/k denote its divided differences, and let c be a positive constant independent of the discretization parameters h and k.

Remark 3.1. In the numerical simulations presented in the next section, vhand E~ are composed of continuous and piecewise affine functions, that is,

100 where R is assumed to be a polyhedral domain and we denote by 'Th a regular family of triangulations of compatible with the partition of the boundary F = 8 R into FD, FF and rc.

4-t

co.

Let be an appropriate approximation of the initial condition A fully discrete approximation of Problem VP, based on the forward Euler scheme, is the following. Problem Vphk. Find a discrete displacement field uhk= {u~~)~'=, c vh and a discrete damage field = {i,hk)kO c E~ such that [,hk = 4-oh and, f o r n = l , . . . ,N ,

chk

where ukk E vh is given. Using classical arguments of nonlinear variational equations, we obtain the existence of a unique solution to Problem vphk(see2' for the details). Theorem 3.1. Assume that (8)-(11) hold. Then there exists a unique solution (uhk,chk) c Vh x Eh to Problem VPhk. Remark 3.2. We notice that Problem VPhk is solved at each time step as follows. First, the discrete "initial condition" ukk is the solution of the problem,

This is a nonlinear variational equation which is solved by using a penaltyduality algorithm. Next, for each n E (1,.. . ,N),the discrete damage field is obtained solving the discrete variational equation (22) by employing Cholesky's method since it leads to a linear system. Moreover, the discrete displacement field is the solution of the nonlinear variational equation (23). Again, a penaltyduality algorithm is employed. We assume the following additional regularity of the damage field,

The aim of this section is to estimate the numerical errors u, in-
-

u;k and

101 Theorem 3.2. Assume that (8)-(13) and the additional regularity (24) hold. Let (u, <) E V XE and (uhk,chk) c Vh x Eh be the respective solutions to Problems V P and v p h k . Then, we have the following error estimates c Eh, for all wh = { ~ j h } j N _ ~c vh and

{tjh}E1

where ukk E

vh is given.

Error estimates (25) are the basis for the analysis of the convergence rate of the algorithm. As an example, let R be a polyhedral domain and denote by Iha regular triangulation of R compatible with the partition of the boundary I' = aR into rD,rFand rc. Let vh and Eh be defined by (20) and (21), respectively, and assume that the discrete initial condition is obtained by

<;

where rh: C(n) + Eh is the standard finite element interpolation operator. Moreover, the "discrete initial condition" ulk is defined as ukk = IIhu(0) where IIh = : [C(n)ld + vh. The following corollary is then derived, establishing the linear convergence of the algorithm with respect to the discretization parameters h and

k. Corollary 3.1. Let the assumptions of Theorem 3.2 hold. Under the following regularity conditions u E C([O,TI; [H2(R)ld)n W 1 * m ( T; ~ , V), E ~ ~ (T;0Y), nc([o, TI; H ~ ( R ) )n ~ ~ (T ;0E), ,

<

the numerical algorithm introduced i n Problem vphkis linearly convergent;

102 that is, there exists c > 0, independent of h and k , such that,

4. Numerical example

In this section, we report some numerical results on a twc-dimensional test problem to show the performance of the numerical method discussed in the previous section. The long memory operator has the form,

where d is the two-dimensional elasticity tensor under the plane stress hypothesis, that is

Here, E and r are Young's modulus and Poisson's ratio of the material which occupies R, respectively, and Jap is the Kronecker symbol. Moreover, @ : Sd -+ Sd is a truncation operator defined by

Here, L > 0 is a given constant. We notice that the existence of L is justified taking into account that the small displacement theory is used. In this example, value L = 1000 is employed. Note that this choice of function G shows that the more damage the material undergoes (5 decreases), the less memory effects arise and the more the material behaves like a purely elastic material. Recall also the definition of the von Mises norm for a plane stress field 7 = (rap) as,

The damage source function is given by

where Q,* (a) = min{~(u).E(u), q * } for some constant q*

W is a function defined by

> 0 and q : R H

103 Here, XI, X2 and X3 are process parameters whose values in the simulations below are

Also, the value 6 = is chosen. The following normal compliance function is employed in the simulations,

where r + = max(0,r) and p is a positive constant which represents a deformability coefficient (value p = l0l0 is taken). Finally, we use affine elements for the finite element spaces v h and E~ (see the definitions (20)-(21)) . We considered a rectangular domain [O, 21 x [O, 31. The body was fixed on = [O, 31 x (01, the contact occured on I'c = (0) x [O,1] and surface tractions were prescribed on (2) x [O, 31, while (0) x [l,21 U [O, 31 x (2) was traction-free (see Fig. 2). The following data were used in this example:

It-

t-

Fig. 2. Physical setting.

In Fig. 3 the von Mises stress norm is plotted over the deformed configuration at initial time (left-hand side) and at final time (right-hand side). Since the force is assumed to be constant, the effect of the memory can be seen: at final time the body tends to recover the stress-free state.

104 .

,

,. : 1

.

..

I

,

.,

: 4

-

-

.-..:.

- .", . .

I1.7 1

-

1 3

- 7

2

,

.

. 1

.:t ! :.r::

-

.,,

-.

.

:::.

.,I1:.., ., .

(

I

i ,

Fig. 3. Von Mises stress norm at initial and final times over the deformed configuration.

Also, in Fig. 4 the damage field at final time is shown over the deformed configuration. As can be seen, the most stressed areas coincide with the most damaged ones.

Fig. 4.

Damage a t final time over the deformed configuration.

Acknowledgement This work was partially supported by MCYT-Spain (Project BFM200305357). It is also part of the project "New materials, Adaptive Systems and their Nonlinearities; Modelling, Control and Numerical Simulation" carried out in the framework of the european community program "Improving the Human Research Potential and the Socio-Economic Knowledge Base" (Contract No. HPW-CT-2002-00284).

105 References 1. M. Fk6mond and B. Nedjar, Damage in concrete: the unilateral phenomenon, Nuclear Eng. Deszgn 156 (1995), 323-335. 2. M. Fr6mond and B. Nedjar, Damage, gradient of damage and principle of virtual work, Internat. J. Solids Structures 33 (1996), 1083-1103. 3. E. Bonetti and M. FrBmond, Damage theory: microscopic effects of vanishing macroscopic motions, Comp. Appl. Math. 22 (3) (2003), 313-333. 4. E. Bonetti and G. Schimperna, Local existence for Fr6mond1smodel of damage in elastic materials, Comp. Mech. Thermodyn. 16 (4) (2004), 319-335. 5. M. Campo, J.R. Fernbdez, K.L. Kuttler and M. Shillor, Quasistatic evolution of damage in an elastic body: numerical analysis and computational experiments, Appl. Numer. Math. (to appear). 6. M. Campo, J.R. Fernbndez and T.-V. Hoarau-Mantel, Analysis of two frictional viscoplastic contact problems with damage, J. Comput. Appl. Math. 196 (2006), 180-197. 7. 0. Chau and J.R. FernBndez, A convergence result in elastic-viscoplastic contact problems with damage, Math. Comp. Modelling 37 (2003), 301-321. 8. 0 . Chau, J.R. FernBndez, W. Han and M. Sofonea, A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage, Comput. Methods Appl. Mech. Engrg. 191 (2002), 5007-5026. 9. M. Carnpo, J.R. Fernbndez, W. Han and M. Sofonea, A dynamic viscoelastic contact problem with normal compliance and damage, Finite Elem. Anal. Des. 42 (2005), 1-24. 10. W. Han, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. Comput. Appl. Math. 137 (2001), 377-398. 11. K. T. Andrews, J.R. Fernbndez, and M. Shillor, Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod, IMA J. Appl. Math. 70 (6) (2005), 768-795 12. I. Figueiredo and L. Trabucho, A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity, Internat. J. Engrg. Sci. 33 (1) (1995), 45-66. 13. A. Rodriguez-Arbs, J.M. Viaiio and M. Sofonea, A class of evolutionary variational inequalities with Volterra-type term, Math. Models Methods Appl. Sci. 14 (4) (2004), 557-577. 14. M. Sofonea, A. Rodriguez-Arb and J.M. Viaiio, A class of integro-differential variational inequalities with applications to viscoelastic contact, Math. Comput. Modelling 41 (11-12) (2005), 1355-1369. 15. G. Duvaut and J.L. Lions, Inequalities in mechanics and physics, SpringerVerlag, Berlin, 1976. 16. A. C. Pipkin, Lectures in viscoelasticity theory, Applied Mathematical Sciences 7, George Allen & Unwin Ltd. London, Springer-Verlag New York, 1972. 17. O.C. Zienkiewicz and R.L. Taylor, The finite element method, McGraw-Hill, London, 2, Solid Mechanics, 1989. 18. K.L. Kuttler, Quasistatic evolution of damage in an elastic-viscoplastic ma-

106 terial, Electron. J. Differential Equations 147 (2005), 1-25. 19. K.L. Kuttler and M. Shillor, Quasistatic evolution of damage in an elastic body, Nonlinear Anal. Real World. Appl. (to appear). 20. M. Campo, J.R. Fernhdez and A. Rodriguez-Arbs, A quasistatic contact problem with normal compliance and damage involving viscoelastic materials with long memory, Submitted.

107

MEMORY EFFECTS ARISING IN THE HOMOGENIZATION OF COMPOSITES WITH INCLUSIONS L. FAELLA Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale Universitb degli Studi di Casino, via G. Di Biasio n.43, C a s i n o (FR), 03043, Italy. E-mail: [email protected]

S. MONSURRO Dipartimento di Matematica ed Informatica, Universitb degi Studi di Salerno, via Ponte don Melillo, Fisciano (SA), 84084, Italy. E-mail: [email protected] We want to describe the asymptotic behavior, as E + 0,of the solution of a first order evolution problem set in a domain of Rn with periodic inclusions of size E and prescribing a jump on the interface proportional to the conormal derivatives by means of a function of order E ? . The limit behaviors obtained are different according to the values of the parameter y. In particular, for y = 1, a linear memory term appears in the homogenized problem.

AMS Subject Classification: 35B27, 35k20.

1. Introduction Let R = R1, U Rz, be a domain of Rn made up of a connected component RIE and a disconnected one R2,, that is union of &-periodicconnected inclusions of size E . We want to describe the asymptotic behavior, as E -+ 0, of the solution u, = (ul,, ~ 2 , )of the following first order evolution problem, according to

108 the different values of the parameter y E R,

where AE(x) := A(x/E), with A periodic, bounded and positive definite matrix field, hE(x) := h ( x / ~ ) ,with h a positive, bounded and periodic function and where ni, denote the unitary outward normals to Ri,, for i = l,2. Problem (1) models the heat diffusion in a two-component composite with a thermal barrier on the interface, whose influence in the heat propagation varies with y. Intuitively, letting y increase corresponds to increment the influence of the thermal barrier on the homogenized material. The idea which generated this paper is to complete the study of the homogenization of problems involving composites with inclusions and having a jump condition on the interface that was developed by S. Monsurrb in13 and P. Donato and S. Monsurrb in7 in the elliptic case and by P. Donato, L. Faella and S. Monsurrb in8 in the hyperbolic one. In this work, coherently with the results of the previous ones, we obtain different limit problems in the cases y < 1 and y = 1. As already evidenced in,137 and,8 the case y > 1 will not be considered since for these values of the parameter we have no boundedness of the solutions (we refer to H.C. Hummel inlo for the elliptic case). For y < 1 and under natural assumptions on the data (see (24)), we prove that

P,Eul, ---\ u1 weakly* in Lm(O, T ;Hi(R)), upE eZul weakly* in Lm(O,T ;L2(R)),

(2)

-

where Pf is a suitable extension operator (see Lemma 3.1) and denotes the zero extension to the whole of R. Convergences (2) mean that ul, converges to the same limit ul of Pful,, up to a constant 92 representing the proportion of material occupying the inclusions. This function u l is the unique solution of the following homog-

109 enized problem, which depends on y

The matrix A; is a constant and positive definite and is different in the three cases y < -1, y = -1 and -1 < y < 1, see Theorem 2.1. For y < -1 and y = -1, we prove that h -

+

h -

A6Vu1, A6Vu2,

-

A;VU~ weakly* in LW(O,T ;[ L ~ ( R ) ] ~(4))

which gives the limit behavior of the union of the two components R1, and R2E ' For -1 < y < 1, convergence ( 4 ) can be improved, since we have

-

{ A'G

AEVulE A:Vul weakly* in LW(O,T ;[L2(R)ln), 0 weakly* in LW(0,T ;[L2(R)jn), -A

(5)

where the single contributions given by R I Eand 522, can be identified separately. In the case y = 1, see Theorem 2.2, we still have

--

P,Eul, u1 weakly* in LW(O,T ;H i (a)), u2, u2 weakly* in ~ ~ T ( ;L0 2 (, R ) ) ,

(6)

but now the couple (ul,u2) turns out to be the unique solution of the homogenized problem

+

Olui - div (A;Vul) ch(92ul - u2)= f l in Rx]O,T[, uh - ch(92ul - u2) = f 2 ul = O on dRx]O,T [ , 2~1(~,0)= ug ~ ,(

in Rx]O,T[, (7)

x , o ) = u ~ in R,

where A: is the same obtained for -1 < y < 1, O1 = 1 - 92 and Ch = & h ( ~ ) d>~ 0.y Convergences (5) remain valid. As proved in Section 3, this means that the limit function ul is the solution of the following first order evolution problem

&,

+

Olu; - div ( A ; v u ~ ) ~h62lll- c;& on dRx]O,T[ ul = 0 UI(O) =

g

K ( t , s)ul(s)ds = F in Rx]O,T[,

in R,

(8)

110 where K is an exponential memory kernel given by

and

Moreover, u2 is given by

Hence, for y = 1, the limit problem can be regarded as a problem with a linear memory effect, due to the presence of term cft62 K(t, s)ul(s) ds, K being a primary memory kernel.

Sot

We give the variational formulation of problem (1) and state the main homogenization theorems in Section 2. The outlines of the proofs of Theorems 2.1 and 2.2 are given in Section 3. For similar problems in the elliptic case we refer also to R. Lipton in,ll R. Lipton and B.Vernescu in,12 J.L. Auriault and H. Ene in,l t o Pernin in14 and E. Canon and J.N. Pernin in,2 to H. Ene and D. Polisevski in.g Some correctors results are given by P.Donato in.6 2.

Statement of the problem and main results

Let 0 be an open and bounded set in Rn, {E) be a sequence of positive real numbers that converges to zero and Y =]O, ll[x ...x]O, ln[ be the reference cell, with

y=yluE where Yl and Y2 are two nonempty open sets such that Yl is connected and Y2 has Lipschitz continuous boundary I?. For any k E Zn, let

where kl = (klll, .., knl,) and i = 1,2. For any fixed E, we denote by K, the set of the n-tuples such that is included in Cl, namely

&qk

111 and by

the two components of R and the interface, respectively. We assume that

therefore RIE is connected and R2, is union of e-a disjoint translated sets of &Y2and clearly 852 n rE= 0. In the sequel, we denote by X , the characteristic function of any open set w c Bn, m,(v) = v dx, and by " the zero extension to Bn of functions defined on RiE or Y,, for i = 1,2.

6

It is known (see for instance4) that xni,

-

Bi := I' weakly in ~ ~ ( 0 i) =~1,2.

(13) IYI Now, let us recall the definitions of the two normed spaces VE and H;

with the norm

and, for every y E R ,

H;

:= {v = (01, v2)

1 v1 E VE

and v2 E H ~ ( R ~ , ) } ,

(14)

IIvIIL; := IIV~lIIZ2(n,.~+ I l v ~ ~ l l Z ~ (+n~2' .l~l v ~- ~ 2 1 1 S z ( r ~ ) .

(15)

with the norm

Next proposition is contained in the proofs of Lemmas 2.7 and 2.8 of.13 Proposition 2.1. For every fixed E the norms of H; and VE x ~ ' ( 0 2 ~ )

are equivalent. In order to introduce the coefficient matrix, we take a , p E R with 0 < a < ,B and we denote by M ( a , p, Y) the set of the n x n Y-periodic matrix-valued functions in Loo(Y) such that

for any X E Rn and a.e. in Y.

112 We assume

and we set, for any E

> 0,

Moreover we consider an Y-periodic function h such that

and set

hE( x ) := h

(z)

The aim is to study, as the parameter y varies in R, the asymptotic behavior as E -+ 0 of the solutions u , = (ul,, u2,) of the problems:

where ni, is the unitary outward normal to RiE, i = 1,2. We suppose that

u,":= (u,",,u&)E V Ex

PI)

(R2E)

and

The variational formulation of problem (20) is: '

Find u , = ( u l E u2,) , in W Esuch that

+

(uiE1v l ) ( v c ) ,ve t ("'E1v1)(Hl(~2c))r,H1(~2e) + Jnl, A E V u l E. V v l d x JQzr A E V w E. Vv2dx+ +E? Jr. hE( u i E- ~ 2 . ) (vi - 212) d c = = Jal, f i a . vldx + Jan. f E . v2dxi for every (vl , v2) E V Ex H i ( 0 2 E ) , in D' (01T ) 1 . ulE( x ,O ) = u~O, in RlE, u 2 ~( x ,0 ) = UZOE in Q 2 a . +

+

where

W E= v = (

{

~ 1 ~ E~ L~ 2 (0, ) T

;V E )x L2 (0,T ;H'

such that v' E L2(0,T ;L2 ( 0 1 . ) )

X

(02~))

,

)

L2 (0,T ;L2 ( 0 ~ ~ ) )

113 equipped with the norm

Under the assumptions

-

lI~:llx;

c,

U0 := (u:,U;) weakly L2 (a)x L2 (a) with U i E Ht ( R ) if y < 1.

U,O

where C

(24)

> 0 is a constant independent of E and

-

(f?;,g )

( f i tf 2 )

weakly in L2 (0,T ;L2 ( 0 ) )xL2 (0,T ;L2 ( R ) ) (25)

the limit behavior of problem (20),for y as follows:

< 1, can be completely described

Theorem 2.1. (case y < 1). Let A" and h' be defined by (17) and (19) respectively and suppose that (Zl), (22), (24) and (25) hold and let u, be the solution of the problem (20) with y < 1 . Then, there exists an extension operator Pf E L(Lw(O,T ;H k ( f l l , ) ) Lw(O, ; T ;~ ~ ( f l ) for ) , k = 1,2, such that

Pfule --\ ~1 weakly * in Lw(O,T ;Hd(R)), ~ 2 , 0 2 ~weakly ~ * in Lw(O,T ;L 2 ( R ) ) ,

(26)

-\

-

P,EU;, U ; weakly * in Lw(O, T ;L 2 ( R ) ) , uZE 0 2 ~ ; weakly* in Loo(0,T ;L ~ ( R ) ) .

where 02 is given by (13). Moreover, if -1

{

CV

AeVule A€=

--

(27)

< y < 1,

A;Vul weakly * in Lw(O,T ; [ L ~ ( R ) ] ~ ) , (29) 0 weakly* in L ~ ( o , T[L2(R)jn). ;

The function ul is the unique solution in L2(0,T;H i ( R ) ) , with u$ in L2(0,T ;H: ( R ) ) ,of the problem

+

in Rx]O,T[, u$ - div (A:Vul) = f~ f 2 ul = 0 on dRx]O,T[, in R. ul(0) = U: U;

+

(30)

114 For y < -1 the homogenized matrix A; is defined by

'A;X = m y ( t A V W A ) with W x E H 1 ( Y ) solution, for any X

E

Rn, of

-div ( t A V W x )= 0 i n Y, W x - X . y Y - periodic & . f y (Wx - A - y)dy =o. For y = -1 the matrix A: is defined by

with (w:, w:)

E

H 1 ( Y l )x H1(Y2)solution, for any X E Rn, of

-div ( t A I V w i ) = 0 in Yl, -div ( t A 2 ~ ~= : )0 in Y2, t A 1 ~ w .: nl =t A2vw: . n2 on I?, -tAIVw: - nl = h ( w i - w:) , w: - X . y Y - periodic, , my* (w: - Y ) = 0, ni being the unit outward normal to Yi, i = 1,2. For -1 < y < 1 the homogenized matrix A: is defined by '

with wx E H 1 ( Y l ) is the solution, for any X

E Rn,

of

-div ( t A V w x )= 0 i n Y, (tAVwx). nl = 0 on r, Y - periodic, WA-X y j&, .fY ( W A - Y ) dy = 0-

(36)

a

Theorem 2.2. (case y = 1). Let AE and hE be defined by (17) and (19) respectively and suppose that (21), (221, (24) and (25) hold and let uE be the solution of the problem (20) with y = 1 . Then, there exists an extension operator Pf E C(Lm(O,T ;~ ~ ( 0Lm~(0,~ T ;H ) ~)( R; ) )for , k = 1,2, such that u1 weakly* in Lm(O, T ;H; ( R ) ) , Pful, & u2 weakly* in LW(0,T ;L 2 ( R ) ) ,

-

{

P ua,

A

u weakly * i n Loo(O,T ;L 2 ( R ) ) , o2u; weakly* in Lw(O, T ;L 2 ( R ) ) .

115

{

-

AEVulE A;Vul weakly* in Lw(O, T; [L2(R)ln), ;A ' 0 weaklv* in Lw(O, [L2(R)In).

where the couple (u1,u2) is the unique solution L2(0,T;Hi(R)) x L2(0,T; L2(R)), with (21'1,u;) in L2(0,T; H: (R)) x L2(0,T ; H? (a)), of the problem

+

ch(82ul - u2) = f l 81ui - div (A:Vul) uh - ~ ~ ( -8u2)~= 2f2 ~ ~in Rx]O, T[, ul = O on dRx]O, T[, u0

ul(x,O)=+

~2(~,0)=U20

in Rx]O, T[, (37)

in R,

where Oil for i = 1,2 are given by (IS'), ch = J, h(y)du, homogenized matrix A{ is defined by (35) and (36).

> 0 and the

Remark 2.1. The extension operator Pi in Theorems 2.1 and 2.2 is the one introduced by Cioranescu and P. Donato in.3 3. Outline of the proof of the main results

Here, we give only an idea of the proof of the main results that can be obtained arguing similarly to the hyperbolic case studied in,8 with opportune modifications. First, we recall suitable extension operators. Then, we apply an abstract theorem to get the existence and uniqueness of the solution of problem (3) and we give uniform a priori estimates. Successively, we define two classes of test functions needed when applying Tartar's method. Finally, we outline the proof of the identification of limit problems (30) and (37). 3.1. Extension opemtor

Let us introduce some extension results proved by D. Cioranescu and J. Saint Jean Paulin in,5 S. Monsurrb in13 and D. Cioranescu and P. Donato in,3 needed in the sequel:

Lemma 3.1. i) There exists a linear continuous extension operator Q1 belonging to L(H1(Yl); H1(y)) n L(L2(Yl);L2(Y)), such that, for some positive constant C

Il

IlQi~iLZ(Y)

I CIIV~IIL~(Y~)

116 and

for every vl E H1 (Yl) . ii) There exists a linear and continuous extension operator Q2 E L(H1(Y2); H;,,(Y)), such that, for some positive constant C

for every v2 E H1(Y2). iii) There exists an extension operator Qi belonging to L(L2(R1,); L2(R))f l L(VE;Hi(R)) such that, for some positive constant C (independent of E )

and

for every vl E VE. iv) There exists a linear continuous extension operator Pf belonging to C(Lm(O, T ;H ~ ( R ~ , ) Lm ) ; (0, T ;H ~ ( R ) ) ,k = 1,2, such that, for some positive constant C (independent of E )

for any cp E Loo(O,T ;Hk(RiE)).

3.2. A priori estimates

Let us state the result on the existence and the uniqueness of the solution of problem (20) that can be proved by means of an abstract result based on Galerkin's method (see for instance1'). Theorem 3.1. Let T €10, +m[, y E R, H$ be defined by (14) and (15) and

A" and hE by (17) and (19). Under assumptions (21) and (22) problem (23)

117 admits a unique solution. Moreover, there exists a constant Co, independent of E , CO= C O ( a , P , T )where a and p are given i n (16), such that

Fkom this last theorem, due to the boundedness of the H;-norm of the initial condition U: (assumption (24)) and (25), we deduce the following uniform a priori estimates from which we obtain, up to a subsequence, the convergence of the extensions of the solutions given below. Proposition 3.1. Let A" and hE be defined by (17) and (19) respectively and suppose that (21), (22), (24) and (25) hold. Let u, be the solution of (23) with y 5 1. Then there exists a constant C, independent of e, such that

-

Moreover there exists a subsequence, still denoted by E, such that

HA

Pful, ul ul, -, u l ul, -+ O1ul u;, -+ u2

weakly* i n Lw(O, T; (R) ) weakly* in CO(O,T; L~ (R) ) weakly* i n Lw(O, T; L2 (R) ) weakly * i n Lw(O, T; L2 (R) )

~ f u ; ,A u;

weakly* in Lw(O, T; L2 (R) )

u;,

weakly* i n Loo(O,T; L2 (R) )

w

-+

Olu;

N

u2"

-' u2

weakly* i n Lw(O, T ;L2 (R) )

3.3. Oscillating test functions

To identify the limit problems, by Tartar's method (see15), we recall the definitions of two classes of suitable test functions. The first one, used in the classical homogenization of the stationary heat equation in a perforated domain with a Neumann condition on the boundary of the holes (see5), is the following:

118 where Q1 is given in Lemma 3.1 and x of problem (36). It is known (see5), that

--

w;l-X.x w,fx(x) w

tAVwi

l =~ A-y-wx(y), with wx(y) solution

X .x

weakly in H1 ( 0 ) strongly in L2 ( 0 )

tA;X

weakly in [ L (R)] ~

,

with A; given by (35). The second class of test functions, introduced in17is defined by means of the solution (xiA,x&) of the following problem '

,

-div (tAV~:A)= -div (tAX) in Yl, -div ( t A V ~ 4 A=) -div (tAX) in Y2, - A. y) . n l =t AV (XiA- X y) . n2 on r, tAV tAV (xfx - A. y) - n l = - ~ ? + l h(xfA- x i A )on I?, xiA Y - periodic, m y , (xfx) = 0.

(43)

Clearly, these functions depend on E , since the flux through r is proportional t o the jump of the solutions by a coefficient of order E?+'. We point out that the asymptotic behavior, as E -+ 0, of these functions depends on the value of y. This explains why in Theorems 2.1 and 2.2 the expression of A; changes according to y. We set

where the extension operators Qi, for i = 1,2, are defined in Lemma 3.1 and xrA,i=1,2, are solutions of (43) and

P. Donato and S. Monsurrb in7 and13 proved that the functions defined in (44) and (45) satisfy the following convergences: Proposition 3.2. Let wFA and q i be defined by (4.2) and (4.3). Then, w,fX(x) w&(x) for i = 1,2. Moreover, if y 5 -1

+

--

A. x X x

-

t

~

weakly in H1 (R) strongly in L2 (R)

(46)

weakly ; ~ in [L2(R)In,

(47)

119 with A: defined by (31) and (33) for the cases y tively.

ex

-\

tA;X

E-2kL-E 2 r]2X

-\

0

< -1 and y = -1 respec-

weakly i n [L2( f l ) I n weakly i n [L2(O)]

, ,

with A: defined by (35). 3.4. Limit problems.

Let u, be, for every value of y, the solution of problem (20), and set := ((1E)J 2 E ) = ( A E V ~ l eA E,v u 2 E ) .

(49)

By Proposition 3.1 we get, up to a subsequence,

2

--\

Ji

weakly* in Lm(O, T ; [ ~ ~ ( f l ) ] " for ) , i = 1,2.

(50)

Our aim is to identify J1 , J2 as well as the limits ul and u2 defined in (39). We emphasize that these limit functions change according to the different values of y. The first step is to show that J 1 J2 satisfies the following equation

81u:

+ u', - div ( J 1 +

J2) = fl

for every y 5 1. After, we prove that the sum I1

J1

+ + f2

in D1(O,T ) ,V v

E D(R),

(51)

+ J2 is such that

+G = A;vul,

(52)

for every y. Therefore, by (50), we obtain convergence (28) for a subsequence. Moreover, in view of (51),we have that

BIU',

+ uh - div ( A ; V u l ) = fl + f2

in D1(O,T ) , b' v E D(S2).

(53)

Let us point out that identity (52) holds for every y,but in the case -1 < y 5 1, it specifies as follows:

ti = A;VUI and

that is convergences (29),up to a subsequence.

(54)

120 Next step is to describe the limit function u2 in terms of u1. Here, we need to treat separately the two cases y < 1 and y = 1. For y < 1, we prove that

which gives (for a subsequence) convergence (26). This, together with (53), provides the limit equation in (30). For y = 1 we prove that u l and u 2 verify the 0.d.e.

This, together with (53) gives the coupled equations in (37). Finally, always in the spirit of,7 it can be proved that ul and u2 satisfy the initial conditions stated in (30) and (37), respectively.

Remark 3.1. Let us observe that Problem (30), for y < 1, has an unique solution, this show that convergences in Theorem 2.1 hold for the whole sequences. In the case y = 1, we explicitely also observe that u2 can be computed in terms of ul, namely we have

where

K (t, s) = e c h ( S - t ) . Replacing this expression in the first equation of (37) we deduce that solution of the problem (8), where

F (x, t) = f l (x, t)

+

(57) u1

is

+ ch J d t ~ ( tS), f2 (x, s ) ~ s .

t

Since K(ul) = So K(t, s)che2ul is a linear and continuous operator from

$

) ) itself and the initial data belong to H,'(0), problem L2(0,T; ~ ~ ( 0into (8) admits an unique solution. Then we have that also convergences in Theorem (2.2) hold true for the whole sequences.

Acknowledgments. This work is part of the European project "New materials, Adaptive systems and their Nonlinearities. Modelling, Control and Numerical Simulation", contract number HPRN-CT-2002-00284SMART-SYSTEMS. It was partially realized during the permanence of the authors at the Laboratoire J.L. Lions of the University Paris VI.

121 References 1. J.L. Auriault and H . Ene , Macroscopic modelling of heat transfer i n composites with interfacial thermal barrier (Internat. J . Heat and Mass Tranfer 37, 2885-2892, 1994). 2. E. Canon and J.N. Pernin, Homogenization of diffusion i n composite media with interfacial barrier ( R e v . Roumaine Math. Pures Appl. 44, no. 1, 23-36, 1999). 3. D. Cioranescu and P. Donato, 1989, Exact internal controllability i n perforated domains, J . Math. pures et appl., 68, 185-213. 4. D. Cioranescu and P. Donato, 1999, A n Introduction to Homogenization, Oxford Lecture Series i n Mathematics and Its Applications, 17. 5. D. Cioranescu and J . Saint Jean Paulin, 1979, Homogenization i n open sets with holes, J. o f Math. An. and Appl., 71, 590-607. 6. P. Donato, in Some corrector results for composites with imperfect interface (Rendiconti di Matematica, Serie VII, (26) Roma, 189-209, 2006). 7 . P.Donato, S.Monsurx-b, Homogenization of two heat conductors with interfacial contact resistance (Analysis and Applications, vol 2, n.3, 247-273, 2004). 8. P.Donato, L.Faella, S.Monsurrb, Homogenization of the wave equation i n composites with imperfect interface (preprint No.8 DMI, University o f Salerno, 2006, submitted). 9. H . Ene and D. Polisevski, Model of diffusion i n partially fissured media ( Z A M P 53, 1052-1059, 2002). 10. H.C. Hummel, Homogenization for Heat Dansfer i n Polycristals with Interfacial Resistances (Appl. An. 75 (3-4), 403-424, 2000) 11. R . Lipton, Heat conduction i n fine scale mixtures with interfacial contact resistance, (Siam J. Appl. Math., vol. 58, n. 1, 55-72, 1998). 12. R . Lipton and B. Vernescu, Composite with imperfect interface (Proc. Soc. Lond. A 452, 329-358, 1996). 13. S. Monsurrb, Homogenization of a two-component composite with interfacial thermal barrier ( A d v . in Math. Sci. and Appl., Vol 13, No. 1, pp. 43-63, 2003). 14. J.N. Pernin, Homog6n6isation d'un probltme de diffusion e n milieu composite B deux composantes (C.R. Acad. Sci. Paris 321, s6rie I , 949-952, 1995). 15. L. Tartar, 1978, Quelques remarques sur 11homog6n126sation,Functional Analysis and Numerical Analysis, Proc. Japan-Rance Seminar 1976 (Fujita ed.), Japanese Society for t h e Promotion o f Science, 468-482. 16. L. Tartar, 1990, Memory effects an homogenization, Arch. Rational Mech. Anal., 121-133. 17. E. Zeidler, 1990, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear monotone operators, Springer-Verlag, New York [etc.].

122

NUMERICAL EXPERIMENTS ON THE CONTROLLABILITY OF THE GINZBURG-LANDAU EQUATION R. GARZON and V. VALENTE Istituto per le Applicaioni del Calcolo "M.Picone", CNR Viale del Policlinico 137, 00161 Rome, Italy E-mail: garzonOiac.rm.cnr.it, [email protected] The paper deals with the numerical approach to the open problem of the asymptotic controllability of the Ginzburg-Landau equation. Controllability t o steady state solutions is considered. The conjugate gradient method for the minimization of the functional associated to the linear problem, the fixed point algorithm and finite difference schemes for the numerical approximation of the equations are implemented. Several numerical experiments both in the scalar and vector case are carried out also to investigate on the controllability of some blow-up phenomena. Keywords: Controllability; nonlinear parabolic problems; numerical simulation.

1. Introduction a n d physical motivation

Let R be a bounded open set of R2and T a positive number, we consider the Ginzburg-Landau evolution equation in the unknown m : 52 x R+ -+ Rd (1 5 d 5 3) depending on a small positive parameter E

with associated boundary and initial conditions m = m 0 on aR x (0, T ) ,

m(., 0) = m0 in R.

(2)

We address the controllability problem for the Ginzburg-Landau equation. Theoretical aspects of controllability for nonlinear parabolic problems including those ones connected to the Ginzburg-Landau equation, have been studied by several authors; the open problem remains the asymptotic controllability as E -+ 0. The aim of this paper is the numerical investigation of this problem.

123 The equation (1) is connected to the gradient flow for the functional

This implies that the stable steady states of (I), (2) are the minimizers of (3). For theoretical results on the Ginzburg-Landau equation and the characterization of the limit as E --, 0 we refer to [4], [15]. The interest for this problem arises from the applications in materials science. Indeed equation (1) is involved in the analysis of several phenomena: the phase transitions (d=l), the dynamics of vortices in superconductors (d=2), the evolution of singularities for certain nematic liquid crystals (d=3). In particular for d=3 the Ginzburg-Landau equation (1) and its asymptotic behavior (as e --, 0) is involved in the existence of the heat flow equation of harmonic maps

It is known that for the equation (4) regular initial data on the sphere (that is lmOl= 1) may develop singularities in a finite time to in the sense that

Indeed the result of partial regularity due t o Struwe [15] asserts that for a given datum mOE H1(R, S2), there is a global weak solution m to the system (4), (2) on the sphere, satisfying the energy inequality

which is Cw(R x [0,T[) away from finitely many points (xk,tk), 0 4 k 4 i, and it is unique in the class of nonincreasing energy. An initial datum which develops singularity in finite time, has been constructed in [5] when R is a disk of Kt2. Can we control such evolution phenomena by means of an internal action? Actually the control problem is very difficult, a first step is to look, as it is customary in the study of problems concerning harmonic maps, at the problem (I), (2) obtained by replacing the constraint Iml = 1 by a term depending on a small positive parameter E . Then, in order to satisfy the constraint Iml = 1, one looks at the asymptotic solution as e --, 0. In the next sections we formulate the control problem. Then we describe the numerical approach to the problem in particular for the scalar case d = 1. In the last section results of the numerical simulation both in scalar and vector case are reported.

124 2. Formulation of t h e controllability problem

Let w be a nonempty subset of R and X, its characteristic function, we assume an initial datum m0 E H1(R; Rd) and consider the equation

where Q is the cylinder R x (0, T ) and f E L2(Q;Rd) is the unknown control function to be determined in order to drive the solution to a suitable state at time T. Proposition 2.1 For every f E L2(Q;IRd) and m 0 E H1(R; Itd) there exists a unique solution m E H 2 t 1 ( ~IRd) ; to ( 5 ) , (2) where

Proof. Indeed, putting x,f E f W ,let f,W, m: be smooth approximations of the functions f Wand m0 respectively, depending on a positive parameter T such that as T -, 0 f,W-+fW i n ~ ~ ( Q ; l R ~ )m:-+mo , in~l(R;R~). From Theorem 7.1 in [12, Chap. VII], for each E > 0 one has that there exists a unique classical solution, denoted by m,, to (5), (2) corresponding to the data f,W, m:; moreover the following inequality holds

and hence, passing to a subsequence, for compactness we get

where m weakly solves (5). Further regularity for the limit function m can be deduced directly from the equation (5) which gives A m E L2(Q;Itd) and hence one has m E H2y1(Q;Itd) from well known estimates (see for example [ll,Chap. I]). The uniqueness follows from the maximum principle for the equation of the modulus of the difference between two supposed solutions. Indeed supposed that m1 and m 2 are two solutions of ( 5 ) , (2) and introduced y = m1 - m 2 one has

and hence 1 1; - A1 1 -2 ~ - 1l l2 5 0. The desired uniqueness result follows from the above inequality applying the maximum principle to the inequality for the function IY12e-2tl".

125 Now we introduce the function m* which is the solution of the stationary problem

Am* -

lm*I2- 1 m*=O i n R E

m* = m0

on dR.

We can consider the following problems of the controllability to stationary solutions.

Exact Controllability Problem: given m0 in H1(R;IRd)and T > 0 find the control function f in L2(Q;IRd) such that the solution to (5), (2) satisfies m(T) = m*. The system ( 5 ) , (2) is said to be locally controllable at time T to the stationary solution of (7) if there exists a positive constant y such that for any initial data m0 satisfying llmO- m*IIH;cn;wdl < 7,there exists a control function f and such that the solution m of (5), (2) satisfies m(T) = m* a.e. in R. Approximate Controllability Problem : given m0 in H1(R;IRd), T > 0 and fixed (T > 0, find the control function f E L2(Q;IRd) such that the solution to the problem (5), (2) satisfies Ilm(T) - m*11L2(n;Rd) 5 (T. The problem of controllability to steady states can be reduced to a null controllability problem. Indeed introducing the function y = m - m* one has

where

and the associated boundary and initial conditions y =0

on dR x (0,T),

= m0 - m*

y(., 0) =

in R.

(10)

We remark that the vectorial function G(m*,y) satisfies the condition G(m*,y) . y L -E-' 1 1.' Indeed one can easily check that G(m*,y) . y =

+

+

-

+

m*I2 (lyI4 31yI2(y m*) 31y. m*I2)- ly12] and [I~1~1m*1~ (IyI4+ 31yI2(y.m*) 3 1 ~m*I2)2 (lyI2- d l y l l y rn*I)'.

+

The controllability of the parabolic equations in the linear and nonlinear case is widely studied in the last years from a theoretical point of view, we quote among others the pioneer papers by J.L. Lions [13],[14]and in the

126 framework of the qualitative controllability of nonlinear parabolic equation we quote [2], [6], [7]. On the local controllability problem for GinzburgLandau equation, we refer to the results established in [2]and, in particular, in [3] where the local controllability to the stationary solutions of the phasefield model is proved by two control forces localized on the same subdomain; see also [l]where the local controllability to the trajectories of phase-field models is obtained by one control force acting on a single equation of the system. As a common praxis the nonlinear controllability problem is reduced to a sequence of linear controllability problems. The desired result for the nonlinear system follows then from a fixed point argument. So we introduce, for 1 = 0, I.., the systems

where a = aij, with i, j = 1,..., d , defined by aij(m*, z) = E-'(/z

+ m*I2 - l)bij + E-'(z~ + 2m;)mf,

i , j = I, ..., d (12)

with boundary and initial conditions yLf' = 0

on aS1 x (0, T),

yl+'(., 0) = yo

in 0,

(13)

and for each fixed 1 we solve the null controllability of the linear system (111, (1217 (13). For a suitable choice of the stationary solution m* (for example in the case d = 3 one may consider the state m* = (0,0,1)) the linearized system reduces to a system in cascade with d uncoupled equations which can be independently studied (see also [9], [lo]).For that reason in the next session we describe the control numerical procedure for a linear scalar equation, the extension to a cascade system can be easily done. 3. The scalar controllability problem

We recall some results and methodologies related to the resolution of the scalar controllability problem. Let y be the solution of the scalar linear problem

+

Q

Y(.,0) = YO

on C in S1

yt - AY a(m*,W)Y= f X , in

{y=o and a(m*,w) = &-'(1wI2

+ 3)m*I2+ 3wm* - 1) E LM(Q).

127 The proof of the null controllability for the linear problem is standard. Let p be the solution of the backward problem

consider the functional J(pO), firstly introduced by J.L. Lions (see for example [14]) for the approximate controllability, defined by

and let pU0be the unique minimizer of J(pO) over all the functions p0 E L2(R). Then the control function f = f, = p,~,, where p, is the solution to (15) with given final datum pU0,drives the solution of the problem (14) to the final state y,(T) and Ily,(T)II Lz(n) a. Moreover as a -+ 0 there exists a subsequence ak vanishing as k -+ CQ such that

<

f,,

+f

y,, (T)

-+

weakly in L2(w x (0,T))

(17)

0

(18)

weakly in L2( a ) .

The main ingredient for the convergence is the observability condition

where C = C(R,w,T, IlallLmcQ))is the constant of observability defined also by means of the Carleman estimates (see [8]) and

Since the minimizer p i verifies the equation

applying (19) we find

The above estimate allows to pass to the limit as a goes to zero and get to (171, (18) The functional (16) can be written as

128 where z is the solution of the homogeneous direct problem (14) (that is the solution of (14) with vanishing control). The first variation of (21) defines the gradient G(pO)of the functional J(PO)

where in the above equation we have denoted by problem

4S the

solution to the

+t--Aq5+a(rn*,w)$=O i n R x (s,T) on 69 x (s, T)

(4-0

in 0

$(.I s) = PU(S)XW and the second variation gives 0

0

0

('WP )v ,q ) ~ z ( n= )

where v, q are the solutions of the backward problem with initial datum v0 and q0 respectively. The functional is coercive (from the observability estimate (19)) and convex since ( ' ~ l ( ~ ~ ) v ~ , 0, hence there exists a unique minimizer p: which verifies

>

for each q0 E L2(R).

THEMINIMIZATION METHOD: The crucial point of the linear controllability problem is the minimization of the functional (21). We propose for the numerical investigation of the problem the Conjugate Gradient Method (CGM): we fix p': E L2(R) and consider the sequence

where the descent direction is given by

129 and the step length by the following expression,

The rate of convergence of the (CGM) depends in particular on the the subset w and requires many iterations when the volume of w decreases. 4. The finite difference approximation

In this section we look at the computational aspects of the problem. The numerical algorithm we propose, requires the implementation of the conjugate gradient method (internal iterative method) for the minimization of the functional (21) and a fixed point iterative procedure (external iterative method) for the adjustment of the nonlinear term. We propose the discretization on space and time simultaneously of the equations that we are treating. Let assume R be a rectangle and consider an uniform partition of R in rectangles R i j with edges hl > 0 and h2 > 0, one has x i = i h 1 , i ~ Z = { 0 , 1 ..., , I),

y j = j h z , jE.7={0,1,..., J )

h;)ll2/&. Moreover an uniform mesh for the time and assume h = variable with step size 6 > 0 gives tn=n.6,

with n ~ N = { 0 , 1 ..., , N)

and T = N d

and we define +Cj = $(xi, yj, t,); when not specified the indices i, j, n vary in the sets Z, .7 and N respectively. We introduce the discrete operators Aha and Vhs, s = 1,2,

and implement the following iterative procedures. External iterations. Let us denote by 1 (1 = 0, I...) the index which accounts for the external iteration process due to the nonlinearity of the problem, so given yt; (ytf arbitrary chosen) we may introduce

130

.zZz

and the approximation of the solution of the direct homogeneous problem at the level set 1 given by

Internal iterations. Fixed $:,j = consider the discrete functional

$zj ( k ) ,with $&(o) arbitrary given, we

+ c:~:

'j$,' ':2 - $Ejhih2 where $yj is computed according to the following explicit finite difference scheme

For applying the numerical conjugate gradient method we need the following gradient approximation

with ($&,,)

computed, for any T = 0 , ...,N - 1, according to the scheme

Finally we define

+tj( k ) = $ t j ( k

- 1) - p ; v t j ( k - 1)

where v k j ( k ) and p; are the approximations of the direction dk and the parameter pk respectively, computed according to ( 2 4 ) , (25) and k is the iteration index of the minimization method with k = 1,...K and K associated to the stopping criterium

131 The convergence of the numerical approach derives from the stability and consistence of the proposed finite difference scheme for the repeated numerical resolution of linear parabolic problems. We recall that the classical stability condition for the heat flow equation requires 6-I ~ f 2hy2. = Actually in our situation that condition can be adopted until e is large enough with respect to the space step h. For small values of e a more restrictive condition on the steps of the discretization, which accounts also for E , has to be imposed according to the Lemma below which preserves the exponential behavior. Finally we denote by = (K) the minimizer of the functional J~ and with qhits minimum. Let be the solution of the discrete adjoint 01 system with final data &;;, we take = (xu)iYj and solve the direct system

>

+: +zj

+ti

f$

-

The fixed point iterative algorithm continues until the index L satisfies the requirement

where € 2 is a small positive fixed number. We want to point out that the numerical computation of the gradient at each level of internal iterative approach, requires the numerical resolution of N parabolic problems. As a consequence small discretization parameters lead to large times of computation for the whole controllability problem.

Gtj,

Lemma 3.1 Let e > 0 and q a bounded function in R x [0,TI.Then, for each hl, h2 and 6 positive numbers which verify the condition

the numerical solution computed by

~

132 verifies

where IIV h un 11 2l z ( n ) = x : = l 1lvhsunll:2(n) of E , 6 and h,.

and C is a constant independent

Proof. For sake of simplicity we consider the mono-dimensional case i.e.

R c Bs,with s = 1, so the above scheme reduces to

Firstly we multiply equation (30) by ur, summing in n and i we obtain

Since the last term is negative we get

having put

= {sup,,i Iq?l); hence

' ur), sum in n and i and set Now we multiply equation (30) by ( u ~ +-

133 Now, rearranging the terms in A2 and taking into account the Dirichlet homogeneous boundary conditions, we have

that is,

Moreover for the term A3 we get the following inequality N-1 I-I

We are now in a position to prove our thesis. If 6 satisfies (28) we have

and hence from the relationship Al

< A2 + IA31, we have

and the above inequality with (31) leads to (29). The extension of the proof to the two-dimensional case requires only technical difficulties.

134 Remark 3.1 The exponential behavior of IIuNII$.(,) follows from the discrete analogous of the Gronwall inequality. Indeed from (29) 2 2 2 IIu NIIlz(n) + 6lIVhuN llp(q 5 ll~Oll;z(n) + JIIVh 0 lllz(a) (32) +C&-lC~L;6 ( ( I I u ~ I I % + (~~I)I V ~ U ~ I I : ~ ~ ~ ) ) passing through the d n e decomposition of IIUII:~(~, + 6 l l ~ ~ u l l : ~ ( ~ ,

and the Gronwall Lemma applied to (32) leads t o

where (? -+ C as 6 -+ 0; and hence since 6

< h2 we get

5. Numerical simulation

In this section we will discuss the numerical results obtained from the application of the algorithm described above. The numerical experiments are related to the case d = 1 and d = 3. In both cases we consider the controllability to stable steady solutions, specifically, m* = 1 (for d = 1) and m* = (0,0,1) (for d = 3). In all the tests we have taken, fl = (0,l) x (0,l) and hl = h2 = h. The discretization parameters h and 6 will be specified in every test and chosen according to the stability condition. Some tests concerning the local and approximate numerical controllability are carried out. 5.1. The case d=l

Here we consider the numerical local controllability applied to the scalar Ginzburg-Landau equation (d = 1) with initial datum = sin(xr) sin(yr) .0.1

(33)

All the plots show the behaviors in time of the L2(fl)-norm of the computed control function f and the L 2(R)-norm of the computed corresponding state y. Two examples are considered concerning the influence of the dimension of the subset w and the effect of the parameter E. In both examples we assume m* = 1 in fl, the final observation time T = 0.05, the space Moreover we asstep considered is h = 0.05 and the parameter a = €2 = sume the following stopping criteria (see (26), (27)) €1 =

135 In Figure 1 we consider the case when the control is applied in a strip of R. Three experiments are reported corresponding to the cases w = wi, i = 0,1,2 with wo=R; w l = ( 0 . 1 5 , 0 . 8 5 ) ~ ( 0 , 1 ) ; w2=(0.25,0.75)~(0,1). The plots are related to the last iteration of the external iterative method. In the upper graphic of the Figure 1it is shown the plots in time of 11 f 11 L z ( n ) and in the lower graphic it is shown the plots in time of 11 yJIL~ ( 0 ) . More precisely, we have considered 6 = 0.00025 for the case w = wo (line a) and b = 0.0001 for the cases w = w l (line b) and w = w2 (line c). Notice that the computed control shapes loose their monotonic behavior when the volume of w decreases and in this case a finer time step 6 is required in order to stabilize the computed control solution. I

,

.

.

Fig. 1.

.

E

.

= 0.5

,

Fig. 2.

E

= 0.1

Fig. 3.

E

= 0.05

In the Figures 2-3 are again shown in the upper graphic the plots in time of 11 f I ( L 2(n) and in the lower graphic it is shown the plots in time of 11 yll z ( n ) for different values of E . We have considered again 6 = 0.00025 for the case w = wo (line a) and 6 = 0.0001 for w = w2 (line b). The L2(R)-norm of the control decreases when a decreases, moreover the control is concentrated near the final time T for small values of the parameter a both in the case w = wo (line a) and in the case w = w p (line b). In all tests the state goes to zero, more quickly as a is small enough. The behavior in a of the computed control problem solution is strongly connected to the choice of m* E 1 which is a solution of the static problem for each E > 0. The numerical

136 results show the expected effect of the parameter a which forces, as a -, 0, the L2(R)-norm of the state y to vanish for t > 0. Hence, less control is needed for small values of a since the action control has the same objective of driving y to 0 (that is, m to m*). 5.2. The case d=3

Here we look at the numerical controllability of the system (d = 3) where we have assumed m* = (0,0,1). Note that with this choice of m*, the linear system (11) turns out a cascade system, so we can solve three scalar problems a t each iteration of the nonlinear procedure. All the numerical tests reported below are obtained controlling the system in w = 0. Two examples related to different choices of the initial datum are carried out. More precisely in the first example below we show the results of numerical local exact controllability starting from an initial datum m0 on the sphere such that (1 m0 - m* 11 (R;W3) is small enough. In the last example we assume an initial datum which develops singularities in finite time as E -+ 0 (see the theoretical result in [5]).The aim is to show numerical results of exact and approximate controllability when blow-up phenomena for the limit problem occur. We fix in both examples below the following stopping criteria for the gradient L2(R)-norm of the minimization algorithm and for the Lm(R)-norm of the difference between two successive states of the fixed point algorithm respectively: €1 = and €2 =

EXAMPLE 1. We consider the following initial datum components

= m0 - m* of

In Figure 4 we report the plots of the L2(R)-norms of the control f = (fl, f2, f3), the associated state y = (yl, y2, y3) and the function Iml concerning the numerical experiments obtained when all the components and T = 0.05. In of the control act on the system for values a = 2 . this example we fix h = 0.05 and 6 = 0.00025. Notice that differently from the scalar case the norm of the control does not decrease when E -+ 0. This behavior is confirmed in Figure 5 where the are compared. The and a = 2 . plots of the controls for E = 2 . plots of Figure 5 are obtained adopting a finer mesh step h = 0.0125 and 6 = 0.00001. Much more control is required in driving the state to zero in the short time T = 0.0005. As consequence of the control action the solution is moved away the sphere (see the plots of Iml in Figures 4-5) before to get

137

Fig. 4. Profiles versus time of IlfllL2(n) (on the left), IlyllL~cn)(in the center) and IlmllLz(n)(on the right) for w = 0,E = 2 . (continuous line), compared with the uncontrolled solution (dot line).

(continuous line) Fig. 5. On the left profiles versus time of IlfllLzcn) for E = 2 . and E = 2 . (broken line). Profiles of IlyllLzcn)(inthe center) and IlmllLzcn)(onthe right) for E = 2 . (continuous line) compared with the uncontrolled state (dot line).

the given final state on the sphere. In all the tests of this example we have fixed the parameter a = 0.

EXAMPLE 2. The experiments shown in this example are related to the controllability of solutions with initial datum, introduced in [5],which develops singularities for the limit problem ( E + 0)

+

where r = J(s - i ) 2 (y - i ) 2 and f (r) = 87rr(r - 1 ) for r 5 112 and f ( r ) = 0 otherwise. In order to reproduce numerically the singularities a small parameter E and a small space step h are required. For that we fix in this example h = 0.0125 and as in the previous examples 6 = 0.00001. The numerical experiments show that we can control the solution before 11 Vmll oo reaches its maximum value. In this example we have also asked for an approximate controllability property, since less control is needed to drive the solution to an approximate desired state and then, since we have a parabolic system, we can take advantage of the dissipation property to finally drive the solution to

138 m*. In Figure 6-7 we report the plots of the numerical experiments for the

Fig. 6. Profiles versus time of IlfllL2(n) (on the left), IIVmllLm(n) (in the center) and I1yllL2(,) (on the right) for w = R, E = 2 . and u = 0.

exact and approximate controllability respectively. All the components of the control act on the system through w = R for a time T=0.0005, whereas the experiment is considered for a total time To=0.035. More precisely, in each figure we report on the left the L2(R)-norm of the computed control f , on the right the comparison of the L2(R)-norm of the associated state y = m -m* with the uncontrolled state and in the center the LM(a)-norms of the V m when the control acts and with vanishing control. We can observe that more control is needed to satisfy the local exact controllability property as we could assume. It is evident the effect of the control on the

3 2 YD

1

0'

t

2

S

S

4 X

to-.

%

control

control 0.5

1

1.5

2

25

3

8.3 x 10-

0

05

3

1.5

2

IS

3

3.5 .to-'

Fig. 7. Profiles versus time of Ilf llL2(n) (on the left), IIVmllLm(n)(in the center) and IlyllLn(s2)(on the right) for w = R, E = 2 . and u = 0.1

function IIVmllLmcnl, its peak is pulled down both in the case of the local exact (Figure 6) and the approximate (Figure 7) controllability. In the last case the process can be continued without control assuming as initial datum the controlled final state at T = 0.0005 as it is shown in Figure 7.

139 Acknowledgement The work is supported by the European Community's Human Potential Programme under contract HPRN-CT-2002-00284 (SMART-SYSTEMS).

References 1. F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force. S I A M J.Contro1 Optim. 42 (2003) 1661-1680. 2. V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim., 42 (2000) 73-89. 3. V. Barbu, Local controllability of the phase field system. Nonlinear Analysis, 50 (2002) 363-372. 4. F. Bethuel, H. Brezis and F. HBlein, Ginzburg-Landau Vortices. Progress i n Nonlinear Differential Equations and Their Applications 13, Birkhauser, 1994. 5. K.C. Chang, W.Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geometry, 36, 507-515 (1992). 6. E. Fernhndez-Cara , Null controllability of the semilinear heat equations. ESAIM Control Optim. Calc. Var., 2 (1997) 87-103. 7. E. Fernhdez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincare' Anal. Non Line'aire, 17 (2000) 583-616. 8. A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes. Ser. 34, Seoul National University, Korea, 1996. 9. R. Garzdn and V. Valente, Numerical aspects of controllability for the Ginzburg-Landau equations associated to the dynamics of smart systems. Proceedings of I1 ECCOMAS Thematic Conference on Smart Structures and Materials, Lisboa 18-21 July 2005. 10. R. Garz6n and V. Valente, On the numerical controllability of the GinzburgLandau equation. Proc. VIII SIMAI Congress, Ragusa 22-26 May, 2006. 11. O.A. Ladyzenskaja, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, 1968. 12. O.A. LadyZenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic ripe, Trans. Math. Monographs 23, American Mathematical Society, 1968. 13. J.L. Lions, ContrGlabilitB Exacte, Perturbations et Stabilisation de Systemes DistribuBs, tome 1-2, Masson, Paris, 1988. 14. J.L. Lions. Remarks on approximate controllability. J. Anal. Math., 59 (1992) 103-116. 15. M. Struwe, Geometric Evolution Problems, in Nonlinear Partial Differential Equations i n Differential Geometry, Park City UT (1992) 257-339, IAS/Park City Math. Ser. 2, Amer. Math. Soc. Providence, RI, (1996).

140

HOMOGENIZATION OF THIN PIEZOELECTRIC PERFORATED SHELLS MAFUUS GHERGU

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P. 0. BOX 1-764, RO-014 700, Bucharest, Romania E-mail: [email protected] GEORGES GRISO

Laboratoire Jacques-Louis Lions, Universitt Pierre et Marie Curie (Paris VI), 4, Place Jussieu, 75252 Paris, fiance E-mail: georges.grisoOwanadoo.fr BERNADETTE MIARA

Labomtoire de Moddlisatzon et Simulation Num&rique, ESIEE, 2, Boulevard Blaise Pascal, 91360, Noisy-Le-Gmnd, France E-mail: b.miamOesiee. fr We consider a composite piezoelectric material whose reference configuration is a thin shell with fixed thickness. We assume that the structure has periodically distributed perforations. Using the periodic unfolding method we establish the limit constitutive law by letting the size of the microstructure going to zero.

Keywords: Computational solid mechanics; Homogenization; Piezoelectricity; Shells.

1. Introduction

We present in this paper the homogenization of a thin piezoelectric perforated shell with constant thickness. We assume that the shell has a periodically distributed microstructure of size E containing holes. Periodicity of the shell is viewed as periodicity with respect to the curvilinear parametrization of its middle surface. Generally speaking, piezoelectric materials produce an electric field when exposed to a strain caused by an imposed mechanical force and conversely, an applied electric field produces a mechanical stress. The piezo-

141 electric behavior of the material is given by the couple (u", cpE) where u" represents the elastic displacement field and cp" is the scalar electric potential. Our purpose in this paper is to determine the limit model when the size E of the microstructures goes to zero. The main feature here consists in the presence of the periodic perforations that determine a more complicated limit constitutive law. The model we use here extends the Koiter's one5 in the case of linearized elasticity. The periodic unfolding method introduced by Cioranescu, Damlamian and Grisol and the Korn's inequality for perforated domains are used to derive the limiting model. The interest is also due to the fact that the two dimensional linearized change of metric and curvature tensors present different order of derivatives of their arguments. This gives rise to local problems which are different than the global one.

2. Two dimensional model of shell 2.1. Reference configuration

We consider a two dimensional thin shell with the middle surface S and having constant thickness 2t > 0,that is, a body whose reference configuration is the set @(Rx [-t, t]), where R c R2 is a smooth, connex domain with C2 boundary and @ : R x [-t,t] + R3 is a C3-injective map. We denotea by hap, b{ the covariant and the contravariant components of the curvature tensor of the surface S. Also consider y = (yap) and p = ( p , ~ ) the two dimensional linearized change of metric and curvature tensor. Their components are given by

where d, := d/dx, represents the derivative with respect to x, and denote the two dimensional Christoffel symbols for the surface S.

aLatin indices and exponents take their values in the set {1,2,3), Greek indices and exponents (except E ) take their values in the set {1,2) and the summation convention with respect to repeated indices and exponents is used. Boldface letters represent vectorvalued functions or spaces. The summation convention is also assumed for repeated indices.

142 2.2. Perforated d o m a i n a n d equilibrium equations

Let Y = [O, I[' be the reference cell and let Y * be the part of Y which is occupied by the material. Denote by E > 0 the size of the elementary microstructure that contains the holes. Let fiE be the smallest union of EY*cells that contains R, that is,

where ICE = {E E 2'; E(Y+<)no# 0 ) . The periodically perforated domain RE is then defined by RE = R n f i E . The two dimensional model of piezoelectric shell is characterized by the following four tensors (see Haene14): the membrane elastic tensor ( c F T ) ,which is symmetric and positive definite; the flexural elastic tensor (cgPaT),which is symmetric and positive definite; the dielectric tensor (dap), which is symmetric and positive definite; the piezoelectric tensor (eaPu), which is symmetric in the sense that eaPa = ePaa.

The components of the four above tensors belong t o L m ( R ) . Let us consider the functional space V ( R E )= V ( R E x) V ( R E x) W ( R E ) , where

Let p E ~ ~ ( 5 be 2 )the resultant of the applied mechanics forces. The elastic displacement field uEE V ( R E )and the electric potential cpE E V ( R E satisfy ) the equilibrium equations

where c&, e& and d& are defined as

143 3. Periodic unfolding operator

For all x E R2 let [ x ] ybe the unique integer combination such that x [ x ] yE Y and set { x ) =~ x - [xIy E Y . Then, for all x E R2 and E > 0 we have

For all v E L1( R E )extended by zero in fiE \ RE we define

In this way, 'TE: L1(RE) L1(R x Y * )is a linear operator and for all v , w E L1(R")we have 'TE(vw)= 'TE(v)'TE(w). 4. Main results

For a function u = U ( X ,y ) we denote by V x u and V,u the gradient of u with respect t o x and y variables. Also &,,u stands for the derivative of u with respect to yi. For k = 1 , 2 we denote by H$,,(Y*) the set of all functions of H k ( y * )with vanishing mean value, extended by Y-periodicity. Assume that there exist four tensors C M , C F , e, d such that for all ( x ,y) E R x Y * we have

Set V,,, ( Y * )= Hi,, (Y*) x H;,, (Y*) x Hier(Y*).

Theorem 4.1. Let ( u " ,cp') E V ( R E )x V ( R E )be the unique solution of (2). Then, there exist (u,cp)E V ( R ) x H J ( R ) and two corrector fields ii E L 2 ( R ,V p e r ( Y * ) ) ,q E L2(R,H;,,(Y*)) defined by the following strong convergence

' T E ( u E-+ ) u

strongly in L 2 ( R x Y * ) , strongly in L ~ ( R x Y*),

' T E ( ~ " )(P

+ ~ ~ ~ , ~ ( ; strongly i i ) in p a P ( u ) + r,p,,(U) strongly in

T ' ( Y ~ P ( u ' ) ^lap(") ) -+

7 ' ( p a p ( u E ) )4 ~"(VXcpE 4) Vxcp VyF

+

L~(R x Y * ) , (3) L~(R x Y*), strongly in L ~ ( R x Y*).

Moreover, (u,cp, ti,q) is the unique solution of the following variational problem posed for all v E V ( R ) , F E L 2 ( R ,v,,,(Y*)), $ E H J ( R ) ,E~

144

where

The correctors a and T are expressed as a linear combination of basis ?(la), (TPB,B)EP,(P,Tja) E Vper(Y*) x H;~,(Y *) as follow^: functions

-a0 *(la Furthermore, the three pain of local functions ( F ~ , < ~ ' ) , (h ,t? ) and (zu,vu) are respectively the unique solution of the following three variational problems posed for all (a, E Vper(Y*)x H&,(Y*):

4)

145 where

Concerning the global homogenized problem we have: Theorem 4.2. The limit elastic field u E V ( a )and the limit electric field cp E HG(S2) are the unique solution of the global homogenized variational problem

(ZK(u,v )

+ EK (v,(~)).\/;ldx = J,p

(-EK(U,$) + d ~ ( ~ , $ ) ) . \ / ; l d= x 0,

V v E V(a),

v.\/;ldx,

(7)

v$ E H i (a),

where Z K , E K , d K are defined by

In (8) we have introduced the homogenized tensors E M , are defined as follows: -ape7

1

cZAP($d;

- IY*I EaPo~ - 1 CM

- a P o ~'F

~

i"P"

;i"P

-

P

-

U

= =

c, E F , E, i,d which

+ S A , , ~ ( ~+~e~a P) A) a A , y P T ,

caPA~ IY*I ./Ye F r~PLL,Y(guT), 1 u T IY*I F ('Adp f r ~ P , ~ ( l E u r ) ) ,

&* PAP

(AlJy. e a p x ( d i+ aX,y7u)+ c Z ~ ~(zU), S A ~ , (9) ~

+,

JY.

C;~*~~,,~(T),

&. ( - e X P a s ~ P , y ( r+p )daA(df + &,,$)).

A complete proof of the above results can be found in.3 Acknowledgements This work has been supported by the European Program "Smart System" HPRN-CT-2002-00284.

146 References 1. A. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, S 6 . I335 (2002) 99-104. 2. M. Ghergu, G. Griso, H. Mechkour and B. Miara, HomogBnBisation de coques minces piBzoBlectriques perforhes, C.R. Me'canique 333 (2005) 249-255. 3. M. Ghergu, G. Griso and B. Miara, Homogenization of thin piezoelectric perforated shell, submitted. 4. Ch. Haenel, Analyse et simulation numhrique de coques piBzoBlectriques, Ph.D. Thesis, UniversitB Pierre et Marie Curie, Paris (2000). 5. W.T. Koiter, On the foundations of the linear theory of thin elastic shell, Proc. Kon. Ned. Akad. Wetensch. B73 (1970) 169-195.

147

DAMAGED SUPPORT IDENTIFICATION IN ALUMINIUM CURTAIN-WALLS USING NEURAL NETWORKS P. NAZARKO and L. ZIEMIANSKI

Department of Stmctuml Mechanics, Rzeszow University of Technology, W . Pola 2, 35-595 Rzeszow, Poland E-mails: [email protected]; [email protected] Ch. EFSTATHIADES and C. C. BANIOTOPOULOS

Institute of Steel Structures, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece E-mails: chefst9civil.auth.g~; ccb9civil.auth.gr

G. E. STAVROULAKIS Department of Production Engineering and Management Technical University of Crete, GR-73132 Chania, Greece and Department of Civil Engineering, Carlo Wilhelmina Technical University 0-381 06 Braunschweig, Germany E-mail: gestavr9dpem.tuc.g~ One of the possible reasons of aluminium curtain-wall systems failure is the total or partial destruction of connections with their bearing structure. The damage identification approach proposed in this paper deals with the structural monitoring and formation of a Neural Network (NN) to identify possible connections faults. Finite Element (FE) models of a typical damaged and undamaged aluminium curtain-wall were built and deflections of columns for those structures were computed a t several control points. The optimal number of required gages and their placement were investigated. The collected results are used t o create a Damage Parameters Database ( D P D ) and train NNs in order to approximate the relation between the input and output data. The regularization techniques (jitter) was employed to improve the generalization properties of the network. Several network architectures are studied and the effectiveness of selected training algorithms is analyzed. The obtained results show that NNs can be an efficient and low-cost method for the identification of imperfections in aluminium curtain-walls.

Keywords: Curtain-walls; Structural Health Monitoring; Damage identification; Finite Elements Method; Neural Networks.

148 1. Introduction

Glass-aluminium curtain-walls became one of an important building component since they greatly improve the serviceability and appearance of the respective structures. These kind of facades started to be designed as a part of the principal load-bearing structure. In consequence, a great research effort is given to study the structural behavior and improve designs of such systems (from the financial and safety point of view) since in certain cases the cost of the facade exceeds the fifteen percent of the total cost of a structure.l Curtain-walls are currently being used in various shapes and types, not only in new buildings (Fig. la) but also during the renovation of existing structures (Fig. lb), mainly in order to improve the physical properties of the buildings. Safety and reliability of structures like curtain-walls require the use of faults warning system. The early damage location is very important especially for high rices and buildings placed on earthquake areas. Structural Health Monitoring (SHM) is the crucial factor and is able to help engineers avoid unfortunate effects and take in time decisions for structural repairs. Another important issue is the interpretation of structural response recorded from a large number of gages and often gigabytes of data. Recognizing the structural state is possible owing to development of efficient software with intelligent algorithms. In recent years, the development of damage detection and assessment

(a)

(b)

Fig. 1. Glass-aluminium curtain-wall systems. (a) In a new building. (b) In the renovation of an existing building.

149 techniques become an interesting topic of research not only for civil infrast r ~ c t u r e but , ~ also for mechanical systems, aircrafts and aerospace struct u r e ~As . ~ a matter of fact the integrity of the structure is a very important aspect, because it may reduce maintenance cost and, in extreme events, collapse (e.g. after infrequent but high forces like earthquakes, wind gusts, accidental impact). Ideally, health monitoring of civil infrastructure consist of determining by measured parameters the location and severity of damage in a structure as they happen. Currently, these methods can only determine whether or not damage is present,4 whereas information about the extent of damage is not sufficiently accurate. During the past decade, the concept of Artificial Intelligence has offered some useful methods, especially for the solution of so-called inverse problems. In this area, Artificial Neural Networks (ANNs) are usually applied for damage identifi~ation.~-'OThe employment of such novel techniques allow us predict the structural health during the service, without the need to interrupt or terminate the normal system operational life. One of the most probable position for a fault in a glass-aluminium curtain-wall system is the connection of the facade with its bearing structure. In most cases, these connections are easily accessible during the construction of the facade, but they are not accessible during the operational life of the building. Due to this fact we need herein a global system that is ready to find local changes related with the damage of a single support. The correct selection of the structural parameters related to damage, is very important for the success of the task. The network training is usually carried out based on values like mode shape, natural frequencies, displacements, acceleration spectra, strains, etc.'' The optimal positioning of the sensors is also a critical factor for the effectiveness of monitoring techniques. Taking into account the cost of sensors, it is uneconomical to install for example strain-gauges on every part of structure. Accordingly a simple analysis of sensor placement will be provided. Some preliminary results related with identification of supports failure in the curtain-wall system will be presented in this paper. First, the data obtained from FE models will be preprocessed and the optimal placement of control points will be studied. Then, NNs will be trained and tuned in order to detect the position of a single support where a bolt is missing. Next, regularization techniques will be employed to improve the generalization ability of the networks. Finally some detection results for damage introduced at two locations will be presented.

150 2. Formulation of the problem

Localization of partially destructed supports based on neural prediction is the main objective of this study. Using FEM the structural deflections were computed for damaged and undamaged models. It was proved that certain failure cases cause changes in the measurements, but from the other hand these variations have rather a local influence. The second issue is the optimal selection of control points - their number and position. There are 52 supports in the considered model of the curtainwall. Probably the fastest way to achieve information about connections failures is placement of one gage for every support. For our model it is not a problem, but in applications there exist structures with thousands connections. Due to this fact we would like to use as less of gages as possible. First, the FE model of a curtain-wall was built and millions of deflection values were stored. In this way the relationship between changes in boundary conditions and measured parameters was defined. It was assumed in the present paper that only the horizontal displacements under static wind load and structure loads will be measured. The control points were located at the midpoints of each column and displacements related to different damage patterns were computed. Next, it leads (together with related target vectors) to the formulation of the Damage Parameters Database (DPD). Based on the achieved changes in the structure behavior due to fault occurrence, the NNs were trained to assess the positions of damaged supports. There are two main tasks considered herein: (i) Damage appears in single support. (ii) Damage appears at two random supports in the same time. 3. Numerical computations 3.1. Model description

The FE model of the glass-aluminium curtain-wall is used to compute the required data for NNs training. It is composed of a typical grid of 1 meter height and 1 meter width respectively and comprises vertical aluminium members (mullions) interconnected by discontinuous horizontal aluminium members (transoms).l2 Mullions and transoms use beam elements (2 nodes, 6 degrees of freedom) while the glass panels were constructed with surface elements (plates with 9 nodes). The whole system of the curtain-wall is supported on the bearing structure (the building) with connections (supports in our model), where it is assumed

151 that the mid supports are partially, whereas the end supports are fully restrained. The glass panels are continuously supported over all sides of the grid.

3.2. Connections The connections of the curtain-wall with its bearing structure consist of the brackets and the fixings (bolts) as it is shown on the simple cross-section through a typical support point (Fig. 2).

Part of the bearing structure

!

Aluminium mullion

I 1I

i

1

r_/

Aluminium Transom and Glass Panels

!

! ! 1

i

Fig. 2. Connection of a curtain-wall to a bearing structure. (a) Cross-section of a typical type of bracket. (b) A bolt in the bracket with slotted holes.

Fixings (bolts) depend on a number of requirements like load values, loads nature, thickness of members etc. The "bolt 1" has a freedom to slightly move in the W axis direction and "bolts 2" have a freedom to slightly move in the XX axis direction. When a fault is introduced to the structure (e.g. a bolt partially slacked out), the boundary conditions of a support in the FE model change. These boundary conditions for an end-support connection are shown in Table 1. The curtain-wall is linked with the main structure with point connections in certain distances all over the covered area. The bolted connections used in glass-aluminium curtain-wall systems can not assume to be either

152 Table 1. Boundary conditions for an end support connection related with bolts numbers on Fig. 2a. -

Boundary condition

--

Bolt 1 Undamaged Damaged (slacked out)

-

--

Bolt2 Undamaged Damaged (slacked out)

-

Restrained Restrained Restrained Restrained Restrained Restrained

Restrained RYd/RYdl Rzmax RxxmaX Restrained Restrained

Restrained Restrained Restrained Restrained Restrained Restrained

Restrained Restrained R~mas Restrained RYym,, Restrained

completely rigid or pinned but in most cases are semi-rigid. The damage in these connections can be represented by a reduction of their rigidity. The damage is introduced to the structure by decreasing the initial stiffness of any connection of the curtain-wall and the main structure. Depending on the kind of the fault of the connection the reduced rigidity can be half of the original value or even much less. By following this idea, we generated data defining the relationship between changes in boundary conditions, i.e. faults in the connections with the supporting structure and the different deflections values at the midpoints of the mullions of the curtain-wall (exactly 117 control points shown on the Fig. 3b).

Fig. 3. The grid of the curtain-wall model. (a) The supports coordinates. (b) The numbering of the mullions midpoints (control points).

153 3.3. Results of the analysis

A parametric analysis was performed by calculating the displacements for all 117 midpoints of the mullions (Fig. 3b) and various boundary conditions. The first data set consisted of 53 cases: single faults of 52 connections (supports) plus the case of the intact (healthy) structure. A table with 53x117 values is prepared taking into account all cases for the same degree of fault and the same loads magnitude (Table 2). An exemplary pattern of Table 2. Some results of the parametric analysis of the glassaluminium curtain-wall. Mullions midpoints deflections

imm1

Deflections in certain control points due to the fault of single support (coordinates in [m]) none (0,O) (1,O) (0,3) (1,3) . . . (1'49)

deflections due to existing damage in one of the supports (marked as 3-3 on Fig. 3a) is presented in Fig. 4.

Fig. 4.

The deflections computed for the glass-aluminium curtain-wall with a fault.

154 4. Applications of A N N s

The fault location problem in the herein proposed approach will be solved based on ANNs prediction. Feed-forward networks consisting of one and two hidden layers were designed and tested. For each damage case, the number of 12 to 20 units, related with selected control points on the structure, was allocated in the input layer while the output layer consisted of damaged s u p port position (two units related with damaged support coordinates). ANNs simulations were implemented using Matlab and the Neural Networks Toolbox. Log-sigmoid transfer function, written as f (N) = 1/(1+ exp(-N)), was used to activate the neurons in both the hidden and the output layer. Training functions trainrp and trainlm according to the resilient backpropagation algorithm were used for the training of the networks in this preliminary study. Evaluation of the neural approximation accuracy was done based on the most popular Mean Square Error (MSE) defined as: 1

M S E = fi

[ti - oil2

where ti and oi are the target and output vectors for every iteration respectively, and V = L,T denotes the number of elements in target vector (learn and test respectively). A second auxiliary factor, called Success Ratio (SR), is used as well:

where NBep is the number of patterns within the Bep area and V is the number of patterns in the considered sets. For estimation of neural prediction the following relative error was used:9 ep = 1 - y:/tPI

.loo%

(3)

where t;, y: are respectively the target and neurally computed i-th outputs for the p-th pattern. 4.1. Data preprocessing The network input data ID and output data OD sets were normalized to the range [0.1, 0.91 for all cases. Next, they were divided to learning and testing vectors with ratio of 70%ID and 30%ID respectively. In the early studies the selection of testing data set was carrying out randomly during each time of simulation. It leads to multifold ~alidation.~ From the other hand, due to the relatively small number of damage patterns in the first set,

155 it easily causes to non-uniform testing patterns distribution and produces high randomness. Due to this fact manual selection that covers a complete data set was carried out in parallel in order to avoid the mentioned effects. It was assumed that every third support will be included to the testing set. In this study better precision was obtained for training with pre-selected constant training vector. Concerning the network 16-3-2 and MSE results for case with random test data selection its values were higher about 0.009 and 0.027 for x and y coordinates prediction respectively. It was consider herein for this case and the following sections of this paper that only manual selection of test data will be done and respective results will be shown. First, the networks input vector xi was defined in different ways to provide the most useful data form for neural prediction and three follow formulas were in use: xi = di xi = do - di xi = (do - di)/do where di, do are displacements for i-th damage case and undamaged structure respectively. It was suspected that by including the information about undamaged structure will improve the prediction accuracy. Results of these simulations (Fig. 5) showed that slightly lower level of error values was achieved for the first input vector. Introducing the information about undamaged structure (inputs no. 2 and 3) did not improve the precision of indication the damaged support coordinates. It may be easily seen especially for the learning rate (marked by circle) where for the same conditions (networks architectures) the error level is significantly higher. Several tests related with various size of the hidden layer showed that the best results were achieved for the smallest networks, herein 16-3-2. It is a consequence of the small number of training patterns, because networks with large number of neurons in the hidden layer may easier overfit the learning data set and give worse results for testing set (network generalization is impossible in this case). The validity of a neural network-based damage detection approach also depends on the initial weight and biases, order of input elements, choice of transfer function, training algorithm, etc.1° Thus, to achieve sufficient efficiency, training procedure was repeated at least 50 times and the respective average errors results are mainly shown in this paper.

156 H.coor. "xu(1)

W

H.coor. "xu (2)

0.06

H.coor. "xu (3)

a max test

3 4 5 6 7 8

3 4 5 6 7 8

3 4 5 6 7 8

V.coor. "y"(1)

V.coor. "y"(2)

V.coor. "y" (3)

neurons number in the hidden layer Fig. 5. Damage identification error (MSE)for networks (16-H-2) trained with different input vectors and various size of the hidden layer H; manual selection of testing set.

4.2. Various size of the input vector

F'rom the brief analysis of DPD it was concluded that deflections in the center of the structure are more frequent and relatively high, while on the border of the structure they are smaller and less frequent. The second hint is that we should attach to the input vector values computed for the middle points of the columns. An objective of the test presented here was the optimal selection of control points, their numbers and their position on the aluminum curtainwall structure. It has been already assumed that the set of input data will consist of deflections measured at the midpoint of each column. The studied input combinations are collected in Table 3. The size of network input vectors leads to different architectures. Additionally, the influence of the number of neurons in the hidden layer on the Table 3. Assumed sets of control points. Total number of control points

Considered combinations of control points

157 precision of the prediction was evaluated. Our investigations were started from networks with only one hidden layer and small number (three units) of neurons in this layer. Then, the number of neurons and hidden layers was increased. Results of this study are shown on the graphs below (Fig. 6). Like before the output of the networks consisted of the two coordinates of the damaged support. H.coor. "x" (12)

H.coor. "xu (16)

H.coor. "x" (20)

0.1

max test

0.04

3 4 5 6 7 8

3 4 5 6 7 8

3 4 5 6 7 8

V.coor. "y" (12)

V.coor. "y" (16)

V.coor. "y" (20)

0.1

p 0.06 0.05 W

345,678 3 4 5 6 7 8 ' 3 4 5 6 7 8 neurons number In the hidden layer

(4

(b)

(c)

Fig. 6. MSE of fault location for various inputs and number of neurons in hidden layer. (a) Networks trained with 12 control points. (b) Networks trained with 16 control points. (c) Networks trained with 20 control points.

Quite poor number of patterns (here 52) leads to better results when networks with small number of neurons in the hidden layer are employed. It seems also that the optimal input length should consist of 16 displacement values. Another conclusion is that the x coordinate was predicted with better precision than the second coordinate y. Neural Networks with two hidden layers were also tested. There were no significant differences in the group of trained architectures. MSE values of damaged support identification for few networks were collected in the Table 4. It may be assessed that the lowest value of error parameter was achieved for the network 16-4-3-2, but the loss of the network 16-3-2 is very small particularly in testing results. As it was mentioned above, small architectures are privileged for this part of the DPD. Based on exemplary results provided by the network 16-3-2 (adequate

158 Table 4. Average MSE values of damaged support location.

MSE

Network

- Learning

MSE - Testing X Y

Y

X

Note: a Exemplary identification results after one training simulation.

error values were distinguished by superscript 'a' in the Table 4)' a graph with the predicted failure position was created (Fig. 7). Empty circles mean Damaged support identification 0

c + ':

1

8 0

Fig. 7.

1

0 2

3

< 4

7 5

6

8 7

< 8 8 9 1

Horizontal coordinate x [m]

8 0

8 1

1

8 1

target L target T learn test c.polnts

1 2

Identification of fault location by network 16-3-2.

there the real supports position and the location of control points where deflections were measuring has been marked as well by a star. Additionally position of supports that were included to the testing set was marked by the lighter circles. Learn results after 2000 training epochs were shown by 'plus' while testing set was presented by 'x'. It may be seen that precision for some damaged support locations is not sufficient enough and network prediction accuracy should be improved further. Results obtained in this section are shown also on the SR plot (Fig. 8) and it corresponds in fact to the cumulative probability function. For in-

159 Success ratio plot

0,

Fig. 8. Success Ratio plot for learn and test data sets provided by network 16-3-2.

stance when the relative error is not greater than i.e. B e p = 10% the identification is approximately on the level of SR = 35% and SR = 61% for testing and learning sets respectively. It means that 35% (or 61%) of 52 patterns will be well predicted by the neural network with accuracy lepl I 10%. 4.3. Networks generalization improvement

The main objective in this study is to achieve the most possible network generalization with good identification accuracy. The main reason that limits the networks performance is the small number of patterns. Relation between the input and output is more difficult to be learnt in this case. The second issue is that function that we are trying to simulate should be smooth. It means that a small change in the inputs should mostly produce a small change in the outputs. One of the regularization techniques that may improve networks generalization is based on the addition of small amounts of artificial noise (jitter) into the inputs during training. In other words, if we have two cases with similar inputs, the desired outputs will usually be the similar. As long as the amount of jitter is sufficiently small, we can assume that the output will not change enough t o be of any consequence, so we can just use the same target value. It should be noted that too high jitter will obviously produce garbage, while too little will have little effect. In this section the optimal value of standard deviation (std) of noised

160 input will be studied. It was assumed that noise will have random entries chosen from normal distribution with mean zero, variance one and various value of standard deviation (in the range 0.0001 to 0.1). The network 16-3-2 trained with trainrp algorithm was employed here. Selection of patterns t o the training set followed by the scheme with the follows supports numbers (4 5 10 11 16 17 21 24 27 30 33 36 39 42 43 49 51). Finally, the number of patterns to network training increased from 52 t o 520, where the first 52 patterns are related to data without noise. One important thing is that jitter was added after dividing the DPD into learning and testing data sets. In that way, the sizes of these vectors were equal to 350 and 170 samples respectively. Results of these simulations were shown on Fig. 9. Comparing to networks training with only 52 patterns (Fig. 6b) Prediction of the x-coor. (trainrp) 1 I8

Prediction of the y-coor. (trainrp)

I

Po-.

1o

-~

o-~

I

lo-'

standard deviation

Fig. 9. MSE of neural fault location with various standard deviation of noise.

it is difficult to observe the improvement of MSE values due to various std level. The lowest error values for the vertical coordinate (worse predicted) were achieved with std equal t o 0.0003. Next, networks simulations with various sizes of the hidden layer and considered std levels were carried out. It was observed that there is no significant improvement of damage position prediction and error parameters for various networks. Even introducing of networks with two hidden layers did not give satisfactory results. Two the best results are compared in the Table 5. An example of damaged support location for network 16-32 was shown on Fig. 10. With these results corresponds also the SR plot

161 Table 5. Average MSE values of damaged support location.

MSE

Network

- Learning

MSE

Y

X

- Testing Y

X

Damaged support identif~cation g o b 0 8-

g

*+

I 1

2

3

+,

0

+

I 0

o

flw

.-..

I

target L etargetT + learn x test c.points 0

@pPo

+

re

+

i

y

y*

*

I:

o

o

+

*

Y * +

.yo +

I

* # i 0

I

4 5 6 7 8 9 1 0 Horizontal coordinatex Irn]

I

O

I 1

1

1

I 2

Fig. 10. Damaged support location predicted by the network 16-3-2 trained with noised input (std = 0.0003).

presented on the next graph (Fig. 11). In this case the number of patterns predicted with error value lepl 5 10% is approximately equal to 42% and 58% for testing and learning data sets respectively. It seams that we achieved here small prediction improvement for about 7% that is related with training patterns. As a comparison, the exemplary results for networks training with jitter and random selection of training data set were shown on the Fig. 12. Assessment of damaged support position looks pretty good here due to the fact that learning and testing patterns are very similar to each other. Unfortunately this solution says nothing about the network generalization. 4.4. Identification of two damaged supports

The basic DPD was extended here for a data related with damage located at two supports in the same time. The matrix consists of 1326 deflections shapes for the same conditions (level of damage and loads) like before. It

162 Success ratio plot

I

0,

Fig. 11. Success Ratio plot for learn and test data sets provided by network 16-3-2 trained with noised input (std = 0.0003).

Damaged support identification 0

target L

+ learn r test * c.polnts

' kc'a S i C ; *

&%*a

9

0

1

2

3

r

w

4 5 6 7 8 9 1 0 Horizontal coordinate x [m]

1

1

1

2

Fig. 12. Damaged support location predicted by the network 16-3-2 trained with noised input and randomly selected training set.

should be mentioned that performing the computation for many damage cases is time consuming or almost impossible (for all combinations). Due to this fact extended set of patterns was created by superposing of the deflections obtained for failure in single support. It followed the formula:

,+ d j - do

2 . .- d .

9 -

for

i#j

(7)

where i, j = 1 , 2 . . .52 are the support numbers and d means deflections for certain cases {i,j, 0) (zero for undamaged structure).

163 In this way high affinity with values computed from the FE model was achieved for all damage cases except those where two damaged supports were close to each other. This situation is simply shown on Fig. 13 where

Fig. 13. Comparison of deflections superposed in Matlab and computed from FE model. (a) Damaged supports (2,3) and (3,3). (b) Damaged supports (3,3) and (4,6).

for certain control points, the deflections computed from the model have greater magnitude but from the other hand the shape of the deflection surface is correct. For the needs of the current analysis it was assumed that these differences will have no meaning and detection of damaged supports will be carried out. For the new set of patterns several NNs architectures were tested. The input vector consisted of 16 values measured at selected control points (Table 4) while the output vector had 4 values related with damaged supports coordinates {XI, y l , x ~yZ). , An optimal size of the hidden layer was studied for trainrp and tminlm algorithms. The MSE level for coordinates prediction is very similar for the first and the second damaged support, so the Fig. 14 shows these results only for assessed position of the first failure. Networks training was held up after 5000 and 1000 epochs for the trainrp and trainlm algorithms respectively. The selection of training set was performed randomly and the size of certain sets was equal to 928 (70%) and 398 (30%) patterns for learning and testing respectively. Better accuracy of the damaged support prediction was obtained for networks trained based on trainlm algorithm where the optimal size of the hidden layer is close to the number 14 in this case. When the second algorithm was used this number is approximately equal to 18. The exact error values for described cases were collected in Table 6. Statistically the MSE has one of the lowest levels reached during all simulations. Unfortu-

164 Pred~ctionof the xl-cwr (tralnrp)

Prediction of the yl-coor. (tralnrp)

Pred#cl#on of the yl-coor. (tramtm)

Predicllonof the xl-coor. (tralnlm)

mm test 0 07

0 07

"?v~"

001-

O'lb'tk

neurons 8n the hldden layer

18'22'2630 34 38 42 46

8

neurons ~nthe hidden layer

10 12 14 15 neucons ~nthe hldden layer

8

10 12 14 16 neurons ~nthe hldden layer

Fig. 14. MSE values for various size of the hidden layer. (a) Prediction of X I coord. (trainrp). (b) Prediction of yl coord. (trainrp). (c) Prediction of xl coord. (trainlm). (d) Prediction of yl coord. (tminlm). Table 6. Average MSE values of two damaged support location. Network

xl 16-18-4 16-14-4

0.0106 0.0042

MSE yl

- Learning

0.0207 0.0064

x2

Y2

xl

0.0102 0.0046

0.0221 0.0062

0.0123 0.0058

MSE yl

- Testing

0.0275 0.0110

x2

Y2

0.0121 0.0062

0.0292 0.0100

nately the graph with hits (Fig. 15) where single simulation results were Damaged support identification

0

1

2 3 4 5 6 7 8 9101112 Horizontal coordinate x [m]

Fig. 15. Damaged supports location predicted by the network 16-14-2 trained with tmznlm algorithm and randomly selected training set.

165 shown does not have satisfactory precision. It was confirmed also on the SR plot (Fig. 16) and supports coordinates prediction for the error value Success Ratio plot

Fig. 16. SR plot for the network 16-14-4 trained with trainlm algorithm.

(ep(5 10% is approximately equal t o 48% and 52% for testing and learning respectively. It is the similar error level like in the first task.

5. Conclusion and final remarks In this paper, an approach of a structural health monitoring and fault detection in curtain-walls was presented. Several models of a damaged structure and the respective numerical simulation were studied. An initial position of control points was analyzed and their optimal number was found. The regularization technique (jittering) employed here improved networks generalization that increased the identification accuracy. Position (coordinates) of damaged supports was predicted by ANNs. The identification results were burdened with an error, therefore the presented activities were an attempt to increase the accuracy. It should be noted, that the number of patterns in the first task was rather poor for effective network learning. The strong dependence of the identification accuracy on the training set size has been already observed earlier.'' In this publication it was mentioned that the number of training cases may determine the precision which a network can offer. Increasing the damage cases from 50 to 80 caused an absolute error decreasing from 27.7% t o 8.6%. Due to this fact, more damage patterns were computed and

166 the problem was extended t o identification of two damaged supports. Unfortunately the task complexity has also increased and the obtained results have not sufficient accuracy. The final conclusion is that the static analysis, especially measurement of deflections is not enough t o describe the structural behaviour for damage detection. It happens mainly because the introduced damage causes local changes in the measured values. A proposal for the future work and improvement of damaged support identification may be an employment of dynamic analysis i.e. dynamic loads or impact loads.

Acknowledgments The financial support of the European Commission t o the present research activity is gratefully acknowledged ("Smart systemsn-HPRN-CT2002-00284).

References 1. Ch. Efstathiades, M. Zygomalas and C. C. Baniotopoulos, Glass-Aluminium Curtain-Wall System: Optimization with respect to Serviceability Criteria, Proceedings of the Xth International Conference for Metal Structures (Timisoara, Romania, 2003). 2. A. A. Mufti, Str. Health Monit. 1(1), (2002), pp. 89-103. 3. E. Keller and A. Ray, Str. Health Monit. 2(3), 257-267 (2003). 4. P. C. Chang, A. Flatau and S. C. Liu, Str. Health Monit. 2(3), (2003), pp. 191-203. 5. C. C. Baniotopoulos, Int. J. of Pres. Vessels and Piping 75, (1998), pp. 43-48. 6. M. Engelhardt, G. E. Stavroulakis and H. Antes, PAMM - Proc. Appl. Math. Mech. 3, (2003), pp. 511-512. 7. G. E. Stavroulakis and H. Antes, Comput. Mechan. 20, (1997), pp. 439-451. 8. G. E. Stavroulakis, Inverse and Crack Identification Problems in Engineering, (Kluwer Academic Publishers - Springer, Dordrecht, 2000). 9. Z. Waszczyszyn and L. Ziemianski, Parameter Identification of Materials and structures, in CISM Courses and Lectures No. 469, eds. Z. Mroz and G. E. Stavroulakis, (SpringerWien New York, 2005), pp. 256-340. 10. L. Ziemianski and G. Harpula, The use of neural networks for damage detection i n eight storey frames, Engineering Applications of Neural Networks Proceedings of the 5th International Conference, ed. A. Marszalek (Warsaw, Poland, 1999), pp. 292-297. 11. Z. Su and L. Ye, Str. Health Monit. 4(1), (2005), pp. 57-66. 12. C. C. Baniotopoulos, E. Koltsakis, F. Preftitsi, P. D. Panagiotopoulos, Aluminium Mullion-Ransom Curtain Wall Systems: 3 0 F.E. Modeling of their Structural Behavior, Proc. ISCAS 4th Intern. Conf. on Steel and Aluminium Str., (Helsinki, 1998).

167

MATHEMATICAL RESULTS ON THE STABILITY OF QUASI-STATIC PATHS OF ELASTIC-PLASTIC SYSTEMS WITH HARDENING A. P E T R O V Weierstrafl-Institut fur Angewandte Analysis and Stochastik, Mohrenstmfle 39, 10117 Berlin, Germany E-mail: petrovOwias-berlin. de J. A. C . MARTINS

Znstituto Superior Tdcnico, Dep. Eng. Civil and ICIST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal E-mail: jmartinsOcivil.ist.utl.pt M. D. P. MONTEIRO MARQUES Centro de Matemdtica e Aplica~6esfindamentais and F.C.U.L., Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal E-mail: mmarquesOptmat.fc.ul.pt In this paper, existence and uniqueness results for a class of dynamic and quasistatic problems with elastic-plastic systems are recalled, and a stability result is obtained for the quasi-static paths of those systems. The studied elasticplastic systems are continuum 1D (bar) systems that have linear hardening, and the concept of stability of quasi-static paths used here takes into account the existence of fast (dynamic) and slow (quasi-static) times scales in the system. That concept is essentially a continuity property relatively t o the size of the initial perturbations (as in Lyapunov stability) and relatively t o the smallness of the rate of application of the forces (which plays here the role of the small parameter in singular perturbation problems).

Keywords: differential inclusions, plasticity, hardening, existence, stability

1. Introduction Martins and co-workers studied in [1,2] the relation that exists between, on one hand, dynamic and quasi-static problems and, on the other hand, the theory of singular perturbations. More precisely, they performed a change of variables in the governing system of dynamic equations that consists of

168 replacing the (fast) physical time t by a (slow) loading parameter X whose rate of change with respect to time, e = dX/dt, is eventually decreased to zero. In this manner they obtained a system of equations or inclusions defining a singular perturbation problem, i.e. a problem where the highest order derivative with respect to the loading parameter appears multiplied by the time rate of change (e) of that parameter. The variational formulations for elastic perfect plastic and elastic-viscoplastic systems were established by Duvaut and Lions [3].Hardening effects were introduced in the formulations by Johnson [4,5], who proved existence of a strong solution and, under some assumptions, a regularity result for the velocity field. In the present paper, the definition of stability of quasi-static paths given in [2] is adapted to the present continuum case. And we establish that, similarly to the finite dimensional elastic-plastic systems with hardening discussed in [6],the dynamic evolutions remain close to a quasi-static path when they start sufficiently close to that quasi-static path and the load is applied sufficiently slowly. The structure of the article is the following. In Section 2, the mathematical formulations for dynamic and quasi-static elastic-plastic systems with hardening are presented, and in Section 3, existence and uniqueness results are recalled, which use the theory of m-accretive operators [3,7-lo]. The proof of stability of a quasi-static path is presented in Section 4. 2. Governing equations

We consider an elastic-plastic bar with linear kinematic hardening that has the length L along the x axis. Geometrical linearity is assumed. The governing dynamic equation can be non-dimensionalized by using the nondimensional time (T)and load parameter (A, X = A1 E T ) , yielding

+

e2u" - u, (u, r ) = f (x, A),

(1)

where u, r , f are the non-dimensional axial displacement, stress in the plastic element, and applied force per unit length along the bar, respectively; u is the stress in the elastic-plastic element, which depends on u and r ; and the subscript x denotes a derivative with respect to x. The extension e is the derivative in space of the non-dimensional generalized displacement u, and it can be decomposed into elastic, ee, and plastic, ep, parts:

169 The stress u is related to the elastic part of the extension by means of Hooke's law,

Therefore (3) leads to

u ( u ,r ) = Dux

+ DH-'r

where D = (E-'

+ H-l)-'.

(4)

Carrying ( 4 ) into ( I ) , we obtain E ~ U " - Dux,

-D H - ' ~= ~ f.

(5)

The behavior of the plastic element is characterized by the non-dimensional inequality and flow rule:

The governing dynamic equations (5),together with the conditions (6) can be put in the form of a singular perturbation system of first order differential equation and inclusion. For that purpose, let C denote the following closed convex set in L2(0,L )

and let sign-'(r) be the normal cone to C at r E L2(0,L). Then we observe that (6) can be written in the differential inclusion form:

(ep)' E sign-' ( r ).

(8)

Relations (3) lead to

(ep)' = d-l (EU;- r') where D = E

+ H.

(9)

Substituting (9) in ( 8 ) ,we get

Eu; - r'

E

sign-' ( r ) .

We now introduce the following spaces

vo

'H = ~ ~ L), ( 0 v, = ~ ' ( 0L ), , = H;(o, L ) , W = { ( u , r )E Vo x C : u = D(ux H-'r) E V ) .

+

(10)

170 We will denote the norm in Fl (resp. V) by I . I (resp. I I 11) and the scalar product in Fl by ( . , . ). From (5) and (10) we finally obtain the governing dynamic system EU' - v

= 0, - DH-l~, = f ,

EV' - DU,,

(11)

-' Eu; - r' E ~ s i ~ n(r), together with the Dirichlet boundary conditions and the initial conditions The corresponding quasi-static system is then (let

E =0

in (11))

with the Dirichlet boundary conditions = 0 on (0, L )

x (A1,A2),

(15)

and the initial conditions Note that, consistently with the above, the quasi-static displacement rate with respect to the physical time vanishes (V E 0). Note that the dynamic system (11)-(13) can be written in the equivalent variational form: Find (u, r) E W such that V(u*,r*) E W, 1 1 (E2.", u*) Z(ux,11): -(r, u:) = (f, u*),

+

(rl,r - r * )- (u:,

+2

T

- r*) 5

(17)

0,

with the initial conditions (13). The corresponding variational formulation of the quasi-static problem (14)-(16) is: (Find (ii,P) E W such that V(ii*,T*)E W, 1 ;i(7,B:) = ( f , ~ * ) ,

+

(r', T - r*) - (G:,

(18)

r - T*) 5 0,

with the initial conditions (16). Finally note that if X is a space of scalar functions, the bold-face notation X d will denote the space xd.

171 3. Existence and uniqueness of solution for the dynamic

and the quasi-static systems We observe that the dynamic and the quasi-static systems introduced in Section 2 can be rewritten in a form that may be studied with the theory of m-accretive operators. Recall that existence and uniqueness of solution to the differential inclusion problem

can be obtained from the following Proposition:

Proposition 3.1. Assume that A is an m-accretive operator in the Hilbert space y, g belongs to w ~ ~X 2(; y x ) and ~ x1, E ?)(A). Then there exists a unique solution x of (19) belonging to W ~ T( X"I ,X 2 ; Y ). The reader can find a detailed proof of this Proposition in [5]or in [4]. By applying Proposition 3.1, we prove existence and uniqueness of solution for the dynamic system (11)-(13) and for the corresponding quasi-static system (14)-(16).Differentiating with respect to x the first equation in the system ( l l ) ,performing a change of unknown function by using e = u, and denoting x = (D1I2e,v , D-li2r), we get the inclusion (19a) with

+

Let us define 2 = { ( e , r ) E 'Hz : De DH-lr E V ) . First we check by direct estimate that A is a monotone operator (see [7,11]).Second, if ( g l , f , g 2 ) E 'H3 and ( v ,e, r ) E V x 2 , there exists h ( r ) E sign-'(r) E 7-f for which the resolvent equation ( 1 A)(D1I2e,v , ~ - l / ' r )3 ~( g l , f / E , g Z ) T is equivalent to solving the system

+

172 This is equivalent to solve for v E V the following equation:

The form is coercive, and existence of a solution follows. The components of (e, r) E 2 are obtained directely from the first and third terms in (20) respectively. Hence, we conclude that A is m-accretive. For more details, see [8,9].Observing that with p = 3 and Y = V x 2, Proposition 3.1 yields the following Corollary:

Corollary 3.1. Assume that f belongs to W1~"(l~,X2;'H)and that (13) holds. Then there exists a unique solution (v, e,r) of (11)-(13) such that (v, e, r ) and (v', e', r') belong respectively to Lw (Xi, X2; V) x LY(X1, X2; 'FI) and Ly(X1, X2; 'H) and (De DH-'r) belongs to LM(X1,Xz; V).

+

Remark 3.1. According to Corollary 3.1 and since e = u,, u = 0 on (0, L), u and u' belong respectively to L" (A1, X2;VO)and Lw(X1, X2;Vo). In what concerns the quasi-static problem, we differentiate the first identity of (14) with respect to X to get

with Dirichlet boundary conditions. We deduce that (21) has a unique solution ii'. On the other hand, iih H - ~ P ' depends linearly and continuously on f', i.e.

+

Inserting this in the inclusion in (14) leads to

The sub-differential d q ( ~ )= sign-'(f) is an m-accretive operator since p(P) is a proper convex and lower semi-continuous function. For F, A x = Hsign-l(F), g = DBf' with p = 1in (19) and y = V, we apply Proposition 3.1 and we obtain the following Corollary:

Corollary 3.2. Assume that f belongs to W1rM(X1,X2;'FI) and (16) holds. Then there exists a unique solution ( i i , ~ )of (14)-(16) such that (ii,F) and (ii', F') belong respectively to L" (Xi, X2; Vo) x Lw (Xi, X2; 'H) and LM(X1,X2; Vo) x Lm(X1,X2, 'H) and (DG, DH-lf) belongs to Loo(X1,X2; V).

+

173 4. Stability of quasi-static p a t h s of elastic-plastic systems

In Section 4.1, we adapt the definition of stability of a quasi-static path [2,6,12] to the present elastic-plastic problem with hardening, which can be seen as the limit of a sequence of elastic-visco-plastic problems. In Section 4.2, we introduce such elastic-visceplastic problems and we recall existence and uniqueness results for them. In Section 4.3, a priori estimates on the elastic-visco-plastic system are obtained which, in Section 4.4, lead to the proof that the dynamic and the quasi-static solutions remain close to each other if they start sufficiently close, and the loading rate e is sufficiently small. From now on we assume, without loss of generality, that E = H = 1. 4.1. Definition of stability of a quasi-static path

The mathematical definition of stability of a quasi-static path at an equilibrium point is presented in the context of the governing dynamic system (11)-(13) and the quasi-static system (14)-(16). Definition 4.1. The quasi-static path (ii(X),f(X)) is said to be stable at X1 if there exists 0 < AX 5 X2 - XI, such that, for all 6 > 0 there exists p(6) > 0 and F(6) > 0 such that for all initial conditions u1, v1, r1 and .til, f 1 and all E > 0 such that lull2

+ 1ulX- .tilXl2+ Irl -

f1I2

< p(6) and e 5 ~ ( 6 ) ,

the solution (u(X), v(X), r(X)) of the dynamic system (11)-(13) satisfies

+

+

Iv(x>l2 Iux(X) - iiX(X)l2 Ir(X) - f(X)I2 for all X E [A1,

X1

< 6,

+ AX].

For more details, the reader is referred to [2]. 4.2. Existence and uniqueness of solution for the

elastic-visco-plastic systems We introduce here the elastic-visceplastic systems: 1 1 where J p ( r p ) = - (rP- p r ~ j C r p ) ,(23) P

with the Dirichlet boundary conditions

174 and the initial conditions

Here projc denotes the projection on the convex C. The variational formulation of the problem (23)-(25) is the following: Find (u,, r,) E W such that V(u*, r*)E W, 1 1 ( E ~ U ; , ~ * )-2( ~ p x , ~ ; ) I(r,,u;) = ( f , ~ * ) ,

+

(rh,r*) - (~:,,r*)

+

+ 2(3,(7-,),7-*)

(26)

= 0,

with the initial conditions (25). Note that this elastic-visco-plastic problem is an Yosida regularization of the original elastic-plastic problem. For a similar approximation in the corresponding finite-dimensional system see [6]. Let us define v, = EU',.

Proposition 4.1. Assume that f belongs to W1~w(X1,X2;?i) and that (25) holds. Then there exists a unique solution (u,,v,,r,) of (23) -(25) such that (u,, v,, r,) and (u',, vh, rh) belong respectively to Ly(X1, X2; V O x) Lm(X1, Xz; 7f) and Ly(X1, X2; 7f) and (u,,, +r,,) belongs to Lm(X1, Xz; 7f). Moreover, as p tends to zero, (up, up, r,) converges strongly to its limit.

+

Idea of the proof. We regularize (26) in the space variable. Then a priori estimates and the Galerkin method (cf. [13]) lead us to the desired result. The reader can find a detailed proof in the Appendix of [ll]or in [3]. Observe that this Proposition can be also proved using the theory of maccretive operators. 4.3. A priori estimates

Lemma 4.1. Assume that (25) holds and f belongs to W19"(X1, X2; 'H). Then independently of p > 0, for all X belonging to (Xi, Xz), v,(X), u,,(X) and r,(X) are bounded in 'H. Proof. This estimate results from the application of Gronwall's lemma to energy estimates. Choosing u* = 2uL and r* = r, in (26), and adding both identities, we obtain ~ ( E ~ U2:~,:)

+ (ups, u:,) + (TI,r,) + 2(3,(r,),

r,) = 2(f, u:).

(27)

175 Observing that (Jp(r,), r,) is non negative we conclude from (27) that

We integrate (28) over (XI, A), X E [XI,Xz], and since vp = &uL,we get

Integrating by parts in time the right hand side of (29), we obtain

We estimate the product (2,y) by lz12/2yi+yily12/2, and, choosing different values for yi, i = 1,2,3, in different terms, we have

where

On the other hand, the Poincarh inequality (see [14,15]) shows that there exists a strictly positive constant c such that

Using (31) in (30) and choosing yl = 7'3 = 2c and 7'2 = 1 in (30), we may infer that 21vp(X)l2

+ 2141(,.l

+ Ir,(4I2

1

5 .I+ 2

JIl

Iupx12dt.

(32)

By classical Gronwall's lemma, we get

As the last term on the right hand side of (32) is now easily estimated, we finally obtain

from which the desired result follows.

176 Lemma 4.2. Assume that (25) holds and f belongs to L2(X1, X 2 ; 'H). Then for all X belonging to ( X I , X z ) , v,(X) -+ v(X) strongly in 'H, u,,(X)

-+

u,(X) strongly in 'H,

r P(A) --, r(X) strongly i n 'H, as l - ~tends to 0.

Proof. These convergence properties are obtained by energy estimating the difference between the elastic-visco-plastic system and the elastic-plastic system with hardening. Choosing u* = u;-u' and u* = ul-u; respectively the first identities in (23) and (17), and adding both identities, we get

Observing that the second identity in the system (23) implies that

Carrying (35) into (34) and integrating over (A1, A), X E [ X I , X z ] , and using the initial conditions ( 2 5 ) and (12) lead t o the following identity

Since (J,(r,), r , - r ) is non negative, v, = EU; and v = EU', then we may deduce from (36) that

The conclusion follows from Lemma 4.1. The finite dimensional (Galerkin) aproximation of the above elastovisco-plastic problem is following. We let { w j ) z l be a complete orthonormal sequence in 'H whose elements belong to H2(0,L). Let u,, = EL1gin(X)wi(x)and r,, = CZ1hin(X)wi(x)satisfying the following variational formulation For all U* = Cy=lgrn(X)wi(x)and T* = EL1hL(X)wi(x), ( E ~ ~ ; ,7 ,u*)

(rL,,,r*)

+

1

- ($',,,r*)

7

u:)

+

1 (T,,

> u:)

+ 2(Z,(r,,),r*)

=

(f u*)

= 0,

9

7

(37)

177 with l i ~ , , xy=l gin(X1)wi(x) = 211, limn-+, and limn,, EL1hin(Xl)wi(z)= T i -

gin(Xl)wi(x) =

U1

Lemma 4.3. Assume that (25) holds and f belongs to W21w(X1, X 2 ; 'FI). Then there exists a subsequence, still denoted by vhn, such that

vCn

-

v; weakly

*

in LW(X1,X z ; 'FI).

(38)

Moreover there exists a positive constant c(X1,X 2 ) that depends on the interval of X and such that

Proof. This estimate results from the energy estimate, Gronwall's lemma and the proof can be completed by a classical Galerkin method. We drop now the subscript n. Differentiating the governing system (37)with respect to A, taking u* = 2~'u; and r* = ~ ' r and ; finally adding both identities, we get

+

+

2 (2 U~I,l l ,E 2 u 11p ) (u;,, E'u;,) (r;, e2r;) i2((3,(r,))I,~~rL)= 2 ( f l , ~ ~ u ; ) . The monotonicity of r,

H

(40)

&(r,) leads to

Then we deduce from (40) that

We integrate (41) over ( X I , A), X E [ X I , X z ] , and since v, = EU;, we get

On one hand, we subtract the first equation in (23) at X 1 to the first one in (14) at X I . From (25),we deduce that

IEV

1

2

( X I )1 5

I ( u I ~ ~ + ~ 1 %-) (

~

+

1'.

~ 11 % )~

~

(43)

Moreover the initial condition r,(A1) = rl E C implies that 3,(rl) = 0 and then the second identity in (23) leads to the following identity 2

l&r;(Xl)12= 1 ~ 1 x 1

(44)

178 On the other hand, we integrate by parts the right hand side of (42), and we estimate the product (z, y) by lzI2/2yi yi 1 y12/2, and, choosing different values for yi, i = 1,2,3, we get

+

.

.

Since v = EU' then the Dirichlet boundary conditions and the Poincar6 inequality show that there exists a strictly positive constant c such that 1v,(t)l2 5 clvPz(t)l2, vt E ( h , ~ ~ ) . Carrying (46) into (45), choosing yl = y3 = c and 7 2 = 1, we have

(46)

Introducing (43), (44) and (47) in (42), we obtain

where

By classical Gronwall's lemma, it is clear that lv,,(~)1~6 2g(Ai, E)exp(Xn - Xi).

(49) Therefore the last term on the right hand side of (48) is now easily estimated. We finally obtain

which proves the Lemma. Proposition 4.2. Assume that f belongs to W2700(X1,Xz; 'H) and that (13) and (16) hold. Then there exist yi > 0, i = 1'2, such that

+ I r ( 4 - +)I2 5 71(11v11I2 1 + Irl ~ -1f1I2~+ I ( ~ l z z + rlx) - (Glxz + v1z)I2) + 6 7 2 -

Iv(Wl2 + luz(X) - a x ( 4 I 2

+ Iulz - ~

(50)

179 Proof. This result follows from an energy estimate of the difference between the dynamic elastic-visceplastic system and the quasi-static elasticplastic system. Choosing u* = u; - a' and r* = (r, + T)/2 in (26), c* = u-1 - u; and f * = (f r)/2 in (18), and adding the resulting ex-

+

pressions, we obtain the following inequality:

Since T E C then 3,(T) = 0, and due to the monotonicity of 3,, we get ( 3 (TP ~

~p

- f)

= (3,

Using (52) in (51) and since v,

(r,) - 3, (T),T, - T) 2 0.

= eu;,

(52)

we infer that

We integrate (53) over (A1, A), X E [A1, X2] and we obtain Iv.(X)12

1

1

+ ZIupx(X)- ~ x ( X )+l ~II~p(X)- T(X)I2

+~~(i'-a:,r,-r)dcsc(~~)+2

(54)

where c(X1) = lull

2

1 1 + -lulx 2 - iilXl2+ 2

-17.1

- fl12.

Let us observe that

where

Carrying (55) into (54) and using Cauchy-Schwarz's inequality we have

180 where X

hp,n(Xl,X) = l 1 ( i t - fik,rp- r ) d c + 2 Introducing (39), the estimate obtained in Lemma 4.3, in (56), we deduce that there exist yi > 0, i = 1,2, such that

+

<

- fix(X)I2+ Ir(X) - r(X)I2 2hp,,(X1, A) yl ( l l ~ 1 1 1 ~ lv(X)l2 + I~x(X) 2 - 2 + I ~ l x- u1x1 Ir1 - ~ l l I('111xx rlx) - ('11lXX rlx)I2) E72. The conclusion follows then from Lemma 4.2.

+

+

+

+

+

4.4. Stability of a quasi-static path

In order to prove the stability result, it is convenient to compare the dynamical solution (v, u, T) with another dynamical solution (fi, 6, F) that solves (23) with the Dirichlet boundary conditions (12) and the initial conditions

Let us remark that the variational formulation of that problem is the following: Find (G, F) E W such that V(G*,F*) E W, 1 - 1 ( E ~ G ~ ~ , G l(ux,u:) *) -(F,G:) = ( f , F ) , 2 (Ft,F - F*) - (GL,F - F*) 5 0,

+

+

(58)

with the initial conditions (57).

Lemma 4.4. Assume that (25) and (57) hold and that f belongs to L2(X1,X2; 7f). Then

+ Iux(X) - G , ( X ) ~+~Ir(X) 5 21v1 - fi(X1)I2+ (ulx- filXl2+ -

21v(X) - 6(X)I2

(TI

-

F(X)I2

~ 1 1 ~ .

(59)

Proof. Once again we use energy techniques to compare two elastic-plastic problems with hardening that have the same boundary conditions but different initial conditions. Choosing u* = ut - Gt and G* = Gt - ut in (17) and (58), respectively, we have

On the other hand, taking r* = F and F* = r in (17) and (58), respectively, we get (TI

- f l , r - 7)

5 (r - 7, u:

- ii;).

(61)

181 Carrying (61) into (60) and since v = eu' and fi = eii', we obtain

We integrate (62) over (A1, A), X E [A1, A2], and using the initial conditions (13) and (57), leads to the result in the Lemma.

Proposition 4.3. (Stability). Assume that (25) and (57) hold and that f belongs to L ~ ( XA2; ~ ,7-t). Then there exist y > 0 such that for 0 < E < 1,

+ Iux(A) - G x ( ~ )+l ~I+)

Iv(A)I2

5~

l2

+

( 1 ~ 1lulX- ~

1

-r ( U 2

+ Irl - r1I2+ e). ~

1

~

Proof. The stability result follows from the estimates obtained in Proposition 4.2 and Lemma 4.4. Let us remark that (59) leads to the following inequality

where

+

+

c(A1) = 21v1 - fi(X1)I2 Iulz - iilXl2

17-1 -

r1I2.

On the other hand, choosing u = ii, v = fi and r = F in (50) and using the fact that ii(A1) = til and .-(A1) = Fl, we obtain

+

+

(A) +.-(A) -~ ( ~ 1 1 ~ lfi(x)12 I I . ~ ~-~ G,(A)(~

+E Y ~ .

(64)

Introducing (64) in (63), we get lv(A)12

+ Iux(A) - iiX(A)l2+ Ir(A) - r(A)I2 < yllIfi(A1)112+ 2c(A1) + ~ 7 2 .

Since e'(A1) and GL(X1) are bounded in 7-f and fi(X1) = E C ~then the Proposition follows.

Acknowledgements This work is part of the project "New materials, adaptive systems and their nonlinearities; modelling control and numerical simulation" carried out in the framework of the European Community Program "Improving the human research potential and socio-economic knowledge base" (contract nOHPRN-CT-2002-00284). J. A. C. Martins, M. D. P. Monteiro Marques and A. Petrov were partially supported by F.C.T.

182 (Fundas50 para a Ciencia e a Tecnologia) / P O C T I / F E D E R a n d by Project POCTI/MAT/40867/2001 "Estabilidade de Traject6rias Quase-Esttiticas e Problemas d e Perturbas50 Singular". A. Petrov thanks also t h e Research Center "MATHEON" (project C18) for generous support.

References 1. B. Loret and a. M. J. A. C. SimGes, F. M. F., Flutter instability and illposedness in solids and fluid-saturated porous media, in Material Instabilities in Elastic and Plastic Solids (Udine, 1999), ed. H. Petryk, CISM Courses and Lectures, Vol. 414 (Springer, Vienna, 2000) pp. 109-207. 2. J. A. C. Martins, F. M. F . SimGes, A. Pinto d a Costa and I. Coelho (2004). 3. G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics (SpringerVerlag, Berlin, 1976). Translated from the Fkench by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219. 4. C. Johnson, J. Math. Pures Appl. (9) 55, 431 (1976). 5. C. Johnson, J. Math. Anal. Appl. 62, 325 (1978). 6. J. A. C. Martins, M. D. P. Monteiro Marques and A. Petrov (2005). 7. H. Brbzis, Ope'rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (North-Holland Publishing Co., Amsterdam, 1973). North-Holland Mathematics Studies, No. 5. Notas de Matem6tica (50). 8. R. E. Showalter and P. Shi, J. Math. Anal. Appl. 216, 218 (1997). 9. R. E. Showalter and P. Shi, Comput. Methods Appl. Engrg. 151, 501 (1998). 10. E. Zeidler, Nonlinear functional analysis and its applications. 111(SpringerVerlag, New York, 1985). Variational methods and optimization, Translated from the German by Leo F. Boron. 11. J . A. C. Martins, M. D. P. Monteiro Marques and A. Petrov (2006). 12. J. A. C. Martins, N. V. Rebrova and V. A. Sobolev (2004). 13. R. Dautray and J.-L. Lions, Analyse mathe'matique et calcul nume'rique pour les sciences et les techniques. Vol. 8INSTN: Collection Enseignement. [INSTN: Teaching Collection], INSTN: Collection Enseignement. [INSTN: Teaching Collection] (Masson, Paris, 1988). ~volution:semi-groupe, variationnel. [Evolution: semigroups, variational methods], Reprint of the 1985 edition. 14. H. BrBzis, Analyse fonctionnelleCollection Mathematiques Appliqubes pour la Maitrise. [Collection of Applied Mathematics for the Master's Degree], Collection Mathbmatiques Appliqubes pour la Maitrise. [Collection of Applied Mathematics for the Master's Degree] (Masson, Paris, 1983). Th6orie et applications. [Theory and applications]. 15. A. Kolmogorov, S. Fomine and V. M. Tihomirov, ElLments de la the'orie des fonctions et de l'analyse fonctionnelle (Editions Mir, Moscow, 1974). Avec un compl6ment sur les algkbres de Banach, par V. M. Tikhomirov, Traduit du russe par Michel Dragnev.

183

MATHEMATICAL RESULTS ON THE STABILITY OF QUASI-STATIC PATHS OF SMOOTH SYSTEMS N. V. REBROVA* and J. A. C. MARTINS

Dep. Eng. Civil e Arquitectum, and ICIST, Instituto Superior TBcnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal *E-mail: [email protected] http://wunu.civil.ist.utl.pt V. A. SOBOLEV

Dep. of Differential Equations and Control Theory, Samara State University, Ac. Pavlova str 1, 443011 Samara, Russia http://www.~~u.samara.m In this paper we present sufficient conditions for stability of quasi-static paths of finite dimensional systems that have a smooth behavior. The concept of stability of quasi-static paths used here is essentially a continuity property relatively t o the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces (which plays here the role of the small parameter in singular perturbation problems). These conditions are applied t o a mechanical system with a convex potential energy.

Keywords: Stability of quasi-static paths; Singular perturbations; Differential equations.

1. Introduction

The governing equation for the dynamic evolution of mechanical systems is Newton's law, i.e. force equals mass times accelemtion. A classical approximation for the equations that govern the slow evolution of mechanical systems is to neglect inertia effects and take the balance equations as static equilibrium equations, i.e. force equals zero. The slow evolutions made up of the successive equilibrium configurations are called quasi-static evolutions. The relationship of this issue with the theory of singular perturbations has been established in Ref. 4, where the existence of fast (dynamic) and slow (quasi-static) time scales is recognized: a change of variables is performed that replaces the (fast) physical time t by a (slow) loading parameter

184 X whose rate of change with respect to time, E = dX/dt, is eventually decreased to zero. In the finite dimensional case this change of variables leads to a system of differential equations or inclusions defining a singular perturbation problem, i.e. a dynamic problem where some terms with the highest order derivative with respect to X appear multiplied by the small parameter E . The quasi-static problem is governed by the reduced equations obtained by letting E = 0 in the singularly perturbed system. In this context mathematical concepts of stability and attractiveness of quasi-static paths were presented in Ref. 5 and sufficient conditions for attractiveness or instability were proved. As might be expected such results rely on the existence of strictly negative or strictly positive real parts of the tangent operator t o the fast flow a t the initial position of the quasi-static path. In this paper we investigate the stability of quasi-static paths and, for that purpose, we shall use functions with properties close to Lyapunov functions, though the presented concept of stability differs from Lyapunov stability:

+

we consider solutions on a bounded interval [XI, XI AX] of the slow independent variable while in Lyapunov stability theory the interval of the independent variable is infinite; classical Lyapunov theory considers the distance between two solutions while in the present discussion the quasi-static path in general is not a true solution of the dynamic problem. On the other hand Lyapunov functions with negative definite derivatives along the solution are used in some works on singular perturbations (see Refs. 2,3,6 for example) to prove Tikhonov type results (that is the solution of the associated system is attractive, or the eigenvalues of the corresponding matrix have negative real parts). We shall investigate here the case where the derivatives of the Lyapunov type functions are allowed to be non positive. Consider the system in the form:

where x and y are m- and n-vectors, respectively, ' denotes a, E > 0 is a small parameter, and f and g are vector functions of corresponding d

185 dimensions. The initial conditions are: x(X1) = x l , y(X1) = y1

If we set c

=0

we obtain the reduced problem 3' = f (A, 2 , g , 0 ) , 0 = g(X,37Y)7

for which some initial condition is specified:

Let y = +(A, 2 ) be a solution of the second equation in (3). Substitute y = +(A, 3 ) into the first equation of (3). Then we obtain the system:

with the initial condition (4). In what follows we shall investigate the distance between the solution ( x ( X ,E ) , y(X, E ) ) of the system (1)-(2) and the solution ( 3 ( X ) ,g(X)) of the reduced system (3)-(4). If this difference can be controlled on some finite interval [A1,X1 AX] by the size of the parameter E and the closeness of the initial values ( x l , y l ) to (2', y(X1)) we say that the quasi-static path of the system (1)-(2) is stable. Formally,

+

Definition 1.1. The quasi-static path x = 3(X), y = g(X) is said to be stable at A1 if there exists 0 < AX 5 X2 - X I , such that, for all 6 > 0 there exists p(6) > 0, E(S) > 0 such that for all initial conditions x l , y l , 31 such that 11x1 - 3111

+ l l ~-l %(Xl)Il < P(6)7

the solutions of problem (1)-(2) x

for all X E [ X I , XI

= x(X, e ) , y = y(X, E ) satisfy

+ I~Y(X, + AX] and E E (0,E(6)]. Ilx(k&)-

(6)

- ?7(X>II< 6,

(7)

2. Sufficient conditions for the stability of the quasi-static

path. Assume that:

(I) The functions f , g together with their derivatives up to the second order are uniformly continuous and bounded on all the variables in some region:

186 and let L be a Lipshitz constant for the function f . (11) The solution jj = $(X,Z) of the second equation of (3) is isolated, that is there exists a positive number q such that in the neighborhood 11y - $(A, x) 11 5 q no other solution exists. Moreover

for all X E [A1, Xz], x E Rm,where M I is some constant. Consider the auxiliary equation

where A*, x* are considered as parameters. And let ijo be its steady state :

Denote $ = y j - go and assume also that

(111) There exists a function V($, Go) such that

where W is a continuous and strictly increasing function on [0,co),such that W(0) = 0. The function V is a Lyapunov-type function for the problem governed by the auxiliary equation (9) with the steady state (10). We shall prove now that the assumptions (I), (11) and (111) are sufficient to guarantee the stability of the quasi-static path according to the Definition 1.1. Denote g(X1) by 8' and for $ = y - 8 and Go = jj consider the equality

av(w(s)7 )' - av('(s)'

a8

)'

,$A(s,~ ( s )+) &(s, ~(s)).')) ds.

187 The first product in the integral is non-positive due to (13). Thus,

+dz(s7 Z(s))f(s,?(s),d(s7 z(s)),o)llds. Applying ( l l ) , (8) and (12) we obtain

where the constant A bounds the function f . Applying Bihari inequality (see Ref. 1) we get

where

and .tois a constant value lower than J. Denoting cl = F(V($(X1),g l ) ) we obtain the following estimate

Put Eo = MlMz(l+ A)AX and denote w = max{&) A)AX I s I MlMz(l+ A)AX cl. Then

+

+

on M1M2(l

Now, consider the difference (x - z):

which, by taking (8) into account yields

Ilx(x) - Z(x)ll 5 Ilx(X1) - ~ ( A l ) l l +ELAX +L i:(llx(s) -

+ IY(s) - 9(s)ll)ds.

Putting together (14) and (15) we derive

(15)

188

and by Gronwall lemma, we finally get

The inequality (16) shows that the difference between the dynamic solution and the quasi-static path is controlled by the initial conditions and by E.

Proposition 2.1. If conditions (I)-(111) are satisfied, then the quasi-static path x = %(A),y = g(X) of the system (1) - (2) is stable at XI. 3. A mechanical problem with a convex potential energy

In this section we consider the application of the previous theory to a second order evolution problem which, in a mechanical context, involves a system with p degrees of freedom. For simplicity, after appropriate non-dimensionalization and change of independent variable, this mechanical system has a diagonal unit mass matrix, a potential energy of internal forces U(u) and a vector of external applied forces h(X). u(X) E Rp is the vector of generalized displacements, X E I = [A1, AS] denotes the load parameter (slow time variable) and E is the small loading rate. Again ()' denotes a derivative with respect to X and

au(~> denotes the gradient of U, that is a vector with components -. du dui So, the governing equation is: with initial conditions

Denoting section:

EU'

by v and putting the problem in the form of the previous

we observe that in this problem there is no slow variable x, the fast variable a u (F ~ ) h ( ~ ) ) with is y = { u , v ) ~and g(X, y) = {v, ~ , n = 2p. In what

+

189 follows, a subscript u will denote the subvector of the fist p components of n-dimensional vectors. We assume the following hypotheses

(1) The function h is bounded together with its first and second derivatives

( 2 ) The function U is a function with continuous first and second partial derivatives with respect to all ui and satisfies the inequality

where q is some constant. (3) For the matrix of second derivatives and p such that

there exist constants k

(4) There exists an isolated on I solution C = +,(A) of the equation

We shall now check that the conditions ( I ) - (111)of the previous section hold when the above assumptions 1 -4 hold. Condition ( I ) is satisfied due to Conditions 2 and 1. Condition (11) is satisfied due to Condition 4 and the properties of function U. In order to prove Condition (111)we start by proving that u and v are bounded. For that purpose, multiply equation (17) by u' and integrate it. E~ (u",u')

E ~ ( uu') ',

+ (avazl(u),u') = (h(X),u'),

+ 2U(u)- ~ ~ ( u ' ( Xul(XI)) l ) , - 2U(u(X1))= 2 JL (h((l,ul(l))dl,

) 2U(.u(X))I 2U(Ul))+ ( ~ 1v1) , + (h(X), u(X)) (v(X),W ) + -(h(Xi),W ) - 2 J;~( h l ( ~~)(,1 ) ) d l .

(23)

190 Then,

where cl = 2U(ul)

+ ((v1(I2 + cllulll + c2AX. Note, that

In the and either 2Lllu(X)11q- cllull > 0 and growing or llull I (&)A. latter case u is obviously bounded. If 2LJlu(X)119 - cllull > 0 then we denote J ( X ) = H(IIuII) = 2LIIu(X)IIQ- cllull and rewrite inequality (25) as X

J ( h ) 5 cl

+ 2 c l 1 H-l(J(l))dl.

Since H a is growing function, we can apply Bihary inequality:

Jzl

X_d'B(s) with some constant xo where G ( x ) = G - l ( G ( c l ) 2cAX) by a, we can write

+

2Lllu(X)II9- cllull

< x. Denoting the constant


The last inequality shows that u(X) is bounded by H-'(a). Hence,

l l v ( ~ ) 15 1 ~cl

+~ c H - ' ( ~ ) A A .

Now we consider the auxiliary problem

with the steady state {Go,

and we construct the function V with $ = {ii - Go, GIT

191 Note, that V can be rewritten as follows:

where 0 5 8 5 1. Obviously, from (28) it follows that V satisfies (11). Since u and v are bounded then is bounded and (12) holds (M3u is some constant):

a2u(3~0'uu.)

Finally, the inequality (13) also holds:

According to Proposition 2.1 the quasi-static path of the system (17) is stable. 4. Final remarks

The singular perturbation nature of the problems studied here and the similarity (or better, the compatibility) that exists between the classical concept of Lyapunov stability and the present concept of stability of a quasistatic path are quite clear. This is why in the Introduction of this paper we stressed the differences between those two concepts, as well as the difference between the application made here of Lyapunov-type functions and their application in the theory of singular perturbations. The differences stressed earlier have consequences on the conditions and on the proof of stability presented in Section 2. As in singular perturbation theory, it is the equation that governs the fast flow that is relevant for the stability. Since we are working in a finite interval of the slow time (load parameter) the contribution of the equation that governs the slow flow to the stability is only through the trivially obtained continuity with respect to the initial data. The Lyapunov type function that is needed here is the Lyapunov function for an auxiliary autonomous differential equation which is the equation of the fast flow with

192 the slow time (A) and the slow dependent variable ( x ) considered as "frozen" parameters (A* and x * ) . Though that auxiliary system (9) is an autonomous one, the corresponding Lyapunov function will depend on those parameters through the steady state GO. Since along a finite interval of the slow time A this steady state coincides with the slowly evolving quasi-static state, this dependence causes additional terms in the estimate of the derivative of the Lyapunov function along the trajectories of the original system. Condition (12) together with conditions (I) and (8) allow us t o control these terms. Note also that the inequality (13) is usually stated in the singular perturbation theory in a stronger manner than ours: the derivative of the Lyapunov function is bounded above by -cll$/I, which means that the auxiliary system is asymptotically stable. This stronger condition leads t o control the additional terms by the smallness of the parameter c.

Acknowledgments V.A. Sobolev is supported by Programm 22 of General Committee of RAN and Programm 16 of the Departament of Energetics, Mechanics, Engineering and Control of RAN.

References 1. I. A. Bihari. A Generalization of a Lemma of Bellman and its Applications to Uniqueness Problems of Differential Equations. Acta Math. Acad. Sci. Hung., 7(1), 81-94, 1956. 2. A. Fradkov, B. Andrievsky, Singular Perturbations of Systems Controlled by Energy-Speed-Gradient Method. in 43d IEEE Conference on Decision and Control, 2004 3. I. S. Gradstein, Application of Lyapunov's Stability Theory to the Differential Equations with Small Multipliers before Derivatives. Math. collect., Vol 32, NO 2, 263 -286, 1953. 4. B. Loret, F. M. F. Simdes, J. A. C. Martins. Flutter Instability and Illposedness in Solids and Fluid-saturated Porous media. In: Material Instabilities in Elastic and Plastic Solids, H. Petryk, ed. International Centre for Mechanical Sciences, Courses and Lectures No. 414. Springer-Verlag, Wien, New York, 2000. 5. J. A. C. Martins, N. V. Rebrova, V. A. Sobolev. On the (in)stability of quasistatic paths of smooth systems: definitions and sufficient conditions.Math. Methods Appl. Sci, 29, No. 6, 741-750, 2006. 2004. 6. A. A. Martynyuk, V. G. Miladzhanov, Stability of singularly perturbed systems under structural perturbations. General theory. Int. Appl. Mech. 39, No.4, 375-401, 2003.

193

SENSITIVITY ANALYSIS OF ACOUSTIC WAVE PROPAGATION IN STRONGLY HETEROGENEOUS PIEZOELECTRIC COMPOSITE E. R O H A N ~and ~ ~B. M I A R A ~ a Department

of Mechanics, Faculty of Applied Sciences, University of West Bohemia, Univerzitni 22, 30614 Plzed, Czech Republic E-mail: rohanokme. zcu. cz Labomtoire de Mod6lisation et Simulation Numdrique, ESIEE, $ boulevard Blaise Pascal, 93160 Noisy-le-Grand, France E-mail: m i a r a b o e s i e e . f r

We consider a composite medium made of weakly piezoelectric inclusions periodically distributed in the matrix which is made of a different piezoelectric material. The medium is subject to a periodic excitation with an incidence wave frequency independent of scale E of the microscopic heterogeneities. Using the unfolding method of homogenization we obtain the limit homogenized model which describes acoustic wave propagation in the piezoelectric medium when E -+ 0. In analogy with the purely elastic situations (phononic crystals), the resulting model is featured by the existence of acoustic band gaps. We present the homogenized material coefficients and discuss their sensitivity with respect t o model parameters. This study is aimed at developing modelling tools t o be used in optimal design of new acoustic devices for "smart systems". Keywords: Homogenization; Acoustic waves; Band gap; Piezoelectric composite; Sensitivity.

1. Introduction

Modelling of acoustic wave propagation in heterogeneous media gains growing interest as an attractive problem from both theoretical and practical points of view due to numerous applications in designing devices related to sound propagation. In the context of mathematical modelling, the method of homogenization was proposed to study the heterogeneous elastic media (the so-called phononic crystals) in1 and recently treated in,'y3 where the unfolding operator method was used and numerical results were reported. The phononic crystals - bi-phasic elastic media with periodic structure and with large contrasts in elasticity of the phases - are called often the

194 phononic band-gap materials due to their essential property to suppress propagation of elastic waves in certain frequency ranges, as confirmed by measured transmission, e.g.4 Similar phenomena in the propagation of the electromagnetic field were studied even before in the context of the photonic crystals, cf.12 The phononic crystals, or composites may be used in modern technologies to generate frequency filters, as beam splitters, sound or vibration protection devices (for noise reduction), or they may serve as waveguides. In general, there are several ways how these materials can be used and designed to achieve a desired - optimal - signal transmission, or attenuation; they can be controlled "actively" by external fields (using mechanical forces, or electric field), or they may be designed using microstructure-related parameters to obtain a "passive" control due to the feedback inherited from their material and structural composition at a sub-macroscopic level. Since for optimal use of these composites with periodic structure we need to predict their behaviour at both macroscopic and microscopic scales, the method of homogenization provides an extremely useful tool which provides a direct and mathematically rigorous link between material models relevant to each of the scales considered. For elastic composites an existence of band gaps for certain wavelengths was shown in3 as the consequence of the non positivity of the limit "hcmogenized mass density". In the present paper we formulate the problem of acoustic wave propagation in a piezoelectric strongly heterogeneous composite to pursue the same approach based on the two-scale homogenization. The main theoretical results on the limit homogenized model are presented briefly (the details and a deeper study of the band gaps will be reported ins). Due to our research interests in the optimal material design for the piezoelectric composites, as the main focus of the paper we issue the sensitivity analysis of the effective material characteristics. The problem of wave propagation is introduced in Section 2 for periodic structures made of two different piezoelectric materials with large contrasts in all their (relevant) material parameters, except of material density, which is assumed to be of the same order in both the materials. By virtue of the homogenization method we study the limit behaviour of the model for the geometrical scale of the heterogeneities tending to zero; the limit model is presented in Section 3, the scale-decoupling procedure is discussed in Section 4. Finally, in Section 5 we develop the sensitivity analysis of the homogenized coefficients w.r.t. a general model parameter.

195 2. Problem setting

We consider a piezoelectric medium whose material properties, being attributed to material constituents vary periodically with position; the period is denoted by E. Throughout the text all quantities varying with this microstructural periodicity are denoted with superscript €. 2.1. Definition of the strongly heterogeneous material

The material properties are related to the periodic geometrical decomposition which is now introduced. We consider an open bounded domain R c R3 and the reference cell Y =]0,1 [3 with inclusion c Y, whereby the matrix part is f i = Y \ E . Using the reference cell we generate the decomposition of R as follows Rq=

U E ( Y z + ~ )w, h e r e K E = { k ~ ~ / e ( k + CE R) ) , kEIKC

- -

so that R = Rt U R; U P, where rEis the interface r E= RE1 n 05. Properties of a three dimensional body made of the piezoelectric material are described by three tensors: the elasticity tensor czjkZ,the dielectric tensor dij and the piezoelectric coupling tensor giij, where i, j , k = 1,2,. . . , 3 . As usually we assume both major and minor symmetries of cijkz (ctkl = c ; = ~ ctlij), ~ ~ symmetry of d:j, i.e. d t = d;, and the following one of g ; . . : g;,. =g;.. 23 3 3%' We assume that inclusions are occupied by a "very soft material" in such a sense that there the material coefficients are significantly smaller than those of the matrix compartment, except the material density, which is comparable in both the compartments; as an important feature of the modelling, the strong heterogeneity is related to the geometrical scale of the underlying microstructure by coefficient E ~ : ~ ' ( 5 )=

{

p1 in Rt , in a;,

giij in Rf, c2gEij in R;,

ctjkl czjkl(x) = {E2c;j,1 dzrj (x)=

in R f , in n;,

d:j in Rf, ~ ~ in d 05. : ~

2.2. Problem formulation

We consider stationary wave propagation in the medium introduced above. Although the problem can be treated for a general case of boundary conditions, for simplicity we restrict the model to the description of clamped

196 structures loaded by volume forces and subject t o volume distributed electric charges. Assuming a synchronous harmonic excitation of a single frequency w

where f = (fi), i = 1,2,3 is the magnitude field of the applied volumic force and q is the magnitude of the distributed volumic charge, in general, we should expect a dispersive piezoelectric field with magnitudes (u', cpE)

This allows us to study the steady periodic response of the medium, as characterized by field (uE,cpE)which satisfies the following boundary value problem:

where the stress tensor uE = (u;) defined by constitutive laws

and the electric displacement DE are

uzj = c?.jklekl(uE)- ggijdk(pE, Di

= giijekl(uE)

+ dildlcpE.

(2)

The problem (1) can be weakly formulated as follows: Find (u', pE) E H;(R) x H t (R) such that

for all (v, +) E H; (R) x H; (0). 3. Homogenized model

We apply the unfolding method of homogenization5 to obtain a limit model when E -+ 0. Since the detail development of the model will be given in

197 a forthcoming publication,8 we just record the resulting system of coupled equations which describe the structure behaviour a t two scales, the "macroscopic1' one and the "microscopic" one. However, for the reader's convenience we recall the basic framework of the periodic unfolding operator. For all z E R3 let [z] be the unique integer such that z - [z] E Y, recalling Y ~ 1 0I ,[ ~for ; more general situations of Y being a parallelepiped the definition of [z] can be adapted. We may write z = [ t ]i- { z ) for all z E R3, so that, consequently, for all E > 0 we get the unique decomposition

Based on this decomposition, we can define the periodic unfolding operator I,() : v E L2(R;R) -t L2(R x Y; R); for this we shall consider functions v E L~(R; R) as extended to L1(R3;R) by zero (vanishing) outside R, i.e. v = 0 in R3 \ R. Now the unfolding operator is defined by

Using the unfolding operator, it can be shown that I,(ue) --\ u1 + x y 2u 2 p1 +xy2 p2 weakly in L2(R x Y), weakly in L2(R x Y) and similarly I,(9") where xy, is the characteristic function of the inclusion, Y2. 3.1. Coupled system of homogenized equations

The coupled system of equations is defined in terms of three piezoelectric fields (ull

$4E

HA(^) x H;(R)

(c,(P) E L ~ ( RH;(Y)) ;

1

x ~ ~ (H$(Y)) 0 ;

,

(4)

(u2,p2) E L ~ ( ~ ; H ; ( Y ~x)L2(fl; ) ~ t ( ~ 2, ) ) where by H$(Y) and H$(Y) we denote the space of those functions in H1(Y) and H1(Y), respectively, which are Y-periodic. In the sequel we shall use the following abbreviations: by df and diy we denote d/dxi and d/dyi, respectively, whereby we adhere this meaning also in the context of the symmetric gradients (strains) ;e and eyj. Further, for averaging in Z c Y we use & = IYI-lJz.

Limit coupled system of equations. The piezoelectric response of the composite medium relevant to the macroscopic level is given in terms of

198 ( u l ,c p l ) ; this field is coupled with others in (4) by means of the following variational equalities

+

S,Ll -

c;jr-r[e;r(ul) + ~ L ( G )e;(vO) I

Jn 31 g ~ ~ ~ e ; ( [a;pl v o ) + alp] = S,f

Jn 3E,

v O,

y1

(5)

gLij [e; ( u l )+ e b (u)Ia;@

+ Jn {

dtj[a;pl

+ q p ]a:",!JO=

y1

for all v0 E Hi(S-2) and for all

Jn

q?bO,

?bO E Hd (a).

Microscopic equations for perforated domain. The corrector functions ( C ,p) satisfy variational equations imposed in Yl c Y , thus in the "perforated domain" :

for all 8 E H $ ( Y ) and $ E H $ ( Y ) . The response ( u ,p) can be expressed in terms of ( u l ,cpl).

Microscopic equations for inclusions. In inclusions Y 2 , the "macroscopic response" given by ( u l ,p l ) is corrected using filed ( u 2p2) , which depends on ( u l ,c p l ) through the following equations

+

L2 ~

~( u 2 ~erj ) ( 6 )~-

~g&ebe (i)a:l p 2 = ~f

.

(7)

to be satisfied for all 6 E H i ( & ) and $ E H6(Y2). Note that ( 5 ) depends on the solution of ( 7 ) exclusively by virtue of the inertia term involving u2.

199 4. Decoupling scales in the limit problems

In this section we present decoupling procedure; using the linearity of all the equations involved, this consists in introducing such local microscopic problems associated to (6) and (7), that their solutions are independent of the macroscopic response presented by (ul, PI). This, in turn, allows for computing the homogenized coefficients which constitute the macroscopic (global) problem.

4.1. Local problem i n the inclusion

The system (7) exhibits self-oscillating solutions due to the inertial term in (7)1, however, the response of the electric field is instantaneous (we recall the assumption of sufficiently low frequencies w , which permit neglecting the coupled electro-magnetic field oscillations). In order to decouple the Iocal response from the global one presented by (ul, cpl), we proceed as follows:

(1) we define a particular solution cp2P w.r.t. the r.h.s. in (7)2, (2) and consider a modified system (7) with zero r.h.s. in its second equation, as derived from (7)2; its solution can then be expressed in terms of the eigenelements defined below.

For brevity in what follows we employ the following notations:

whereby analogical notations are used when the integrations apply over Yl.

200 4.1.1. Particular solution Let us define satisfying

(p2P

= q(x)@(y) where @ E Hh(Y2) is the unique solution

v$ E ~

1

hence also dy2 ( l p 2 P , $) = q(x)

$J

yz

V$

,

i ( ~ 2 )

E

H ; ( Y ~ ).

(9)

4.1.2. Spectral problem The spectral problem associated with (7) reads as: find eigenelements [AT;( z r , p r ) ]where , z' E Hi(Y2) and p' E HA(Y2), r = 1,2,.. . , such that ~ Y Z( z T ,v ) - ~ Y ( Z v ,P')

= Xreyz ( z T ,v )

Vv E H ; ( Y ~ ) ,

~YZ(~',$)+~Y~(P',$)=O ~$EH;(Y~), with the orthonormality condition imposed on eigenfunctions z r :

(10)

+

(11) ay2 ( z r ,z S ) dy2 (pr,p S ) = Arey2 ( z r , z S ) Ar6,,. The orthogonality in (11) follows easily on rewriting (10) for v = z S and $ = P', ( z T 1z S )- gyz ( z S pr) , = ATeyz( z r ,z s ) , gyz ( z S ,P') + dy2 ( p S 1pr) = 0, so that on eliminating gyz ( z S ,pr) one obtains ~ Y Z

+

aY2 ( z r ,z S ) dy2 ( p s l pT) = Ar@yZ( z r ,z S ) !

= ASeY2 ( z S ,z r )

Moreover, the ellipticity of ay2 (., 1,2, ....

a)

and dy2 (., .) yields A'

.

> 0 for all r

=

4.1.3. Solution decomposition With the eigenelements ( z r ,pr) defined in (10)-(11)and having computed ( p 2 P , we look for the solution to (7) in the decomposed form:

u 2 ( x ,Y ) =

C~'(x)z'(Y) 1

201 On substituting (12) into system (7) we arrive at -w

2

ey,

(u2, v)

+ ay2 (u2, v) - gy2 (v, v ~ ~ ) gy2 (u2, $) + d y ,

$) = o V$

( ( P ~ ~ ,

E

fft(y2). (13)

By virtue of (12) also (13)2 holds, since both u 2 and (p2H are introduced in terms of the same coefficients ar. In order to treat (13)1, we substitute therein v = zS, which yields [-w2ey2 (zr, zs)

+ ay2 (zT,za) - gy2(zs, pT)]

r

= f ( x ) . k 2 v +w2u1(x)

-12

(14)

p2v +q(x)gy2 (v,8)

Using (10)l and ( l l ) , we obtain the left-hand side of (14) in the form

Car[-w2eyZ (zT,za) + ay2 (zT,z a ) - gy2 (zS,pr)] = as (A"- w2)

,

r

which furnishes a simple expression for coefficients a T :

4.2. Homogenized coefficients a n d the macro-model

First we shall focus on the homogenized coefficients which are responsible for the dispersive properties of the homogenized model, since they depend on the incident wave frequency. Whereas this first group of the "frequencydependent coefficients" depends just on the material properties of the inclusion (except of the material density), the second group of coefficients is related exclusively to the matrix compartment.

In order to express the homogenized coefficients involved in the macroscopic momentum equation (to be discussed later on), we need to evaluate ,9'u2 =

C [f (x) . Sy, z ' 8 L 2 p2zT+ w2u1(x) - l2 P2ZT8L2 P2ZT r>l

202 which is substituted in the 1.h.s. of (5). To simplify the notation we introduce the eigenmomentum rnr = (m:),

mr =

Jy,p2zr

Having collected in (5) all terms involving accelerations -w2u1, forces f and imposed charge q, we recover the following tensors, all depending on w2: Mass tensor A* = (A;j)

Applied load tensor B* = (Bij)

Applied charge tensor Q* = (Qf)

4.2.2. Coeficients related to the perforated matrix domain

As mentioned above, the second group of the homogenized coefficients is defined independently of the inclusions material. In other words, the elasticity CGkl, piezoelectricity Giij and dielectricity Dil homogenized coefficients can be recovered in the same form which is defined for periodically perforated piezoelectric material. Moreover, due to the nature of (6) they do not depend on the frequency. Below we summarize some results which follow as consequences of the homogenization treated in7 for a piezoelectric bi-phasic composite. In order t o compute C*, G* and D*,we must solve the local microscopic problems for the corrector functions; these are now listed. (1) Find (xij, .lrij) E H$+(Yl) x H i (Yl), i,j = 1,. . . , 3 such that

+

ayl (Xij nij,v) - gy, (v, rij) = O , VV E H$+(Yl), gyl (xij llii, @) dy1 (rij, @)= 0 , V@ E H i (YI) ,

+

+

where l l i j = (112) = (yjbik);

(19)

203 (2) Find

(xk,a k ) E H$.+(Yl)x H$(Yl),

i, j = 1,. . . , 3 such that

+

ay1 (xk,v) - gy1 (v, rk nk)= 0 , Vv E H;(Y~) , xk,$)+dy1(ak+nk,$) =O,V$EH$(Y~),

(20)

where Hk = yk. Using the corrector basis functions just defined we compute the homogenized coefficients:

4.3. Global equations - the macromodel

It has been outlined above, how the limit homogenized system is decoupled by virtue of the local solutions defined in Y and using the homogenized coefficients which are involved in the macroscopic problem derived from (5). Now we are ready to state the macroscopic (global) equations; we find ( u l , (pl) E Hi(S-2) x Hh (a) such that

and

As an important feature of the limit macroscopic equations, its inertia term is defined in terms of the w-dependent homogenized mass tensor ATj. It was proved in3 for the elastic media that there exist intervals of frequencies for which the limit problem admits only an evanescent solution; these intervals are called the acoustic band gaps. More precisely, such intervals are indicated by negative definiteness, or negative semi-definiteness of A&(w2);

204 while the first case does not admit any oscillating solution, in the latter one the admissibility of an oscillating response depends on the assumed direction of amplitudes of propagating waves. Thus, some frequencies may result in a strongly anisotropic behaviour of the homogenized medium. Similar conclusions can be derived also in the present situation with the piezoelectric coupling. The detailed study of this phenomenon will be pursued in the further research. 5. Sensitivity analysis of t h e spectral problem

The frequency-dependent homogenized coefficients depend on the spectral problem (10) through eigenvalues AT and the eigenmomenta m r , which are defined in terms of eigenfunctions zr. In order to cope with optimal material-design problems and with the sensitivity of the band gaps in particular, as discussed above, we need to analyse the sensitivity of A,tj with respect to model parameters; these may be introduced as material parameters of the composite constituents, or as geometry-related design parameters, e.g. the shape of inclusion. In this section we investigate general sensitivity formulae for eigenvalue problem (10). With this result in hand it is then possible to treat subsequent analysis of homogenized coefficients (16)-(18). In what follows by 6f we denote a variation of function f w.r.t. some model parameters, whereby for example 6c(., -) is the variation of a bilinear form c ( - ,.). On differentiating in (10) we obtain for all v E HA(Y2) and $ E H:(Y2) Sayz (zr, v)

+ ayZ(6zr, v) - 6gy2 (v, pr) - gy2 (v, 6pr)

= 6Areyz (zr, v)

+ ArSevz (zr, v) + Areyz(6zr, v) ,

(23)

bg~z(zr, $1 +gyz (6zr, $) +6dy2 (pr, $) +dy2 (6pr, $) = 0 . Since also 6p3 E H:(Y2), we can use $ = 6pr in

so that for any r, s 2 1

In (23)1 we use v = as, then we add the resulting equation with (24), thus obtaining

+

+

(zr, z3) ay2 (bar, z3) - 6g~z(zS, pr) dy2 (p3, 6pr) =6Ar~y2(zr, z3) Ar6~yZ (ar, z3) Arey2(6zr, zS) .

+

+

(25)

Above the last left-hand side term can be eliminated using (23)2, which rewritten with substitution $ = pS yields

205 Hence, using (26) in (25) we have that

+

dayz ( z r , z S ) aYz (har,a s ) - bgy, ( z S ,pr) -

+

+

( 6 z r ,ps) Sgyz (z', pS) Sdy2 (P', pS)] ATSeyZ( z r , z S ) AreyZ(Sz', z S ) .

+

+

= SArey2 ( z r ,z S )

(27)

Finally, we use

which follows by (10)1 , therein with test function v replaced by 6zr E HA (Y2); on substituting this identity in (27) we obtain the principal sensitivity formula

(AS - A')

@y2(6%')z S )= Sdyz

+

(pr,pS) SgYz ( z S ,pr) + Sgyz ( z r ,p S )

- bay2 ( z r , a s )

+ JAreyz (z', z S )+ A ' S ~ y 2 ( a r , z S )

(28) Using this formula, for all r 1 we can determine sensitivities ( S A T ,S z r ) , assuming well separated eigenvalues, i.e. AT # As for any r # s.

>

5.1. Sensitivity of eigenvalue AT

In (28) we consider r = s, so that the term on the 1.h.s. containing differential Szr vanishes. Thus, using the orthonormality (1I ) , we obtain the eigenvalue sensitivity in the form

SAT = bay2 ( a r , z r ) - 26gy, ( a r ,pr) - bdy, (p', p')

- Ar6py2 ( z r , a')

. (29)

5.2. Sensitivity of eigenfunction 3

Due to compactness of eigenfunctions { z S } s 2 1in HA(Y2)we may use the following decomposition of the unknown sensitivity:

6zr=x<;zk,

where

[ 6 = e y Z ( z k , z r ) , k = 1 , 2 ,... .

k>l

(30)

Therefore, the sensitivity Szr can be determined in terms of coefficients {<;}k>l as follows. For r # s from (28) we compute =

1

AS - A ' 16dy2 (pry P") + 6gyZ( z s ,P')

+

+ Sgy, ( z r ,pS) - Say, ( z r ,a s )

+SArey2 ( z r , z S ) Ar6ey2 ( a r , z S ) ], s # r.

(31)

206 In order to determine [F we employ the orthonormality property; we differentiate (11) for r = s, so that using (30) we get

6. Conclusion

In this paper we introduced problem of acoustic wave propagation in a class of piezoelectric composites constituted by two materials with large difference in their material properties; this forms an important assumption of the present model, where the material parameters of individual constituents are related to the geometrical scale of heterogeneities. The two-scale homogenized limit model was presented, which will allow for further studies of the homogenized material, especially in the context of the acoustic band gaps. In particular, we have in mind optimal designs of such piezo-phononic structures; the results on the eigenvalue problem sensitivity, as presented in Section 5, will allow 1) to derive the sensitivity of homogenized coefficients defined in Section 4.2.1 and 2) to derive the sensitivity of the bang gap bounds. Consequently, it is possible to treat the associated problem of optimal band gaps distribution w.r.t. t o model parameters using gradient based numerical methods. Similar studies have already been pursued in the elastic whereby the sensitivity analysis of the homogenized piezoelectric material coefficients was reported in.lo As one of further steps in this focus, the sensitivity results should be extended to allow for treatment of band gap structures with multiple eigenmodes, see (10); such situations characterize the so-called strong band gaps, ~ f . thus, , ~ they may arise in well designed structures for some merits of their optimization.

Acknowledgement. This work has been supported by the European Community's Human Potential Programme under contract "Smart Systems" number HPRN-CT-2002-00284 and in part also by project MSMT 1M06031 "Materials and components for environmental protection". References 1. J.L. Auriault and G. Bonnet, Dynamique des composites elastiques periodiques, Arch. Mech., 37 (1985), 269-284. 2. A. ~ v i l a ,G. Griso, B. Miara, Bandes phononiques interdites e n e'lasticite' line'arise'e, C . R. Acad. Sci. Paris, Ser. I 340, 2005, 933-938. 3. A. ~ v i l aG.Griso, , B. Miara, E. Rohan, Multi-scale modelling of elastic waves,

207 Theoretical justification and numerical simulation of band gaps. Multiscale Modeling & Simulation", SIAM journal, submitted (2006). 4. S. Benchabane, A . Khelif, A . Choujaa, B . Djafari-Rouhani and V . Laude, Interaction of waveguide and localized modes i n a phononic crystal. Europhys. Lett., 71 ( 4 ) , (2005) 570-575. 5. D. Cioranescu, A . Damlamian, G . Griso. Periodic unfolding and homogenization. C . R. Acad. Sci. Paris, Ser. 1 335 (2002) 99-104. 6. B. Miara and E. Rohan, Shape optimization of phononic materials. 77th Annual Meeting o f the Gesellschaft fiir Angewandte Mathematik und Mechanik e.V. ( G A M M ) , Berlin, (2006), t o appear in PAMM. 7. B . Miara, E. Rohan, M. Zidi and B . Labat, Piezomaterials for bone regeneration design - homogenization approach, Jour. o f the Mech. and Phys. o f Solids, 53 (2005) 2529-2556. 8. B. Miara, E. Rohan and G . Perla, Phononic problem i n strongly heterogeneous piezoelectric composite. Forthcoming paper (2006). 9. E. Rohan, O n the strongly heterogeneous elastic media. Direct upscaling. Sbornfi seminee: Modelovani a mBfeni nelinearnich jevou v mechanice. ZCU, S K O D A V f z k u m , NeEtiny (2006), 189-198. 10. E. Rohan and B . Miara, Homogenization and shape sensitivity of microstructures for design of piezoelectric bio-materials. T o appear in Mechanics o f Advanced Materials and Structures (2006). 11. E. Rohan and B. Miara, Sensitivity analysis for optimal shape design of phononic structures. Forthcoming paper (2006). 12. E. Yablonovitch, 1993.Photonic band-gap crystals. J. Phys. Condens. Matter, volume 5, pages 2443-2460.

208

NEW RESULTS ON THE STABILITY OF QUASI-STATIC PATHS OF A SINGLE PARTICLE SYSTEM WITH COULOMB FRICTION AND PERSISTENT CONTACT F. SCHMID

Weierstmss Institut for Applied Analysis, Mohrenstrasse 39 10117, Berlin, Germany E-mail: [email protected] J.A.C. MARTINS and N. REBROVA

Instituto Superior Tecnico, Dep.Eng.Civi1 and ICIST, Av.Rovisco Pais, 1049-001 Lisboa Portugal E-mail: jmartinsOcivil.ist.utl.pt In this paper we announce some new mathematical results on the stability of quasi-static paths of a single particle linearly elastic system with Coulomb friction and persistent normal contact with a flat obstacle. A quasi-static path is said to be stable a t some value of the load parameter if, for some finite interval of the load parameter thereafter, the dynamic solutions behave continuously with respect to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces, E (as in singular perturbation problems). In this paper we prove sufficient conditions for stability of quasi-static paths of a single particle linearly elastic system with Coulomb friction and persistent normal contact with a flat obstacle. The present system has the additional difficulty of its non-smoothness: the friction law is a multivalued operator and the dynamic evolutions of this system may have discontinuous accelerations.

Keywords: Coulomb friction; Quasi-static; Persistent contact; Stability

1. Introduction

The study of the stability of frictional contact systems has deserved an increasing attention ( Shevitzl , Adly & Goeleven2 , Van de Wouv & Leine3 , ~ r o g l i a t o, ~Sinou et a1.5 , Duffour & Woodhouse6) due to its relevance in many engineering applications ( Ibrahim7 , Kinkaid et a1.8 , Sinou et al.') as well as in geophysics ( Gu et al.1° , Scholzl1).

209 The concept of stability that one has in mind in many mechanical situations is the concept of Lyapunov stability, which, in particular, can be used to study the stability of the equilibrium configurations under constant applied loads(dynamic trajectories with zero velocity and acceleration). In what concerns the non-smooth friction problems, a discussion on the attractiveness of equilibrium sets with the application of LaSalle's principle can be found in Van de Wouw and Leine3 , while the works of Shevitzl and ~ r o g l i a t odevelop ~ non-smooth Lyapunov functions. A related but different issue is the stability of quasi-static paths of mechanical systems under slowly varying applied loads. In general, the concept of Lyapunov stability cannot be applied to quasi-static paths because such paths are not, in general, true solutions of the original governing dynamic equations (Loret et a1.12). But the "stability of quasi-static paths" can be related to the theory of singular perturbations (see again 12):the physical time t can be recognised as a fast (dynamic) time scale and a loading parameter A, whose rate of change with respect to time, E = dX/dt, is arbitrarily small, can be recognised as a slow (quasi-static) time scale. Changing the independent variable t into X in the governing system of dynamic differential equations or inclusions, one is led to a system in which some of the highest order derivatives with respect to X appear multiplied by the small parameter E. In this manner, following the mathematical definition of stability of quasi-static paths proposed by Martins et al.13 ,I4 a quasi-static path is stable at some point if, in some subsequent finite interval of the load parameter, any dynamic trajectory does not deviate from the quasi-static one more than some desired amount, provided that the initial conditions for the dynamic evolution are sufficiently close to the quasi-static path, and the loading is applied sufficiently slowly. After the study of some smooth cases and some problems that have a not very severe non-smoothness (the elastic-plastic problems with linear hardening)14 , this paper applies the same definition to a class of linearly elastic problems with friction that has a more severe non-smoothness: discontinuous acceleration and friction forces. The structure of the article is the following. In Section 2, the governing dynamic and quasi-static equations and conditions are presented, and the definition of stability of quasi-static paths is recalled. In Section 3, existence results for dynamic and quasi-static problems with persistent frictional contact are recalled and refined. Section 4 contains an auxiliary result on the regularity of the solution of the quasi-static problem, which is shown t o have a derivative with bounded variation. This result is essential to esti-

210 mate, in Section 5, a contribution that involves the product of the inertia term in the dynamic equation with the derivative of the quasi-static solution. The main result of this paper, the stability of the quasi-static path, is then proved in the final Section 5. 2. Governing equations and definition of stability of the

quasi-static path We consider a linear elastic system with two degrees of freedom: a single particle system. Its configuration is determined by the displacement u E IR2 of the particle. In the following we write ut and u, for the tangential and normal displacement components, respectively. This is motivated by the assumption that the particle cannot penetrate a rigid obstacle and this restriction is modelled by the inequality u, 2 0. The evolution of the system is described in terms of the load parameter A, which is linked via the small load rate parameter E > 0 to the physical time t: A = ~ t The . elastic behaviour is modelled by the 2 x 2 positive definite stiffness matrix K , while the applied and the reaction forces acting on the particle are represented by the vector functions with values in IR2, f (A) and r(A), respectively:

The derivative d( )/dA is denoted by ( ) I . krthermore, p 2 0 denotes the coefficient of friction, and in the whole article we assume that for some given time A > 0 we have

The equation of motion in the dynamic case is

where, without loss of generality, we assume a unit mass. The equation of motion in the quasi-static case reads Ku(A) - f (A) = r(A).

(34)

The unilateral contact conditions satisfied by the solutions are given by

211 Introducing the set-valued sign function -1 forsO

we can formulate the Coulomb friction law as follows

In the whole article we assume that we are in situations of persistent contact, so that un = 0. Then, in the dynamic case, the equations (3-a), (4) and (5) lead to the dynamic problem with persistent contact: For given uo, vo E R find ut E W29"([0, A], R) satisfying the initial conditions

and such that, for all X E [0,A],

To distinguish the dynamic solution from the quasi-static one, the latter is denoted by fit. By taking E = 0 in (7) we formally get the corresponding quasi-static problem with persistent contact, which reads: For given iio E R, find Et E W1~"([O, A], R) satisfying

and such that, for all X E [0,A],

Note that for X = 0 this immediately implies some restrictions on the initial condition Go. We can now introduce the

Definition 2.1 (stability of a quasi-static path). Let iit be a quasistatic path, i.e. a solution of the quasi-static problem (9)-(11). W e call the quasi-static path iit stable at X = 0, i f there exists some positive interval of loading parameter values 0 < AX I A, such that for every 6 > 0, we can find constants Cini(6) > 0 and C,(6) > 0, such that for each parameter E and initial conditions uo and vo at X = 0 with

1 ~ 0 1 + IUO

- Eel < Cini(6) and

E

< C,(6)

(12)

212 the dynamic solution ut of (6)-(8) remains near the quasi-static path i n the following sense

for all X E [0,AX]. 3. Existence of solutions In the article of Martins et al.15 it is shown in a quite more general situation, that for initial conditions with positive normal reaction (i.e. kntuo-fn(0) > 0) there exists some A, E (0, A] for which a solution of (6)-(8) exists in the interval [0,A,]. In order to guarantee existence of solution up to an arbitrary given load parameter A > 0, we need a stronger assumption on f that holds on the whole interval [0,A]. Lemma 3.1 (Existence of a dynamic solution). There exists a constant C > 0 that depends on all data except the external normal force fn, such that, for each normal force fn satisfying -fn(X) > C,

for all X

E

[0,A],

(14)

a solution of (6)-(8) exists i n [0,A]. R e m a r k 3.1. By classical results from the theory of differential inclusions we know that for general f E C2([0,A], R2) there exists a solution ut E W29" ([o,A], R) of the inclusion (7) with the initial conditions (6). See for example Aubin16 , Page 98, Theorem 3. The rest of the Proof consists of using energy estimates to show that under assumption (14) on the external normal force f,, this solution ut automatically satisfies (8) for all X E [0,A]. The full proof can be found in1? . L e m m a 3.2 (Existence of a quasi-static solution). Assume that p > 0 and the quasi-static initial condition ti0 E IR satisfies

and let

hold. Then there exists a constant C > 0 that depends o n all data except the external normal force fn, such that, for each normal force fn satisfying -fn(X)

> C, for allX

E

[O,A],

(17)

213 there exists a quasi-static solution iit E w ~ ~ " ( [ oA],R2) , of the problem (9)-(11). Furthermore the solution satisfies

Remark 3.2. The proof follows directly from a result in Mielke & Schmid18 , where existence of a quasi-static solution even without the limitation of persistent contact was proven. Persistent contact is shown under the assumption -fn > C analogous to the prove of Lemma 3.1. ~ l a r b r i n ~ l ~ has shown that if (16) does not hold, one cannot expect in general the existence of a continuous solution to (9)-(11). In the following we assume that fn always satisfies a condition of the form (14) and we focus on the inclusions (7) and (10). 4. Variation of the derivative of the quasi-static path

A short calculation shows that the inclusion (10) is equivalent to the following sweeping process formulation where the set C(X) is defined by

and the corresponding normal cone in u is denoted by Nc(u). In the following we denote by II : 0 = TO < 71 < . < TN, = A, with Nn E N,a partition of the interval [0,A], and we denote by II[O, A] the set of all partitions of [0,A]. The variation of a function of one variable, f : [0,A] -+ Rn, is defined as Nn

var(f;O,A) :=

sup llf(?)-f (~j-1)11ncn[o,~i j=l

We say that f is of bounded variation if var(f; 0, A)

< oo holds.

Lemma 4.1 (bounded variation of the derivative). Assume that the moving set C(X) c R is defined by C(X) := [g(X),h(X)] with functions g, h E C1([O, A], R) satisfying var(g'; 0, A) < oo and var(hl; 0, A) < oo, and also h(X) - g(X) > 0 for all X E [0,A]. Then there exists a solution of

214 Further u is differentiable from the right for all X E [0,A ) , i.e.

ul(X)= lim

u(X

+h ) - u(X)

h Additionally the right derivative u' is a right continuous function with bounded variation, var(ul;0 , A ) 5 var(gl;0 , A ) + lgl(0)l var(hl;0 , A ) + Ih1(0>l. h10

+

Remark 4.1. The proof of this Lemma adapts to the present context of a particle with non-prescribed normal force, some arguments used by Marques2' and Martins et a1.21 for cases with prescribed normal force. The full proof can be found again in17 . 5. Stability of the quasi-static path

From Lemma (3.1) and (3.2) we know that there exist solutions ut,iit : [0,A] + IR of the dynamic problem (6)-(8) and of the quasi-static problem ( 9 ) - ( l l ) ,respectively. First we rewrite the inclusions (7) and (10) by using the functions p, p : [0,A] + [- 1,1] as follows

Note that due to our assumption on fn we have Fn = (kntiit - fn) >.c > 0 , ] ,(-1, I ] ) . Consequently the and then d X )= p ( kkttt,'at (t r( x) -)f-tf(,*()x ) ) E Witm ( [ 0 A right derivative of ?(A) that we will denote by P1(X) exists for almost all X E [0,A ] . By differentiating the first line in (22) we have

To simplify the formula we use the fact that p' # 0 implies .:(A) = 0 due to the right continuity of the right derivative i i i ( X ) and the inclusion in (22). We deduce the following estimate

>

Hence, due to f n = (kntfit - fn) c > 0 , Ip'I is uniformly bounded. Subtracting the equations in (21) and (22) leads us to

+ ktt(ut - fit) = p (prn - p ~ n )= ~

E~U:

( -pp)rn

+pp(m -

fn).

(24)

215 Before multiplying the above equation by (ui -ti:), we observe that the inclusion in (21) is equivalent to -ui(y - p) I 0, for all y E [-I, 11, and (22) is equivalent to -ti:@ - p) 5 0, for all f j E [- 1,1].Choosing y = p and f j = p we get the monotonicity condition (u: - ti:)(p - p)

I 0.

Multiplying then (24) by (ui - Ci), we are immediately led to the estimate This estimate is rewritten after some rearrangements as

(26) Next we integrate the above estimate and further use the estimate (23) on /5/ to obtain

for some finite constant C > 0 defined by estimate (23). Next we apply Gronwall's Lemma to estimate (ut(A) - tit(^))^. In a first step we divide both sides by which is positive due to the assumption (16). We omit the exact and lengthy result and we represent it in the following simplified form. There exist positive constants C1 < oo and C2 < oo, depending on the data K, p and f only, such that

v,

I ClG(&,A) exp ((224 (ut (A) - tit X holds, with G(E,A) := E~ Jo uy(s)tii(s) ds $uf (uo - . i i ~ ) We ~ . can use this to estimate the first integral on the right hand side of (28) by (ut(s) - E ~ ( s ) )ds ~ 5 CIG(E,X) with C3 depending on K , p , f and A only. Using this estimate there exists a constant C4(K, p, f , A) such that

+

+

216 holds for all X E [O, A]. The remaining task is t o estimate the integral E~

u:I(s)&(s) ds.

This uses the results proved in Lemma 4.1 and adapts the argument in21 t o the present case of non-prescribed normal force. The full proof can be found in the article17 . We now summarise our results in

Theorem 5.1 (Stability of the Quasi-Static Path). Let the stiffness matrix K be positive definite and the coefficient of friction p > 0 be such that ktt > ,uJk,tJ holds. I n addition, let the initial condition Go of the quasi-static problem satisfy Ikttao - ft(0)l 5 p(kntaO - fn(0))+, and let the external force f satisfy f E C1 ([o,A],IR2) , v a r ( f l ;0 , A ) < oo and inf {- fn(X) : X E [0,A ] ) 2 C, for some constant C > 0 that depends o n all data except f, (see (14) ) . Then the dynamic (6)-(8) and the quasi-static (9)-(11)problems with persistent contact have solutions and the quasi-static path is stable at time 0 in the sense of definition 2.1. References 1. D. Shevitz and B. Paden, IEEE Transactions on Automatic Vontrol39 ( 9 ) , 1910 (1994). 2. S. Adly and D. Goeleven, Journal de Mathkmatiques Pures et Applique'es 8 3 ( I ) , 17 (2004). 3. N. Van de Wouw and R. Leine, Nonlinear Dynamics 35 ( I ) , 19 (2004). 4. B. Brogliato, Systems and Control Letters 51, 343 (2004). 5. J. Sinou, F. Thouverez and L. Jezequel, International Journal of Nonlinear Mechanics 38 ( 9 ) , 1421 (2003). 6. P. Duffour and J. Woodhouse, Journal of Sound and Vibration 271 (1-2), 365 (2004). 7. R. Ibrahim, A S M E Applied Mechanics Reviews 47, 209 (1994). 8. N. Kinkaid, 0 . O'Reilly and P. Papaclopoulos, Journal of Sound and Vibration 267 ( I ) , 105 (2003). 9. J. Sinou, F. Thouverez and L. Jezequel, Journal of Vibration and AcousticsT+-ansactions of the A S M E 126 ( I ) , 101 (2004). 10. J. Gu, J. Rice, A. Ruina and S. Tse, J Mech Phys Solids 32, 167 (1984). 11. C. Scholz, Nature 391, 37 (1998). 12. B. Loret, F. Sim6es and J. Martins, in: Petryk H (eds) Material Instabilities i n Elastic and Plastic Solids. International Centre for Mechanical Sciences, Courses and Lectures 414 (2000). 13. submitted. 14. J. Martins, M. Monteiro Marques, A. Petrov, N. Rebrova, V. Sobolev and I. Coelho, J Phys Conf-Inst of Physics (2005). 15. J. Martins, M. Monteiro Marques and A. Petrov, Z A M M Z. Angew. Math. Mech. 85, 531 (2005).

217 16. J.-P. Aubin and A. Cellina, Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 264 (Springer-Verlag, Berlin, 1984). Set-valued maps and viability theory. 17. submitted (2006). 18. Coming soon. 19. A. Klarbring, Ingenieur Archiv 60, 529 (1990). 20. M. D. P. Monteiro Marques, European Journal of Mechanics A/Solids 13, 273. 21. J. Martins, M. Monteiro Marques and N. Rebrova, On the stability of a quasistatic paths of a linearly elastic system with friction, in Analysis and Simulation of Contact Problems, eds. P.Wriggers and U.Nackenhorst (SpringerVerlag, 2006). 4th Contact Mechanics International Symposium, Hannover, Germany,July 4-6, 2005.

218

NUMERICAL EXPERIMENTS ON SMART BEAMS AND PLATES GEORGIOS E. STAVROULAKIS Dept. of Production Engineering and Management, Technical University of Crete, Chania, Greece and Carolo WiIheImina Technical University, Braunschweig, Germany GEORGIA A. FOUTSITZI Dept. of Finance and Auditing, Technological Educational Institute of Epirus Preveza. Greece EVANGELOS P. HADJIGEORGIOU Dept. of Material Science and Engineering, University of Zoannina,, Greece DANIELA G. MARINOVA Dept, of Applied Mathematics and Informatics, Technical University of Sofia, Bulgaria EMMANUEL C. ZACHARENAKIS Dept. of Civil Engineering, Technological Educational Institute of Crete Heraklion, Greece CHARALAMPOS. C. BANIOTOPOULOS Dept. of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece Smart composite beams and plates with embedded piezoelectric sensors and actuators are considered. After a short presentation of the mechanical models and their discretization, we focus on problems of active structural control and identification. In particular we solve, using various algorithms, robust optimal control problems and damage identification tasks.

1. Introduction The use of active control techniques in smart structures is an area of intensive research area. Vibration control of composite beams and plates including piezoelectric sensors and actuators is studied. A simplified model that decouples

219 the multi-physics problem is adopted [I-41. The control is based on linear feedback. Since there are always differences between the physical plant, that is controlled, and the model on which the controller design is based (for instance, neglected higher frequency dynamics, damage, etc.) robustness is an important goal for any applicable controller [5-61. The performance specifications, which the control system must llfill and the class of uncertainties for which the control system must be robust against, determine the robust controller for any particular vibration control problem. In this study a vibration control problem in flexible structure (smart beam) is considered and the performance specification is stated in terms of a disturbance attenuation requirement for particular class of external disturbances acting on the structure. In particular this contribution outlines H2 and H,robust controllers for the active vibration control of flexible structures using piezoelectric patches as sensors and actuators. The considered robust control design methodologies lead to linear time invariant feedback controllers. The controllers are designed to acheve optimal performance for a nominal model and maintain robust stability and robust performance for a given class of uncertainties. This is acheved by the solution of two algebraic Ricatti equations, while in classical structural control one such equation arises. A more general nonlinear feedback controller can be constructed with the help of intelligent computational tools. Neural, fuzzy and hybrid control applications are briefly mentioned here. Finally, the existence of actuators and sensors with the corresponding wiring on the smart systems makes possible the consideration of structural health monitoring and damage identification schemes. Some existing results are mentioned at the end of this chapter. This text is based on the cited, original publications of the authors; it's purpose is to demonstrate that smart systems design is a really multidisciplinary field with a large number of theoretical and practical questions, most of them open and suitable for further research. 2. Simplified modeling of composite smart structures In the smart beam of Figure 1, the control actuators and the sensors are piezoelectric patches symmetrically bonded on the top and the bottom surfaces of the host beam. Both piezoelectric layers are positioned with identical poling directions and can be used as sensors or actuators. [7,8,9]. The linear theory of piezoelectricity is employed. Furthermore, quasi-static motion is assumed, which means that the mechanical and electrical forces are balanced at any given instant.

220

-.I

Beam L

Actuator layer

*

I

ontr troller p

Figure I . Laminated beam with piezoelectric sensors, actuators and the schematic control system.

The linear constitutive equations of the two coupled fields read: (01=

[el((&) [ d J PI) -

(1)

I

I

where {a)6x1 is the stress vector, is the strain vector, {D)3,i is the electric displacement, {E)3x1 is the strength of applied electric field acting on the surface of the piezoelectric layer, is the elastic stiffness matrix, [dl3x6is the is the permittivity matrix. Eq. (1) describes the piezoelectric matrix and inverse piezoelectric effect (which is exploited for the design of the actuator). Eq. (2) describes the direct piezoelectric effect (which is used for the sensor). Additional assumptions are used for the construction of the simplified model: (a) Sensor and actuator (SIA) layers are thin compared with the beam thickness. (b) The polarization direction of the SIA is the thickness direction (z axis). (c) The electric field loading of the SIA is uniform uni-axial in the x-direction. (d) Piezoelectric material is homogeneous, transverse isotropic and elastic. Therefore, the set of equations (1) and (2) is reduced as follows

The electric field intensity

EZ can be expressed as

221

where V is the applied voltage across the thickness direction of the actuator and hA is the thickness of the actuator layer. Since, only strains produced by the host beam act on the sensor layer and no electric field is applied to it the output charge fiom the sensor can be calculated using eq. (4). The charge measured through the electrodes of the sensor is given by

where

Sq

is the effective surface of the electrode placed on the sensor layer.

The current on the surface of the sensor is given by

The current is converted into open-circuit sensor voltage output by

where G, is the gain of the current amplifier. Furthermore, we suppose that (5) bending-torsion coupling and the axial vibration of the beam centerline are negligible and (6) the components of the displacement field {u) of the beam are based on the Timoshenko beam theory which, in turn, means that the axial displacement is proportional to z and to the rotation t,u(x,t) of the beam cross section about the positive y-axis and that the transverse displacement is equals to the transverse displacement w(x,t) of the point of the centroidal axis *z=O). The strain-displacement relationshps read

The simpler Euler-Bernoulli theory which considers zero transverse shear deformation Y, has also been tested. The kinetic energy of the beam with the layers can be expressed as

222

on the assumption that the host beam and piezoelectric patches identical densities. The strain (potential) energy is given by

If the only loading consists of moments induced by piezoelectric actuators and since the structure has no bending-twisting couple then the first variation of the work has the form

where 8 is the first variation operator, by the actuator layer and is given by

is the moment per unit length induced

Using Hamilton 's principle the equations of motion of the beam are derived. For the finite element discretization beam finite elements are used, with two degrees of freedom at each node: the transversal deflection wi and the rotation yli. They are gathered to form the degrees of freedom vector Xi = [wi y i ] .After assembling the mass and stiffness matrices for all elements, we obtain the equation of motion in the form

where M and K are the generalized mass and stiffness matrices, F, is the generalized control force vector produced by electromechanical coupling effects,

223

A is the viscous damping matrix and F,,, is the external loading vector. The computer implementation in MATLAB follows the lines of [lo]. It should be mentioned here that bending theories for plates can be constructed analogously. Furthermore, a three-dimensional finite element model of a composite beam, without the simplifications introduced here, is presented in the Chapter by M. Betti et al. in the present Volume. The main objective is to design robust control laws for the smart beam bonded with piezoelectric SIA subjected to external induced vibrations. For this purpose the following state space representation will be used:

as it is common in control problems for general dynamical systems. Here

x is the state vector, A is the state matrix, B1 and B2 are allocation matrices for the disturbances w (corresponding to external forces F,,,) and control u

(corresponding to F,). The initial conditions are assumed to be zero. The identity matrix is denoted by I. 3. Controlled system

Let us consider that the measurements have the following form:

The control law is a linear feedback of the form

where K is the unknown controller gain. The objective in this study is to determine the vector of active control forces u(t) subjected to some performance criteria and satisfying the dynamical equations (15)-(17) of the structure, such that to reduce in an optimal way the external excitations and to meet the above mentioned requirements. The investigations may be implemented in the time domain as well as in the frequency domain. The problem for vibration suppression is solved by both LQR and Hz, Hinf optimal performance criteria. LQR is a state space method, while Hz is a frequency domain approach.

224 3.1. Linear Quadratic Regulator In this section the E2 performance problem in the time domain is studied [ll]. The following quadratic cost function is minimized

The free parameters Q and R represent weights on the different states and control. They are the main design parameters. J represents the weighted sum of energy of the state and control. We require that Q be symmetric semi-positive definite and R be symmetric positive definite for a meaningful optimization problem. The problem (15), (18) is known as LQR problem and belongs to the powerful machinery of the optimal control. Assuming full state feedback, the control law is given by U = -KLQR X,

(19)

with constant control gain

The constant matrix P is a solution of the Riccati Equation

Under technical assumptions existence and uniqueness of the above controller is guaranteed. The closed loop system is given by

LQR method is designed to satisfy specified requirements for steady state error, transient response, stability margins or closed loop pole location. An advantage of the linear quadratic formulation of the problem is the linearity of the control law, which leads to easy analysis and practical implementation. Another advantage is good disturbance rejection and good tracking. The gain and phase margins Imply good stability. All these preferences are met when a complete knowledge of the whole state for each time instance is available. If a limited number of measurements are available and they are supposed to be corrupted by some measurement errors the effectiveness of LQR deteriorates. In this case, first the system is reconstructed by the available measurements, and then the optimal control problem is based on this reconstructed system.

225 3.2. H-2 Control The major problem with LQR is the lack of robustness. Too much emphasis on optirnality and not enough attention to the model uncertainty leads to control that fail to work in real environment. Robustness with respect to external disturbances or uncertainties of the system or loading is the main reason why the authors started studying techniques dealing with feedback properties in frequency domain. We assume that the exogenous signals are fixed or have fixed power spectrum. Since the vibration control is stated in terms of a disturbance attenuation request for a particular class of external disturbances, the H2 robust control methodology is particularly suited. Unlike the standard LQG approach which is based on a nominal model [12], the H2 technique is based on an uncertain system model. Let us divide the system inputs in two groups: exogenous input w and command signals u that are the output of the controller and becomes the input to the actuators driving the plant. The plant outputs are also categorized in two groups: the measurementsy that are fed back to the controller and the regulated outputs z we are interesting in controlling. The plant (15)-(17) can be represented in the more general state space form as

Suppose that the measures y are corrupted and the regulated outputs z are controlled. w, u, y, and z are continuous-time signals. Let T, denote the linear time invariant system from w to z and is its transfer function. We get as a performance criterion the minimization of the H2 norm of Tm

F'

-

over all internally stabilizing controllers K. The Hz norm of F:, minimizes the worst case root mean square value of the regulated variables when the disturbances are unit intensity white processes. This circumstance allows for a state space solution to the frequency domain optimization problem. Under some assumptions it can be shown that there exists a unique controller K2 which minimizes Trnwith the following transfer matrix representation [6]

226

where X and Yare the solutions of the two Riccati equations

A ~ x + x ~ - x B , B , T x + c ~ c=~O

AY

,

+ Y A -~YC;C,T + B,B; = o

(25)

for a stable matrix A. Applications on smart beams have been presented in [8]. 3.3. Uncertainty Modelling and Robust Control

Uncertainty denotes the difference between the model and the reality. The H, approach begins with an uncertain system model for the plant to be controlled. In this section we will consider an uncertainty introduced by varying the nominal plant parameters. Disk-shaped regions on the real axis approximate the variations in the structure system. A multiplicative uncertainty as shown in Figure 2 is assumed.

Figure 2. Introduction of uncertain in the dynamical system.

Let us suppose that the three actual physical parameters M, A , and K in the eq. (14) are lie within known intervals. In particular, the actual mass M is within p~

-

percentages of the nominal mass M , the actual damping value A is within p~ percentages of the nominal value A , and the spring stiffness K is within p~ percentages of its nominal value of K . Further, real perturbations are introduced:

227

which are assumed to be unknown within the values (-1 < 8, Thus, the actual physical parameters of the system take the form

,SA,SK5 1) .

The uncertainty in the matrices M", h and K can be represented by matrix functions called upper linear fractional transformations (LFT) in the perturbations AM,AA and AK respectively [6]

Thus, the considered control design problem will be formulated in a LFT framework. The LFT in (28) have a nominal mapping that are perturbed by AM, AA, AK while the other members of the matrices describe how the perturbations affect the nominal maps. This way the system can be rearranged as a standard one via "pulling out the A's". For thls purpose, we first isolate the uncertainty parameters and denote the inputs of AM,AA, AK as y ~y ,~y~, and their outputs as U M , UA,U K . The outputs uA= [uM,U A , uK]h m the perturbations are added to the yh, yK] to the perturbations are added to system's inputs and the inputs y~ = bM, the system's outputs (see Figure 2). The model for the uncertain system is finally obtained in the following matrix form

228

and represents an LFT of the natural uncertainty parameters dM, a,, dK. The matrix H is a distribution matrix defining the locations of the control forces. The matrix G in eq. (29) is known from the nominal parameters of the system. The system model uncertainty matrix in eq. (29), denoted by A, is a structured matrix.

It is H,norm bounded,

I AI ,

5 1, has a block diagonal structure and influences

on the input/output connection between the control u and the output y in a way that can be represented as a feedback by the upper LFT

y = F, (G,A)u .

(31)

Further we consider the perturbed system (29). The performance criterion is to keep the errors as small as possible in some sense for all perturbed models. The performance specifications will be specified in some requirements on the closed loop frequency response of the transfer matrix between the disturbances and the errors, within the H, theory. The robust stability and robust performance criteria can be treated in a unified framework using LFT and the structured singular value (SSV) ,uh We shall consider the real parametric uncertainty with normbounded dynarnical uncertainty. For the robust stability analysis the controller K can be viewed as a known system component and absorbed into an interconnection structure P together with the plant G,, marked by a dashed line in Figure 2. According to the Nyquist

229 criterion, if the matrices P and A are stable then the interconnection system is stable if and only if det(1- PA) # 0 [ 6 ] . For the robust stability we are interested in finding the smallest perturbation A, real and norm bounded )IAII, < 1 (that is ensured by means of eq. (24)) in the sense of maximal singular value F(A), such that destabilizes the closed loop framework i.e.

det(1- PA) = 0

(32)

The matrix function SSV is defined as

1 PA(P) = min(F(A) : A E D,det(1 - PA) = 0)

(33)

SSV )UA is bounded by the spectral radius p(P) of the matrix P as lower bound and by is the maximal singular value a ( P ) of the matrix P as follows

The interconnection system is well-posed and internally stable for all norm bounded perturbations A if and only if

Hence, the peak value on the ,uAplot of the frequency response determines the size of the perturbations for whch the loop is robustly stable. The quantity

is a stock of stability with respect to the structured uncertainty influenced P. The robust stability is not the unique feature required for the system with parameter perturbations. Often, exogenous influences acting on the system lead to errors in tracking and regulating. Therefore, we need to test the robust performance of the system. The nominal performance of a system is characterized by using the H, norm of some transfer matrix, here we take the weighted sensitivity transfer matrix of the closed loop. We assume that for good performance the following relationship is satisfied

230 The weighting matrix W, is taken such that to suppress the influence of the disturbance on the output. Further details and an application of damage-induced uncertainties of smart beams are given in [ 131. 3.4. H-infinity Control

To obtain a best possible performance in the face of the uncertainties a robust H, optimal control is considered. The implementation of H, control theory is motivated by the inability of the H2 theory to directly accommodate plant uncertainties. Let us present the considered uncertain system (23) by the diagram in Figure 3.

Figure 3. General framework for the H, control problem.

where the exogenous input

w = [uA d p

includes all signals coming to the

system and the error z = [yA ep includes all signals characterizing the system response. Therefore, the system can be represented by the equation

The aim of this section is to design an admissible controller, which. stabilizes internally the system and minimizes the H, norm of the closed loop transfer matrix fiom w to z. The closed loop transfer matrix from w to z is given as a lower LFT in K

z = F, (P,K)w

(39)

Then the optimal Hm control design problem can be formulated as: min The transfer matrix F, (P,K ) contains measures of nominal performance and stability robustness. Its Hm norm gives a measure of the worst case response

231 of the system over an entire class of input disturbances. The optimal H, controller as just defined is not unique for a MIMO system (in contrast with the standard Hz theory, in which the optimal controller is unique). Knowing the optimal H, norm is useful theoretically since it sets a limit on what we can achieve. In practice a suboptimal solution may be useful too: For given Y > 0, find an admissible controller K,(s)such that the H, norm of the closed loop transfer matrix is less than y. Theoretically the optimal controller leads to a difficult, possibly nonconvex optimization problem for whlch many theoretical and algorithrmc questions remain open.

3.5. Nonlinear and Intelligent Control The advantage of classical control, which covers the study of all previously outlined methods, is the availability of mathematical tools for the design of the controller and the study of it's properties, like stability, robustness etc. Nevertheless, one should mention that most beneficial properties are based on the knowledge of the whole dynamical system, which is usually nonrealistic. At this stage an estimator, like a Kalman-filter one, is introduced. The quality and reliability of h s estimator defines the effectiveness of the whole control system. Furthermore, a serious disadvantage is the adoption of a linear feedback. Nonlinear control laws may be more suitable. The tools provided by the classical control for the design of nonlinear controllers are less developed. Therefore nonlinear controllers are mainly based on intelligent and soft computing tools. Without details, we mention several possibilities of using intelligent control in smart structures. 1 . Neural networks can be trained to approximate every nonlinear mapping. They can be used for the approximation of the inverse dynamical mapping of a system Subsequently the trained network is used to suppress vibration of the system. For h s application a large number of representative measurements, or data from modeling, is required for the training and testing of the neural network system 2. Fuzzy inference rules systematize existing experience and can be used for the rational formulation of nonlinear controllers. The feedback is based on fuzzy inference and may be arbitrary nonlinear and complicated. Knowledge or experience on the controlled system is required for the application of this technique. Since the linguistic rules are difficult to be explained and formulated for multi-input, multioutput systems, most applications are based on multi-input, singleoutput controllers. 3. Hybrid techniques that combine the best of every world have also been proposed. For example the required details of a fuzzy Inference system can be tuned by means of examples and neural networks of genetic optimization.

232 4. Inverse and Identification Problems Nondestructive evaluation techmques are often based on dynamic excitation and changes of the response due to an internal defect [14, 151. It is generally accepted that the suitability of the method is case-dependent and that the interpretation of the results strongly depends on the experience of the user. Output error minimization provides a suitable vehicle for an objective study of the arising inverse problems [14, 161. Unfortunately this approach requires the integration of highly sophisticated structural analysis and optimization software and, in addition, may lead to nonclassical nonconvex optimization problems with the possibility of many local minima. Investigations on crack identification problems for two-dimensional elasticity problems (plane stress model) led to meaningful results. Beyond classical optimization or the powerful but expensive genetic optimization, inversion techniques based on neural networks and filter algorithms have also been proposed and tested [14, 17, 181. Recently, an extension to defect identification problems for plates in bending has been attempted. First results, using genetic optimization, demonstrate that this approach is useful [ 191. Two general classes of problems can be identified in this area: 1. Structural health monitoring, where one tries to identify changes of the structural system related to possible damages, cracks etc, and secondly one tries to correlate these changes with concrete sources. Ambient or service loads and corresponding measurements are used for this task. The usefulness of having an instigator of structural integrity and a warning for possibly dangerous changes is obvious. 2. Parameter and defect identijkation is a more complicated task, since one tries to find, in addition, the cause and size of the structural changes. To this end, usually additional test loadings are required, focusing on specific parts of interest in the structure under investigation. The challenge is that smart systems have already integrated sensors and actuators. Therefore one is able to introduce suitably designed test loadings and use the measurements in order to solve both above mentioned problems. The design of the experiments and the post processing of huge amounts of measurements is, by no means, a trivial task. Current research effort focused on the study of the problem for specific structural systems, like beams and plates in bending etc.

233 5. Representative numerical results 5.1. Vibration Suppression of a Smart Piezocomposite Beam 30-5

, FREE

coFLLED

NOMI?

smu:wE

,

,

Figure 4. Vibration of a composite cantilevered beam. Free vibration with and without control, with red (grey) dotted and magenta (grey) solid lines respectively. Controlled vibration with and without damage, with blue (dark) dotted and blue (dark) solid lines.

The effectiveness of a control scheme applied on a composite cantilevered beam for the case of a beam without and with a small damage is schematically shown in Figure 4. A suitable design of the controller makes the controlled system robust and less sensitive to changes of it's mechanical parameters, which is possible due to damage, cracks, delaminations or fatigue of the composite. Details can be found in [13]. 5.2. Damage Identification for a Plate in Bending using Genetic Algorithms

For a plate in bending we consider a dynamical loading and the introduction of a small damage. A suitable error norm transforms the defect identification problem to an optimization problem. Typical contours of the error are shown in Figures 5a. The appearance of local minima and one global minima, exactly at the position of the real defect, is observed. Therefore genetic optimization is used for the solution of the inverse problem. One of the results is documented in Figure 5b. More details can be found in [19].

234

Figure 5. Error function for a damage at position (3,3) (a) and corresponding defect identification using genetic optimization 0).

Acknowledgments The work reported here has been co-funded by the European Social Fund and Greek National Resources, EPEAEK 11-ARCHIMIDIS, at the TEI of Crete. Networking activities have been supported by the European Union Research and Training Network (RTN) "Smart Systems. New Materials, Adaptive Systems and their Nonlinearities. Modeling, Control and Numerical Simulation", with contract number HPRN-CT-2002-00284. References 1. M.A. Trindade, A. Benjeddou and R. Ohayon, Piezoelectric active vibration control of damped sandwich beams, Journal of Sound and Vibration, 246(4), 653-677,2001. 2. V. Balamurugan and S. Narayanan, Finite element formulation and active vibration control study on beams using smart constrained layer damping (SCLD) treatment, Journal of Sound and Vibration,249(2), 227-250,2002. 3 . G. Foutsitzi, D.G. Marinova, E. Hadjigeorgiou and G.E. Stavroulakis, Finite element modeling of optimally controlled smart beams, In: Eds. G. Venkov and M. Marinov, Proc. Of the 2gth Intern. Summer School 'Applications of Mathematics in Engineering and Economics', 8-1 1 June 2002, Sozopol,

235 Bulgaria, Bulvest 2000 Press and Faculty of Applied Mathematics and Informatics, Tech. Univ. of Sofia, pp. 199-207, Sofia 2003. 4. H. Irschik, A review of static and dynamic shape control of structures using piezoelectric actuation. Comput. Mech. 26(2002). 5. K.G. Arvanitis, E.C. Zacharenakis, A.G. Soldatos, and G.E. Stavroulalus, Selected Topics in Structronic and Mechatronic Systems (Series on Stability, Vibration and Control of Systems), New trends in optimal structural control, 321 (2003). 6. K. Zhou, Essentials of robust control, Upper Saddle River, New Jersey: Prentice-Hall, 1998. 7. G.E. Stavroulakis, G. Foutsitzi, , E. Hadjigeorgiou, D.G. Marinova and C.C. Baniotopoulos, Design of smart beams for suppression of wind-induced vibration, In: Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing, B.H.V. Topping, (Editor), CivilComp Press, Stirling, United Kingdom, paper 114,2003. 8. G. Foutsitzi, D.G. Marinova, E. Hadjigeorgiou and G.E. Stavroulakis, Robust Hz vibration control of beams with piezoelectric sensors and actuators, International Conference "Physics and Control", vol 1, St. Petersburg, Russia, 157-162,2003. 9. O.J. Aldraihem, R.C. Wetherhold and T. Sigh, Distributed Control of Laminated Beams: Timoshenko Theory vs. Euler-Bernoulli Theory, J. Intelligent Mat. Syst. & Struct., 8, 149-15, 1997. 10. Y.K. Kwon and H. Bany, The finite element method using MATLAB, Boca Raton: CRC Press, 1997. 11. B. Shahlan, M. Hassul, Control system design using MATLAB, PrenticeHall, NJ, 1994. 12. I.R. Petersen, H.R. Pota, Minimax LQG optimal control of a flexible beam, Control Engineering Practice, 2003. 13. D.G. Marinova, G.E. Stavroulakis, E.C. Zacharenakis: Robust Control of Smart Beams in the Presence of Damage-induced Structural Uncertainties. International Conference PhysCon 2005 August 24-26, 2005, Saint Petersburg, Russia. 14. G.E. Stavroulakis, Inverse and crack identification problems in engineering mechanics. Kluwer Academic Press, Dordrecht, Boston, London (2000). 15. Z. Mroz, G.E. Stavroulakis (Eds.): Parameter identification of materials and Structures. CISM Lecture Notes Vol. 469, Springer, Wien, New York, (2005). 16. G. Rus and R. Gallego R, Engineering Analysis with Boundary Elements Optimization algorithms for identification inverse problems with the boundary element method. 26(4):315-327 (2002). 17. G.E. Stavroulakis, M. Engelhardt, A. Likas, R. Gallego and H. Antes, Journal of Theoretical and Applied Mechanics, Polish Academy of

236 Sciences, Neural network assisted crack and flaw identification in transient dynamics, 42(3):629-649 (2004). 18. E.P. Hadjigeorgiou, G.E. Stavroulakis and C.V. Massalas, Shape control and damage identification of beams using piezoelectric actuation and genetic optimization, International Journal of Engineering Science, 44(7):409-42 1 , (2006). 19. D.G. Marinova, D.H. Lukarski, G.E. Stavroulakis, Emmanuel C. Zacharenakis: Nondestructive identification of defects for smart plates in bending using genetic algorithms. I11 European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et.al. (eds.) Lisbon, Portugal, 5-8 June 2006.

237

ON MODELING, ANALYTICAL STUDY AND HOMOGENIZATION FOR SMART MATERIALS A. TIMOFTE

Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 101 17, Germany E-mail: timofte9wias-berlin.de amtimofte9yahoo.com We discuss existence, uniqueness, regularity, and homogenization results for some nonlinear time-dependent material models. One of the methods for proving existence and uniqueness is the so-called energetic formulation, based on a global stability condition and on an energy -. balance. As for the two-scale homogenization we use the recently developed method of periodic unfolding and periodic folding. We also take advantage of the abstract r-convergence theory for rate-independent evolutionary problems.

Keywords: Ferroelectrics; Shape memory alloys; Energetic formulation; Evolutionary variational inequalities; Homogenization; Existence; Uniqueness.

1. Introduction

The models analyzed here concern three types of materials of high interest for applications: shape memory alloys (SMA), ferroelectric materials, and a class of rate-independent systems within the theory of elastoplasticity with hardening. All of them work in the framework of small deformations and quasistatic approximation for the elastic or electrostatic equilibria. The last two are rate-independent, while in the first (SMA) so are the hysteretic flow rule for the phase transformation and the linear constitutive elasticity, but not the heat equation. For both rate-independent models we will apply the energetic method as introduced in Ref. 21 (for a survey see Ref. 18). In each case the energetic formulation will be explicitely described. In Section 2 we consider a thermomechanical model of shape memory alloys. This model (see Ref. 5) takes into account the non-isothermal character of the phase transformations, as well as the existence of the intrinsic dissipation. For the governing equations we prove existence, uniqueness and regularity in several functions spaces.

238 In Section 3 we discuss rate-independent engineering models for multidimensional behavior of ferroelectric materials. These models capture the non-linear and hysteretic behavior of such materials. We show that these models can be formulated in an energetic framework based on the elastic and the electric displacements as reversible variables, and on internal irreversible variables such as the remanent polarization. Quite general conditions on the constitutive laws guarantee the existence of a solution. Under more restrictive assumptions uniqueness of the solutions holds. Section 4 is devoted to the homogenization for a class of rateindependent systems described by the energetic formulation. The associated nonlinear partial differential system has periodically oscillating coefficients, but has the form of a standard evolutionary variational inequality. Thus, the model applies to standard linearized elastoplasticity with hardening. Using the recently developed methods of two-scale convergence, periodic unfolding and periodic folding, we show that the homogenized problem can be represented as a two-scale limit, which is again an energetic formulation, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. 2. Shape Memory Alloys

This section is devoted to the mathematical study of a thermomechanical model describing the macroscopic behavior of shape memory alloys. The analyzed model takes into account the non-isothermal character of the phase transition, as well as the existence of the intrinsic dissipation. The model is published in Ref. 5, but a description of it can also be found in Refs. 31,33. A variant which neglects the intrinsic dissipation was studied in Refs. 3,4. The newest model from Ref. 5 is founded on a free energy which is a convex function with respect to the strain and to the martensitic volume fraction and concave with respect to the temperature. In the circular cylindrical case, uniqueness of solutions in a large class of spaces, as well as their existence in the space of continuous functions were established in Refs. 31,32. Existence, uniqueness and regularity of solutions in various functions spaces were proved in Ref. 28. We next give a brief description of the mathematical problem and of our main results on it. The first law of thermodynamics, the balance of momentum in its quasistatic form, the evolution equation for the internal variables (the volume fraction of martensite), together with the second principle of thermodynamics (the entropy inequality), lead to a partial differential equations system. In the circular cylindrical case the problem reduces

239 to the following ordinary differential system:

a = E(E - gp)

'If p = 0,

then a 5 a+ and

j
p < 1,

(8): 0 5 P 5 1 , {

then a- < a < a + and

{Po*

a = a> O ~ a = a +

If

p = 1,

then a 2 a- and

j>o+a>a+ , P(0) = 0, 0(0) = 0, c(0) = 0, o(0) = 0 \

The unknown data are: the temperature 0 at the surface of the body, the total fraction f,3 of the martensite in the body, and the axial elongation 6 of the sample in the Ox3 direction. The stress a is supposed to be given. All these are real functions only depending on the time variable t 2 0. The constants r,I?,L, C, E, g,p, q, To,T,, AT are all positive, To > T,, r < LIC, and a* := p(To - Ta 6 +PAT) q. Some comments are necessary in order to understand the mathematical problem raised by ( I ) : 1. The known data is an arbitrarily given continuous function a : J + R ( J is an interval with min J = 0) such that a(0) = 0. The system (I) is initially considered for unknown functions P, 0, E : J + R having lateral derivatives everywhere on J, since they should satisfy (X), (E) with respect to these. If p is strictly increasing on some open subinterval Joc J, thena {t E JO I jf(t) > 0) is dense in Jo,and so cr = a+ = p(To-Ta+O+pAT)+q on Jo,by (E). Consequently a should have lateral derivatives on Jo. This poses a serious compatibility problem for our system if the given a does not have lateral derivatives (e.g. if a is continuous but nowhere differentiable). 2. If a is such that ,&(to) > 0 and P(to) = 1, then ,O cannot be differentiable at to, since p 5 1. This may happen even if a is analytic on J, and so /? can be less regular than a. This is the reason to insist on lateral differentiability. 3. There exist strictly increasing continuous functions u : J + R, such that Jot ti(s)ds = 0 # u(t) - u(0) for every t > 0. Since the usual derivative sometimes fails to characterize continuous and almost everywhere differen-

+

5it(t)and ub(t)denote the forward and the backward derivatives of

26

at t .

240 tiable functions, its presence in ( I ) may not guarantee the uniqueness of solutions. 4. Since for arbitrarily given a a pronounced non-differentiability of solutions may occur, it would be natural to study ( I ) in the space C ( J ) of all real continuous functions on J, with the derivative in the sense of distributions. This is related to serious difficulties: what is the meaning of in ('H) and of a(t) in (E), if is a distribution but not a function? In order t o remove the derivatives of @ from (E), we introduced in Ref. 31 a new notion. A point t E Jo (Jo an interval) is said to be an increment point for u E C(Jo), if and only if for every neighborhood V o f t , we have t l < t2 and u(t1) < u(t2) for some t l , t2 E V n Jo.Let M+(u) denote the set of all increment points of u and set M-(u) := M+(-u). If X ( J ) is any of the spaces AClo,(J), Liplo,( J),D ~ ( J ) D? , (J), D?(J), DNo(J), Af (J), Ab(J),Al(J), endowed with its natural derivative (see the list below for details), then an equivalent form of (£) for @, 8 e X ( J ) is @(t)> 0 + a(t) 2 a-(t) @(t)< 1 + a(t) 5 a+(t) t E M+ (@) o(t) = a+(t) t E M-(@) + a(t) = a-(t).

*

If @, 0 E C ( J ) satisfy ( E ) qJ), then @ must be locally monotone (see Ref. 31, Cor.4.2, p.455). If we write ('H) on every interval Joof monotonicity for @, 0 we can then consider the following equation in distributions on Jo:

where ro:=

l?, if

p is increasing on Jo,

-r, else.

The system ( I ) may be considered for any of the functions spaces and derivatives listed below (see Ref. 31 for the definition of an abstract derivation structure X ( J ) and for the corresponding system (7)x(J,). List of functions spaces a n d associated derivatives 1) C ( J ) , with the derivative in the sense of distributions in ~ ' ( j ) We . have

) Let us recall that u E C ( J ) the natural inclusions C ( J ) C ~ ( Cj V1(j). 0 is increasing if and only if u' E V1(J) is positive. 2) BKoc(J) := {u E C(J) 1 u has locally bounded variation}, with the derivative in the sense of distributions. 3) AClo,(J) := {u E C ( J ) 1 u is locally absolutely continuous}, with the

241 derivative almost everywhere. 4) Liplo,(J) := {u E C ( J ) 1 u is locally Lipschitz), with the derivative almost everywhere. 5) For every fixed at most countable subset A of J, consider the spaces: a) D ~ ( J ) := {U E C ( J ) I u is differentiable to the right on J \ A) (respectively D: (J)), with the forward (respectively backward) derivative on J \ A. b) D:(J) = Df ( J ) n D ~ ( J ) with , both forward and backward derivatives. 6) D N O ( ~ :=) {u E C ( J ) I the set of non-differentiability points of u is at most countable), with the usual derivative where this one exists. 7) a) Af(J) := {u E C ( J ) 1 u is forward-analytic), with the forward derivative. A function u E C ( J ) is said t o be forward-analytic at t E J \ {sup J), iff u is analytic on some It, s) c J (s > t). We call u a forward-analytic function, iff u is forward-analytic at every t E J \ {sup J). b) Ab(J) := {u E C ( J ) I u is backward-analytic), with the backward derivative (definitions are similar to those for Af(J)). c) A1(J) := Af(J) n Ab(J), with both forward and backward derivatives. Our problem is the following: for a fixed X ( J ) i n t h e above list and for a given a E X ( J ) with a(0) = 0, we wish to investigate the existence of solutions p, 8, E E X ( J ) of the system ( 7 ) x ( 4 . The constitutive equation a = E ( e - gp) and the condition ~ ( 0 = ) 0 from (T)X(j) can be ignored, since for p, 8 E X ( J ) satisfying all other conditions, we get a solution of (T)X(j) with E = g p E X ( J ) . Therefore, every solution of ( I ) x (J) is given by a pair (p, 8) of functions from X(J). In Ref. 31 the following result is proved.

+

Proposition 2.1. Let X ( J ) be an abstract derivation structure. For /I, 8 E C(J), the following statements are equivalent: (a) (b)

(p,8) is a solution of (p,8) is a solution of

J) J)

. and

p, t9 E X ( J ) .

Now let a E X ( J ) be fixed, such that a(0) = 0. Since every solution of ( I ) x ( ~also ) satisfies (7)c(J), we deduce that (';T)x(J) is compatible if and only if for the unique solution (P,8) of ( I ) c ( J) (see Ref. 32, Th.3.1, p.543) we have p, 8 E X ( J ) . Hence, for our problem, regularity of solutions (that is p, t9 E X ( J ) whenever a E X ( J ) ) is equivalent to their existence. Let X ( J ) be any of the spaces

242 endowed with its natural derivative from the above list. Our main result is the following (for the proof, see Ref. 28): Theorem 2.1. For any given c E X ( J ) , the system ( 7 ) x (has J )a unique solution. 3. Ferroelectric Materials

Here we give a general description of a class of time-dependent models for ferroelectric materials. Our class of models is inspired by the engineering models from Refs. 13-15,25,29. However, we will rephrase the theories there in such a way that it can be formulated in terms of two energetic functionals, namely the stored energy E and the pseudo-potential R for the dissipation. Thus, we will be able to take advantage of the recently developed energetic approach to rate-independent models, (see Refs. 11,17,20,21and the survey 18). The basic quantities in the theory are the elastic displacement field u : R -+ IRd and the electric displacement field D : lRd + lRd. Here, the electric displacement is also defined outside the body, as interior polarization of a ferroelectric material generates an electric field E and displacement D in all of IRd via the static Maxwell equation in lRd. Commonly, the polarization P is used for modeling, and is defined via

where €0 is the dielectric constant (or permetivity) in the medium surrounding the body R. Our formulation stays with D, since it leads to a simple and consistent thermomechanical model. In addition we use internal variables Q : R -+ IRdo which, for instance, may be taken as a remanent strain G , , or a remanent polarization P,,,. The stored-energy functional has the form

where W is the Helmholtz free energy and E(U) is the infinitesimal strain tensor given by

The nonlocal term a ( x , VQ) in E usually takes the form $IVQI2 with k > 0. This term penalizes rapid changes of the internal variable by introducing a

243 length scale which determines the minimal width of the interfaces between domains of different polarization. The external loading C(t) depends on the process time t and is usually given by

where Ee,t, fvol and f,,,f are applied, external fields. For the dissipation potential R we take the very simple form

where R(x, .) : IWdo 4 [0, m) is convex and positively homogeneous of degree 1. Note that the dissipation potential only acts on the rate Q = &Q of the internal variable. The classical way to describe dissipation in ferroelectrics is a switching function of the form

This is equivalent to our dissipation potential R by the relation

To formulate the rate-independent evolution law we use the thermomechanically conjugated forces

where (T is the stress tensor and E the electric field. The elastic equilibrium equation and the Maxwell equations read

+

- div (T fvoi (t , = 0 div D = 0 and curl(E - EeXt(t, 0

)

a))

in R, = 0 in I W ~ ,

where curl E is defined as VE- ( v E ) ~for general dimensions. The evolution of Q follows the force balance law:

where aR(x, .) is the subdifferential of the convex function R(x, .).

244 We now want to rewrite these relations, as equations in function spaces. For this purpose we introduce a suitable state space y = 3x Q as follows. The space F contains the functions u and D,and takes the form

3 = 'H x L : ~ ~ ( R ~where ), L:,(Rd)

I div$

:= ( 4 E L2(IRd;IRd)

=0)

and 'H is a closed affine subspace of H1(R; Rd). The space Q contains the internal state functions Q and is taken to be W1?QQ (R;IRdQ)for a suitable QQ > 1. Using the well-known fact (cf. Ref. 30, Th.1.4) that the total space L2(Rd;Rd) decomposes in two orthogonal closed subspaces L,: (Rd) and LzUr1(Rd)= ( 4 E L'(R~;Rd) I curl $ = O ), we obtain the following result. Proposition 3.1. Let DDE(t, u, D , Q ) [ ~ denote ] the Giiteaw derivative of E in the direction 5 . Then

Thus, we implement the Maxwell equations by the condition DDE(t,u, D, Q) = 0 in a suitable function space. Similarly, the elastic equilibrium is obtained by D,E(t, u, D, Q) = 0. The full problem may be written as

where the last above equation corresponds to the dissipative force balance. In fact, our theory is not based on the force balance (1). Instead, following Refs. 18,21, we use a weaker formulation only based on energies. This energetic formulation avoids derivatives of E and of the solution (u, D, Q). Under suitable smoothness and convexity assumptions the energetic formulation is equivalent to (1). We call (u, D , Q) an energetic solution of the problem associated with E and R , if for all t E [0,T] the stability condition (S) and the energy balance (E) hold: (s) E(t, u(t), ~ ( t )Q(t)) ,

I £(t, Q,5,a ) + ~ ( 6 - ~ ( t for ) ) all Q,5,Q;

(El E(t, u(t), D(t), Q(t)) +

Sot R ( Q ( ~ ) )ds)

= E(O, u(O), D(O), Q(0)) - S,t(i(s),

( 4 ~ 1D(s))) , d

~ .

245 In Refs. 22,23 we showed that (S) & (E) has solutions for suitable initial data, if the constitutive functions W, a, and R satisfy reasonable continuity and convexity assumptions. Under stronger conditions we also proved uniqueness of solutions. We now provide conditions on the constitutive functions W, a and R, in order to get the existence result. The first assumption concerns the domain and the Dirichlet boundary: R c Rd is a connected bounded open set with Lipschitz boundary F, and r'i, a measurable subset of I', such that 1da > 0. (BO) The function R : R x RdQ [O, 00) satisfies

lrDir

R E c ' ( ~ x x d a ) and 3cR,CR > 0 VV E Rdq : cRIVI 5 R(x,V) 5 CRIVI. (B1) Vx E R : R(x, .) : IRdQ

-+

[O, 00) is 1-homogeneous and convex.

(B2)

The functions W and a have to fulfill the following three conditions: W : R x It:,"," x Rd x RdQ-+ [0,oo) is a Caratheodory function,

(B3)

which means that the function W(., E , D,Q) is measurable on R for each (E,D, Q), and that the mapping W(x, ., .) is continuous on R:;,"xRd xRdQ for a.e. x E R. a,

a : IRdQ x d

4

IR is convex and

V(x,Q) E R x RdQ : W(x,.,.,Q) :",:It

x Rd + R is convex.

(B5)

For the external loading d ( t ) we assume

e E c1([o, T I ,(H;~, (R;R ~ ) ) x* L&,(IR~)*).

(B6)

Let us consider the following functions spaces:

Here the subscripts "weak" and "strong" indicate the use of the weak or strong (norm) topology in the corresponding Banach spaces. The functional E is defined as above on [0,TI x 3 x 2 , where E(t, u, D, Q) takes the value or if the integrand is not in L1(R). +oo if Q 6 W19~(R;RdQ) We can now state our existence theorem.

246 Theorem 3.1 (Existence theorem). If the assumptions (B0)-(B6) hold, then for each stable initial condition (uo, Do, Qo) E F x 2, the energetic problem (S) & (E) has a solution (u, D, Q ) : [(),TI -+ 3 x 2, such that (u(O),D(O), Q(0)) = (uo, Do,Qo), and (u, D l Q) E Loo([O,TI;

(52; IRd) x L;,(IW~) x ~ " ~ ( 5 IRd~)). 2;

4. Homogenization for rate-independent systems

Our aim is to provide homogenization results for evolutionary variational inequalities of the type: ( - 4 - e(t), v - q )

+ %(v) - X(q) 2 0

for every v E (2.

(2)

Here (2 is a Hilbert space with dual a*, the continuous linear operator A : (2 -+ (2. is symmetric and positive definite, the forcing e lies in cl([O,T],Q*), and the dissipation functional 31 : (2 -+ [0,oo) is convex, lower semi-continuous and positively homogeneous of degree 1, i.e., 31(yq) = y31(q) for all y 2 0 and q E (2. The last property of 31 leads to rate independence. The problem (2) has many different equivalent formulations. For our purposes the so-called energetic formulation for rate-independent hysteresis problem is especially appropriate, cf. Refs. 18,20. This formulation is solely based on the energy-storage functional E : [0,T ] x (2 -+ R defined via

and on the dissipation functional 31. Thus, homogenization of an evolutionary problem can be reduced to some extend to homogenization of functiona l ~We . formulate our rate-independent evolutions systems and we provide existence and uniqueness theorems for the initial and expected two-scale homogenized problems. We present some r-convergence results and finally our main homogenization theorem. The notion of two-scale convergence has been introduced by Nguetseng (see Ref. 27) in 1989 and developed by Allaire in 1992 (see Ref. 2). Ref. 16 provides an overview of the main homogenization problems studied by this technique. The periodic unfolding method recently introduced (2002) by Cioranescu, Damlamian and Griso in Ref. 8, reduces the two-scale convergence to a weak convergence in an appropriate space. This concept is now applied in a variety of quite different applications in continuum mechanics,

247 see e.g., Refs. 1,10,26,34,35.To the best of our knowledge there is no theory for nonsmooth evolutionary systems like the variational inequalities here. Throughout, the domain fl will be a bounded open subset of Rd. For the semi-open unit cell Y = [0, l ) d , we have UXEZd(A Y) = IZd and (A Y) n (p Y) = 8 for A, p E lEd with X # p. jFrom now on we will assume that p E (1, co). Let us recall the definition of the classical two-scale convergence.

+

+

+

Definition 4.1. Let (v,), be a sequence in LP(R). One says that (v,), two-scale converges to V = V(x, y) in LP(R x Y) (we write v, 4 V), if for any function $ = $(x, y) in C r ( R ; Cg",,Y)), one has

The periodic unfolding operator ?; was introduced in Ref. 8 and then used for homogenization of nonlinear integrals in Refs. 6,7. On the full space Rd, it is defined by

We next introduce the notions of wealc/strong two-scale convergence. Definition 4.2. Let V E Lp(R x Y). A bounded sequence (v,), in LP(R)

-

(w2): weakly t w ~ s c a l econverges to V (we write v,

I,v,

V (weakly) in L P ( I Wx~Y).

(s2): strongly two-scale converges to V (we write v,

&v,

9 V), if

+V

3 V), if

(strongly) in L P ( R ~x Y).

Clearly, the above weak two-scale convergence is stronger than the classical. 4.1.

E

problem

Let us consider: R c Rd, a connected bounded open set, with Lipschitz boundary r, Y = [0, l ) d C Rd, unit periodicity cell, u : R -+ Rd, displacement, z : R + Rm, internal variable. For every E > 0, define the energy functional E, and the dissipation functional 3, by

248

where 1 e(u) = - ( V u + v u T ) E R ~ , " , " : = { ~ E R [ ,~=~a ~T ). 2

The tensors C, W, B defined on Rd are Y-periodic, and take values in: C(y) E ~

~ order m tensor, 4 ~B(y)~E ~ i n ( RIt:,","~), ,

W(y) E

We work under the hypotheses stated below. Assumptions for C, W, B: for all y E Eld and z E Rm, we have

(for some constant C > 0). Assumptions for p:

I

p : Rd x Rm -, [O,m), p(., v) Lebesgue measurable and Y-periodic, for every v

(H,)

i Rm,

p(y , -) l-homogeneous and convex for a.e. y E Rd,

& lvl 6 p(y, v) for a.e. y E Rm and every v E Rm, Ip(y, v) - p(y, v')I 5 Clv - v'l for a.e. y E Rd and all v, v' E Rm.

Let us consider the Hilbert space IZ = HhDi,(0)dx L2(R)m. We call q, = (u,, z,) : [0,T ] --, 9 an energetic solution of the problem associated with E, and X,, if for every t E [0,TI the stability condition (SE) and the energy balance (E") hold:

We now state our existence and uniqueness result for (S") & (EE).

249 Proposition 4.1. Let fext E CLip([O, T I , ( H ; ~ , ~ ( R ) ~Then ) * ) . for all e > 0 and stable (uz,zz) E Q , there is a unique solution (u,, z E )E CLip([O,T I , 8 ) of (S") & ( E E ) ,with (u,( 0 ) ,zE(0))= ( u z ,2 2 ) . Moreover, we have &-independentLipschitz bounds for the solutions, that is, for some constant cl > 0 we have

4.2. Two-scale homogenized problem

We now formulate the problem ( S ) & ( E ) , which will turn out to be the two-scale homogenized problem for ( S E )& ( E E ) . Let Q = H x Z , where

H = H;,,,(R)~ x L ~ ( RH; ; ~ ( Y ) ) z~ = , L ~ ( RL; ~ ( Y )=) L~~ ( R xY ) ~ . Here, H t v ( Y ) = { U E Hke,(Y) I Sy U ( y ) d y= 0 ) . For all Q = (U,2) in Q , with U = (Uo,U l ) ,let us define the two-scale functionals E and R

+

+

where S ( U ) = e x ( & ) e,(Ul), which means G(U)(x,y) = ex(Uo(.))(x) e, (Ul(., .>>(Y>The energetic formulation for the two-scale homogenized problem ( S ) & ( E ) reads: for every t E [O,T],the stability condition ( S ) and the energy balance ( E ) hold, that is,

We next state our existence and uniqueness result for the problem ( S )& ( E l Proposition 4.2. Let fext € CLip([O, T I , ( H ; ~ , ~ ( R ) ~ Then ) * ) . for every stable QO = (U', 2') E Q , the problem ( S ) & ( E ) has a unique solution Q = (U,2) E c L i p ( [ O , TI,Q ) , with Q ( 0 ) = QO.

250 The convergence of E, and R, to E and R can be viewed as a type of t w ~ s c a l eMosco convergence, i.e., F-convergence in the weak and in the strong topology (see Ref. 19). Here we rely on the unfolding operator and on the folding operator in order to construct suitable recovery sequences, also called realizing sequences in Ref. 12. The crucial tool for proving the convergence of the solutions q, to the energetic solution Q associated with E and R is the abstract F-convergence theory developed in Ref. 19. There, the simple theory relies on the fact that the dissipation functions converge continuously in the weak topology; yet this is not the case in our situation. However, we are able to use the quadratic nature of the energies allowing some cancelations in differences of energies. For instance, E, (t, q,) - E, (t, q,) converges t o E(t, Q) - E(t, o ) , if q, has the "weak" two-scale limit Q and q, - <, has a strong(!) two-scale limit Q Our homogenization theorem is based on the notion of two-scale crossconvergence.

a.

Definition 4.3. Let Q = (U, Z) E Q, with U = (Uo, UI). A sequence q, = (u,, z,), in !J is called two-scale cross-convergent to (U, Z), if U,~Uo, We write this as (u,, z,)

~

u

,

~

~

u

O

+

~

z~ , Ug z~ .

,

'x2(U, 2 ) .

We can now formulate our main result stating that (S) &(E) is the two-scale homogenized problem for (SE)& (EE) (see Ref. 24 for the proof).

Theorem 4.1. Let q, = (u,, z,) be the solution of (SE) & (EE). For the initial data, assume that: qz = (u:, z:) 0

is stable for every E

0 cx2

Q0 = (UO,ZO)E 9: = (us, z,) &E(o,92) -* E(o, QO).

> 0,

Q,

Then (q,), two-scale cross-converges to the unique solution Q = (U, Z) of the two-scale homogenized problem (S) & (E), with Q(0) = QO.

Acknowledgments The research was supported by the European Union via the Research Training Network "Smart Systems" (HPRN-CT-2002-00284).

251 References 1. Alber, H.D., Evolving microstructure and homogenization, Contin. Mech. Thermodyn. 12 (2000), 235-286. 2. Allaire, A., Homogenization and two-scale convergence, S I A M J. Math. Anal. 23 (1992), 1482-1518. 3. X. Balandraud, E. Ernst and E. Sobs, Relaxation and creep phenomena in shape memory alloys, Part I: Hysteresis loop and pseudoelastical behavior, Z. Angew. Math. Phys. 51 (2000), 171-203. 4. X. Balandraud, E. Ernst and E. S o b , Relaxation and creep phenomena in shape memory alloys, Part 11: Stress relaxation and strain creep during phase transformation, 2. Angew. Math. Phys. 51 (2000), 419-448. 5. X. Balandraud, E. Ernst and E. S d s , Monotone strain-stress models for Shape Memory Alloys hysteresis loop and pseudoelastic behavior, Z. Angew. Math. Phys. 56 (2005), 304-356. 6. Cioranescu, D., Damlamian, A., De Arcangelis, R., Homogenization of nonlinear integrals via the periodic unfolding method, C.R. Acad. Sci. Paris Ser. 1 339 (2004), 77-82. 7. D. Cioranescu, A. Damlamian, R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method, S I A M J. Math. Anal. 37 (2006), no. 5, 1435-1453. 8. D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. 1 335 (2002), 99-104. 9. D. Cioranescu, P. Donato, An introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17,Oxford University Press, (1999). 10. C. Eck, P. Knaber, S. Korotov, A twescale method for the computation of solid-liquid phase transitions with dendritic microstructure, J. Comput. Phys. 178 (2002), 58-80. 11. G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with non-convex elastic energies, J. reine angew. Math., 595 (2006), 55-91. 12. V. V. Jikov, S. M. Kozlov, 0 . A. Oleinik, Homogenization of differential operators and integral functions, Springer-Verlag, (1994). 13. M. Kamlah, Ferroelectric and ferroelastic piezoceramics-modelling of electromechanical hysteresis phenomena, Continuum Mech. Thermodyn., 13 (2001), 219-268. 14. M. Kamlah and Q. Jiang, A model for PZT ceramics under uni-axial loading, Forschungszentrum Karlsruhe, FZKA 6211, (1998). 15. M. Kamlah and Z. Wang, A thermodynamically and microscopically m e tivated constitutive model for piezoceramics, Comput. Materials Science, 28 (2003) 409-418. 16. D. Lukkassen, G. Nguetseng, P. Wall, Two-scale convergence, International Journal of Pure and Applied Mathematics 2 (2002), 35-86. 17. A. Mainik and A. Mielke, Existence results for energetic models for rateindependent systems, Calc. Var. PDSs 22 (2005), 73-99. 18. Mielke, A., Evolution for rate-independent systems. In Handbook of differen-

252 tial equations. (ed. by Dafermos, C. and Feireisl, E.). Evolutionary Equations Vol. 11, Elsevier B.V. 2005, 461-559. 19. A. Mielke, T . Roubicek, U. Stefanelli, Gamma limits and relaxations for rateindependent evolution equations, WIAS Preprint 1156 (2006). 20. A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl. (NoDEA) 11 (2004) 151-189. 21. Mielke, A., Theil, F., Levitas, V., A variational formulation of rateindependent phase transformations using an extremum principle, Arch. Rational Mech. Anal. 162 (2002), 137-177. 22. A. Mielke, A. M. Timofte, An energetic material model for time-dependent ferroelectric behavior: existence and uniqueness, Math. Meth. Appl. Sciences, 29 (2006), 1393-1410. 23. A. Mielke and A. M. Timofte, Modeling and Analytical Study for Ferroelectric Materials, Mechanics of Advanced Materials and Structures 13 (2006), 457-462. 24. A. Mielke and A. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, submitted. 25. R. McMeeking and C. Landis, A phenomenological multi-axial constitutive law for switching in polycrystalline ferroelectric ceramics, Internat. J. Engrg. Sci 40 (2002) 1553-1557. 26. S. Nesenenko, Homogenization and regularity in viscoplasticity, PhD Thesis, Logos Verlag Berlin, (2006). 27. Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608-623. 28. T . Ratiu, A. Timofte and V. Timofte, Existence, uniqueness and regularity of solutions for a thermomechanical model of shape memory alloys, Math. Mech. Solids (in press). 29. H. Romanowski and J. Schroder, Coordinate invariant modelling of the ferroelectric hysteresis within a thermodynamically consistent framework. A mesoscopic approach. In Trends i n Applications of Mathematics to Mechanics, Y . Wang and K. Hutter (eds), Shaker Verlag (2005), 419-428. 30. R. Temam, Navier-Stokes equations, volume 2 of Studies i n Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, third edition, (1984). 31. A. Timofte, V. Timofte, Uniqueness theorem for a thermomechanical model of shape memory alloys, Math. Mech. Solids 6 (2001), 447-466. 32. A. Timofte, V. Timofte, Existence theorem for a thermomechanical model of shape memory alloys, Math. Mech. Solids 6 (2001), 541-545. 33. A. Timofte, V. Timofte, Qualitative results concerning the behavior of shape memory alloys, Rev. Roumaine Math. Pures Appl. 47 (2002), 121-133. 34. A. Visintin, Two-scale Stefan problem, Nonlinear analysis and applications (Warsaw, 1994) 7 (1996), 405-424. 35. A. Visintin, Two-scale model of phase transitions, Phys. D 106 (1997), 66-80.

253

THE CARDIOVASCULAR SYSTEM AS A SMART SYSTEM M. TRINGELOVA Fakulta aplikovanSch vdd, Katedra mechaniky, Z6padoEeskd univerzita v Plzni, 306 14 Plzeil, Czech Republic E-mail: mtringe1Okme.zcu.c~ P. NARDINOCCHI Dip. Ingegneria Strutturale e Geotecnica, Universitci di Roma "La Sapienza", Via Eudossiana 18, 1-00184 Roma, Italy E-mail: paola.nardinocchi~uniromal.it

L. TERESI* and A. DI

CAR LO^

SMFMODiS, Universiti "Roma n e " , Via Corrado Segre 6, 1-00146 Roma, Italy *E-mail: teresiOuniroma3.it t E-mail: adicarloOmac. com Our work aims a t modelling and simulating the growth processes that allow the cardiovascular system to adapt t o the overall body development and t o changing physiological (and pathological) conditions. Within the cardiovascular system, we pay particular attention to the heart and the aorta. A healthy aortic wall succeeds in keeping a homeostatic stress level in spite of long-standing alterations in pressure or flow by triggering growth and remodelling processes that change its stress-free shape and its structure. A normal heart grows in response t o the gradually increasing haemodynamic loading exerted on myocardial fibres. Postnatal cardiac growth is a form of volume-overload hypertrophy, produced essentially by a progressive myocardial cell enlargement, with no cell proliferation involved. In our continuum model, growth is basically conceived of as the time evolution of the stress-free configuration of the tiny fragments into which the modelled tissue may be subdivided in imagination. It is governed by a novel balance law-the balance of accretive couples-independent of, but constitutively coupled with, the standard balance of forces. Keywords: Cardiovascular system; Adaptability; Growth and remodelling.

254 1. The Cardiovascular System a n d i t s Adaptability

The primary function of the cardiovascular system is the transport of oxygen, carbon dioxide, nutrients, waste products within the body. Since the body undergoes major changes during its life cycle, the capability of the cardiovascular system to accommodate vastly increasing body demands is a vital requirement. The system consists primarily of the heart, which serves as the pump, the blood, which serves as the conducting medium, and the vasculature, which serves as the conduit through which the blood flows1 We focus on the heart and the aorta-the main artery of the body, supplying oxygenated blood to the circulatory system. In particular, we strive to model the growth process through which the aortic wall keeps a homeostatic stress level (Section 3.1) and the postnatal hypertrophy--either physiological or pathologicalLof the cardiac muscle (Section 3.2). Both of these processes take place on a time scale-weeks to years-which is extremely long if compared with the heartbeat time scale (see Fig. 3). Nevertheless, understanding the basic machinery of the cardiac cycle is crucial to our endeavour, since physiological adaptation and pathological developments are both triggered by the values attained during a cardiac cycle by rather gross mechanical quantities, such as blood pressure and heart volume. This is demonstrated by an overwhelming physiological and medical evidence, even though the detailed feedback mechanisms are still largely ~ n k n o w n . ~ The following two subsections collect some basic information on the structural and functional properties of the heart and the aorta, respectively, as an essential preliminary to mathematical modelling. For a fairly recent overview on the biomechanics of cardiovascular development, the reader is referred to the survey paper by Taber.3

1 . l . Structure and function of the heart

The heart is a smart composite structure. Atria and ventricles have a different microstructural organization; also heart valves and ventricle walls possess a different microstructure; moreover, the heart walls are comprised of three different layers, with diverse structure and function. The inner layerendocardium-and the outer layer-epicardzum-are both very thin, each being about 100 pm thick; the middle layer, called myocardium, constitutes the bulk of the cardiac tissue and endows it with the ability to pump blood.

255 1.1.1. The myocardium The myocardium is composed of cardiac myocites and fibroblasts, surrounded by an extracellular matrix. In adults, cardiac myocites are typically 10-20 pm in diameter and 80-125 pm in length; their cytoplasm contains mainly myofibrils (1-2 pm in diameter). Each myofibril consists of a string of contractile units, called sarcomeres, each of which is about 2 pm long. Each sarcomere consists of hundreds of filamentous protein aggregates (myofilaments). Thick myofilaments are composed of several hundred molecules of myosin; thin myofilaments are composed of two helically interwound polymers of actin. Cardiac myocites are active components: they contract and relax at a high frequency producing the heartbeats; on a much longer time scale, they can grow bigger (hypertrophy). Muscle contraction is initiated by the release of calcium ions which are sequestered in the surrounding sarcoplasmic reticulum. A smooth ratcheting action (with a speed of about 15 pmls) results from the shortening of the sarcomere, operated by the action of the act* myosin cross-bridges that release, move forward, and reattach. The release of calcium ions is triggered by an action potential that spreads from the cell membrane. The passage of the electrical signal from cell to cell is facilitated by the fact that myocites are highly interconnected. The biochemical mechanisms that drive the long-term hypertrophy process are more intriguing and far less understood. Some coarse-grained feedback mechanisms are hypothesized in Sec. 3.2, as a first step in our modelling effort. The passive myocardium exhibits a nonlinear, essentially incompressible

Fig. 1. View of the interior of the left ventricle, showing the mitral valve (reproduced from Ref. 4). The flap is opened in the posterolateral wall.

256 (visco-) elastic response, as is typical of all soft tissues. An important feature to be accounted for is the complex distribution of the predominant orientation of muscle fibres across the myocardium. In fact their orientation, while staying everywhere tangent to the wall, changes notably with position: in the equatorial region, for instance, the predominant muscle fibre direction changes from about -65' in the sub-epicardial region to nearly 0" in the mid-wall region to about +65' in the sub-endocardial region, all relative to the circumferential direction. This transmural splay of fibre directions causes the heart to twist during the cardiac cycle.l Another peculiar feature that cannot be disregarded is the existence of a major self-stress that contributes to a more uniform intramural stress distribution under physiological conditions. The self-stress is distributed across the heart wall in a way closely related to the fibered structure of the myocardium, as shown by a series of tests performed by Omens and h n g 5 on rat left ventricles about twenty years ago and recently reported by ~ u m p h r e ~ Evidence .' of compressive circumferential residual stress in equatorial rings of myocardium tissue excised from the left ventricle was obtained by effecting radial cuts that left the rings open and stress-free. A comparison of the length of sarcomeres in this stress-free configuration with the one they had in the intact configuration showed that, while the relaxed length of sarcomeres varies significantly from the endocardium to the epicardium in the unloaded (but self-stressed) myocardium configuration, it is nearly uniform across the wall in the stress-free configuration.

1.1.2. The cardiac cycle The main heart function is to ensure blood pumping into both the high pressure systemic circuit and the low pressure pulmonary circuit (see Fig. 2). The cardiac cycle may be split into four stages (see Fig. 3). The first is the diastolic filling (comprising phases 6,7 and 1 in Fig. 3), which is started by the opening of the valve connecting the left atrium with the left v e n t r i c l e the mitral valve. This stage is characterized by a dilation of the left ventricle caused by the higher pressure in the atrium and progressively slowed down by the increasing apparent stiffness of the ventricle wall. During the diastolic filling, the pressure in the left ventricle does not change significantly, achieving a maximum of 15 mmHg (= 2 KPa), versus an aortic pressure of 90 mmHg. The contraction of myocardial fibres makes the mitral valve close, thus starting the second stage of the cardiac cycle-the isovolumic contraction (phase 2 in Fig. 3). The left-ventricle pressure rises rapidly dur-

257 PULMONARY CIRCUIT

lung -

pulmonary artery right atrium

pulmonary \rein

left atrium aorta

venae cavE

left ventricle

right ventricle

SYSTEMIC CIRCUIT

Fig. 2. A heart-centred schematic picture of blood circulation (reproduced from Ref. 6): the upper part schematizes the pulmonary circuit, the lower part the systemic circuit; the left side carries oxygen-poor blood, the right side oxygen-rich blood.

PHASES

LV Press: left-ventricle pressure

LA Press: left-atrium pressure LVEDV: left-ventricle end-diastolic volume LVESV: left-ventricle end-systolic volume

electrocardiogram heart sounds

cardiac cycle (reproduced from Ref. 7)

258 ing this stage, and the aortic valve opens when it exceeds the aortic pressure. On the contrary, the volume of the left ventricle does not change, as it stays filled with blood. The stage of systolic ejection (comprising phases 3 and 4 in Fig. 3) starts with the opening of the aortic valve. Most of the cardiac output is provided in the first quarter of this stage, before the pressure peaks at about 120 mmHg. Then, when the pressure drops to about 100 mmHg, the aortic valve closes, starting the fourth stage: the isovolumic relaation (phase 5 in Fig. 3). During this stage the volume of the left ventricle keeps its minimum value, while the left ventricular pressure drops fast to the value of the atrial pressure, triggering the opening of the mitral valve and starting a new cardiac cycle. 1 . 2 . The aorta as a prototype large vessel

Arteries are roughly classified as elastic or muscular. Elastic arteries are larger and are located close to the heart, like the aorta; muscular arteries are smaller and closer to the arterioles. The walls of both elastic and muscular arteries consist of three different layers: intima, media, and adventitia. The intima, which is the innermost of the threes and consists of a single layer of endothelial cells, is extremely thin with respect to the arterial wall, at least in large arteries. Its contribution to the mechanical properties of the wall may only become significant in old age or under degenerative conditions: atherosclerosis, in fact, implies a thicker and stiffer intima. The middle layer-the media-is characterized by a complex mixture of smooth muscle cells, elastin and collagen fibrils. Vascular smooth muscles modify the distensibility of large arteries and regulate the lumen size in medium and small arteries. Consistent with their different roles, smooth muscles are organized differently in large (elastic) and in small (muscular) arteries. In elastic arteries, vascular smooth muscles are organized in musculcelastic fascicles-that is, alternate layers of smooth muscle and elastin (up to 70 layers in the human aorta). In muscular arteries, elastin is absent and smooth muscles form a single layer embedded in a matrix of connective tissue. The outermost layer-the adventitia-consists primarily of collagen fibres arranged in helical structures. It contributes significantly to the strength of the arterial wall. The mechanical properties of an arterial wall are approximately homogeneous within each single layer, but not overall. In fact, the difference in stiffness exhibited by different layers plays an important role in the adaptive response of the arterial wall to perturbations from its homeostatic state. The differential growth of the layers originates a non uniform self-stress dis-

259 tribution within the wall. Such self-stress (also called residual stress) has been extensively investigated, due to its importance in stress redistribution and vascular remodelling. When excised from the vascular tree, an arterial segment shortens drastically. A further radial cut releases most of the residual stress in an unloaded intact arterial ring, allowing it to open up, the opening angle being a rough measurable indicator of the preexisting self-stress (see Refs. 8, 9 and our discussion in Sec. 3.1). The aorta, like the other large arteries, exhibits a highly nonlinear behaviour under large strains. A constitutive model of such behaviour should account for both its passive response, due to elastin and collagen, and its active response, due to muscle fibres. Following the fluctuations of the intramural pressure, the arterial wall undergoes instantaneous elastic deformations, essentially determined by its passive response. Despite its complexities, such a response is by no means the most characteristic mechanical property of the arterial wall as a living tissue. When the normal conditions are altered for a sufficiently long time, the passive response may be supplemented by a suitable contraction of the vascular smooth muscles, that constitutes the primary adaptive mechanism aiming at keeping the flow-induced shear stress and the wall stress distribution at their baseline values. Of course, such a goal is achieved at the expense of an altered contractile state of smooth muscle cell^.'^^" Therefore, alterations in pressure or flow which hold out over long periods of time (days, weeks) trigger a different adaptive response, consisting in growth and remodelling processes that modify the stress-free shape and the structure of the vessel wall. The passive mechanical response of arterial walls has been extensively investigated through uniaxial, biaxial, and torsion tests. On the contrary, very few data are available on the active response of vascular smooth muscles. Moreover, these data come only from uniaxial tests performed on arterial rings or helical strips excised from an artery following the local orientation of the muscle fibres. Therefore, they cannot account for the effects of the complex three-dimensional arrangement of these fibres. Even less is known of the long-term adaptive response implying growth and remodelling. 2. Gross Mechanics of Bulk Growth

We concentrate on the growth phenomena which are key to the adaptability of the cardiovascular system-and, more generally, of all smart living systems. In Section 3 we offer the results of our preliminary simulations of the adaptive remodelling of arterial walls (Section 3.1) and sketch the conceptual underpinning of our incipient project aiming at modelling post-

260 natal cardiac hypertrophy (Section 3.2). The present section is devoted t o a condensed, but reasonably self-contained, presentation of the theoretical framework common to both models, based on the dynamical theory of growth introduced by DiCarlo and ~ u i 1 i ~ o t t iWe . l ~ follow the lines of Ref. 13, where an abridged account of the theory of bulk growth was given as a preliminary to the treatment of surface growth." A comprehensive account of experimental and theoretical issues in the biomechanics of growth and remodelling may be found in the review papers by Taber15 and Cowin.16 2.1. Kinematics: gross and refined motions

We regard a growing piece of tissue-which we call the body in the rest of Sec. 2-as a smooth manifold 9 (with boundary d B ) , and call (gross) placement any smooth embedding

of the body into the Euclidean place manifold 8, whose translation space will be denoted by V8. Tangent vectors on the body manifold itself are called line elements. The set of all line elements attached to a single bodypoint b E 28 is called the body element a t b, and denoted T b 9 (the tangent space to 28 at b). The union of all body elements is denoted T 9 (the tangent bundle of 9 ) . The body gradient Vp of a placement p is a tensor field on 9, whose value at any given point b, denoted by Vplb,maps linearly the body element T b 9 onto V8. We call element-wise configuration--or, with a much shorter monosyllable, stance-any tensor field of this kind, be it a gradient or not. Therefore, a stance is any smooth mapping

such that the restriction PITb&@is a linear embedding, for all b E 9. If a stance happens to be the gradient of a placement, we say that it is induced by that placement: all placement induces a stance, but a general stance is not induced by any placement, not even locally. In order t o distinguish growth from passive deformation, we postulate that, at each time T E 9 (the time line, identified with the real line), there exists a dynamically distinguished stance P(r)-smoothly depending aThe same theory encompasses also modes of material remodelling that are not treated here nor in Refs. 12, 13, such as the evolution of the elastic stiffness due to microstructural rearrangements: see Ref. 14 for a treatment of the remodelling of anisotropic bone tissue.

261 on time-which we call prototypal stance or, briefly, prototype. The assignment of a gross placement and a prototype to each time defines a refined motion (p, IP'). The idea to refine the gross motion in this way dates back to ~ r ~ n e and r ' ~~ e e , " who introduced the notion of an "intermediate" configuration in the sixties, to distinguish between elastic and visco-plastic strains. Much later Rodriguez, Hoger and McCulloch imported that notion into biomechanics, reinterpreting it as the "zero-stress reference state" of a growing body element, to quote verbatim from their 1994 paper.1g The velocity realized along the refined motion (p, P) at the time T E 9 is the pair of fields (a superposed dot denoting time differentiation): ( $ ( T ) , @(T)P(T)-') : B

+ V8 x

(V8gVg).

(3)

The linear space of test velocities 2,comprising all smooth fields

will play a central role in Sec. 2.2. The gross velocity of body-points is given by the vector field v, while the tensor field V gives the growth velocity of the corresponding body elements. 2.2. Dynamics: brute and accretive forces; balance principle

A force is primarily a continuousb linear real-valued functional on the space of test velocities, whose value we call the working expended by that force. We assume that the total working-i.e., the working expended by the sum of all forces at play-expended on any test velocity (v,V) E Z admits the following integral representation:

where the integrals are taken with respect to the bulk volume and surface area of body elements in their prototypal configuration-to be called prototypal volume and prototypal area, for short-, and

Dv := (vv)IP'-'

(6)

denotes the prototypal gmdient of v (shorthand for 'Euclidean representation of the body gradient Vv mediated by the prototypal configuration'). b ~ o n t i n u i t yof force functionals is physically essential; t o make it meaningful, the space of test velocities should be endowed with the structure of a topological vector space. We gloss over this issue, leaving up to the mathematically conscious reader the task t o complete the space of smooth test fields Z in a topology appropriate t o Eq. (5).

262 Because of the compound structure of test velocities, the force functional splits additively into a brute force, dual to v, and an accretive force, dual to V. Another important splitting-to be discussed in Sec. 2.4-is between the inner working, given by the first bulk integral in Eq. (5),and the outer working, given by the remaining sum. The brute bulk-force per unit volume b and the brute boundary-force per unit area to, take values in V&; the inner (outer) accretive couple per unit volume & (2) and the brute Piola stress S -also a specific c o u p l e t a k e values in V&@V8. All balance laws are systematically provided by the universal balance principle stating that, at each time, the total working expended on any test velocity should be zero, i.e., all forces at play should sum up to the null functional. Via standard localization arguments, this yields the local statements of balance:

+

balance of brute forces: Div S b = 0 on 93 & S n,, = t,, balance of accretive couples: A'

+ A0 = 0

on 8 2 ; (7)

on 93.

(8)

In Eq. (7), Div S is the only vector field satisfying the scalar identity div (sTv) = (Div S) v

+ S . DV

for each vector field v, while the divergence of a vector field is the trace of its prototypal gradient: div v = (Dv) . I , I being the identity on V8; n,, is the outward unit normal to the facets in 893 in their prototypal configuration. Once brute forces and accretive couples are constitutively related to the refined motion, the balance principle rules out all refined motions that do not satisfy Eqs. 7 and 8 at all times. 2.3. Energetics

To parametrize the state of the body, an additional energetic descriptor is needed. We postulate the existence of a real-valued free energy measure, such that the energy available to any part 9 of 93 is given by Q ( 9 ) = J9 $, where the density $ is the free energy per unit prototypal volume (the integral being taken with respect to the bulk volume of body elements in their prototypal configuration). A body-part is a body-like subset of 93. 2.4. Constitutive issues: theory and recipes

The constitutive theory of inner forces rests on two main pillars, altogether independent of balance: the principle of material indifference to change in observer, and the dissipation principle. Both of them deliver strict selection

263 rules on admissible constitutive recipes for the inner force. None of them applies to the outer force, which has to be regarded as an adjustable control on the motion. In fact, while the inner force represents the interactions among the (few) degrees of freedom resolved by the theory, the outer force represents their interactions with the (myriad) degrees of freedom in the universe whose evolution is not described by the motion-however refined. Not all the degrees of freedom left unresolved by the theory are necessarily localized outside of the body it considers, since its resolution is limited not only in breadth, but also-and possibly more importantly-in depth. Therefore, the crucial innerlouter dichotomy encompasses, but does not coincide with, the more obvious distinction internallexternal t o the body. A force is inner or outer with respect to the theory, not to the body: what appears as an outer force within a given theory may always-in p r i n c i p l e be accounted for as inner within a broader, deeper and more cumbersome theory. In an all-embracing theory there would be no outer force at all. In our theory of the biomechanics of growth, the outer accretive couple A0 plays a primary role, representing the mechanical effects of the biochemical control system, smartly distributed within the body itself: ignoring the chemical degrees of freedom does not make negligible their feedback on mechanics. 2.4.1. Material indifference to change in observer The group D of changes in observer we consider consists of all smooth maps of the time line 9 into the group of isometries of 8, Isom 21 0x V8, 0 being the group of orthogonal transformations of V8. Taking for granted the action of D on the set of gross motions, we extend it t o the set of refined motions as follows: a change in observer (Q, ii) E D transforms (p,P) E !& intoIthe corresponding refined motion (F, defined by

F),

for each b E 9, T E 9. In Eq. (9), xo(7) E 8 is the centre of the spherical isometry (G(T),O), and Zo(r) :=x,(T) ii (7). Notice that D acts trivially on prototypal stances: this trivial extension, while strongly motivated, is by no means obvious and has momentous consequence^.^ Consequently, the velocity (7, T) corresponding to (v, V ) is

+

?(b) = Q(T)v(b)

+ G(T) + %(r) (F(b,

T)

- Z0(r)) ,

T(b) = V(b) , (10)

with G(T) :=g0(r) - G(T) ko(r), and %(T) :=6(7) 6(7lT. 'For the crippling consequences of a different, nontrivial extension, see Ref. 20.

264 The principle of material indifference to change in observer requires that, under an arbitrary change in observer (Q,ii) E 0,the working expended over each body-part on each test velocity (v,V) by the inner force constitutively related to each refined motion (p, P) at any given time should be equal to the working expended on the corresponding velocity (7, (at the same time, over the same body-part) by the inner force related by the same constitutive prescription to the corresponding refined motion (5@). ,A parallel requirement of the same principle is that the free energy Q should be constitutive prescribed in such a way as to be invariant under all change in observer. Since arbitrarily small parts can be taken around any bodypoint, these identities localize at each body-point b E 33. The above principle rules out non-symmetric values of the brute Cauchy stress T := (det F)-' S F ~where , the warp

v)

F

:= Dp = (Vp) P-'

(11)

measures how the gross stance, i.e., the body gradient of the gross placement, differs from the prototypal stance. If we further assume that the response of the body element at b filters off from (p,P) all information other than plb, Vplb, and PIb, we obtain the following reduction theorem: there are constitutive mappings &,, and such that

&

where the constitutive list

&,

eb reduces to

the rotation R and the stretch U being, respectively, the orthogonal and the right positive-symmetric factor of the warp: F = R U . 2.4.2. Dissipation principle We call power expended along a refined motion at any given time the opposite of the working expended by the inner force constitutively related to that motion on the velocity realized along the motion at the given time. Hence, the power expended measures the working done by a putative outer force balanced with the constitutively determined inner force. The dissipation principle we enforce requires that the power dissipated-defined as the difference between the power expended along a refined motion and the time derivative of the free energy along that motion-should be non-negative, for all body-parts, at all times. This localizes into: $ + $ I . ( ~ P - ' ) 5 S - ( D # ) - #.(@IF-'),

(14)

265

+

it being intended that S, d and are given by Eq. (12). The second term on the left side of Eq. (14) stems from the fact that the prototypal-volume form, say w, evolves in time as dictated by the growth velocity PIP-':

2.4.3. Constitutive assumptions: free energy and inner force Our main constitutive assumption concerns the free energy. We posit that, for each body-point b at any given time T, +(b,r) depends solely on the value F(b, r ) of the warp at that time (in fact, only on the value U(b, r ) of the stretch, because of Eqs. 12, 13): there exists a map cpb such thatd

The requirement that Eq. (14) be satisfied along all refined motions is fulfilled if and only if for each b (which will be dropped from now on) the mappings and iisatisfy:

s

where d denotes differentiation. The Eshelby couple

IE := F ~ $ - ~ I

(17)

bridges between brute and accretive mechanics; the extra-energetic re+ + sponses S, A are restricted by the reduced dissipation inequality

to be abided by along all refined motions, at all times (C being given by Eq. (13)). We regard all dissipative mechanisms extraneous to growth to be

+

negligible, assuming the extra-energetic brute stress to be null: S = 0. Then, we make Eq. (18) satisfied in the most facile-though scarcely warrantedway, letting the extra-energetic accretive couple be simply proportional to the growth velocity through a prescribed negative scalar factor:

the resistance to growth c being positive: c > 0. In our preliminary simulations, we specify the free-energy map cp thinking of a putatively homogenized material, in order to avoid detailing the d ~ e the e paper21 by Epstein for an insightful discussion of the implications of Eq. (15).

266 micro-geometry, a common practice in similar application^.^^-^^ In particular, following Humphrey and Yin,23>24 we envisage to model the passive myocardium response assuming that cp splits additively into an isotropic component cpm, accounting for the matrix surrounding the fibres, and a transversely isotropic component cpf, accounting for the oriented collagen fibres: cp = cpm cpf. Our first results on growth-induced self-stress in arteries, summarized in Sec. 3.1, have been obtained in the most simplistic way, disregarding anisotropy altogether: cpf = 0 , and identifying the isotropic component as neo-Hookean:

+

where the single scalar parameter p > 0 is a shear modulus; Eq. (20) has to be complemented by the incompressibility constraint: det F = 1 . 2.4.4. Constitutive assumptions: outer force In the intended applications, the brute bulk-force plays a negligible role: we assume b = 0. The brute boundary-force t,, represents blood pressure and contact interactions with surrounding tissues. The key assumption is the one concerning the outer accretive couple A', whose constitutive prescription should hopefully short-circuit the complex-and ill-understoodsensing/actuating mechanobiological functions that control growth. We offer a preliminary, crude recipe we have being using for simulating the adaptive remodelling of arterial walls (see Sec. 3.1 and Ref. 22); the contrivance of an analogous recipe for postnatal cardiac hypertrophy is still in progress (see Sec. 3.2). Inspired by the stress-dependent growth laws proposed by ~ a b e r , we ~ ~posit - ~ a~ target Cauchy stress TO (as detailed in Sec. 3.1) and assume

@- ~ , A0 = cp I - (det F) F ~ T F

(21)

in order that the outer accretive couple compensates for the Eshelby couple if and only if the target stress is met:

IE+ AO= F ~ ( -T TO) F-T.

(22)

3. Mathematical Models of Adaptive Growth 3.1. Growth-induced residual stress in large arteries

As mentioned in Sec. 1.2, the fact that in vivo arteries are highly selfstressed is revealed by two salient phenomena. Soon after an arterial segment is excised and removed, its undergoes a conspicuous longitudinal

267 shortening, the ratio between the in vivo and excised lengths ranging from 1.4 to 1.7. Furthermore, when a shallow ring-shaped segment is cut in the radial direction, the ring springs open into a sector (see Fig. 4c). A coarse evaluation of the preexisting residual stress is obtained by measuring the opening angle, defined as the angle subtended by two segments drawn from the midpoint of the inner wall to the tips of the open section. The opening angle varies widely with the organ in which the blood vessel is located, with its size and shape, and with tissue remodelling. The opening angle is larger where the vessel is more curved, or thicker. For example, the opening angle

Fig. 4. Photographs reproduced from Ref. 8: (a) cross section of a rat pulmonary artery fixed in vivo a t normal pressure; (b) after excision; (c) after stress release. Our simulation (see text): (d) opening angle for a rat aorta = 58'.

268 of a normal rat artery is about 160" in the ascending aorta, 90' in the arch, 60" in the thoracic region, 80" in the a b d ~ m e n . ~ We try to relate the opening angle measured ex vivo with the growth experienced in vivo by solving numerically the evolution equation @' =

(FT(T - TO) F-T) P

(23)

deriving from Eqs. (8), (12), (16), (17), (19), (22), coupled with the standard equation for nonlinear elasticity deriving from Eqs. (7), (11) , (12), (15), (16), (20). This is done on a 2D computational domain, the annular cross section of a cylindrical vessel mimicking a rat aorta: lumen radius = 0.2 mm, wall thickness = 50 pm (data taken from Ref. 27), shear modulus p = 170 KPa. We assume this configuration to be initially stress-free and solve a cascade of four problems: (i) passive response under intramural pressure = 16 KPa and longitudinal stretch = 1.6; (ii) active response (2D growth) obtained by integrating Eq. (23) from the solution to problem (i) to a steady state, TO being pinpointed by two criteria: (1) to have a constant hoop component, and (2) to satisfy Eq. (7); (iii) passive response under removal of pressure and longitudinal tension (simulated excision) from the steady state reached in (ii); (iv) same as in (iii) for the cut annulus (see Fig. 4d). 3.2. Towards a gross mechanics of cardiac hypertrophy

To attack the complex problem of modelling the growth of the human heart from infancy to adulthood, we concentrate first on its most important component, the left ventricle. Three months after birth, cardiac myocites stop proliferating; however, they get longer and wider by synthesizing more proteins. This slow process, that develops in response to the increasing haemodynamic loading due to the overall body development and to physical e~ercise,~' is a form of volume-overload h y p e r t r ~ p h y It . ~is ~ absolutely physiological, even though it admits dangerous, possibly lethal, pathological variants. The basic tenet of our model is that cardiac hypertrophy is chiefly driven by brute mechanical circumstances associated with the myocardial pump function, with other factors playing a minor role. As a secondary, provisional hypothesis, we admit that the hypertrophic process can be described-at least to a first approximation-by the sole evolution of the stress-free geometry of the cardiac tissue, without any modification of its microstructure and fun~tionality.~ Within the cardiac cycle (see Fig. 3), two events are eSeemingly,this hypothesis has to be removed to cover pathological forms of hypertrophy.

269 critical in determining the pumping efficiency of the left ventricle: the closure of the mitral valve (separating phase 1 from 2: notice the spike in the electrocardiogram), and the opening of the aortic valve (separating phase 2 from 3: the left-ventricle pressure crosses over the aortic pressure). The closure of the mitral valve at the end of the diastolic filling and the successive systolic ejection are operated by the isometric contraction of myocardial fibres. Now, the optimum overlapping between actin and myosin filaments in sarcomeres is obtained when the sarcomere length lies between 1.8 and 2.2 pm: if the sarcomeres were longer (or shorter), the force exerted by the contraction of myocardial fibres would be smaller. Taking it for granted that the ventricle size has to increase with age and physical exercise to accommodate a larger blood volume, the only way to keep the sarcomeres at their optimal relaxed length is to multiply their number by synthesizing longer myofibrils within the preexisting cardiac myociteswhose number does not increase, as already noted. In order for the aortic valve to open, it is necessary that the left-ventricle pressure exceeds the aortic pressure. Now, the value attained by the pressure inside the ventricle at the end of the isovolumic contraction is directly correlated with the stress generated within the ventricle wall by the myocardial ~ o n t r a c t i o n .A~ ~crude estimate of this correlation is readily provided by a simplistic application of Laplace's formula for a pressurized thin-walled spherical container, establishing that the ratio (surface tension)/(intramural pressure) equals the ratio (container diameter)/(wall thickness). This motivates the assumption-consistent with abundant empirical data-that the ventricle wall grows thicker as it grows wider. Therefore, cardiac hypertrophy has to multiply sarcomeres also transversally to myofibrils, packing more of them in parallel within each single myocite. References 1. J. D. Humphrey, Cardiovascular Solid Mechanics. Cells, Tissues, and Organs (Springer, New York, 2002). 2. R. S. Chadwick, Biophysical Journal 39, 279288 (1982). 3. L. A. Taber, Annu. Rev. Biomed. Eng. 3, 1 (2001). 4. www.columbiasurgery.org/pat/cardiac/valve.html. 5. J. H. Omens and Y. C. Fung, Circ. Res. 66, 37 (1990). 6. www.webschoolsolutions.com/patts/systems/lungs.html.

7. C. Courneya, www. cellphys .ubc .ca. 8. Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues, 2nd edition (Springer, New York, 1993). 9. K. Hayashi, Mechanical properties of soft tissue and arterial walls, in Biomechanics of Soft Tissues in Cardiovascular Systems, eds. G. A. Holzapfel and

270 R. W. Ogden, CISM Courses and Lectures, N.441 (Springer, Wien, 2003) pp. 15-64.

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Acknowledgements T h e work of one or more of the four authors was supported by different funding agencies: t h e Fifth European Community Framework Programme through t h e Project HPRN-CT-2002-00284 ("Smart Systems"), GNFMINdAM (the Italian Group for Mathematical Physics), MIUR (the Italian Ministry of University and Research) through the Project "Mat hematical Models for Materials Science" and others, IMA (Institute for Mathematics and its Applications, Minneapolis, MN), Universite Paris 12 Val de Marne.

271

AUTHOR INDEX

Auricchio, F., 1 Baniotopoulos, C. C., 61, 147, 218 Barboteu, M., 15 B e i r h d a Veiga, L., 30, 45 Betti, M., 61 Cagnol, J., 76 Campo, M., 93 Chapelle, D., 30 Di Carlo, A., 253

Nardinocchi, P., 253 Nazarko, P., 147 Niiranen, J., 45 Ouafik, Y., 15 Paris, I., 30 Petrov, A., 167 Reali, A., 1 Rebrova, N. V., 183, 208 Rodriguez-Arbs, A., 93 Rohan, E., 193

Efstathiades, Ch., 147 Faella, L., 107 Fernbndez, J. R., 15, 93 Foutsitzi, G. A., 218 Garzbn, R., 122 Ghergu, M., 140 Griso, G., 140 Hadjigeorgiou, E. P., 218 Marinova, D. G., 218 Martins, J. A. C., 167, 183, 208 Miara, B., 140, 193 Monsurrb, S., 107 Monteiro Marques, M. D. P., 167

Sansalone, V., 76 Schmid, F., 208 Sobolev, V. A., 183 Stavroulakis, G. E., 61, 147, 218 Stefanelli, U., 1 Stenberg, R., 45 Teresi, L., 253 Timofte, A., 237 TringelovB, M., 253 Valente, V., 122 Zacharenakis, E. C., 218 Ziemianski, L., 147