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

. LONDON . SINGAPORE . B E l J l N G

SHANGHAI

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. TAIPEI . C H E N N A I

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ISBN- 13 978-981-270-392-7 ISBN-10 981-270-392-6

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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 /


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 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 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- 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




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+ 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

+


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



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=xl

(30)

Therefore, the sensitivity Szr can be determined in terms of coefficients {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

jo+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, - 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.

10. A. Rachev and K. Hayashi, Annals of Biomedical Engineering 27,459 (1999). 11. J. D. Humphrey and E. Wilson, J. Biomechanics 36, 1595 (2003). 12. A. DiCarlo and S. Quiligotti, Mech. Res. Comm. 29, 449 (2002). 13. A DiCarlo, Surface and bulk growth unified, in Mechanics of Material Forces, eds. P. Steinmann and G. A. Maugin, Advances in Mechanics and Mathematics, Vol. 11 (Springer, New York, 2005) pp. 5344. 14. A. DiCarlo, S. Naili and S. Quiligotti, C. R. Me'canique, doi: 10.1016/ j .crme .2006.06.009(2006). 15. L. A. Taber, Appl. Mech. Rev. 48, 487 (1995). 16. S. C. Cowin, Annu. Rev. Biomed. Eng. 6, 77 (2004). 17. E. Kroner, Arch. Rational Mech. Anal. 4, 273 (1960). 18. E. H. Lee, J. Appl. Mechanics 36, 1 (1969). 19. E. K. Rodriguez, A. Hoger and A. D. McCulloch, J. Biomechanics 27, 455 (1994). 20. S. Quiligotti, Theor. Appl. Mech. 28/29, 277 (2002). 21. M. Epstein, On material evolution laws, in Geometry, Continua 0 Microstructure, ed. G. A. Maugin, (Hermann, Paris, 1999) pp. 1-9. 22. P. Nardinocchi and L. Teresi, Stress driven remodeling of living tissues, in Proceedings of the FEMLAB Conference 2005, (Stockholm, Sweden, 2005). 23. J. D. Humphrey and F. C. P. Yin, J. Biophysical Society 52, 563570 (1987). 24. J. D. Humphrey and F. C. P. Yin, Circ. Res. 65, 805 (1989). 25. G. A. Holzapfel, T. Gasser and R. W. Ogden, J. Elasticity 61, 1 (2000). 26. G. A. Holzapfel, T. Gasser and R. W. Ogden, J. Biomech. Engineering 126, 264 (2004). 27. L. A. Taber and D. W. Eggers, J. Theor. Biol. 180, 343 1996. 28. L. A. Taber, J. Theor. Biol. 193, 201 (1998). 29. L. A. Taber, J. Biomech. Engineering 120, 348 (1998). 30. R. Crepaz, W. Pitscheider, G. Radetti and L. Gentili, Pedriatic Cardiology 19, 463 (1998). 31. W. Grossman and F. C. P. Yin, The American Journal of Medicine 69, 576584 (1980). 32. D. D. Streeter, R. N. Vaishnav, D. J. Petel, H. M. Spotnitz, J. Ross and E. H. Sonnenblick, Biophysical Journal 10, 345362 (1970).

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