Hysteresis operators

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Hysteresis operators

Collaborator: O. Klein, P. Krejcí, J. Sprekels

Cooperation with: M. Brokate, H. Schnabel (Technische Universität München), P. Colli (Università di Pavia, Italy), K. Kuhnen (Universität des Saarlandes, Saarbrücken), E. Rocca (Università di Milano, Italy), M. Sorine (INRIA Rocquencourt, France)

Description:

Phase transitions often manifest a strong hysteretic behavior due to irreversible changes in the material structure. Hysteresis operators represent a tool for modeling and efficient description of such processes, where an information on the evolution of the macroscopic state is more relevant for applications than the dynamics of each individual element in the microstructure. Within this project, the following main research directions have been pursued in 2004.

1. Mathematical modeling and control of piezoelectric actuators

A long-term cooperation with K. Kuhnen from the Saarland University at Saarbrücken is based on applications of generalized Prandtl-Ishlinskii hysteresis operators to developing fast algorithms for real-time control of piezoelectric sensors and actuators. The research in the past was focused on questions of stability of the model as well as numerical stability of the algorithms, see [10, 11]. In 2004, emphasis was put on the problem of parameter identification for the underlying hysteresis operator. As a first step, the paper [12] used the linear offline error model. The real-time identification problem was solved as a discontinuous projected dynamical system which was transformed into a continuous system involving hysteresis operators as solution operators of differential inclusions. The project will continue in 2005 with an extension of this approach to adaptive online identification methods for existing as well as refined hysteresis models.

2. Models for fatigue of elastoplastic materials

The evolution of microcracks in an elastoplastic material is generally accepted to be the cause for accumulated fatigue and damage. In order to avoid complicated considerations about the microcrack growth, the Gurson model suggests to introduce one scalar parameter related to the total relative volume of voids in the material as a measure for macroscopic fatigue. Experiments give a parameter range in which this model gives fairly reliable engineering results, see [1]. From the mathematical point of view, the Gurson model consists of an elastoplastic stress-strain law with yield surface which evolves in time as a function of the fatigue parameter which is itself in turn a function of the plastic strain. In other words, the material law can be written in the form of a rate-independent quasi-variational inequality with state-dependent constraint. While criteria for the existence of solutions to such problems were known, the problem of uniqueness and stability of solutions was completely open. A method leading to the proof of existence, uniqueness, and Lipschitz continuity of the solution mapping for a general class of quasi-variational inequalities was suggested in [3] in cooperation with M. Brokate and H. Schnabel from Munich. First estimates show that this method is applicable to the Gurson model within the relevant parameter range.

3. Nonlocal phase transition models

The system of partial differential equations proposed in [6] and [7] as a model for nonlocal phase separation, which accounts for long-range interactions between particles in a multiphase system, has been reformulated as an abstract evolution equation of the form

$\displaystyle {\frac{{d}}{{d t}}}$ u + A( $\displaystyle \partial$ $\displaystyle \Phi$ (u) + B(u)) $\displaystyle \ni$ g

with a linear unbounded non-invertible operator A with non-trivial null space which stands for the diffusion term, a continuous nonlinear operator B which plays the role of the nonlocal component, and a singular convex potential $\Phi$ which represents the geometric constraints for the order parameter. The main result of the joint paper [5] with P. Colli and E. Rocca is the derivation of a sufficient condition between the null space of A and the subdifferential of $\Phi$ under which the initial-value problem is well posed. An example shows that if the condition is not fulfilled, then even a local solution may fail to exist. Another nonlocal model for phase transitions, where also the temperature evolution is taken into account, has been investigated as part of this project, and results on its well-posedness and asymptotic stability were published in [14].

4. Mechanical models for heart muscle fiber

The mechanics of the muscle fiber is particular by the fact that its material characteristics are not constant, but may vary extremely quickly as a result of electric excitations which release fast chemical reactions in the cells. In particular, in the heart, phases of active contraction, active relaxation, and passive relaxation change periodically in the normal regime. For example, during contraction, the elasticity coefficient increases by several orders of magnitude with a strong hysteresis effect. A numerical model for this behavior was proposed by a French group of engineers, biologists, and mathematicians in [4]. The model was investigated in cooperation with the research group coordinated by M. Sorine from the point of view of mathematical consistency. It was proved in [13] that the corresponding system of partial differential equations has a unique global strong solution, this solution remains bounded under bounded external excitation and, in agreement with experimental observations, tends to an equilibrium in the passive relaxation regime, that is, if no excitation is present. A simplified model where variations along the fiber are neglected shows that elasticity in this equilibrium does not vanish.

5. Outward-pointing hysteresis operators

It has been shown in [8, 9] that one can derive new uniform estimates for the solutions to some partial differential equations involving hysteresis operators, if these operators are ``outward-pointing hysteresis operators'' or even ``strongly outward-pointing hysteresis operators''. For scalar Prandtl-Ishlinskii operators and generalizations of these operators, appropriate conditions that allow to check if these operators are pointing outwards have been formulated in [8, 9], i.e. it has been shown that a Prandtl-Ishlinskii operator is strongly pointing outwards if and only if its initial loading curve is unbounded. The work [2] in progress shows also for some class of Preisach operators that the condition of the unbounded initial loading curve is sufficient to ensure that the operator is pointing outwards. But, considering, for example, appropriate generalized play operators, one realizes that there are strongly outward-pointing Preisach operators with a bounded initial loading curve. Therefore, some further investigations will be done to complete the characterization.

References:

H. BAASER, D. GROSS, Crack analysis in ductile cylindrical shells using Gurson's model, Internat. J. Solids Structures, 37 (2000), pp. 7093-7104.
M. BROKATE, O. KLEIN, P. KREJCI, in preparation.
M. BROKATE, P. KREJCI, H. SCHNABEL, On uniqueness in evolution quasivariational inequalities J. Convex. Anal., 11(1) (2004), pp. 111-130.
D. CHAPELLE, F. CLÉMENT, F. GÉNOT, P. LE TALLEC, M. SORINE, J.M. URQUIZA, A physiologically-based model for the active cardiac muscle, in: FIMH 2001 (Functional Imaging and Modeling of the Heart), First International Workshop, Helsinki, Finland, November 15-16, 2001, Proceedings, T. Katila, I.E. Magnin, P. Clarysse, J. Montagnat, J. Nenonen, eds., vol. 2230 of Lecture Notes in Computer Science, Springer, 2001, pp. 128-133.
P. COLLI, P. KREJCI, E. ROCCA, J. SPREKELS, Nonlinear evolution inclusions arising from phase change models, WIAS Preprint no. 974, 2004.
H. GAJEWSKI, On a nonlocal model of non-isothermal phase separation, Adv. Math. Sci. Appl., 12 (2002), pp. 569-586.
H. GAJEWSKI, K. ZACHARIAS, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), pp. 11-31.
O. KLEIN, P. KREJCI, Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations, Nonlinear Anal. Real World Appl., 4 (2003), pp. 755-785.
, Asymptotic behaviour of evolution equations involving outwards pointing hysteresis operators, Physica B, 343 (2004), pp. 53-58.
P. KREJCI, K. KUHNEN, Inverse control of systems with hysteresis and creep, IEE Proc. Control Theory Appl., 148(3) (2001), pp. 185-192.
, Error estimates for the discrete inversion of hysteresis and creep operators, Math. Comput. Simulation, 61 (2003), pp. 537-548.
, Identification of linear error-models with projected dynamical systems, Mathematical and Computer Modelling of Dynamical Systems, 10(1) (2004), pp. 59-91.
P. KREJCI, J. SAINTE-MARIE, M. SORINE, J.M. URQUIZA, Solutions to muscle fiber equations and their long time behaviour, WIAS Preprint no. 973, 2004.
P. KREJCI, J. SPREKELS, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Adv. Math. Sci. Appl., 14(2) (2004), pp. 593-612.