Hysteresis operators in phase-field equations

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Hysteresis operators in phase-field equations

Collaborator: O. Klein , J. Sprekels

Cooperation with: N. Kenmochi (Chiba University, Japan), P. Krejcí (Academy of Sciences of the Czech Republic, Prague), U. Stefanelli (Università di Pavia, Italy), C. Verdi (Università di Milano, Italy), S. Zheng (Fudan University, Shanghai, China)

Supported by: DFG: ``Hysterese-Operatoren in Phasenfeld-Gleichungen'' (Hysteresis operators in phase-field equations)

Description:

To be able to deal with phase transitions, one has to take into account diffusive effects as well as hysteretic phenomena.

(a) The hysteretic phenomena are modeled by hysteresis operators. A typical example is the multi-dimensional stop operator ${\mathcal S}_Z$ with the characteristic set Z , which is defined for a convex, closed subset Z of $\IR^m$ .For a final time T > 0, this operator maps a pair (u,x⁰) consisting of an input function $u : [0,T] \to X$ and some $x^0 \in Z$ to the solution $x : [0,T] \to X$ to the variational inequality
$\begin{gather} \left\{ \begin{array} {l} u(t) = x(t) + \xi(t), \quad x(t) \in Z... ...ext{for a.e.} \quad t \in (0,T), \ x(0) = x^0,\end{array}\right. \end{gather}$
where $\langle\cdot,\cdot\rangle$ denotes the inner product on $\IR^m$ .The stop operator is used to define the corresponding multi-dimensional Prandtl-Ishlinskii operators
$\begin{gather} \mathcal H \: [u](t) = \int_0^\infty \phi(r) {\mathcal S}_{rZ}[u,x^0_r](t) \,\mathrm{d} r \, ,\end{gather}$
where $\phi \ge 0$ is a real-valued function and, for $r \ge 0$ , $x^0_r \in r Z : = \{ r z \in \IR^m \, : \, z \in Z \}$ is an initial value for the stop operator S_rZ with the characteristic set rZ. In [9], these hysteresis operators are investigated, and for a phase-field system involving multi-dimensional Prandtl-Ishlinskii operators the existence of a unique solution is proved. This is an extension of results in the previous work ([8]), since it is no longer required that the domain Z must be polyhedral. More general classes of multi-dimensional Prandtl-Ishlinskii operators have been studied in [11].

The mathematical modeling of nonlinear thermo-visco-plastic developments leads to the following system
$\begin{gather} \rho u_{tt} - \mu u_{xxt} = \sigma_x + f(x,t), \ \sigma = {\math... ... \nu w_t + {\mathcal H}_3[u_x,w] + \theta {\mathcal H}_4[u_x,w] = 0,\end{gather}$

where u, $\theta$ , $\sigma$ , and w are the unknowns displacement, absolute temperature, elastoplastic stress, and freezing index, respectively, $\rho$ , $\mu$ , C_V, and $\nu$ are positive constants, f, g are given functions, and ${\mathcal H}_1, \dots, {\mathcal H}_4$ , and $\mathcal F$ are hysteresis operators. In [12], this system has been derived, the thermodynamic consistency of the model has been proved, and the existence of a unique strong solution to an initial-boundary value problem for this system has been shown. A large time asymptotic result for this system is presented in [5]. For almost the same problem with a more general boundary condition, similar existence and asymptotic results have been derived. An approach used in [5] to derive uniform estimates for the solutions to partial differential equations involving hysteresis operators has been further investigated in [6]. Moreover, in [6], generalizations of the scalar Prandtl-Ishlinskii operators are introduced and investigated with respect to their thermodynamic consistency.

The system (3)-(6) can be simplified by adding u_xxxx on the left-hand side of (3). The corresponding system has been considered in [13].

(b) Phase-field systems of Penrose-Fife type are models for diffusive phase-transition phenomena, see [14]. For a non-conserved scalar order parameter $\chi$ , one gets the system
$\begin{align} \left( C_V \theta + \lambda(\chi) \right)_t - \kappa_0 \Delta \le... ...arepsilon \Delta \chi + s'(\chi) & = - \frac{\lambda'(\chi)}{\theta}.\end{align}$

In this system C_V, $\kappa_0$ , and $\varepsilon$ are given positive constants, and $\lambda$ , $\alpha$ , s, g are given functions.

In [7], an a posteriori error estimate has been derived for a time-discrete scheme for the system (7)-(8), with $\eta$ being a positive constant, and $\alpha(\theta) = - 1 / \theta$ .

The system (7)-(8) with $\eta$ depending on $\nabla \chi$ can be used to model the anisotropic solidification of liquids. The existence of a solution to this system has been shown in [4].

Following [1, 2], the system (7)-(8) is modified by replacing $- \Delta \chi$ in (8) by an integral operator and allowing $\eta$ to be a function of $\theta$ . For an integrodifferential (non-local) system derived in this way, results concerning global existence, uniqueness, and large-time asymptotic behavior have been derived in [15].

In another modification of the phase-field system, the energy balance (7) is coupled with two order parameter equations:
$\begin{align} w_t - \gamma \Delta w - \kappa_0 \alpha(\theta) \lambda'(\chi) & =... ...repsilon \Delta \chi + \partial I_Z (\chi) + \sigma(\chi) & \ni w_t,\end{align}$
where w is a scalar-valued order parameter, $\chi$ is an order parameter with values in $\IR^m$ ,and $\partial I_Z$ is the subdifferential of the indicator function I_Z of a bounded, closed, convex set $Z \subset \IR^m$ .Formally, this system can be considered as a diffusive approximation of the corresponding multi-dimensional stop operator ${\mathcal S}_Z$ .Results concerning existence, uniqueness, and continuous dependence on data have been presented in [3].

In [10], it has been proved for another phase-field system that the solutions are approximations for the evaluation of the stop operator.

References:

H. GAJEWSKI, On a nonlocal model of non-isothermal phase separation, WIAS Preprint no. 671, 2001, to appear in: Adv. Math. Sci. Appl.
H. GAJEWSKI, K. ZACHARIAS, On a nonlocal phase separation model, WIAS Preprint no. 656, 2001, to appear in: J. Math. Anal. Appl.
N. KENMOCHI, J. SPREKELS, Phase-field systems with vectorial order parameters including diffusional hysteresis effects, WIAS Preprint no. 665, 2001, submitted.
O. KLEIN, Existence and approximation of solutions to an anisotropic phase field system for the kinetics of phase transitions, Interfaces and Free Bound., 4 (2002), pp. 47-70.
$\dito$ ,Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity, WIAS Preprint no. 734, 2002 .
O. KLEIN, P. Krejci, Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations, WIAS Preprint no. 748, 2001 .
O. KLEIN, C. VERDI, A posteriori error estimates for a time discrete scheme for a phase-field system of Penrose-Fife type, WIAS Preprint no. 677, 2001, submitted.
P. Krejci, J. SPREKELS, Phase-field systems and vector hysteresis operators, in: Free Boundary Problems: Theory and Applications II, N. Kenmochi, ed., Gakkotosho, Tokyo 2000, pp. 295-310.
$\dito$ ,Phase-field systems for multi-dimensional Prandtl-Ishlinskii operators with non-polyhedral characteristics, WIAS Preprint no. 648, 2001, to appear in: Math. Methods Appl. Sci.
$\dito$ ,Singular limit in parabolic differential inclusions and the stop operator, WIAS Preprint no. 678, 2001, submitted.
$\dito$ ,On a class of multi-dimensional Prandtl-Ishlinskii operators, Physica B, 306 (2001), pp. 185-190.
P. Krejci, J. SPREKELS, U. STEFANELLI, Phase-field models with hysteresis in one-dimensional thermo-visco-plasticity, WIAS Preprint no. 655, 2001, submitted.
$\dito$ ,One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, WIAS Preprint no. 702, 2001, submitted.
O. PENROSE, P.C. FIFE, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), pp. 44-62.
J. SPREKELS, S. ZHENG, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions, WIAS Preprint no. 698, 2001, submitted.