Physica D 165 (2002) 48–65
On a model for phase separation in binary alloys driven
by mechanical effects
Elena Bonetti
a
, Pierluigi Colli
a
, Wolfgang Dreyer
b
, Gianni Gilardi
a,∗
,
Giulio Schimperna
a
, Jürgen Sprekels
b
a
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy
b
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany
Received 17 July 2001; received in revised form 3 January 2002; accepted 28 January 2002
Communicated by R. Temam
Abstract
This work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of
phase separation driven by mechanical actions in binary alloys like tin/lead solders. The system combines the (quasistationary)
balance of linear momentum with a fourth-order evolution equation of Cahn–Hilliard type for the phase separation, and
it is highly nonlinearly coupled. Existence and uniqueness results are shown. © 2002 Elsevier Science B.V. All rights
reserved.
MSC: 35J25; 35K35; 35R35; 74F20; 74N10
Keywords: Systems of partial differential equations; Existence and uniqueness; Linear elasticity; Mixture effects; Cahn–Hilliard equation
1. Introduction
In many cases binary alloys consist of two coexisting phases. If these alloys are exposed to thermo-mechanical
loads, the interface boundaries are set into motion and drastic changes of the morphology in the m (micron) range
will arise. Phase field models describe the morphology by means of an order parameter that indicates the present
phase at time t and at any point x of the alloy. In the binary tin/lead alloy, which was studied intensively by Dreyer
and Müller (see [6,7]), the tin concentration by itself can be used as a phase field.
The phase field system that was used in the recent paper [6] to study and describe qualitatively phase separa-
tion and coarsening processes under external thermo-mechanical load observed in the binary tin/lead alloy is the
following:
The variables are the fields of
u(x,t) (mechanical) displacement , χ(x,t) (tin) concentration.
∗
Corresponding author. Tel.: +39-0382-50-56-42; fax: +39-0382-50-56-02.
E-mail addresses: bonetti@dimat.unipv.it (E. Bonetti), pier@dimat.unipv.it (P. Colli), dreyer@wias-berlin.de (W. Dreyer),
gilardi@dimat.unipv.it (G. Gilardi), giulio@dimat.unipv.it (G. Schimperna), sprekels@wias-berlin.de (J. Sprekels).
0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.
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