Phase Transitions in Two-Phase Media with the Same Moduli of Elasticity
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
PHASE TRANSITIONS IN TWO-PHASE MEDIA WITH THE SAME MODULI OF ELASTICITY V. G. Osmolovskii St. Petersburg State University 28, Universitetskii pr., Petrodvorets, St. Petersburg 198504, Russia [email protected]
UDC 517.9
For a variational problem of the theory of phase transitions in continuum mechanics with the same moduli of elasticity we obtain explicit formulas for the phase transition temperatures and equilibrium energy. The existence of equilibrium states is studied in some particular cases. Bibliography: 15 titles.
1
Introduction
The energy functional of a two-phase elastic medium occupying a bounded domain Ω ⊂ Rm is defined by (1.1) I[u, χ, t, Ω] = {χ(F + (∇u) + t) + (1 − χ)F − (∇u)} dx. Ω
Here, u(x) = (u1 (x), . . . , um (x)) is an m-dimensional vector-valued function, ∇u is the matrix with entries (∇u)ij = uixj , i, j = 1, . . . , m, χ(x) is the characteristic function supported in a subset of Ω occupied by the phase with index +, and t is interpreted as the temperature. In the quadratic approximation, the energy densities F ± (M ) (here, M ∈ Rm×m is the space of (m × m)-matrices) take the form F ± (M ) = A± (e(M ) − ζ ± ), e(M ) − ζ ± ,
(1.2)
where e(M ) = 12 (M + M ∗ ) belongs to the space Rsm×m of symmetric (m × m)-matrices equipped with the inner product α, β = tr αβ,
α, β ∈ Rsm×m ,
ζ ± ∈ Rsm×m , A± are linear symmetric positive definite mappings from the space Rsm×m to itself. In the theory of elasticity, e(∇u) denotes the strain tensor corresponding to a displacement field u, ζ ± are the residual strain tensors, A± are the tensors of moduli of elasticity.
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 139-147. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0713
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For the domain of the functional (1.1), (1.2) we take ◦
X(Ω) = W 12 (Ω, Rm ),
(1.3)
Z(Ω) is the of all measurable characteristic functions. Such a choice of X(Ω) means that displacement fields are zero on the boundary. For given t by an equilibrium state of a two-phase medium with the energy densities (1.2) we mean a solution to the variational problem I[ ut , χ t , t, Ω] =
inf
u∈X(Ω),χ∈Z(Ω)
I[u, χ, t, Ω],
u t ∈ X(Ω), χ t ∈ Z(Ω).
(1.4)
The solution to the problem (1.4) is called a one-phase solution if χ t ≡ 0 or χ t ≡ 1 and a two-phase solution in the opposite case. It is obvious that u t ≡ 0 for one-phase equilibrium states. As proved in [1], there are the phase transition temperatures t± , t− t+ , independent of Ω and characterized by the fact that the unique solution to the problem (1.4) is one-phase with t ≡ 0 if t > t+ , whereas no one-phase equilibrium state exists if χ t ≡ 1 if t < t− or with χ t ∈ (t− , t+ ). In the case t = t+ = t− , the set of all equilibrium states consists only of the pairs t such that u t ≡ 0 and χ t is an arbitrary element of the set Z(Ω). In the case t− < t+ , u t , χ two-phase equilibrium states can be realized only if u t ≡ 0. The coincidence criterion f
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