A discrete variational scheme for isentropic processes in polyconvex thermoelasticity
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Calculus of Variations
A discrete variational scheme for isentropic processes in polyconvex thermoelasticity Cleopatra Christoforou1 · Myrto Galanopoulou2 · Athanasios E. Tzavaras2 Received: 12 December 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose a variational scheme for the construction of isentropic processes of the equations of adiabatic thermoelasticity with polyconvex internal energy. The scheme hinges on the embedding of the equations of adiabatic polyconvex thermoelasticity into a symmetrizable hyperbolic system. We establish existence of minimizers for an associated minimization theorem and construct measure-valued solutions that dissipate the total energy. We prove that the scheme converges when the limiting solution is smooth. Mathematics Subject Classification 35Q74 · 49M25 · 35L65 · 174B20 · 74A15
1 Introduction We consider the system of adiabatic thermoelasticity, ∂t Fiα − ∂α vi = 0
∂t
∂t vi − ∂α Σiα = 0 r ∂t η = θ
(1.1)
1 2 |v| + e − ∂α (Σiα vi ) = r , 2
Communicated by J.Ball.
B
Myrto Galanopoulou [email protected] Cleopatra Christoforou [email protected] Athanasios E. Tzavaras [email protected]
1
Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus
2
Computer, Electrical, Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia 0123456789().: V,-vol
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describing the evolution of a thermomechanical process y(x, t), η(x, t) ∈ R3 × R+ in Lagrangian coordinates with the spatial variable x ∈ R3 and time t ∈ R+ . A solution to (1.1) consists of the deformation gradient F = ∇ y ∈ M3×3 , the velocity v = ∂t y ∈ R3 and the specific entropy η. The first equation is a compatibility relation, the second describes the balance of linear momentum, while the fourth stands for the balance of energy. One appends to (1.1) the constraint ∂α Fiβ = ∂β Fiα ,
i, α, β = 1, 2, 3,
(1.2)
which guarantees that F is indeed a gradient. Note that (1.2) is an involution, that is it is propagated from the initial data to the solution via (1.1)1 . The remaining variables in (1.1) are the Piola–Kirchhoff stress Σiα , the internal energy e, and the radiative heat supply r . In the theory of adiabatic thermoelasticity, the referential heat flux Q α = 0 and it does not appear in the equations (1.1)3,4 . For simplicity we have normalized the reference density ρ0 = 1. The balance of entropy (1.1)3 holds identically as an equality for strong solutions; by contrast, for weak solutions it is replaced by the Clausius–Duhem inequality [12,13,30] and serves as an admissibility criterion. The system is closed through constitutive relations which, for smooth processes, are consistent with the Clausius–Duhem inequality and describe the material response. For thermoelastic materials under adiabatic conditions, the constitutive theory is determined from the ther
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