Some Variational Principles for Coupled Thermoelasticity

e nonlinear thermoelasticity of type II proposed by Green and Naghdi is considered. e thermoelastic structural model is formulated in a quasistatic range, and the related thermoelastic variational formulation in the complete set of state variables is recovered. Hence a consistent framework to derive all the variational formulations with different combinations of the state variables is provided, and a family of mixed variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. A uniqueness condition is provided on the basis of a suitable variational formulation.


Introduction
Coupled thermomechanical problem arises in a variety of important �elds of application, including casting, metal forming, machining and other manufacturing processes, structural models, and others.
Green and Naghdi (GN) introduced a theory in which heat propagates as thermal waves at �nite speed and does not necessarily involve energy dissipation.Another property of the GN theory of type II is the fact that the entropy �ux vector is determined by means of the same potential as the mechanical stress tensor.Motivated by the procedure presented in [1,2], this paper is concerned with the formulation of variational principles characterizing the solutions of the coupled thermomechanical problem for the GN model without dissipation, that is, type II.e variational characterization of the thermoelastic problem means the identi�cation of a functional whose stationary points are solutions of the problem.Once this functional is known, the solutions of the problem can be identi�ed with certain extrema of the functional.
Following the pioneering work of Biot [3], the variational forms of the coupled thermoelastic and thermoviscoelastic problems for the classical Fourier model have been investigated in several papers; see, for example, [2,[4][5][6][7][8][9] for the case of thermomechanical coupling in dissipative materials.Moreover consistent variational principles for structural problems concerning elastic and elastoplastic models in isothermal conditions are well developed; see, for example, [10][11][12].
On the contrary variational formulations for the GN theory of type II are addressed in a few papers; see, for example, [1,13,14].
In this paper the GN thermoelastic coupled structural model without dissipation is formulated in a suitable form so that we can provide the methodology to build the complete family of all the admissible variational formulations associated with the considered GN model.It is well-known that the GN theory of type II has been developed as a dynamic theory, but an important prerequisite for its use is a thorough understanding of the corresponding problems in which the dynamical effects are disregarded; see, for example, [15].
An advantage of such an approach consists in the fact that the boundary-value problem for the thermoelastic model is formulated in such a way that the model can be cast in terms of a multivalued structural operator de�ned in terms of all the state variables.is operator encompasses in a unique expression the �eld equation, the constitutive relations, the constraint relations and the initial conditions which describe the considered thermoelastic structural model.
e related non-smooth potential can then be evaluated by a direct integration along a ray in the operator domain and depends on all the state variables involved in the model.Appraising the generalized gradient of the non-smooth potential [10] and imposing its stationarity, the operator formulation of the problem is recovered.
Hence a general procedure to derive variational formulations within the incremental framework for the considered GN coupled thermoelastic model without dissipation is formulated.It is then shown how a family of mixed variational formulations, associated with the considered GN model, can be obtained following a direct and general procedure by enforcing the ful�lment of �eld equations and constraint conditions.
e possibility to formulate the coupled GN thermoelastic problem in a variational form has a number of consequences and some bene�cial effects.For instance, the variational framework allows one to apply the tools of the calculus of variations to the analysis of the solutions of the problem.In particular, conditions for the existence (see, e.g., [16,17]) and uniqueness of the solution are based on the variational framework.
Accordingly the condition for the uniqueness of the solution of the considered model is provided by means of a minimization principle.

Thermoelastic Structural Problem
roughout the paper bold-face letters are associated with vectors and tensors.e scalar product between dual quantities (simple or double index saturation operation between vectors or tensors) is denoted by * .A superimposed dot means differentiation with respect to time and the symbol ∇ denotes the gradient operator.
In small strain analysis the theory of thermoelasticity without energy dissipation, as described in Green and Naghdi [18], is considered.Such a theory is based on the introduction of a scalar thermal displacement  de�ned as where  is a point pertaining to a thermoelastic body de�ned on a regular bounded domain Ω of an Euclidean space,  =     represents the temperature variation from the uniform reference temperature   , and   ( 0) is the initial value of the thermal displacement at the time  = 0. Accordingly the time derivative of  is the temperature variation, that is, α =   .e mechanical and thermal parts of the thermoelastic model are herea�er de�ned.
Let  denote the linear space of strain tensors ( ) and  denote the dual space of stress tensors ( ).e inner product in the dual spaces ⟨⋅ ⋅⟩ has the mechanical meaning of the internal virtual work, that is e linear space of displacements  is denoted by .e linear space of forces is ℱ and is placed in separating duality with  by a nondegenerate bilinear form ⟨⋅ ⋅⟩ which has the physical meaning of external virtual work.For avoiding proliferation of symbols, the internal and external virtual works are denoted by the same symbol.Conforming displacement �elds satisfy homogeneous boundary conditions and belong to a closed linear subspace ℒ ⊂ .
e kinematic operator  is a bounded linear operator from  to the space of square integrable strain �elds   .e subspace of external forces ℱ is the dual space of .e equilibrium operator is the continuous operator  ′ from  to ℱ which is dual of .Let ℓ = {   ℱ be the load functional where  and  denote the tractions and the body forces [12,19].
e equilibrium equation and the compatibility condition are given by where ℓ,   ℱ,    and   ,   .e external relation between reactions and displacements is provided by being  a concave function, and the symbol  denotes the sub(super)differential of convex (concave) functions [20].Accordingly, the inverse relation is expressed as where the concave function  * represents the conjugate of  and the Fenchel's relation holds Different expressions can be given to the functional  depending on the type of external constraints such as bilateral, unilateral, elastic, or convex.For future reference the expressions of  and  * are specialized to the case of external frictionless bilateral constraints with homogeneous boundary conditions.Noting that the subspace of the external constraint reactions ℛ is the orthogonal complement of the subspace of conforming displacements ℒ, that is ℛ = ℒ ⟂ , the functional  turns out to be and a direct evaluation shows that its conjugate  * is given by: e proposed framework has the advantage that the formulation of the thermal model is similar to the mechanical one, and the thermoelastic problem turns out to be suitable to build a general variational formulation as shown in the next section.e linear space of thermal displacement  is .e rate of heat �ow into the body by heat sources  and the boundary heat �uxes  belong to the space ℋ ⊂  ′ , dual of , of square integrable �elds on Ω. e external thermal forces are collected in the set ℓ  = {      ′ .e kinematic thermal operator   Lin{  is a bounded linear operator, and a thermal gradient    is said to be thermally compatible if there exists an admissible thermal displacement �eld  such that   .e thermal balance equation is given by  ′   η     , where  ′ is the dual operator of  and  is the entropy �ux vector.
�onstraint conditions can be �t in �eld equations by noting that the external relation between reactive thermal forces  and thermal displacements  is provided by the equivalent relations: being  and  * conjugate convex functionals.e equality (9) 3 represents the Fenchel's relation.�omogeneous constraints on thermal displacement �elds are formulated by considering that thermal displacement �elds belong to the subspace  and are said to be admissible.Reactive thermal forces belong to the orthogonal complement  ⟂ of .en the functional  is the indicator of : and its conjugate  * is given by Accordingly the relations governing the quasistatic thermoelastic structural problem without energy dissipation for given mechanical and thermal loads  and   in the time interval   0  can be collected in the following form: e initial thermal conditions are considered in the following form: where   and   are, respectively, a prescribed initial thermal displacement and temperature in .
e thermoelastic functionals Φ and  provide the thermoelastic constitutive relations of the considered GN model.
In the case of a linear coupled thermoelastic behaviour the expression of the functional Φ is where the �rst term at the r.h.s.above is the isentropic elastic strain energy and the second one represents the thermoelastic coupling.e parameter    is the speci�c heat at constant strain at the reference state with temperature   ,        (  )   is the symmetric and positive de�nite isentropic elastic moduli fourth-order tensor.e second-order thermal expansion tensor  is self-adjoint, that is,  ′  .Moreover the thermal function  is given by where  is the tensor of conductivity moduli, symmetric and positive de�nite.
A solution of the thermoelastic structural problem can be achieved by a �nite element approach (see e.g., �21, 22]) and can be obtained starting from a suitable mixed variational formulation.�ence the de�nition of a general variational formulation which allows one to derive variational principles with different combination of the state variables, without ad hoc procedure, plays a central role.
e time integral of the constitutive relation (12) e operators  and   turn out to be integrable by virtue of the duality existing between the operators , , ,   ,   ,   ,   and  ′ ,  ′ ,  ′ ,  ℱ ,   ,   ′ ,   ′ , the conservativity of ,  and the conservativity of the super(sub)differentials   and   .e related potential can be evaluated by a direct integration along a straight line in the space  starting from its origin to get [23]: From a mechanical point of view, the relation (25) 1 yields the equilibrium equation: the relation (25) 2 provides the compatibility condition: the relations (25) 3-4 yield the constitutive relations: and, �nally, the relation (25) 9 yields the thermal external relation: Hence, performing the super(sub)differentials appearing in (25), the structural problem (18) is recovered.

Mixed Variational Principles.
A family of potentials can be recovered from the potential  by enforcing the ful�lment of �eld equations and of constitutive relations.All these functionals assume the same value when they are evaluated at a solution point of the GN structural problem.
Hence a variational principle in which the external reactions do not appear as independent state variables can be obtained by imposing, in the expression (22) of the potential , the external relations (18) 59 in terms of Fenchel's equalities ( 6) and ( 9 if and only if it is a solution of the thermoelastic structural problem without dissipation (18).
Imposing in the expression (36) of the potential  2 the thermal compatibility condition (18)  if and only if it is a solution of the thermoelastic structural problem without dissipation (18).
From a mechanical point of view, the specialization of the potential  3 to the Cauchy model for a linear coupled thermoelastic behaviour, external frictionless bilateral constrains, and homogeneous thermal boundary conditions is provided.e functionals Φ and Ψ, see ( 14) and (15) Hence it is important to derive a variational formulation in terms of a minimization problem as hereaer reported.
To this end let us note that the constitutive relation (18) 8 can be equivalently expressed in terms of the conjugate Ψ  of the thermoelastic functional Ψ in the following form: and the Fenchel's relation hold: In the case of the Cauchy model with a linear coupled thermoelastic behaviour, the potential Ψ  can be evaluated as the conjugate of ( 15) and is given by Enforcing in the expression of the functional  the thermal constitutive relation (18) 8 in terms of the Fenchel's equality (46) and the external relation (18) 5 in terms of Fenchel's equality (6), it turns out to be if and only if it is a solution of the thermoelastic structural problem without dissipation (18).
Accordingly, if the thermoelastic functional Ψ  pertaining to the GN constitutive model is strictly convex, the functional  4 turns out to be strictly convex, and the GN thermoelastic structural model ( 18) admits a unique solution (if any).
It is worthnoting that, in the linear coupled thermoelastic case, the expression of the functional Ψ * is given by (47) so that it turns out to be a strictly convex functional.erefore the GN thermoelastic structural model ( 18) admits a unique solution.e question of existence of the solution is still an challenge problem; see, for example, [17].

Closure
A variational framework for a class of GN coupled thermomechanical boundary-value problem is presented.e thermoelastic structural model is addressed, and the related general mixed thermoelastic variational formulation in the complete set of state variables is derived starting from the structural model.An advantage of the proposed methodology is that it can be applied to a wide range of structural models, and variational formulations can be obtained following a general reasoning.As a consequence, a family of mixed thermoelastic variational formulations in a reduced number of variables is then contributed.Finally, by appealing to a minimum principle, the connection between uniqueness of the solution and convexity is investigated.
e potential  turns out to be linear in (   ), jointly convex with respect to the state variables (  ) and jointly concave with respect to ( ).e following statement then holds.estationary condition of , enforced at the point ( ) 3 .Hence it turns out to be solution techniques can be exploited, and existence and uniqueness results can be provided by recourse to functional analysis.Actually, uniqueness of the solution is ensured if the functional to be minimized is strictly convex.