Some Variational Principles for Coupled Thermoelasticity

Abstract

The nonlinear thermoelasticity of type II proposed by Green and Naghdi is considered. The 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.

1. Introduction

Coupled thermomechanical problem arises in a variety of important fields 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 finite speed and does not necessarily involve energy dissipation. Another property of the GN theory of type II is the fact that the entropy flux 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. The variational characterization of the thermoelastic problem means the identification of a functional whose stationary points are solutions of the problem. Once this functional is known, the solutions of the problem can be identified 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, 4–9] 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, [10–12].

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 defined in terms of all the state variables. This operator encompasses in a unique expression the field equation, the constitutive relations, the constraint relations and the initial conditions which describe the considered thermoelastic structural model.

The 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 fulfilment of field equations and constraint conditions.

The possibility to formulate the coupled GN thermoelastic problem in a variational form has a number of consequences and some beneficial 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.

2. Thermoelastic Structural Problem

Throughout the paper bold-face letters are associated with vectors and tensors. The 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 𝛼 defined as 𝛼(𝐱,𝑡)=^𝑡₀𝜗(𝐱,𝜏)𝑑𝜏+𝛼_𝑜(𝐱,0), (1) where 𝐱 is a point pertaining to a thermoelastic body defined on a regular bounded domain Ω of an Euclidean space, 𝜗=𝜃−𝜃_𝑟 represents the temperature variation from the uniform reference temperature 𝜃_𝑟, and 𝛼_𝑜(𝐱,0) is the initial value of the thermal displacement at the time 𝑡=0. Accordingly the time derivative of 𝛼 is the temperature variation, that is, 𝛼̇=𝜃−𝜃_𝑟.

The mechanical and thermal parts of the thermoelastic model are hereafter defined.

Let 𝒟 denote the linear space of strain tensors 𝜺(𝐱,𝑡) and 𝒮 denote the dual space of stress tensors 𝝈(𝐱,𝑡). The inner product in the dual spaces ⟨⋅,⋅⟩ has the mechanical meaning of the internal virtual work, that is ⟨𝝈,𝜺⟩=_Ω𝝈(𝐱,𝑡)∗𝜺(𝐱,𝑡)𝑑𝐱. (2) The linear space of displacements 𝒰 is denoted by 𝐮. The linear space of forces is ℱ and is placed in separating duality with 𝒰 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 fields satisfy homogeneous boundary conditions and belong to a closed linear subspace ℒ⊂𝒰.

The kinematic operator 𝐁 is a bounded linear operator from 𝒰 to the space of square integrable strain fields 𝜺∈𝒟. The subspace of external forces ℱ is the dual space of 𝒰. The equilibrium operator is the continuous operator 𝐁^′ from 𝒮 to ℱ which is dual of 𝐁. Let ℓ={𝐭,𝐛}∈ℱ be the load functional where 𝐭 and 𝐛 denote the tractions and the body forces [12, 19].

The equilibrium equation and the compatibility condition are given by ℓ+𝐫=𝐁^′𝝈,𝜺=𝐁𝐮, (3) where ℓ, 𝐫∈ℱ, 𝝈∈𝒮 and 𝜺∈𝒟, 𝐮∈𝒰.

The external relation between reactions and displacements is provided by 𝐫∈𝜕Υ(𝐮), (4) being Υ a concave function, and the symbol 𝜕 denotes the sub(super)differential of convex (concave) functions [20]. Accordingly, the inverse relation is expressed as 𝐮∈𝜕Υ^∗(𝐫), (5) where the concave function Υ^∗ represents the conjugate of Υ and the Fenchel’s relation holds Υ(𝐮)+Υ^∗(𝐫)=⟨𝐫,𝐮⟩. (6) Different expressions can be given to the functional Υ depending on the type of external constraints such as bilateral, unilateral, elastic, or convex. For future reference the expressions of Υ and Υ^∗ 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 Υ turns out to be Υ(𝐮)=_ℒ(𝐮)=0if𝐮∈ℒ−∞otherwise, (7) and a direct evaluation shows that its conjugate Υ^∗ is given by: Υ^∗(𝐫)=_ℒ⟂(𝐫)=0if𝐫∈ℒ^⟂−∞otherwise. (8) The 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.

The linear space of thermal displacement 𝛼 is 𝒜. The rate of heat flow into the body by heat sources 𝑟 and the boundary heat fluxes 𝑞 belong to the space ℋ⊂𝒜^′, dual of 𝒜, of square integrable fields on Ω. The external thermal forces are collected in the set ℓ_𝜃={𝑞,−𝑟/𝜃_𝑟}∈𝒜^′.

The kinematic thermal operator 𝐆∈Lin{𝒜,𝒢} is a bounded linear operator, and a thermal gradient 𝐠∈𝒢 is said to be thermally compatible if there exists an admissible thermal displacement field 𝛼 such that 𝐠=𝐆𝛼. The thermal balance equation is given by 𝐆^′𝐩=𝜂̇+ℓ_𝜃+𝛾, where 𝐆^′ is the dual operator of 𝐆 and 𝐩 is the entropy flux vector.

Constraint conditions can be fit in field equations by noting that the external relation between reactive thermal forces 𝛾 and thermal displacements 𝛼 is provided by the equivalent relations: 𝛾∈𝜕Π(𝛼)⟺𝛼∈𝜕Π^∗𝛾⟺Π(𝛼)+Π^∗𝛾=𝛾,𝛼, (9) being Π and Π^∗ conjugate convex functionals. The equality (9)₃ represents the Fenchel’s relation.

Homogeneous constraints on thermal displacement fields are formulated by considering that thermal displacement fields belong to the subspace 𝒦 and are said to be admissible. Reactive thermal forces belong to the orthogonal complement 𝒦^⟂ of 𝒦. Then the functional Π is the indicator of 𝒦: Π(𝛼)=_𝒦(𝛼)=0if𝛼∈𝒦+∞otherwise, (10) and its conjugate Π^∗ is given by Π^∗𝛾=_𝒦⟂𝛾=0if𝛾∈𝒦^⟂+∞otherwise. (11) Accordingly the relations governing the quasistatic thermoelastic structural problem without energy dissipation for given mechanical and thermal loads ℓ and ℓ_𝜗 in the time interval 𝐼=[0,𝑡] can be collected in the following form: equilibrium:𝐁^′𝝈=ℓ+𝐫,thermalbalanceequation:𝐆^′𝐩=𝜂̇+ℓ_𝜃+𝛾,compatibility:𝐁𝐮=𝜺,thermalcompatibility:𝐆𝛼=𝐠,constitutiverelation:𝝈=𝑑_𝜺Φ,̇𝜺,𝜂constitutiverelation:𝛼=𝑑_𝜂Φ,(𝐠),𝜺,𝜂constitutiverelation:𝐩=−𝑑Ψexternalrelation:𝐮∈𝜕Υ^∗(𝐫),thermalexternalrelation:𝛼∈𝜕Π^∗𝛾. (12)

The initial thermal conditions are considered in the following form: 𝛼(𝐱,0)=𝛼_𝑜(𝐱),𝛼̇(𝐱,0)=𝜃_𝑜(𝐱)with𝐱∈Ω, (13) where 𝛼_𝑜 and 𝜃_𝑜 are, respectively, a prescribed initial thermal displacement and temperature in Ω.

The thermoelastic functionals Φ and Ψ 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 Φ=1𝜺,𝜂2𝔼_𝑠−𝜃𝜺,𝜺^𝑟𝑐^𝑟_𝑉+𝜃𝜂𝐌,𝜺^𝑟2𝑐^𝑟_𝑉𝜂,𝜂, (14) where the first term at the r.h.s. above is the isentropic elastic strain energy and the second one represents the thermoelastic coupling. The parameter 𝑐^𝑟_𝑉 is the specific heat at constant strain at the reference state with temperature 𝜃_𝑟, 𝔼_𝑠=𝔼+𝜃_𝑟(𝐌⊗𝐌)/𝑐^𝑟_𝑉 is the symmetric and positive definite isentropic elastic moduli fourth-order tensor. The second-order thermal expansion tensor 𝐌 is self-adjoint, that is, 𝐌^′=𝐌. Moreover the thermal function Ψ is given by Ψ(𝐠)=12𝜃_𝑟⟨𝐊𝐠,𝐠⟩, (15) where 𝐊 is the tensor of conductivity moduli, symmetric and positive definite.

A solution of the thermoelastic structural problem can be achieved by a finite element approach (see e.g., [21, 22]) and can be obtained starting from a suitable mixed variational formulation. Hence the definition 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.

The time integral of the constitutive relation (12)₆ in the interval 𝐼 is given by 𝛼−𝛼_𝑜=_𝐼𝑑_𝜂Φ𝜃𝜺,𝜂𝑑𝜏=−^𝑟𝑐^𝑟_𝑉𝐼𝜃𝐌∗𝜺𝑑𝜏+^𝑟𝑐^𝑟_𝑉𝐼𝜂𝑑𝜏, (16) where we have set 𝛼_𝑜=𝛼_𝑜(𝐱,0).

Moreover the time integral of the thermal balance equation (12)₂ in the interval 𝐼 is given by _𝐼𝐆^′𝐩𝑑𝜏=_𝐼ℓ_𝜗+𝛾𝑑𝜏+𝜂−𝐌∗𝜺_𝑜−𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜, (17) where we have set 𝜺_𝑜=𝜺(𝐱,0) and 𝜃_𝑜=𝜃_𝑜(𝐱). In fact, the constitutive relation (12)₆, evaluated at the initial time 𝑡=0, provides the equality 𝛼̇(𝐱,0)=𝑑_𝜂Φ(𝜺,𝜂)∣_(𝐱,0)=−𝜃_𝑟𝐌∗𝜺(𝐱,0)/𝑐^𝑟_𝑉+𝜃_𝑟𝜂(𝐱,0)/𝑐^𝑟_𝑉 which yields the relation 𝜂(𝐱,0)=𝐌∗𝜺(𝐱,0)+𝑐^𝑟_𝑉𝜃_𝑜(𝐱)/𝜃_𝑟.

Accordingly the thermoelastic structural problem (12) is rewritten in the following form where the initial conditions (13) are incorporated into the field equations (12): equilibrium:𝐁^′𝝈=ℓ+𝐫,compatibility:𝐁𝐮=𝜺,constitutiverelation:𝝈=𝑑_𝜺Φ𝜺,𝜂,constitutiverelation:𝛼−𝛼_𝑜=_𝐼𝑑_𝜂Φ𝜺,𝜂𝑑𝜏,externalrelation:𝐮∈𝜕Υ^∗(𝐫),thermalbalanceequation:_𝐼𝐆^′𝐩𝑑𝜏=_𝐼ℓ_𝜗+𝛾𝑑𝜏+𝜂,−𝐌∗𝜺_𝑜−𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜,(𝐠),thermalcompatibility:𝐆𝛼=𝐠,constitutiverelation:𝐩=−𝑑Ψthermalexternalrelation:𝛼∈𝜕Π^∗𝛾. (18)

3. General Mixed Variational Principle

Introducing the product space 𝒲=𝒰×𝒮×𝒟×𝒜^′×ℱ×𝒜×𝒢^′×𝒢×𝒜^′ and its dual space 𝒲^′=ℱ×𝒟×𝒮×𝒜×𝒰×𝒜^′×𝒢×𝒢^′×𝒜, the thermoelastic structural problem without dissipation (18) can be collected in terms of the global multivalued structural operators 𝐒and𝐒∶𝒲→𝒲^′ governing the whole problem 𝟎∈_𝐼𝐒(𝐰)+𝐰_𝑜𝑑𝑡+𝐒(𝐰)+𝐰_𝑜,𝐰∈𝒲,𝐰_𝑜,𝐰_𝑜∈𝒲^′. (19) The explicit expression of the structural operators 𝐒, 𝐒 and of the vectors 𝐰, 𝐰_𝑜, 𝐰_𝑜 are given by: 𝐒=𝟎𝐁^′𝟎0−𝐼_ℱ0𝟎𝟎0𝐁𝟎−𝐼_𝒟0𝟎0𝟎𝟎0𝟎−𝐼_𝒮𝟎0𝟎𝟎0𝑑Φ00𝟎0𝟎𝟎0−𝐼_𝒰𝟎𝟎0𝜕Υ^∗0𝟎𝟎0𝟎𝟎𝟎0𝟎0𝐆^′𝟎−𝐼_𝒜′𝟎𝟎𝟎0𝟎𝐆𝟎−𝐼_𝒢0𝟎𝟎𝟎0𝟎0−𝐼_𝒢′−𝑑Ψ0𝟎𝟎𝟎0𝟎−𝐼_𝒜𝟎𝟎𝜕Π^∗,𝟎𝐒=𝟎𝟎𝟎0𝟎0𝟎𝟎0𝟎𝟎𝟎0𝟎0𝟎𝟎0𝟎𝟎𝟎0𝟎𝟎0𝟎𝟎𝟎−𝐼_𝒜𝟎𝟎0𝟎𝟎𝟎0𝟎0𝟎𝟎0𝟎𝟎𝟎−𝐼_𝒜′,𝐰𝟎0𝟎𝟎0𝟎𝟎𝟎0𝟎0𝟎𝟎0𝟎𝟎𝟎0𝟎0𝟎𝟎0𝟎𝟎𝟎0𝟎0𝟎𝟎0_𝑇=𝐮𝝈𝜺𝜂𝐫𝛼𝐩𝐠𝛾^𝑇,𝐰^𝑇_𝑜=−ℓ𝟎𝟎0𝟎−ℓ_𝜗𝟎𝟎0^𝑇,𝐰^𝑇_𝑜=𝟎𝟎𝟎𝛼_𝑜𝟎𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜𝟎𝟎0^𝑇. (20) The operators 𝐒 and 𝐒 turn out to be integrable by virtue of the duality existing between the operators 𝐁, 𝐆, 𝐌, 𝐼_𝒰, 𝐼_𝒟, 𝐼_𝒜, 𝐼_𝒢 and 𝐁^′, 𝐆^′, 𝐌^′, 𝐼_ℱ, 𝐼_𝒮, 𝐼_𝒜′, 𝐼_𝒢′, the conservativity of 𝑑Φ, 𝑑Ψ and the conservativity of the super(sub)differentials 𝜕Υ^∗ and 𝜕Π^∗.

The related potential can be evaluated by a direct integration along a straight line in the space 𝒲 starting from its origin to get [23]: 𝑀=𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾_𝐼10𝐒(𝜁𝐰),𝐰𝑑𝜁−⟨ℓ,𝐮⟩−ℓ_𝜗+,𝛼𝑑𝜏₁₀𝐒(𝜁𝐰),𝐰𝑑𝜁+𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜+,𝛼𝜂,𝛼_𝑜. (21)

Hence it turns out to be 𝑀=𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾_𝐼Φ𝜺,𝜂−Ψ(𝐠)+Υ^∗(𝐫)+Π^∗𝛾−⟨⟩+⟨⟩−⟨⟩−⟨⟩+⟨⟩−ℓ𝝈,𝜺𝝈,𝐁𝐮ℓ+𝐫,𝐮𝐩,𝐠𝐩,𝐆𝛼_𝜗−+𝛾,𝛼𝑑𝜏𝜂,𝛼−𝛼_𝑜+𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜,𝛼. (22) The potential 𝑀 turns out to be linear in (𝐮,𝝈,𝛼,𝐩), jointly convex with respect to the state variables (𝜺,𝜂,𝛾) and jointly concave with respect to (𝐠,𝐫). The following statement then holds.

Proposition 1. The set of state variables (𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾) is a solution of the saddle problem: min_{𝜺,𝜂,𝛾}max_𝐠,𝐫stat_{𝐮,𝝈,𝛼,𝐩}𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾 (23) if and only if it is a solution of the thermoelastic structural problem without dissipation (18).

The stationary condition of 𝑀, enforced at the point (𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾), yields back the thermoelastic structural problem. In fact the stationarity of 𝑀 is: (𝟎,𝟎,𝟎,0,𝟎,0,𝟎,𝟎,0)∈𝜕𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾 (24) which can be rewritten in the following form 𝟎=𝑑_𝐮𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,𝟎=𝑑_𝝈𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,𝟎=𝑑_𝜺𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,0=𝑑_𝜂𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,𝟎∈𝜕_𝐫𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,0=𝑑_𝛼𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,𝟎=𝑑_𝐩𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,𝟎=𝑑_𝐠𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾,0∈𝜕_𝛾𝑀𝐮,𝝈,𝜺,𝜂,𝐫,𝛼,𝐩,𝐠,𝛾. (25)

From a mechanical point of view, the relation (25)₁ yields the equilibrium equation: 𝐁^′𝝈=ℓ+𝐫, (26) the relation (25)₂ provides the compatibility condition: 𝐁𝐮=𝜺, (27) the relations (25)_3-4 yield the constitutive relations: 𝝈=𝑑_𝜺Φ𝜺,𝜂,𝛼−𝛼_𝑜=_𝐼𝑑_𝜂Φ𝜺,𝜂𝑑𝜏, (28) the relation (25)₅ yields the external constraint conditions: 𝐮∈𝜕Υ^∗(𝐫), (29) the relation (25)₆ yields the thermal balance condition: _𝐼𝐆^′𝐩𝑑𝜏=_𝐼ℓ_𝜗+𝛾𝑑𝜏+𝜂−𝐌∗𝜺_𝑜−𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜, (30) the relation (25)₇ yields the thermal compatibility: 𝐠=𝐆𝛼, (31) the relation (25)₈ yields the thermal constitutive relation: 𝐩=−𝑑Ψ(𝐠), (32) and, finally, the relation (25)₉ yields the thermal external relation: 𝛼∈𝜕Π^∗𝛾. (33) Hence, performing the super(sub)differentials appearing in (25), the structural problem (18) is recovered.

3.1. Mixed Variational Principles

A family of potentials can be recovered from the potential 𝑀 by enforcing the fulfilment of field 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 𝑀, the external relations (18)_5,9 in terms of Fenchel’s equalities (6) and (9)₃. Hence it turns out to be 𝑀₁=𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠_𝐼Φ(𝐠)(𝐮)(𝛼)−⟨⟩+⟨⟩−⟨⟩−⟨⟩+⟨⟩−ℓ𝜺,𝜂−Ψ−Υ−Π𝝈,𝜺𝝈,𝐁𝐮ℓ,𝐮𝐩,𝐠𝐩,𝐆𝛼_𝜗−,𝛼𝑑𝜏𝜂,𝛼−𝛼_𝑜+𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜,𝛼, (34) and the following statement holds.

Proposition 2. The set of state variables (𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠) is a solution of the saddle problem: min_{𝐮,𝜺,𝜂}max_𝛼,𝐠stat_𝝈,𝐩𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠 (35) if and only if it is a solution of the thermoelastic structural problem without dissipation (18).

The proof of the mixed GN thermoelastic variational formulation reported in the Proposition 2 is provided in the Appendix. The proofs of the following variational formulations are omitted since they are obtained following a similar reasoning.

Imposing in the expression (34) of the potential 𝑀₁ the compatibility condition (18)₂ follows as 𝑀₂𝐮,𝜂,𝛼,𝐩,𝐠=_𝐼Φ(𝐠)(𝐮)(𝛼)−⟨⟩−⟨⟩+⟨⟩−ℓ𝐁𝐮,𝜂−Ψ−Υ−Πℓ,𝐮𝐩,𝐠𝐩,𝐆𝛼_𝜗−,𝛼𝑑𝜏𝜂,𝛼−𝛼_𝑜+𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜,𝛼, (36) and the following five-field mixed variational principle holds.

Proposition 3. The set of state variables (𝐮,𝜂,𝛼,𝐩,𝐠) is a solution of the saddle problem: min_𝐮,𝜂max_𝛼,𝐠stat_𝐩𝑀₂𝐮,𝜂,𝛼,𝐩,𝐠 (37) if and only if it is a solution of the thermoelastic structural problem without dissipation (18).

Imposing in the expression (36) of the potential 𝑀₂ the thermal compatibility condition (18)₇ follows: 𝑀₃=𝐮,𝜂,𝛼_𝐼Φ()(𝐮)(𝛼)−⟨⟩−ℓ𝐁𝐮,𝜂−Ψ𝐆𝛼−Υ−Πℓ,𝐮_𝜗−,𝛼𝑑𝜏𝜂,𝛼−𝛼_𝑜+𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜,𝛼, (38) and the following three-field mixed variational formulation holds.

Proposition 4. The set of state variables (𝐮,𝜂,𝛼) is a solution of the saddle problem: min_𝐮,𝜂max_𝛼𝑀₃𝐮,𝜂,𝛼 (39) 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 𝑀₃ to the Cauchy model for a linear coupled thermoelastic behaviour, external frictionless bilateral constrains, and homogeneous thermal boundary conditions is provided. The functionals Φ and Ψ, see (14) and (15), are Φ=1𝜺,𝜂2_Ω𝔼_𝑠𝜃𝜺∗𝜺𝑑𝐱−^𝑟𝑐^𝑟_𝑉Ω𝜃𝜂𝐌∗𝜺𝑑𝐱+^𝑟2𝑐^𝑟_𝑉Ω𝜂²Ψ(𝐠)=1𝑑𝐱,2𝜃_𝑟Ω𝐊𝐠∗𝐠𝑑𝐱. (40)

Hence the potential 𝑀₃ becomes 𝑀₃𝐮,𝜂,𝛼=_𝐼𝐻^𝑢(𝐮)+𝐻^th𝜂,𝛼𝑑𝑡−𝐻^th₁𝜂,𝛼−_𝐼𝐻^coup₁𝐮,𝜂𝑑𝑡+𝐻^coup₂(𝛼), (41) under the condition that the displacement and the thermal displacement are conforming and admissible, that is, 𝐮∈ℒ and 𝛼∈𝒦, where 𝐻^𝑢(𝐮)=12_Ω𝔼_𝑠𝐁𝐮∗𝐁𝐮𝑑𝐱−_Ω𝐛∗𝐮𝑑𝐱−_𝜕Ω𝐭∗Γ𝐮𝑑𝐱, (42) is the total potential energy in classical elasticity in terms of the isentropic elastic stiffness, the functionals: 𝐻^th1𝜂,𝛼=−2𝜃_𝑟Ω𝜃𝐊∇𝛼∗∇𝛼𝑑𝐱+^𝑟2𝑐^𝑟_𝑉Ω𝜂²+1𝑑𝐱𝜃_𝑟Ω𝑟∗𝛼𝑑𝐱−_𝜕Ω𝐻𝑞∗Γ𝛼𝑑𝐱,_th1𝜂,𝛼=_Ω𝜂∗𝛼−𝛼_𝑜𝑑𝐱, (43) are the mixed thermal part of the potential, and the functionals: 𝐻^coup₁=𝜃𝐮,𝜂^𝑟𝑐^𝑟_𝑉Ω𝐻𝜂𝐌∗𝐁𝐮𝑑𝐱,_coup2(𝛼)=_Ω𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜𝛼𝑑𝐱, (44) take into account the thermoelastic coupling.

Minimum principles in structural mechanics are relevant to investigate since 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.

Hence it is important to derive a variational formulation in terms of a minimization problem as hereafter 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: 𝐠=𝑑Ψ^∗(−𝐩), (45) and the Fenchel’s relation hold: Ψ(𝐠)+Ψ^∗(−𝐩)=−⟨𝐩,𝐠⟩. (46)

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 Ψ^∗(𝐩)=𝜃_𝑟2_Ω𝐩∗𝐊⁻¹𝐩𝑑𝐱. (47)

Enforcing in the expression of the functional 𝑀 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 𝑀₄=𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝛾_𝐼Φ𝜺,𝜂+Ψ^∗(−𝐩)−Υ(𝐮)+Π^∗𝛾−⟨⟩+⟨⟩−⟨⟩+⟨⟩−ℓ𝝈,𝜺𝝈,𝐁𝐮ℓ,𝐮𝐩,𝐆𝛼_𝜗−+𝛾,𝛼𝑑𝜏𝜂,𝛼−𝛼_𝑜+𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜,𝛼, (48) and the following minimum principle holds.

Proposition 5. The set of state variables (𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝛾) is a solution of the minimum problem: min_{𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝛾}𝑀₄𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝛾 (49) 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 𝑀₄ 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. Therefore the GN thermoelastic structural model (18) admits a unique solution. The question of existence of the solution is still an challenge problem; see, for example, [17].

4. Closure

A variational framework for a class of GN coupled thermomechanical boundary-value problem is presented. The 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.

Appendix

The stationary condition of 𝑀₁, see (34), enforced at the point (𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠), yields back the thermoelastic structural problem. In fact the stationarity of 𝑀₁ is (𝟎,𝟎,𝟎,0,0,𝟎,𝟎)∈𝜕𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠, (A.1) which can be rewritten in the following form: 𝟎∈𝜕_𝐮𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠,𝟎=𝑑_𝝈𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠,𝟎=𝑑_𝜺𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠,0=𝑑_𝜂𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠,0∈𝜕_𝛼𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠,𝟎=𝑑_𝐩𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠,𝟎=𝑑_𝐠𝑀₁𝐮,𝝈,𝜺,𝜂,𝛼,𝐩,𝐠. (A.2) The mechanical interpretation of the relation (A.2)₁ yields the equilibrium equation and the external constraint conditions. In fact there exists a reaction 𝐫∈𝜕Υ(𝐮) such that 𝐁^′𝝈−ℓ=𝐫. (A.3) The relation (A.2)₂ provides the compatibility condition: 𝐁𝐮=𝜺, (A.4) the relations (A.2)_3-4 yield the constitutive relations: 𝝈=𝑑_𝜺Φ𝜺,𝜂,𝛼−𝛼_𝑜=_𝐼𝑑_𝜂Φ𝜺,𝜂𝑑𝜏, (A.5) the relation (A.2)₅ yields the thermal balance condition and the thermal external relations. In fact there exists a thermal reaction 𝛾∈𝜕Π(𝛼) such that _𝐼𝐆^′𝐩𝑑𝜏−_𝐼ℓ_𝜗+𝛾𝑑𝜏−𝜂+𝐌∗𝜺_𝑜+𝑐^𝑟_𝑉𝜃_𝑟𝜃_𝑜=0, (A.6) the relation (A.2)₆ yields the thermal compatibility: 𝐠=𝐆𝛼, (A.7) and the relation (A.2)₇  yields the thermal constitutive relation: 𝐩=−𝑑Ψ(𝐠). (A.8) Hence the structural problem (18) is thus recovered.

Acknowledgments

Research support by “Ministero dell’ Istruzione, dell’Università e della Ricerca” of Italy is kindly acknowledged.