Georgian Mathematical Journal Volume 8 (2001), Number 4, 767–784

SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM OF

A RECTANGULAR PARALLELEPIPED IN TERMS OF ASYMMETRIC ELASTICITY

N. KHOMASURIDZE

Abstract. An effective solution of a number of boundary value and boundary contact problems of thermoelastic equilibrium is constructed for a homogeneous isotropic rectangular parallelepiped in terms of asymmetric and pseudo-asymmetric elasticity (Cosserat’s continuum and pseudo- continuum). Two opposite faces of a parallelepiped are affected by arbitrary surface disturbances and a stationary thermal field, while for the four remaining faces symmetry or anti-symmetry conditions (for a multilayer rectangular parallelepiped nonhomogeneous contact conditions are also defined) are given. The solutions are constructed in trigonometric series using the method of separation of variables.

2000 Mathematics Subject Classification: 74B99.

Key words and phrases: Asymptotic thermoelasticity, Cosserat’s continuum, Cosserat’s pseudo-continuum, boundary value problem, Fourier method, Fourier series.

Introduction

There are a number of papers devoted to problems of elastic equilibrium in terms of asymmetric elasticity. Asymmetric elasticity was first introduced by the Cosserat brothers [1] and was later on developed by Truesdell and Toupin [2]. Papers by Kuvshinsky, Aero, Palmov, Grioli, Mindlin, Tiersten and Koiter are devoted to linear theory for Cosserat’s medium. These and some other papers are mentioned in [3]. Works by Eringen [4], Nowacki [5], Kupradze, Gegelia, Basheleishvili and Burchuladze [6], Carbonari and Russo [7] are also worth mentioning. Linear theory of asymmetric thermoelasticity was developed by Nowacki [5].

As far as we know, there have been no papers dealing with thermoelastic equilibrium of a rectangular parallelepiped in terms of asymmetric elasticity. This issue is the subject of the given work, in which using the general solution introduced by the author as well as the method of separation of variables and double series, the solutions of some boundary value and boundary contact problems of asymmetric thermoelasticity are constructed for a rectangular parallelepiped occupying the domain Ω = {0 < x_{1} < x_{11}, 0 < x_{2} < x_{21}, 0 < x_{3} < x_{31}}. For the lateral faces x_{1} = 0, x_{1} = x_{11}, x_{2} = 0, x_{2} = x_{21} symmetry and antisym-

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

metry conditions are defined while on the faces x_{3} = 0 and x_{3} = x_{31} arbitrary surface and thermal disturbances can be applied. In the case of the rectangular parallelepiped which is multilayer along the coordinate x_{3} we have symmetry and antisymmetry conditions again with x_{1} = 0, x_{1} = x_{11}, x_{2} = 0, x_{2} = x_{21}, on the contact surfaces x_{3} = const nonhomogeneous contact conditions are defined while on the two remaining faces arbitrary surface and thermal disturbances can be applied.

1. If no mass forces are applied, the equilibrium equations for a homogeneous isotropic body in the case of Cosserat’s continuum have the following form [6]:

\begin{matrix}a) ∑_{j} \frac{∂N_{ji}}{∂x_{j}} = 0, \\ b) ∑_{j} \frac{∂M_{ji}}{∂x_{j}} + ∑_{k,j} ε_{ijk}N_{jk} = 0, \\ c) ΔT = 0, \end{matrix}  (1)

where i = 1, 2, 3; j = 1, 2, 3; k = 1, 2, 3; ε_{ijk} is Levy-Civita symbol; x_{1}, x_{2}, x_{3} are Cartesian coordinates, N_{11}, N_{22}, N_{33} are normal stresses, N_{21}, N_{31}, N_{12}, …, N_{23} are tangential stresses, M_{11}, M_{22}, M_{33} are torsion micromoments; M_{21}, M_{31}, M_{12}, …, M_{23} are bending micromoments; T is the change in the temperature of the body; Δ = \frac{∂}{∂x_{1}^{2}} + \frac{∂}{∂x_{2}^{2}} + \frac{∂}{∂x_{3}^{2}} .

The equalities connecting stresses and micromoments with the components of displacement and rotation vectors have the following form [6]

\begin{aligned}N_{ij} & = δ_{ij}[λ div U⃗ − (3λ + 2μ)γT] + (μ + σ_{1})\frac{∂u_{j}}{∂x_{i}} \\ & + (μ − σ_{1})\frac{∂u_{i}}{∂x_{j}} − 2σ_{1} ∑_{k} ε_{ijk}ω_{k}, \\ M_{ij} & = δ_{ij}σ_{2} div ω⃗ + (σ_{3} + σ_{4})\frac{∂ω_{j}}{∂x_{i}} + (σ_{3} − σ_{4})\frac{∂ω_{i}}{∂x_{j}} , \end{aligned}  (2)

where U⃗ = u_{1}l⃗_{1} + u_{2}l⃗_{2} + u_{3}l⃗_{3} (l⃗_{1}, l⃗_{2}, l⃗_{3} are basis vectors in the Cartesian coordinate system), ω⃗ = ω_{1}l⃗_{1} + ω_{2}l⃗_{2} + ω_{3}l⃗_{3} are displacement and rotation vectors, respectively, δ_{ij} is Kronecker’s symbol, λ, μ are classical elasticity characteristics and σ_{1}, σ_{2}, σ_{3}, σ_{4} are asymmetric elasticity characteristics with μ > 0, 3λ + 2μ > 0, σ_{1} > 0, σ_{3} > 0, 3σ_{2} + 2σ_{3} > 0, σ_{4} > 0; λ = \frac{νE}{(1−2ν)(1+ν)}, μ = \frac{E}{2(1−ν)}, where E is elasticity modulus, ν is the Poisson coefficient, γ is the coefficient of linear thermal expansion.

Substituting (2) into (1)a, (1)b we have

grad[(λ + 2μ) div U⃗ − (3λ + 2μ) γ T] − rot[(μ + σ_{1}) rot U⃗ − 2σ_{1}ω⃗] = 0,

grad[(σ_{2} + 2σ_{3}) div ω⃗] − rot[(σ_{3} + σ_{4}) rot ω⃗ − 2σ_{1}U⃗] − 4σ_{1}ω⃗ = 0.

(3)

The second equation of system (3) can be written as

grad[(σ_{2} + 2σ_{3}) div ω⃗] − rot[(σ_{3} + σ_{4}) rot ω⃗ + 2μU⃗] + 2[(μ + σ_{1}) rot U⃗ − 2σ_{1}ω⃗] = 0.

Taking into consideration the last equality we can write

a) grad[(λ + 2μ) div U⃗ − (3λ + 2μ)γ T]

− rot[(μ + σ_{1}) rot U⃗ − 2σ_{1}ω⃗] = 0,

b) div[(μ + σ_{1}) rot U⃗ − 2σ_{1}ω⃗] = −\frac{2σ_{1}}{σ_{2} + 2σ_{3}} [(σ_{2} + 2σ_{3}) div ω⃗],
c) grad[(σ_{2} + 2σ_{3}) div ω⃗] − rot[(σ_{3} + σ_{4}) rot ω⃗ + 2μU⃗]

+ 2[(μ + σ_{1}) rot U⃗ − 2σ_{1}ω⃗] = 0,

d) div[(σ_{3} + σ_{4}) rot ω⃗ + 2μU⃗] = \frac{2μ}{λ + 2μ}[(λ + 2μ) div U⃗

− (3λ + 2μ)γ T] + \frac{2μ(3λ + 2μ)}{λ + 2μ}γT = 0.

(4)

It can be easily seen that (4b) and (4d) are identities.

Without loss of generality we can represent the function T as

T = \frac{2(λ + 2μ)}{(3λ + 2μ)γ} \frac{∂T̃}{∂x_{3}} ,  (5)

where ΔT̃ = 0. No loss of generality will occur if T is expressed by a derivative of the harmonic function T̃ since it is preserved by an appropriate representation of the harmonic function T̃ using the method of separation of variables as in [8].

Introduce the notation

a) (λ + 2μ) div U⃗ − (3λ + 2μ)γ T = D,
b) (σ_{2} + 2σ_{3}) div ω⃗ = D^{∗},
c) (μ + σ_{1}) rot U⃗ − 2σ_{1} ω⃗ = K⃗ = K_{1}l⃗_{1} + K_{2}l⃗_{2} + K_{3}l⃗_{3},
d) (σ_{3} + σ_{4}) rot ω⃗ + 2μ U⃗ = K⃗^{∗} = K_{1}^{∗}l⃗_{1} + K_{2}^{∗}l⃗_{2} + K_{3}^{∗}l⃗_{3}.

(6)

Taking (5) into account, we can write system (4) in the following way

grad D − rot K⃗ = 0,  div K⃗ = −\frac{2σ_{1}}{σ_{2} + 2σ_{3}} D^{∗},
grad D^{∗} − rot K⃗^{∗} + 2K⃗ = 0, div K⃗^{∗} = \frac{2μ}{λ + 2μ}D + 4μ\frac{∂T̃}{∂x_{3}} .

(7)

Finally we obtain the following system

a) \frac{∂D}{∂x_{1}} − \frac{∂K_{3}}{∂x_{2}} + \frac{∂K_{2}}{∂x_{3}} = 0,

b) \frac{∂D}{∂x_{2}} − \frac{∂K_{1}}{∂x_{3}} + \frac{∂K_{3}}{∂x_{1}} = 0,

c) \frac{∂D}{∂x_{3}} − \frac{∂K_{2}}{∂x_{1}} + \frac{∂K_{1}}{∂x_{2}} = 0,

d) \frac{∂K_{1}}{∂x_{1}} + \frac{∂K_{2}}{∂x_{2}} + \frac{∂K_{3}}{∂x_{3}} = −\frac{2σ_{1}}{σ_{2} + 2σ_{3}} D^{∗},

e) \frac{∂D^{∗}}{∂x_{1}} − \frac{∂K_{3}^{∗}}{∂x_{2}} + \frac{∂K_{2}^{∗}}{∂x_{3}} + 2K_{1} = 0,

f) \frac{∂D^{∗}}{∂x_{2}} − \frac{∂K_{1}^{∗}}{∂x_{3}} + \frac{∂K_{3}^{∗}}{∂x_{1}} + 2K_{2} = 0,

g) \frac{∂D^{∗}}{∂x_{3}} − \frac{∂K_{2}^{∗}}{∂x_{1}} + \frac{∂K_{1}^{∗}}{∂x_{2}} + 2K_{3} = 0,

h) \frac{∂K_{1}^{∗}}{∂x_{1}} + \frac{∂K_{2}^{∗}}{∂x_{2}} + \frac{∂K_{3}^{∗}}{∂x_{3}} = \frac{2μ}{λ + 2μ}D + 4μ\frac{∂T̃}{∂x_{3}} ;

(8)

a) \frac{∂u_{1}}{∂x_{1}} + \frac{∂u_{2}}{∂x_{2}} + \frac{∂u_{3}}{∂x_{3}} = \frac{D}{λ + 2μ} + 2\frac{∂T̃}{∂x_{3}} ,

b) (μ + σ_{1}) (\frac{∂u_{3}}{∂x_{2}} − \frac{∂u_{2}}{∂x_{3}} ) − 2σ_{1}ω_{1} = K_{1},

c) (μ + σ_{1}) (\frac{∂u_{1}}{∂x_{3}} − \frac{∂u_{3}}{∂x_{1}} ) − 2σ_{1}ω_{2} = K_{2},

d) (μ + σ_{1}) (\frac{∂u_{2}}{∂x_{1}} − \frac{∂u_{1}}{∂x_{2}} ) − 2σ_{1}ω_{3} = K_{3},

e) \frac{∂ω_{1}}{∂x_{1}} + \frac{∂ω_{2}}{∂x_{2}} + \frac{∂ω_{3}}{∂x_{3}} = \frac{D^{∗}}{σ_{2} + 2σ_{3}} ,

f) (σ_{3} + σ_{4}) (\frac{∂ω_{3}}{∂x_{2}} − \frac{∂ω_{2}}{∂x_{3}} ) + 2μu_{1} = K_{1}^{∗},

g) (σ_{3} + σ_{4}) (\frac{∂ω_{1}}{∂x_{3}} − \frac{∂ω_{3}}{∂x_{1}} ) + 2μu_{2} = K_{2}^{∗},

h) (σ_{3} + σ_{4}) (\frac{∂ω_{2}}{∂x_{1}} − \frac{∂ω_{1}}{∂x_{2}} ) + 2μu_{3} = K_{3}^{∗},

(9)

in 14 unknowns, in particular, the unknowns D, K_{1}, K_{2}, K_{3}, D^{∗}, K_{1}^{∗}, K_{2}^{∗}, K_{3}^{∗}, ω_{1}, ω_{2}, ω_{3}, u_{1}, u_{2}, u_{3} (the temperature T is defined by (1c) under appropriate boundary conditions). Note that identities (8d) and (8h) complement six equilibrium equations (8a,b,c,e,f,g) to obtain two similar systems (8a,b,c,d) and (9e,f,g,h) which consist of four differential equations with the first order partial derivatives in four unknowns D, K_{1}, K_{2}, K_{3} and D^{∗}, K_{1}^{∗}, K_{2}^{∗}, K_{3}^{∗}, respectively (in this case the functions D^{∗} and D, which also appear in the right-hand side of (8d) of system (8a,b,c,d) and in the right-hand side of (8h) of system (8c,f,g,h), are assumed to be known).

The given work deals with the thermoelastic equilibrium of a rectangular parallelepiped (RP) occupying the domain Ω = {0 < x_{1} < x_{11}, 0 < x_{2} < x_{21}, 0 < x_{3} < x_{31}}. The boundary conditions appearing in the paper and defining the class of the boundary value problems under consideration are given below.

\begin{aligned}For x_{1} = x_{1s} : & a) & u_{1} = 0, N_{12} = 0, N_{13} = 0, M_{11} = 0, \\ & & ω_{2} = 0, ω_{3} = 0, \frac{∂T}{∂x_{1}} = 0 ⟺ \\ & & u_{1} = 0, \frac{∂u_{2}}{∂x_{1}} = 0, \frac{∂u_{3}}{∂x_{1}} = 0, \frac{∂ω_{1}}{∂x_{1}} = 0, \\ & & ω_{2} = 0, ω_{3} = 0, \frac{∂T}{∂x_{1}} = 0 \\ or \\ & b) & N_{11} = 0, u_{2} = 0, u_{3} = 0, ω_{1} = 0 M_{12} = 0, \\ & & M_{13} = 0, T = 0 ⟺ \\ & & \frac{∂u_{1}}{∂x_{1}} = 0, u_{2} = 0, u_{3} = 0, ω_{1} = 0, \frac{∂ω_{2}}{∂x_{1}} = 0, \\ & & \frac{∂ω_{3}}{∂x_{1}} = 0, T = 0. \end{aligned} (10)

\begin{aligned}For x_{2} = x_{2s} : & a) & u_{2} = 0, N_{23} = 0, N_{21} = 0, M_{22} = 0, \\ & & ω_{3} = 0, ω_{1} = 0, \frac{∂T}{∂x_{2}} = 0 ⟺ \\ & & u_{2} = 0, \frac{∂u_{3}}{∂x_{2}} = 0, \frac{∂u_{1}}{∂x_{2}} = 0, \frac{∂ω_{2}}{∂x_{2}} = 0, \\ & & ω_{3} = 0, ω_{1} = 0, \frac{∂T}{∂x_{2}} = 0 \\ or \\ & b) & N_{22} = 0, u_{3} = 0, u_{1} = 0, ω_{2} = 0 M_{23} = 0, \\ & & M_{21} = 0, T = 0 ⟺ \\ & & \frac{∂u_{2}}{∂x_{2}} = 0, u_{3} = 0, u_{1} = 0, ω_{2} = 0, \frac{∂ω_{3}}{∂x_{2}} = 0, \\ & & \frac{∂ω_{1}}{∂x_{2}} = 0, T = 0. \end{aligned} (11)

\begin{aligned}For x_{3} = x_{3s} : & g_{s1}N_{33} + g̃_{s1} · u_{3} = f_{s1}(x_{1}, x_{2}), \\ & g_{s2}N_{31} + g̃_{s2} · u_{1} = f_{s2}(x_{1}, x_{2}), \\ & g_{s3}N_{32} + g̃_{s3} · u_{2} = f_{s3}(x_{1}, x_{2}), \\ & g_{s4}M_{33} + g̃_{s4} · ω_{3} = f_{s4}(x_{1}, x_{2}), \\ & g_{s5}M_{31} + g̃_{s5} · ω_{1} = f_{s5}(x_{1}, x_{2}), \\ & g_{s6}M_{32} + g̃_{s6} · ω_{2} = f_{s6}(x_{1}, x_{2}), \\ & g_{s7} \frac{∂T}{∂x_{3}} + g̃_{s7}T = f_{s7}(x_{1}, x_{2}). \end{aligned}  (12)

In (10), (11) and (12) we have s = 0, 1 with x_{10} = x_{20} = x_{30} = 0; g_{s1}, g̃_{s1}, …, g̃_{s7} being the defined constants governed by the conditions g_{s1}·g̃_{s1} ≥ 0, g_{s2}·g̃_{s2} ≥ 0, …, g_{s7} · g̃_{s7} ≥ 0. The conditions imposed on the functions f_{s1}, f_{s2}, …, f_{s7} will be described in below, we can just note that the functions are taken so that on the edges of the RP the compatibility conditions are satisfied. With g̃_{s1} = g̃_{s2} = g̃_{s3} = 0, g̃_{s4} = g̃_{s5} = g̃_{s6} = 0 we have the boundary conditions for the first problem of asymmetric thermoelasticity, with g_{s1} = g_{s2} = g_{s3} = 0, g_{s4} = g_{s5} = g_{s6} = 0 we have the boundary conditions for the second problem of asymmetric thermoelasticity, etc. Note that (12) implies that with x_{3} = 0 one type of conditions can be defined while with x_{3} = x_{31} there can be another conditions. Conditions (10a) and (11a) are called the symmetry conditions, while conditions (10b) and (11b) are called the antisymmetry conditions. We should also note that conditions (10) and (11) are the conditions of the continuous extension of the solution across the corresponding face of the RP. The validity of this statements is a direct consequence of the constructed solutions of the boundary value problems. It is interesting to observe that the technical interpretation of boundary conditions (10) and (11) is the same as of the corresponding conditions in [3].

Theorem 1. The general solution of system (7), (6) or, what is the same, system (8), (9) according to asymmetric theory has the following form

\begin{aligned}ω_{1} & = \frac{∂ψ_{3}}{∂x_{1}} + \frac{κ}{2} \frac{∂φ_{1}}{∂x_{2}} + \frac{1}{2} \frac{∂^{2}φ_{2}}{∂x_{1}∂x_{3}} + \frac{∂ψ_{1}}{∂x_{3}} , \\ ω_{2} & = \frac{∂ψ_{3}}{∂x_{2}} − \frac{κ}{2} \frac{∂φ_{1}}{∂x_{1}} + \frac{1}{2} \frac{∂^{2}φ_{2}}{∂x_{2}∂x_{3}} + \frac{∂ψ_{2}}{∂x_{3}} , \\ ω_{3} & = \frac{∂ψ_{3}}{∂x_{3}} + \frac{1}{2} \frac{∂^{2}φ_{2}}{∂x_{3}^{2}} − \frac{∂ψ_{1}}{∂x_{1}} − \frac{∂ψ_{2}}{∂x_{2}} ; \end{aligned}  (13)

\begin{aligned}u_{1} & = \frac{∂}{∂x_{1}} (φ_{3} − x_{3}φ_{1}) + \frac{∂φ_{2}}{∂x_{2}} \\ & − \frac{σ_{3} + σ_{4}}{2μ} (\frac{∂^{2}ψ_{2}}{∂x_{1}^{2}} − qψ_{2} − \frac{∂^{2}ψ_{1}}{∂x_{1}∂x_{2}} ) + x_{3} \frac{∂T̃}{∂x_{1}} , \\ u_{2} & = \frac{∂}{∂x_{2}} (φ_{3} − x_{3}φ_{1}) − \frac{∂φ_{2}}{∂x_{1}} \\ & + \frac{σ_{3} + σ_{4}}{2μ} (\frac{∂^{2}ψ_{1}}{∂x_{2}^{2}} − qψ_{1} − \frac{∂^{2}ψ_{2}}{∂x_{1}∂x_{2}} ) + x_{3} \frac{∂T̃}{∂x_{2}} , \\ u_{3} & = \frac{∂}{∂x_{3}} (φ_{3} − x_{3}φ_{1}) + κφ_{1} \\ & − \frac{σ_{3} + σ_{4}}{2μ} (\frac{∂^{2}ψ_{2}}{∂x_{1}∂x_{3}} − \frac{∂^{2}ψ_{1}}{∂x_{2}∂x_{3}} ) + x_{3} \frac{∂T̃}{∂x_{3}} + T̃. \end{aligned}  (14)

In (13) and (14) φ_{1}, φ_{2}, φ_{3} and T̃ are harmonic functions, ψ_{1}, ψ_{2}, ψ_{3} are meta-

harmonic functions with Δψ_{1} −qψ_{1} = 0, Δψ_{2} −qψ_{2} = 0, Δψ_{3} − \frac{4σ_{1}}{σ_{2} + 2σ_{3}} ψ_{3} = 0

and q = \frac{4μσ_{1}}{(μ + σ_{1})(σ_{3} + σ_{4})}.

Proof. (8a, b, c) implies

ΔD = 0.

SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 773

Without loss of generality assume

D = κμ\frac{∂φ_{1}}{∂x_{3}} ,  (15)

where Δφ_{1} = 0 (see the comments to (5)) and κ = \frac{2(λ+2μ)}{λ+μ} = 4(1−ν), so taking (15) into account we can write (8a, b, c, d) as

\begin{matrix}a) & \frac{∂}{∂x_{3}} (K_{2} + κμ\frac{∂φ_{1}}{∂x_{1}} ) − \frac{∂K_{3}}{∂x_{2}} = 0, \\ b) & \frac{∂K_{3}}{∂x_{1}} − \frac{∂}{∂x_{3}} (K_{1} − κμ\frac{∂φ_{1}}{∂x_{2}} ) = 0, \\ c) & \frac{∂}{∂x_{2}} (K_{1} − κμ\frac{∂φ_{1}}{∂x_{2}} ) − \frac{∂}{∂x_{1}} (K_{2} + κμ\frac{∂φ_{1}}{∂x_{1}} ) = 0, \\ d) & \frac{∂K_{1}}{∂x_{1}} + \frac{∂K_{2}}{∂x_{2}} + \frac{∂K_{3}}{∂x_{3}} = −\frac{2σ_{1}}{σ_{2} + 2σ_{3}} D^{∗}. \end{matrix}  (16)

From (16a, b, c) we have

K_{1} = \frac{∂φ̃_{2}}{∂x_{1}} + κμ\frac{∂φ_{1}}{∂x_{2}} , K_{2} = \frac{∂φ̃_{2}}{∂x_{2}} − κμ\frac{∂φ_{1}}{∂x_{1}} , K_{3} = \frac{∂φ̃_{2}}{∂x_{3}} .  (17)

Substituting (18) into (16d) we obtain

Δφ̃_{2} = −\frac{2σ_{1}}{σ_{2} + 2σ_{3}} D^{∗}.  (18)

(8e, f, g) implies

ΔD^{∗} − \frac{4σ_{1}}{σ_{2} + 2σ_{3}} D^{∗} = 0.

If we write the solution of this equation as

D^{∗} = 4σ_{1}ψ_{3},

where Δψ_{3} − \frac{4σ_{1}}{σ_{2}+2σ_{3}} ψ_{3} = 0, then the solution of (18) can be represented as

φ̃_{2} = μ\frac{∂φ_{2}}{∂x_{3}} − 2σ_{1} · ψ_{3},  (19)

where Δφ_{2} = 0 (see the comments to (5)). Taking (19) into account we can write (17) as

\begin{aligned}K_{1} & = μ\frac{∂^{2}φ_{2}}{∂x_{1}∂x_{3}} − 2σ_{1} \frac{∂ψ_{3}}{∂x_{1}} + κμ\frac{∂φ_{1}}{∂x_{2}} , \\ K_{2} & = μ\frac{∂^{2}φ_{2}}{∂x_{2}∂x_{3}} − 2σ_{1} \frac{∂ψ_{3}}{∂x_{2}} − κμ\frac{∂φ_{1}}{∂x_{1}} , \\ K_{3} & = μ\frac{∂^{2}φ_{2}}{∂x_{3}^{2}} − 2σ_{1} \frac{∂ψ_{3}}{∂x_{3}} . \end{aligned}  (20)

With (20) in mind (8e, f, g, h) can be written as

\begin{matrix}a) & \frac{∂}{∂x_{3}} (K_{2}^{∗} + 2μ\frac{∂φ_{2}}{∂x_{1}} ) − \frac{∂}{∂x_{2}} (K_{3}^{∗} − 2κμφ_{1}) = 0, \\ b) & \frac{∂}{∂x_{1}} (K_{3}^{∗} − 2κμφ_{1}) − \frac{∂}{∂x_{3}} (K_{1}^{∗} − 2μ\frac{∂φ_{2}}{∂x_{2}} ) = 0, \\ c) & \frac{∂}{∂x_{2}} (K_{1}^{∗} − 2μ\frac{∂φ_{2}}{∂x_{2}} ) − \frac{∂}{∂x_{1}} (K_{2}^{∗} + 2μ\frac{∂φ_{2}}{∂x_{1}} ) = 0, \\ d) & \frac{∂K_{1}^{∗}}{∂x_{1}} + \frac{∂K_{2}^{∗}}{∂x_{2}} + \frac{∂K_{3}^{∗}}{∂x_{3}} = \frac{2κμ^{2}}{λ + 2μ} \frac{∂φ_{1}}{∂x_{3}} + 4μ\frac{∂T̃}{∂x_{3}} . \end{matrix}  (21)

From (21a, b, c) we have

K_{1}^{∗} = \frac{∂φ̃_{3}}{∂x_{1}} + 2μ\frac{∂φ_{2}}{∂x_{2}} , K_{2}^{∗} = \frac{∂φ̃_{3}}{∂x_{2}} − 2μ\frac{∂φ_{2}}{∂x_{1}} , K_{3}^{∗} = \frac{∂φ̃_{3}}{∂x_{3}} + 2κμφ_{1}. (22)

Substituting (22) into (21d) we have

Δφ̃_{3} = 4μ\frac{∂}{∂x_{3}} (T̃ − φ_{1}).

The solution of this equations is

φ̃_{3} = 2μφ_{3}^{∗} − 2μx_{3}(φ_{1} − T̃),  (23)

where Δφ_{3}^{∗} = 0.. If we substitute (23) into (22) then

\begin{aligned}K_{1}^{∗} & = 2μ\frac{∂φ_{3}^{∗}}{∂x_{1}} − 2μx_{3} \frac{∂φ_{1}}{∂x_{1}} + 2μ\frac{∂φ_{2}}{∂x_{2}} + 2μx_{3} \frac{∂T̃}{∂x_{1}} , \\ K_{2}^{∗} & = 2μ\frac{∂φ_{3}^{∗}}{∂x_{2}} − 2μx_{3} \frac{∂φ_{1}}{∂x_{2}} − 2μ\frac{∂φ_{2}}{∂x_{1}} + 2μx_{3} \frac{∂T̃}{∂x_{2}} , \\ K_{3}^{∗} & = 2μ\frac{∂φ_{3}^{∗}}{∂x_{3}} − 2μx_{3} \frac{∂φ_{1}}{∂x_{3}} + 2μx_{3} \frac{∂T̃}{∂x_{3}} + 2μ(4κ − 1)φ_{1} + 2μT̃. \end{aligned}  (24)

Applying rot to (6d) and substituting it into (6c) with rot rot ω⃗ = grad div ω⃗− Δω⃗, in mind we obtain

Δω⃗ − qω⃗ = \frac{1}{σ_{2} + 2σ_{3}} grad D^{∗} − \frac{1}{σ_{3} + σ_{4}} rot K⃗^{∗} + \frac{q}{2σ_{1}} K⃗.

If we project this equation onto the coordinate axes and take into account equality D^{∗} = 4σ_{1}ψ_{3} and formulas (20) and (24) we shall have

\begin{matrix}a) & Δω_{1} − qω_{1} = (\frac{4σ_{1}}{σ_{2} + 2σ_{3}} − q) \frac{∂ψ_{3}}{∂x_{1}} − \frac{q}{2} (κ \frac{∂φ_{1}}{∂x_{2}} + \frac{∂^{2}φ_{2}}{∂x_{1}∂x_{3}} ) , \\ b) & Δω_{2} − qω_{2} = (\frac{4σ_{1}}{σ_{2} + 2σ_{3}} − q) \frac{∂ψ_{3}}{∂x_{2}} + \frac{q}{2} (κ \frac{∂φ_{1}}{∂x_{1}} − \frac{∂^{2}φ_{2}}{∂x_{2}∂x_{3}} ) , \\ c) & Δω_{3} − qω_{3} = (\frac{4σ_{1}}{σ_{2} + 2σ_{3}} − q) \frac{∂ψ_{3}}{∂x_{3}} − \frac{q}{2} \frac{∂^{2}φ_{2}}{∂x_{3}^{2}} . \end{matrix} (25)

The solution of (25a) (25b) will be

\begin{aligned}ω_{1} & = \frac{∂ψ_{3}}{∂x_{1}} + \frac{1}{2} (κ \frac{∂φ_{1}}{∂x_{2}} + \frac{∂^{2}φ_{2}}{∂x_{1}∂x_{3}} ) + \frac{∂ψ_{1}}{∂x_{3}} , \\ ω_{2} & = \frac{∂ψ_{3}}{∂x_{2}} − \frac{1}{2} (κ \frac{∂φ_{1}}{∂x_{1}} − \frac{∂^{2}φ_{2}}{∂x_{2}∂x_{3}} ) + \frac{∂ψ_{2}}{∂x_{3}} , \end{aligned}  (26)

respectively, where Δψ_{1} − qψ_{1} = 0, Δψ_{2} − qψ_{2} = 0. We define ω_{3} from equality (9e)

ω_{3} = \frac{∂ψ_{3}}{∂x_{3}} + \frac{1}{2} \frac{∂^{2}φ_{2}}{∂x_{3}^{2}} − \frac{∂ψ_{1}}{∂x_{1}} − \frac{∂ψ_{2}}{∂x_{2}} + f(x_{1}, x_{2}),  (27)

where f is a function resulting from integration of (9e) and satisfying the following equation and the following conditions:

\frac{∂^{2}f}{∂x_{1}^{2}} + \frac{∂^{2}f}{∂x_{2}^{2}} − qf = 0;

\frac{∂f}{∂x_{s}} = 0 or f = 0 with x_{s} = 0,

\frac{∂f}{∂x_{s}} = 0 or f = 0 with x_{s} = x_{s1},

(28)

where s = 1, 2. The solution of (28) is f = 0.

Substituting (26), (27) and (24) into (9f,g,h) we have

u_{1} = \frac{∂}{∂x_{1}} [φ_{3}^{∗} − \frac{κ(σ_{3} + σ_{4})}{4μ} \frac{∂φ_{1}}{∂x_{3}} ] − x_{3} \frac{∂φ_{1}}{∂x_{1}} + \frac{∂φ_{2}}{∂x_{2}}

− \frac{σ_{3} + σ_{4}}{2μ} (\frac{∂^{2}ψ_{2}}{∂x_{1}^{2}} − qψ_{2} − \frac{∂^{2}ψ_{1}}{∂x_{1}∂x_{2}} ) + x_{3} \frac{∂T̃}{∂x_{1}} ,

u_{2} = \frac{∂}{∂x_{2}} [φ_{3}^{∗} − \frac{κ(σ_{3} + σ_{4})}{4μ} \frac{∂φ_{1}}{∂x_{3}} ] − x_{3} \frac{∂φ_{1}}{∂x_{2}} − \frac{∂φ_{2}}{∂x_{1}}

+ \frac{σ_{3} + σ_{4}}{2μ} (\frac{∂^{2}ψ_{1}}{∂x_{2}^{2}} − qψ_{1} − \frac{∂^{2}ψ_{2}}{∂x_{1}∂x_{2}} ) + x_{3} \frac{∂T̃}{∂x_{2}} ,

u_{3} = \frac{∂}{∂x_{3}} [φ_{3}^{∗} − \frac{κ(σ_{3} + σ_{4})}{4μ} \frac{∂φ_{1}}{∂x_{3}} ] − x_{3} \frac{∂φ_{1}}{∂x_{3}} + (κ − 1)φ_{1}

− \frac{σ_{3} + σ_{4}}{2μ} (\frac{∂^{2}ψ_{2}}{∂x_{1}∂x_{3}} − \frac{∂^{2}ψ_{1}}{∂x_{2}∂x_{3}} ) + x_{3} \frac{∂T̃}{∂x_{3}} + T̃.

If we introduce the notation [φ_{3}^{∗} − \frac{κ(σ_{3}+σ_{4})}{4μ} \frac{∂φ_{1}}{∂x_{3}} ] = φ_{3}, where, naturally, Δφ_{3} = 0, we can say that Theorem 1 is proved.

2. In the case of Cosserat’s pseudo-continuum equations (1) still hold, but instead of (2) we have [3]

N_{ij} = δ_{ij}[λ div U⃗ − (3λ + 2μ)γ T] + μ (\frac{∂u_{j}}{∂x_{i}} + \frac{∂u_{i}}{∂x_{j}} )

− η_{1} ∑_{k} ε_{ijk}Δ(rot_{k} U⃗),

M_{ij} = 2η_{1} \frac{∂}{∂x_{i}} (rot_{j} U⃗) + 2η_{2} \frac{∂}{∂x_{j}} (rot_{i} U⃗),

(29)

where η_{1}, η_{2} are asymmetric elasticity characteristics with η_{1} > 0, η_{2} > 0. Substituting (29) into (1) we have

a) grad[(λ + 2μ) div U⃗ − (3λ + 2μ)γ T]
− rot(μ rot U⃗ − η_{1}Δ rot U⃗) = 0,
b) div(μ rot U⃗ − η_{1}Δ rot U⃗) = 0,

(30)

where (30b) is a known identity. Write (30) as

\begin{cases}grad D − rot K⃗^{∗} = 0, \\ div K⃗^{∗} = 0, \end{cases}  (31)

where D = (λ + 2μ) div U⃗ − (3λ + 2μ)γ T, K⃗^{∗} = K_{1}^{∗}l⃗_{1} + K_{2}^{∗}l⃗_{2} + K_{3}^{∗}l⃗_{3}, K⃗^{∗} = K⃗ − \frac{η_{1}}{μ} ΔK⃗ and K⃗ = μ rot U⃗ = K_{1}l⃗_{1} + K_{2}l⃗_{2} + K_{3}l⃗_{3} = μ rot_{1} U⃗ · l⃗_{1} + μ rot_{2} U⃗ · l⃗_{2} + μ rot_{3} U⃗ · l⃗_{3}. If we write (31) in a scalar form, we have

\begin{matrix}a) & \frac{∂D}{∂x_{1}} − \frac{∂K_{3}^{∗}}{∂x_{2}} + \frac{∂K_{2}^{∗}}{∂x_{3}} = 0, \\ b) & \frac{∂D}{∂x_{2}} − \frac{∂K_{1}^{∗}}{∂x_{3}} + \frac{∂K_{3}^{∗}}{∂x_{1}} = 0, \\ c) & \frac{∂D}{∂x_{3}} − \frac{∂K_{2}^{∗}}{∂x_{1}} + \frac{∂K_{1}^{∗}}{∂x_{2}} = 0, \\ d) & \frac{∂K_{1}^{∗}}{∂x_{1}} + \frac{∂K_{2}^{∗}}{∂x_{2}} + \frac{∂K_{3}^{∗}}{∂x_{3}} = 0. \end{matrix}  (32)

As in the case of Cosserat’s continuum for a RP, we can state boundary conditions defining the class of solvable boundary value problems considered according to pseudo-asymmetric thermoelasticity.

\begin{aligned}For x_{1} = x_{1s} : & a) & u_{1} = 0, N_{12} = 0, N_{13} = 0, \\ & & K_{2} = 0, K_{3} = 0, \frac{∂T}{∂x_{1}} = 0 ⟺ \\ & & u_{1} = 0, \frac{∂u_{2}}{∂x_{1}} = 0, \frac{∂u_{3}}{∂x_{1}} = 0, \frac{∂^{2}u_{1}}{∂x_{1}^{2}} = 0, \frac{∂^{3}u_{2}}{∂x_{1}^{3}} = 0, \\ & & \frac{∂^{3}u_{3}}{∂x_{1}^{3}} = 0, \frac{∂T}{∂x_{1}} = 0 \\ or \\ & b) & N_{11} = 0, u_{2} = 0, u_{3} = 0, \\ & & M_{12} = 0, M_{13} = 0, T = 0 ⟺ \\ & & \frac{∂u_{1}}{∂x_{1}} = 0, u_{2} = 0, u_{3} = 0, \frac{∂^{2}u_{2}}{∂x_{1}^{2}} = 0, \frac{∂^{2}u_{3}}{∂x_{1}^{2}} = 0, \\ & & \frac{∂^{3}u_{1}}{∂x_{1}^{3}} = 0, T = 0. \end{aligned} (33)

It should be noted that in (32a) the fourth, the fifth and the sixth conditions, appearing to the right of the equivalence sign, have been obtained assuming that with x_{1} = x_{1s}, besides the defined boundary conditions, equation (32a) also holds. In its turn, in (33b) the sixth condition to the right of the equivalence sign has been obtained assuming that with x_{1} = x_{1s} equations (32a), (32b) and (32c) differentiated, correspondingly, with respect to x_{1}, x_{2} and x_{3} are true.

\begin{aligned}For x_{2} = x_{2s} : & a) & u_{2} = 0, N_{23} = 0, N_{21} = 0, \\ & & K_{3} = 0, K_{1} = 0, \frac{∂T}{∂x_{2}} = 0 ⟺ \\ & & u_{2} = 0, \frac{∂u_{3}}{∂x_{2}} = 0, \frac{∂u_{1}}{∂x_{2}} = 0, \frac{∂^{2}u_{2}}{∂x_{2}^{2}} = 0, \frac{∂^{3}u_{3}}{∂x_{2}^{3}} = 0, \\ & & \frac{∂^{3}u_{1}}{∂x_{2}^{3}} = 0, \frac{∂T}{∂x_{2}} = 0 \\ or \\ & b) & N_{22} = 0, u_{3} = 0, u_{1} = 0, \\ & & M_{21} = 0, M_{23} = 0, T = 0 ⟺ \\ & & \frac{∂u_{2}}{∂x_{2}} = 0, u_{3} = 0, u_{1} = 0, \frac{∂^{2}u_{3}}{∂x_{2}^{2}} = 0, \frac{∂^{2}u_{1}}{∂x_{2}^{2}} = 0, \\ & & \frac{∂^{3}u_{2}}{∂x_{2}^{3}} = 0, T = 0. \end{aligned} (34)

Remarks similar to those conditions (33) take place.

\begin{aligned}For x_{3} = x_{3s} : & g_{s1}N_{33} + g̃_{s1}u_{3} = f_{s1}(x_{1}, x_{2}), \\ & g_{s2}N_{31} + g̃_{s2}u_{1} = f_{s2}(x_{1}, x_{2}), \\ & g_{s3}N_{32} + g̃_{s3}u_{2} = f_{s3}(x_{1}, x_{2}), \\ & g_{s4}M_{31} + g̃_{s4}K_{1} = f_{s4}(x_{1}, x_{2}), \\ & g_{s5}M_{32} + g̃_{s5}K_{2} = f_{s5}(x_{1}, x_{2}), \\ & g_{s6} \frac{∂T}{∂x_{3}} + g̃_{s6}T = f_{s6}(x_{1}, x_{2}). \end{aligned}  (35)

Comments to conditions (12) also hold for conditions (35) and to make it clear we shall point out to following.

In (33), (34) and (35) we have s = 0, 1; (33a) and (34a) are symmetry conditions and (33b) and (34b) are antisymmetry conditions. Conditions (33) and (34) are the conditions of the continuous extension of the solution across the corresponding face of the RP onto a domain which is a mirror reflection with respect to this face. The validity of the statement is a direct implication of the constructed solution of the boundary value problems.

Theorem 2. The general solution of system (31) according to pseudo-asym-metric theory has the following form:

\begin{aligned}u_{1} & = \frac{∂}{∂x_{1}} (φ_{3} − x_{3}φ_{1}) + \frac{∂φ_{2}}{∂x_{2}} \\ & − η_{1} (η_{1} \frac{∂^{2}ψ_{2}}{∂x_{1}^{2}} − η_{1} \frac{∂^{2}ψ_{1}}{∂x_{1}∂x_{2}} + μψ_{2}) + x_{3} \frac{∂T̃}{∂x_{1}} , \\ u_{2} & = \frac{∂}{∂x_{2}} (φ_{3} − x_{3}φ_{1}) − \frac{∂φ_{2}}{∂x_{1}} \\ & − η_{1} (η_{1} \frac{∂^{2}ψ_{2}}{∂x_{1}∂x_{2}} − η_{1} \frac{∂^{2}ψ_{1}}{∂x_{2}^{2}} + μψ_{1}) + x_{3} \frac{∂T̃}{∂x_{2}} , \\ u_{3} & = \frac{∂}{∂x_{3}} (φ_{3} − x_{3}φ_{1}) + κφ_{1} \\ & − η_{1}^{2} (\frac{∂^{2}ψ_{2}}{∂x_{1}∂x_{3}} − \frac{∂^{2}ψ_{1}}{∂x_{2}∂x_{3}} ) + x_{3} \frac{∂T̃}{∂x_{3}} + T̃. \end{aligned}  (36)

In (36) φ_{1}, φ_{2}, φ_{3} and T̃ are harmonic functions and ψ_{1} and ψ_{2} are metaharmonic functions with Δψ_{s} − \frac{μ}{η_{1}} ψ_{s} = 0, where s = 1, 2.

Proof. (32a,b,c) implies

ΔD = 0.

If we assume

D = κμ\frac{∂φ_{1}}{∂x_{3}} ,  (37)

where Δφ_{1} = 0, then taking (37) into account, we can write (32) as

\begin{matrix}\frac{∂}{∂x_{3}} (K_{2}^{∗} + κμ\frac{∂φ_{1}}{∂x_{1}} ) − \frac{∂K_{3}^{∗}}{∂x_{2}} = 0, \\ \frac{∂K_{3}^{∗}}{∂x_{1}} − \frac{∂}{∂x_{3}} (K_{1}^{∗} − κμ\frac{∂φ_{1}}{∂x_{2}} ) = 0, \\ \frac{∂}{∂x_{2}} (K_{1}^{∗} − κμ\frac{∂φ_{1}}{∂x_{2}} ) − \frac{∂}{∂x_{1}} (K_{2}^{∗} + κμ\frac{∂φ_{1}}{∂x_{1}} ) = 0, \\ \frac{∂K_{1}^{∗}}{∂x_{1}} + \frac{∂K_{2}^{∗}}{∂x_{2}} + \frac{∂K_{3}^{∗}}{∂x_{3}} = 0. \end{matrix}  (38)

It follows from system (38) that a function μ\frac{∂φ_{2}}{∂x_{3}} exists such that

\begin{matrix}K_{1}^{∗} = μ\frac{∂^{2}φ_{2}}{∂x_{1}∂x_{3}} + κμ\frac{∂φ_{1}}{∂x_{2}} , K_{2}^{∗} = μ\frac{∂^{2}φ_{2}}{∂x_{2}∂x_{3}} − κμ\frac{∂φ_{1}}{∂x_{1}} , \\ K_{3}^{∗} = μ\frac{∂^{2}φ_{2}}{∂x_{3}^{2}} , \end{matrix}  (39)

with Δφ_{2} = 0.. The projection of the equality ΔK⃗ − \frac{μ}{η_{1}} K⃗ = −\frac{μ}{η_{1}} K⃗^{∗} onto the coordinate axes with (39) in mind will give

\begin{matrix}a) & ΔK_{1} − \frac{μ}{η_{1}} K_{1} = −\frac{μ^{2}}{η_{1}} \frac{∂^{2}φ_{2}}{∂x_{1}∂x_{3}} − \frac{κμ^{2}}{η_{1}} \frac{∂φ_{1}}{∂x_{2}} , \\ b) & ΔK_{2} − \frac{μ}{η_{1}} K_{2} = −\frac{μ^{2}}{η_{1}} \frac{∂^{2}φ_{2}}{∂x_{2}∂x_{3}} + \frac{κμ^{2}}{η_{1}} \frac{∂φ_{1}}{∂x_{1}} , \\ c) & ΔK_{3} − \frac{μ}{η_{1}} K_{3} = −\frac{μ^{2}}{η_{1}} \frac{∂^{2}φ_{2}}{∂x_{3}^{2}} . \end{matrix}  (40)

From (40a) and (40b) we have

\begin{aligned}K_{1} & = μ^{2}η_{1} \frac{∂ψ_{1}}{∂x_{3}} + μ\frac{∂^{2}φ_{2}}{∂x_{1}∂x_{3}} + κμ\frac{∂φ_{1}}{∂x_{2}} , \\ K_{2} & = μ^{2}η_{1} \frac{∂ψ_{2}}{∂x_{3}} + μ\frac{∂^{2}φ_{2}}{∂x_{2}∂x_{3}} − κμ\frac{∂φ_{1}}{∂x_{1}} , \end{aligned}  (41)

where Δψ_{s} − \frac{μ}{η_{1}} ψ_{s} = 0 (s = 1, 2) and using the identity div(μ rot U⃗) = \frac{∂K_{1}}{∂x_{1}} + \frac{∂K_{2}}{∂x_{2}} + \frac{∂K_{3}}{∂x_{3}} = 0 we can define

K_{3} = −μ^{2}η_{1} (\frac{∂ψ_{1}}{∂x_{1}} + \frac{∂ψ_{2}}{∂x_{2}} ) + μ\frac{∂^{2}φ_{2}}{∂x_{3}^{2}} + f(x, y),  (42)

where f is a function resulting from integration of the equality div(μ rot U⃗) = 0 with respect to x_{3}, which is the solution of boundary value problem (28) if we replace the constant q by the constant \frac{μ}{η_{1}} . It can be easily seen that (28) implies f = 0.

Taking into account (35), (41) and (42) we can obtain the following system

\begin{matrix}\frac{∂u_{1}}{∂x_{1}} + \frac{∂u_{2}}{∂x_{2}} + \frac{∂u_{3}}{∂x_{3}} = \frac{κμ}{λ + 2μ} \frac{∂φ_{1}}{∂x_{3}} + \frac{3λ + 2μ}{λ + 2μ}γ T, \\ \frac{∂}{∂x_{2}} (u_{3} − κφ_{1}) − \frac{∂}{∂x_{3}} (u_{2} + μη_{1}ψ_{1} + \frac{∂φ_{2}}{∂x_{1}} ) = 0, \\ \frac{∂}{∂x_{3}} (u_{1} − μη_{1}ψ_{2} − \frac{∂φ_{2}}{∂x_{2}} ) − \frac{∂}{∂x_{1}} (u_{3} − κφ_{1}) = 0, \\ \frac{∂}{∂x_{1}} (u_{2} + μη_{1}ψ_{1} + \frac{∂φ_{2}}{∂x_{1}} ) − \frac{∂}{∂x_{2}} (u_{1} − μη_{1}ψ_{2} − \frac{∂φ_{2}}{∂x_{2}} ) = 0. \end{matrix}  (43)

from the equalities (λ+2μ) div U⃗ −(3λ+2μ)γ T = D = κμ\frac{∂φ_{1}}{∂x_{3}} and μ rot U⃗ = K⃗. From (43) we have

\begin{aligned}u_{1} & = \frac{∂φ̃_{3}}{∂x_{1}} + μη_{1}ψ_{2} + \frac{∂φ_{2}}{∂x_{2}} , \\ u_{2} & = \frac{∂φ̃_{3}}{∂x_{2}} − μη_{1}ψ_{1} − \frac{∂φ_{2}}{∂x_{1}} , \\ u_{3} & = \frac{∂φ̃_{3}}{∂x_{3}} + κφ_{1}; \end{aligned}  (44)

Δφ̃_{3} = −2\frac{∂φ_{1}}{∂x_{3}} − μη_{1} (\frac{∂ψ_{2}}{∂x_{1}} − \frac{∂ψ_{1}}{∂x_{2}} ) + \frac{3λ + 2μ}{λ + 2μ} γ T.  (45)

If the change in the temterature T is expressed by means of the function T̃ according to formula (5) (note that ΔT̃ = 0), then (45) will result in

φ̃_{3} = φ_{3} − x_{3}φ_{1} − η_{1}^{2} (\frac{∂ψ_{2}}{∂x_{1}} − \frac{∂ψ_{1}}{∂x_{2}} ) + x_{3}T̃.  (46)

Substitution of (46) into (44) shows the validity of Theorem 2.

3. Below we shall give the construction pattern for regular solutions of boundary value problems in asymmetric thermoelasticity. But before we continue our discussion we must define the concept of regular solutions.

In the case of Cosserat’s continuum the solution of system (4) defined by seven functions T, u_{i}, ω_{i} (i = 1, 2, 3) will be called regular if T is twice and u_{i}, ω_{i} is three times continuously differentiable in the domain Ω̃ where Ω̃ is the domain Ω together with the boundaries x_{1} = x_{1s} and x_{2} = x_{2s} (s = 0, 1) and on the surface x_{3} = x_{3s} they together with their first derivatives can be represented by uniformly converging trigonometric series. It is also assumed that the equilibrium equations are true for x_{1} = x_{1s} and x_{2} = x_{2s}.

The differential properties imposed on the functions f_{s1}, f_{s2}, …, f_{s6} from (12) are defined by the regularity requirements imposed on the solution.

In the case of Cosserat’s pseudo-continuum the solution of (30), which is defined by four functions T, u_{i} is called regular if T is twice and u_{i} is five times continuously differentiable in the domain Ω̃ and on the surface x_{3} = x_{3s}, T together with its first derivative and u_{i} together with its derivatives up to the fourth order can be represented by uniformly converging trigonometric series. We also assume that the equilibrium equations and their first derivatives hold for x_{1} = x_{1s} and x_{2} = x_{2s}.

The differential properties imposed on the functions f_{s1}, f_{s2}, …, f_{s5} from (35), similar to the case of Cosserat’s continuum, are defined by the regularity requirement imposed on the solution.

SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 781

4. Let conditions (10b),(11b), (12) be satisfied on the surface of the RP, then

T̃ = ∑_{_{m}=1}^{∞} ∑_{_{n}=1}^{∞} E_{0mn}(x_{3}) sin(mx_{1}) sin(nx_{2}),
φ_{1} = ∑_{_{m}=1}^{∞} ∑_{_{n}=1}^{∞} E_{1mn}(x_{3}) sin(mx_{1}) sin(nx_{2}),
φ_{2} = ∑_{_{m}=0}^{∞} ∑_{_{n}=0}^{∞} E_{2mn}(x_{3}) cos(mx_{1}) cos(nx_{2}),
φ_{3} = ∑_{_{m}=1}^{∞} ∑_{_{n}=1}^{∞} E_{3mn}(x_{3}) sin(mx_{1}) sin(nx_{2}),
ψ_{1} = ∑_{_{m}=1}^{∞} ∑_{_{n}=0}^{∞} Ẽ_{1mn}(x_{3}) sin(mx_{1}) cos(nx_{2}),
ψ_{2} = ∑_{_{m}=0}^{∞} ∑_{_{n}=1}^{∞} Ẽ_{2mn}(x_{3}) cos(mx_{1}) sin(nx_{2}),
ψ_{3} = ∑_{_{m}=0}^{∞} ∑_{_{n}=0}^{∞} Ẽ_{3mn}(x_{3}) cos(mx_{1}) cos(nx_{2}),

(47)

where E_{imn}(x_{3}) = A_{imn} exp(−px_{3}) + B_{imn} exp[p(x_{3} − x_{31})], i = 0, 1, 2, 3, p =

\sqrt{m^{2} + n^{2}}, m = \frac{π_{m}}{x_{11}}, n = \frac{π_{n}}{x_{21}} , A_{imn} and B_{imn} are constant; Ẽ_{smn}(x_{3}) =

Ã_{smn} exp(−p_{0}x_{3}) + B̃_{smn} exp[p_{0}(x_{3} − x_{31})], s = 1, 2, p_{0} = \sqrt{p^{2} + q}, Ã_{smn} and B̃_{smn} are constant; E_{3mn}(x_{3}) = Ã_{3mn} exp(−p_{1}x_{3}) + B̃_{3mn} exp[p_{1}(x_{3} − x_{31})] with p_{1} = \sqrt{p^{2} + \frac{4σ_{1}}{σ_{2}+2σ_{3}}} , Ã_{3mn} and B̃_{3mn} are constant.

Substitution of (47) into (13) and (14) gives

ω_{1} = ∑_{_{m}=1}^{∞} ∑_{_{n}=0}^{∞} ω_{1mn}(x_{3}) sin(mx_{1}) cos(nx_{2}),

ω_{2} = ∑_{_{m}=0}^{∞} ∑_{_{n}=1}^{∞} ω_{2mn}(x_{3}) cos(mx_{1}) sin(nx_{2}),

ω_{3} = ∑_{_{m}=0}^{∞} ∑_{_{n}=0}^{∞} ω_{3mn}(x_{3}) cos(mx_{1}) cos(nx_{2}),

u_{1} = ∑_{_{m}=0}^{∞} ∑_{_{n}=1}^{∞} u_{1mn}(x_{3}) cos(mx_{1}) sin(nx_{2}),

u_{2} = ∑_{_{m}=1}^{∞} ∑_{_{n}=0}^{∞} u_{2mn}(x_{3}) sin(mx_{1}) cos(nx_{2}),

u_{3} = ∑_{_{m}=1}^{∞} ∑_{_{n}=1}^{∞} u_{3mn}(x_{3}) sin(mx_{1}) sin(nx_{2}),

(48)

where ω_{1mn}(x_{3}),ω_{2mn}(x_{3}), ω_{3mn}(x_{3}) are known expressions containing the constants A_{imn}, B_{imn}, Ã_{imn}, B̃_{imn} (i = 1, 2, 3) and depend on x_{3}, m, n and depending on x_{3}, m, n and the elastic characteristics of the RP (these expresseions are not given due to their awkwardness).

The functions T̃, ω_{1}, ω_{2}, …, u_{3} at least formally, satisfy equalibrium equations and boundary conditions (10b) and (11b) (the word “formally” will be omitted as soon as the constants A_{0mn}, B_{0mn}, A_{imn}, B_{imn}, Ã_{imn}, B̃_{imn} (i = 1, 2, 3) have been defined, convergence of the series and uniqueness of the obtained solution have been proved).

After the right-hand sides of formulas (12) have been substituted by the corresponding trigonometric series and the obtained series have been compared with the series represented by formulas (47a) and (48) we shall have two systems of linear algebraic equations for fixed m and n. The first one with a second order amtrix enables one to define the constants A_{omn}, B_{0mn}, while the second one woth a twelfth-order matrix can be used to define the constants A_{imn}, B_{imn}, Ã_{imn}, B̃_{imn} (i = 1, 2, 3). Similar to [8], it is proved that the obtained solution or, to be more precise, the series representing the solution, uniformly converge in the domain \overline{Ω} = \{0 ≤ x_{1} ≤ x_{11}, 0 ≤ x_{2} ≤ x_{21}, 0 ≤ x_{3} ≤ x_{31}\} and thet the resulted regular solution is unique. In particular, the uniqueness of the solution of boundary value problems of asymmetric elasticity, corresponding to thermoelasticity problems considered in the given paper, as well as of a number of other boundary value ptoblems of asymmetric thermoelasticity is proved in [6]. The solution method can be easily extended to the case of the boundary value and boundary value constant problems of asymmetric thermoelasticity considered in the present work.

Thus a regular solution of boundary value problem (4), (10b), (11b), (12) has been obtained for g_{s1} = 0, g_{s2} = 0, …, g_{s7} = 0 and g̃_{s1} = 1, g̃_{s2} = 1, …, g̃_{s7} = 1.

We should also say a few words about the above-mentioned second- and twelfth-order matrices or, to be more exact, of the determinants Δ_{2} and Δ_{12} of these matrices. We shall start with Δ_{12}. Scaling in the expressions for E_{imn} and Ẽ_{imn} (i = 1, 2, 3) can be always performed so the the elements of the twelfthorder matrix should be bounded values for any m̃ and ñ (including the case when m̃ → ∞, ñ = ñ_{0} = const or ñ → ∞, m̃ = m̃_{0} = const, or when both m̃ → ∞ and ñ → ∞). If this is the case, then analizing the system for m̃ → ∞, ñ = ñ_{0}, for ñ → ∞, m̃ = m̃_{0} and for m̃ → ∞, ñ → ∞, one can easily see that for all those cases Δ_{12} ≠ 0, while when m̃ = m̃_{0} and ñ = ñ_{0} by virtue of the uniqueness theorem we have Δ_{12} ≠ 0. The same holds for Δ_{2} too, though the expression for this detertminant can be immediately written out and analysed.

Quite similarly one can find and analyzed a solution of any other problem from the class of boundary value problems (4), (10), (11), (12) with the only difference that different boundary conditions for x_{1} = x_{1s} and x_{2} = x_{2s} in the expressions for T̃, φ_{1}, φ_{2}, …, ψ_{3} (see formulas (47)) will correspond to different trigonometrical functions.

If in (47) we omit the last equality (the expression for ψ_{3}) and in the expression for p_{0} we replace q by \frac{μ}{η_{1}} , then the functions T̃, φ_{1}, φ_{2}, φ_{3}, ψ_{1}, ψ_{2} will define, according to Cosserat’s pseudo-continuum, the solution of boundary value problem (30), (33b), (34b), (35). The comments and generalizations given to boundary value problem (4), (10b), (11b) and (12) for Cosserat’s continuum, exactly hold for the same problem in CosseratTs pseudo-continuum, i.e. problem (30), (33), (35b), (35). The only difference is that unlike in the case of Cosserat’s continuum, where one has to solve systems of linear equations with a matrices of the second and twelfth order, in the case of Cosserat’s pseudo-continuum we have a second-order and tenth-order matrices.

5. According to asymmetric theory, consider a RP multilayer along x_{3} (or MRP in the abbreviated form) occupying the domain Ω_{3}. Ω_{3} is a union of the domains Ω_{31} = {0 < x_{1} < x_{11}, 0 < x_{2} < x_{21}, 0 < x_{3} < x_{31}}, Ω_{32} = {0 < x_{1} < x_{11}, 0 < x_{2} < x_{21}, x_{31} < x_{3} < x_{32}}, …, Ω_{3β} = {0 < x_{1} < x_{11}, 0 < x_{2} < x_{21}, x_{3(β−1)} < x_{3} < x_{3β}}, contacting along the planes x_{3} = x_{3j}, where j = 1, 2, …, β − 1,, and β is the number of layers. Each layer has its elastic and thermal characteristics. With x_{1} = 0 and x_{1} = x_{11} conditions (10) are simultaneously satisfied for all layers while with x_{2} = 0 and x_{2} = x_{21} conditions (11) hold.

If the body occupies the domain Ω_{3}, then conditions (10), (11), (12) are satisfied on its faces with x_{31} replaced by x_{3β} in (12). On the contact planes x_{3} = x_{3j} (x_{3} = x_{3j} is the contact plane of the j-th layer contacting the j+1-th layer) the following conditions

\begin{matrix}T_{j} − T_{j+1} = τ_{j1}(x_{1}, x_{2}), & λ_{j}^{∗} \frac{∂T_{j}}{∂x_{3}} − λ_{j+1}^{∗} \frac{∂T_{j+1}}{∂x_{3}} = τ_{j2}(x_{1}, x_{2}), \\ u_{1j} − u_{1(j+1)} = q_{j1}(x_{1}, x_{2}), & N_{31j} − N_{31(j+1)} = Q_{j1}(x_{1}, x_{2}), \\ u_{2j} − u_{2(j+1)} = q_{j2}(x_{1}, x_{2}), & N_{32j} − N_{32(j+1)} = Q_{j2}(x_{1}, x_{2}), \\ u_{3j} − u_{3(j+1)} = q_{j3}(x_{1}, x_{2}), & N_{33j} − N_{33(j+1)} = Q_{j3}(x_{1}, x_{2}), \\ ω_{1j} − ω_{1(j+1)} = q_{j4}(x_{1}, x_{2}), & M_{31j} − M_{31(j+1)} = Q_{j4}(x_{1}, x_{2}), \\ ω_{2j} − ω_{2(j+1)} = q_{j5}(x_{1}, x_{2}), & M_{32j} − M_{32(j+1)} = Q_{j5}(x_{1}, x_{2}), \\ ω_{3j} − ω_{3(j+1)} = q_{j6}(x_{1}, x_{2}), & M_{33j} − M_{33(j+1)} = Q_{j6}(x_{1}, x_{2}) \end{matrix}  (49)

or

\begin{matrix}T_{j} − T_{j+1} = τ_{j1}(x_{1}, x_{2}), & λ_{j}^{∗} \frac{∂T_{j}}{∂x_{3}} − λ_{j+1}^{∗} \frac{∂T_{j+1}}{∂x_{3}} = τ_{j2}(x_{1}, x_{2}), \\ N_{31j} = Q_{j1}(x_{1}, x_{2}), & N_{31(j+1)} = Q̃_{j1}(x_{1}, x_{2}), \\ N_{32j} = Q_{j2}(x_{1}, x_{2}), & N_{32(j+1)} = Q̃_{j2}(x_{1}, x_{2}), \\ u_{3j} − u_{3(j+1)} = q_{j3}(x_{1}, x_{2}), & N_{33j} − N_{33(j+1)} = Q_{j3}(x_{1}, x_{2}), \\ ω_{1j} − ω_{1(j+1)} = q_{j4}(x_{1}, x_{2}), & M_{31j} − M_{31(j+1)} = Q_{j4}(x_{1}, x_{2}), \\ ω_{2j} − ω_{2(j+1)} = q_{j5}(x_{1}, x_{2}), & M_{32j} − M_{32(j+1)} = Q_{j5}(x_{1}, x_{2}), \\ M_{33j} = Q_{j6}(x_{1}, x_{2}), & M_{33(j+1)} = Q̃_{j6}(x_{1}, x_{2}) \end{matrix} (50)

are defined. The right-hand sides of equalities (49) and (50) are defined functions and λ_{j}^{∗} and λ_{j+1}^{∗} are thermal conductivity coefficients.

When we state the problem of thermoelastic equilibrium of a MRP we give expressions for the functions T̃^{(j)}, φ_{1}^{(j)}, φ_{2}^{(j)}, φ_{3}^{(j)}, ψ_{1}^{(j)}, ψ_{2}^{(j)}, ψ_{3}^{(j)} for the j-th layer bearing in mind conditions (10), (11), and following the technique of the previous section construct a system of 2β equations in 2β unknowns and another system of 12β equations in 12β unknowns. Solvability of the system, convergence of the corresponding series and uniqueness of the obtained regular solutions of the respective boundary contact problems of thermoelasticity are proved (the solution for the MRP is called regular if each of the solutions u_{1j}, u_{2j}, u_{3j} is regular).

Besides the given contact conditions, a number of other contact conditions can be constructed under which the solution of boundary contact problems of thermoelasticity can be likewise effective.

Solutions of boundary contact problems for Cosserat’s pseudo-continuum are considered in a similar way.

In conclusion, we should note that the constructed effective solutions of boundary vale and boundary contact problems, as a rule, become elementary. Indeed, if the functions defined on the surface x = x_{3s} (although we imply boundary value problems the above-stated can be easily extended to boundary contact problems as well) can be represented by finite trigonometric series, then the solutions can be also represented by finite series the summands of which are elementary functions. In particular, this statement can be used to specify and define new elasticity characteristics of a material σ_{1}, σ_{2}, σ_{3}, σ_{4} and η_{1}, η_{2}, naturally, using high-quality measuring instruments.

References

1. E. Cosserat and F. Cosserat, Theórie des corps déformables. Hermann, Paris, 1909.

2. C. A. Truesdell and R. A. Toupin, The classical field theories. Handbuch der Physik, Band 3, Teil 1, Springer, Berlin, 1960.

3. W. Nowacki, Theory of elasticity. (Translated from Polish into Russian) Mir, Moscow, 1975.

4. A. C. Eringer, Theory of micropolar elasticity. J. Math. Mech. 15(1966), 909–923.

5. W. Nowacki, Couple stresses in the theory of thermoelasticity. Bull. Acad. Polon. Sér. Sci. Techn. 14(1966), No. 8.

6. V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. (Translated from Russian) North-Holland Series in Applied Mathematics and Mechanics, v. 25, North-Holland Publ. Co., Amsterdam–New York–Oxford, 1979; Russian original: Nauka, Moscow, 1976.

7. B. Carbonaro and R. Russo, On some classical theorems in micropolar elasticity. Internat. J. Engrg. Sci. 23(1985), 119–129.

8. N. Khomasuridze, Thermoelastic equilibrium of bodies in generalized cylindrical coordinates. Georgian Math. J. 5(1998), No. 6, 521–544.

(Received 18.09.2000; revised 22.09.2001)

Author’s address:

I. Vekua Institute of Applied Mathematics

I. Javakhishvili Tbilisi State University

2, University St., Tbilisi 380043

Georgia