SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM OF A RECTANGULAR PARALLELEPIPED IN TERMS OF ASYMMETRIC ELASTICITY

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1 Georgian Mathematical Journal Volume 8 00) Number 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. 000 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 [] and was later on developed by Truesdell and Toupin []. 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 < x 0 < x < x 0 < x 3 < x 3 }. For the lateral faces x x = x x x = x symmetry and antisym- ISSN X / $8.00 / c Heldermann Verlag

2 768 N. KHOMASURIDZE metry conditions are defined while on the faces x 3 = 0 and x 3 = x 3 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 x = x x x = x 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.. 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]: a) b) j j N ji x j M ji x j c) T + kj ε ijk N jk ) where i = 3; j = 3; k = 3; ε ijk is Levy-Civita symbol; x x x 3 are Cartesian coordinates N N N 33 are normal stresses N N 3 N... N 3 are tangential stresses M M M 33 are torsion micromoments; M M 3 M... M 3 are bending micromoments; T is the change in the temperature of the body; = The equalities connecting stresses and micromoments with the components of displacement and rotation vectors have the following form [6] N ij = δ ij [λ div U 3λ + µ)γt ] + µ + σ ) u j x i + µ σ ) u i σ ε ijk ω k x j M ij = δ ij σ div ω + σ 3 + σ 4 ) ω j x i + σ 3 σ 4 ) ω i x j k ) where U = u l + u l + u 3 l3 l l l 3 are basis vectors in the Cartesian coordinate system) ω = ω l + ω l + ω 3 l3 are displacement and rotation vectors respectively δ ij is Kronecker s symbol λ µ are classical elasticity characteristics and σ σ σ 3 σ 4 are asymmetric elasticity characteristics with µ > 0 νe 3λ + µ > 0 σ > 0 σ 3 > 0 3σ + σ 3 > 0 σ 4 > 0; λ = µ = E ν)+ν) ν) where E is elasticity modulus ν is the Poisson coefficient γ is the coefficient of linear thermal expansion. Substituting ) into )a )b we have grad[λ + µ) div U 3λ + µ) γ T ] rot[µ + σ ) rot U σ ω] 3) grad[σ + σ 3 ) div ω] rot[σ 3 + σ 4 ) rot ω σ U] 4σ ω = 0.

3 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 769 The second equation of system 3) can be written as grad[σ + σ 3 ) div ω] rot[σ 3 + σ 4 ) rot ω + µ U] + [µ + σ ) rot U σ ω] = 0. Taking into consideration the last equality we can write a) grad[λ + µ) div U 3λ + µ)γ T ] rot[µ + σ ) rot U σ ω] b) div[µ + σ ) rot U σ ω] = σ σ + σ 3 [σ + σ 3 ) div ω] c) grad[σ + σ 3 ) div ω] rot[σ 3 + σ 4 ) rot ω + µ U] + [µ + σ ) rot U σ ω] d) div[σ 3 + σ 4 ) rot ω + µ U] = µ λ + µ [λ + µ) div U 3λ + µ)γ T ] + µ3λ + µ) γt = 0. λ + µ 4) It can be easily seen that 4b) and 4d) are identities. Without loss of generality we can represent the function T as T = λ + µ) 3λ + µ)γ T 5) where T = 0. No loss of generality will occur if T is expressed by a derivative of the harmonic function T since it is preserved by an appropriate representation of the harmonic function T using the method of separation of variables as in [8]. Introduce the notation a) λ + µ) div U 3λ + µ)γ T = D b) σ + σ 3 ) div ω = D c) µ + σ ) rot U σ ω = K = K l + K l + K 3 l3 d) σ 3 + σ 4 ) rot ω + µ U = K = K l + K l + K3 l 3. 6) Taking 5) into account we can write system 4) in the following way grad D rot K div K = σ σ + σ 3 D grad D rot K + K div K = µ T D + 4µ. λ + µ 7)

4 770 N. KHOMASURIDZE Finally we obtain the following system a) D K 3 + K b) D K + K 3 c) D K + K d) K + K + K 3 = σ σ + σ 3 D e) D K 3 + K + K f) D K + K 3 + K g) D K + K + K 3 h) K + K + K 3 = µ T D + 4µ ; λ + µ 8) a) u + u + u 3 = D λ + µ + T u3 b) µ + σ ) u ) σ ω = K u c) µ + σ ) u ) 3 σ ω = K u d) µ + σ ) u ) σ ω 3 = K 3 D e) ω + ω + ω 3 = x σ + σ 3 ω3 f) σ 3 + σ 4 ) ω ) + µu = K x 3 ω g) σ 3 + σ 4 ) ω ) 3 + µu = K x ω h) σ 3 + σ 4 ) ω ) + µu 3 = K x 3 in 4 unknowns in particular the unknowns D K K K 3 D K K K 3 ω ω ω 3 u u u 3 the temperature T is defined by c) under appropriate boundary conditions). Note that identities 8d) and 8h) complement six equilibrium equations 8abcefg) to obtain two similar systems 8abcd) and 9efgh) which consist of four differential equations with the first order partial derivatives in four unknowns D K K K 3 and D K K K 3 respectively in this case the functions D and D which also appear in the right-hand side of 9)

5 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 77 8d) of system 8abcd) and in the right-hand side of 8h) of system 8cfgh) are assumed to be known). The given work deals with the thermoelastic equilibrium of a rectangular parallelepiped RP) occupying the domain Ω = {0 < x < x 0 < x < x 0 < x 3 < x 3 }. The boundary conditions appearing in the paper and defining the class of the boundary value problems under consideration are given below. For x = x s : a) u N N 3 M T ω ω 3 = 0 u u u 3 ω T ω ω 3 = 0 or b) N u u 3 ω = 0 M M 3 T = 0 u ω u u 3 ω ω 3 T = 0. For x = x s : a) u N 3 N M T ω 3 ω = 0 u u 3 u ω T ω 3 ω = 0 or b) N u 3 u ω = 0 M 3 M T = 0 u ω u 3 u ω 3 ω T = 0. 0) ) For x 3 = x 3s : g s N 33 + g s u 3 = f s x x ) g s N 3 + g s u = f s x x ) g s3 N 3 + g s3 u = f s3 x x ) g s4 M 33 + g s4 ω 3 = f s4 x x ) g s5 M 3 + g s5 ω = f s5 x x ) g s6 M 3 + g s6 ω = f s6 x x ) g s7 T + g s7 T = f s7 x x ). ) In 0) ) and ) we have s with x 0 = x 0 = x 30 = 0; g s g s... g s7 being the defined constants governed by the conditions g s g s 0 g s g s 0... g s7 g s7 0. The conditions imposed on the functions f s f s... 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 g s = g s = g s3 g s4 = g s5 = g s6 = 0 we have the boundary conditions for the first problem of asymmetric thermoelasticity with g s = g s = g s3 g s4 = g s5 = g s6 = 0 we have the boundary conditions for the second problem of asymmetric thermoelasticity etc. Note that ) implies that with x 3 = 0 one type of

6 77 N. KHOMASURIDZE conditions can be defined while with x 3 = x 3 there can be another conditions. Conditions 0a) and a) are called the symmetry conditions while conditions 0b) and b) are called the antisymmetry conditions. We should also note that conditions 0) and ) 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 0) and ) is the same as of the corresponding conditions in [3]. Theorem. The general solution of system 7) 6) or what is the same system 8) 9) according to asymmetric theory has the following form ω = ψ 3 + κ ω = ψ 3 κ ϕ + ϕ + ϕ + ψ ϕ + ψ ω 3 = ψ 3 + ϕ ψ ψ ; 3 u = ϕ 3 x 3 ϕ ) + ϕ σ 3 + σ 4 ) ψ qψ µ ψ u = ϕ 3 x 3 ϕ ) ϕ + σ 3 + σ 4 ) ψ qψ µ ψ u 3 = ϕ 3 x 3 ϕ ) + κϕ σ 3 + σ 4 ) ψ ψ µ + x 3 T + x 3 T + x 3 T + T. 3) 4) In 3) and 4) ϕ ϕ ϕ 3 and T are harmonic functions ψ ψ ψ 3 are metaharmonic functions with ψ qψ ψ qψ ψ 3 4σ ψ 3 = 0 σ + σ 3 4µσ and q = µ + σ )σ 3 + σ 4 ). Proof. 8a b c) implies D = 0.

7 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 773 Without loss of generality assume D = κµ ϕ 5) where ϕ = 0 see the comments to 5)) and κ = λ+µ) λ+µ 5) into account we can write 8a b c d) as a) K + κµ ϕ ) K 3 b) K 3 c) From 6a b c) we have K κµ ϕ ) ) K κµ ϕ K + κµ ϕ d) K + K + K 3 = σ σ + σ 3 D. = 4 ν) so taking ) 6) K = ϕ + κµ ϕ K = ϕ κµ ϕ K 3 = ϕ. 7) Substituting 8) into 6d) we obtain 8e f g) implies D 4σ σ + σ 3 D = 0. If we write the solution of this equation as ϕ = σ σ + σ 3 D. 8) D = 4σ ψ 3 where ψ 3 4σ σ +σ 3 ψ 3 then the solution of 8) can be represented as ϕ = µ ϕ σ ψ 3 9) where ϕ = 0 see the comments to 5)). Taking 9) into account we can write 7) as K = µ ϕ σ ψ 3 + κµ ϕ K = µ ϕ σ ψ 3 κµ ϕ 0) K 3 = µ ϕ 3 σ ψ 3.

8 774 N. KHOMASURIDZE With 0) in mind 8e f g h) can be written as a) b) c) K + µ ϕ ) K3 κµϕ ) K3 κµϕ ) K µ ϕ ) K µ ϕ ) K + µ ϕ ) d) K + K + K 3 = κµ ϕ + 4µ T. λ + µ ) From a b c) we have K = ϕ 3 + µ ϕ K = ϕ 3 µ ϕ K 3 = ϕ 3 + κµϕ. ) Substituting ) into d) we have The solution of this equations is ϕ 3 = 4µ T ϕ ). ϕ 3 = µϕ 3 µx 3 ϕ T ) 3) where ϕ 3 = 0.. If we substitute 3) into ) then K = µ ϕ 3 µx 3 ϕ + µ ϕ + µx 3 T K = µ ϕ 3 µx 3 ϕ µ ϕ + µx 3 T K 3 = µ ϕ 3 µx 3 ϕ + µx 3 T + µ4κ )ϕ + µ T. 4) Applying rot to 6d) and substituting it into 6c) with rot rot ω = grad div ω ω in mind we obtain ω q ω = σ + σ 3 grad D σ 3 + σ 4 rot K + q σ K. If we project this equation onto the coordinate axes and take into account

9 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 775 equality D = 4σ ψ 3 and formulas 0) and 4) we shall have ) 4σ ψ3 a) ω qω = q q κ ϕ ) + ϕ σ + σ 3 ) 4σ ψ3 b) ω qω = q + q κ ϕ ) ϕ σ + σ 3 ) 4σ ψ3 c) ω 3 qω 3 = q q ϕ. σ + σ 3 3 The solution of 5a) 5b) will be ω = ψ 3 + κ ϕ ) + ϕ ω = ψ 3 κ ϕ ) ϕ + ψ + ψ 5) 6) respectively where ψ qψ ψ qψ = 0. We define ω 3 from equality 9e) ω 3 = ψ 3 + ϕ ψ ψ + fx x ) 7) 3 where f is a function resulting from integration of 9e) and satisfying the following equation and the following conditions: f + f qf = 0; f x s = 0 or f = 0 with x s f x s = 0 or f = 0 with x s = x s where s =. The solution of 8) is f = 0. Substituting 6) 7) and 4) into 9fgh) we have u = [ ϕ 3 κσ ] 3 + σ 4 ) ϕ ϕ x 3 + ϕ 4µ σ 3 + σ 4 ) ψ qψ µ ψ T + x 3 u = [ ϕ 3 κσ ] 3 + σ 4 ) ϕ ϕ x 3 ϕ 4µ + σ 3 + σ 4 ) ψ qψ µ ψ T + x 3 u 3 = [ ϕ 3 κσ ] 3 + σ 4 ) ϕ ϕ x 3 + κ )ϕ 4µ 8)

10 776 N. KHOMASURIDZE σ 3 + σ 4 µ ) ψ ψ T + x 3 + T. If we introduce the notation [ ϕ 3 κσ 3+σ 4 ) 4µ 0 we can say that Theorem is proved. ϕ ] = ϕ3 where naturally ϕ 3 =. In the case of Cosserat s pseudo-continuum equations ) still hold but instead of ) we have [3] N ij = δ ij [λ div U 3λ + µ)γ T ] + µ η ε ijk rot k U) k M ij = η rot j U) + η rot i U) x i x j uj + u ) i x i x j 9) where η η are asymmetric elasticity characteristics with η > 0 η > 0. Substituting 9) into ) we have a) grad[λ + µ) div U 3λ + µ)γ T ] rotµ rot U η rot U) b) divµ rot U η rot U) where 30b) is a known identity. Write 30) as grad D rot K div K 30) 3) where D = λ + µ) div U 3λ + µ)γ T K = K l + K l + K 3 l 3 K = K η µ K and K = µ rot U = K l + K l + K 3 l3 = µ rot U l + µ rot U l + µ rot 3 U l3. If we write 3) in a scalar form we have a) D K 3 + K b) D K + K 3 c) D K + K d) K + K + K 3 = 0. 3) 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

11 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 777 according to pseudo-asymmetric thermoelasticity. For x = x s : a) u N N 3 T K K 3 = 0 u u u 3 u or 3 u 3 T = 0 b) N u u 3 M M 3 T = 0 u u u 3 u 3 u T = 0. 3 u u 3 33) It should be noted that in 3a) the fourth the fifth and the sixth conditions appearing to the right of the equivalence sign have been obtained assuming that with x = x s besides the defined boundary conditions equation 3a) also holds. In its turn in 33b) the sixth condition to the right of the equivalence sign has been obtained assuming that with x = x s equations 3a) 3b) and 3c) differentiated correspondingly with respect to x x and x 3 are true. For x = x s : a) u N 3 N T K 3 K = 0 u u 3 u u or 3 u T = 0 b) N u 3 u M M 3 T = 0 u u 3 u u 3 3 u T = 0. 3 u 3 u 34) Remarks similar to those conditions 33) take place. For x 3 = x 3s : g s N 33 + g s u 3 = f s x x ) g s N 3 + g s u = f s x x ) g s3 N 3 + g s3 u = f s3 x x ) g s4 M 3 + g s4 K = f s4 x x ) g s5 M 3 + g s5 K = f s5 x x ) g s6 T + g s6 T = f s6 x x ). 35) Comments to conditions ) also hold for conditions 35) and to make it clear we shall point out to following. In 33) 34) and 35) we have s ; 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

12 778 N. KHOMASURIDZE respect to this face. The validity of the statement is a direct implication of the constructed solution of the boundary value problems. Theorem. The general solution of system 3) according to pseudo-asymmetric theory has the following form: u = u = ϕ 3 x 3 ϕ ) + ϕ ψ ψ η η η + µψ ϕ 3 x 3 ϕ ) ϕ η η ψ η ψ u 3 = ϕ 3 x 3 ϕ ) + κϕ η ) ψ ψ ) + µψ ) + x 3 T + x 3 T + x 3 T + T. 36) In 36) ϕ ϕ ϕ 3 and T are harmonic functions and ψ and ψ are metaharmonic functions with ψ s µ η ψ s where s =. Proof. 3abc) implies If we assume D = 0. D = κµ ϕ 37) where ϕ then taking 37) into account we can write 3) as K + κµ ϕ ) K 3 K3 K κµ ϕ ) K κµ ϕ ) K + κµ ϕ ) 38) K + K + K 3 = 0.

13 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 779 It follows from system 38) that a function µ ϕ exists such that K = µ ϕ + κµ ϕ K = µ ϕ κµ ϕ K3 = µ ϕ 3 39) with ϕ = 0.. The projection of the equality K µ η K = µ η coordinate axes with 39) in mind will give K onto the From 40a) and 40b) we have a) K µ η K = µ η ϕ κµ η ϕ b) K µ η K = µ η ϕ + κµ η ϕ c) K 3 µ K 3 = µ ϕ. η η 3 K = µ η ψ + µ ϕ + κµ ϕ K = µ η ψ + µ ϕ κµ ϕ 40) 4) where ψ s µ η ψ s = 0 s = ) and using the identity divµ rot U) = K K + K 3 = 0 we can define + K 3 = µ ψ η + ψ ) + µ ϕ + fx y) 4) 3 where f is a function resulting from integration of the equality divµ rot U) = 0 with respect to x 3 which is the solution of boundary value problem 8) if we replace the constant q by the constant µ η. It can be easily seen that 8) implies f = 0. Taking into account 35) 4) and 4) we can obtain the following system u + u + u 3 = κµ λ + µ u 3 κϕ ) u µη ψ ϕ ) u + µη ψ + ϕ ϕ 3λ + µ + λ + µ γ T ) u + µη ψ + ϕ u 3 κϕ ) ) u µη ψ ϕ ) = 0. 43)

14 780 N. KHOMASURIDZE from the equalities λ+µ) div U 3λ+µ)γ T = D = κµ ϕ and µ rot U = K. From 43) we have u = ϕ 3 + µη ψ + ϕ u = ϕ 3 µη ψ ϕ u 3 = ϕ 3 + κϕ ; ϕ 3 = ϕ µη ψ ψ ) + 3λ + µ λ + µ 44) γ T. 45) If the change in the temterature T is expressed by means of the function T according to formula 5) note that T = 0) then 45) will result in ϕ 3 = ϕ 3 x 3 ϕ η ψ ψ ) + x T 3. 46) Substitution of 46) into 44) shows the validity of Theorem. 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 = 3) will be called regular if T is twice and u i ω i is three times continuously differentiable in the domain Ω where Ω is the domain Ω together with the boundaries x = x s and x = x s s ) 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 = x s and x = x s. The differential properties imposed on the functions f s f s... f s6 from ) 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 Ω 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 = x s and x = x s. The differential properties imposed on the functions f s f s... f s5 from 35) similar to the case of Cosserat s continuum are defined by the regularity requirement imposed on the solution.

15 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM Let conditions 0b)b) ) be satisfied on the surface of the RP then T = ϕ = ϕ = ϕ 3 = ψ = ψ = ψ 3 = m= ñ= m= ñ= m=0 ñ=0 m= ñ= m= ñ=0 m=0 ñ= m=0 ñ=0 E 0mn x 3 ) sinmx ) sinnx ) E mn x 3 ) sinmx ) sinnx ) E mn x 3 ) cosmx ) cosnx ) E 3mn x 3 ) sinmx ) sinnx ) Ẽ mn x 3 ) sinmx ) cosnx ) Ẽ mn x 3 ) cosmx ) sinnx ) Ẽ 3mn x 3 ) cosmx ) cosnx ) 47) where E imn x 3 ) = A imn exp px 3 ) + B imn exp[px 3 x 3 )] i 3 p = m + n m = π m x n = πñ x A imn and B imn are constant; Ẽsmnx 3 ) = Ã smn exp p 0 x 3 ) + B smn exp[p 0 x 3 x 3 )] s = p 0 = p + q Ãsmn and B smn are constant; E 3mn x 3 ) = Ã3mn exp p x 3 ) + B 3mn exp[p x 3 x 3 )] with p = p + 4σ σ +σ 3 Ã3mn and B 3mn are constant. Substitution of 47) into 3) and 4) gives ω = ω = ω 3 = u = u = u 3 = m= ñ=0 m=0 ñ= m=0 ñ=0 m=0 ñ= m= ñ=0 m= ñ= ω mn x 3 ) sinmx ) cosnx ) ω mn x 3 ) cosmx ) sinnx ) ω 3mn x 3 ) cosmx ) cosnx ) u mn x 3 ) cosmx ) sinnx ) u mn x 3 ) sinmx ) cosnx ) u 3mn x 3 ) sinmx ) sinnx ) 48) where ω mn x 3 )ω mn x 3 ) ω 3mn x 3 ) are known expressions containing the con-

16 78 N. KHOMASURIDZE stants A imn B imn Ãimn B imn i = 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 T ω ω... u 3 at least formally satisfy equalibrium equations and boundary conditions 0b) and b) the word formally will be omitted as soon as the constants A 0mn B 0mn A imn B imn Ãimn B imn i = 3) have been defined convergence of the series and uniqueness of the obtained solution have been proved). After the right-hand sides of formulas ) 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 à imn B imn i = 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 Ω = {0 x x 0 x x 0 x 3 x 3 } 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) 0b) b) ) has been obtained for g s g s... g s7 = 0 and g s = g s =... g s7 =. We should also say a few words about the above-mentioned second- and twelfth-order matrices or to be more exact of the determinants and of these matrices. We shall start with. Scaling in the expressions for E imn and Ẽ imn i = 3) can be always performed so the the elements of the twelfthorder matrix should be bounded values for any m and ñ including the case when m ñ = ñ 0 = const or ñ m = m 0 = const or when both m and ñ ). If this is the case then analizing the system for m ñ = ñ 0 for ñ m = m 0 and for m ñ one can easily see that for all those cases 0 while when m = m 0 and ñ = ñ 0 by virtue of the uniqueness theorem we have 0. The same holds for 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) 0) ) ) with the only difference that different boundary conditions for x = x s and x = x s in the expressions for T ϕ ϕ... 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 µ η then the functions T ϕ ϕ ϕ 3 ψ ψ

17 SOME PROBLEMS OF THERMOELASTIC EQUILIBRIUM 783 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) 0b) b) and ) 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 3 = {0 < x < x 0 < x < x 0 < x 3 < x 3 } Ω 3 = {0 < x < x 0 < x < x x 3 < x 3 < x 3 }... Ω 3β = {0 < x < x 0 < x < x x 3β ) < x 3 < x 3β } contacting along the planes x 3 = x 3j where j =... β and β is the number of layers. Each layer has its elastic and thermal characteristics. With x = 0 and x = x conditions 0) are simultaneously satisfied for all layers while with x = 0 and x = x conditions ) hold. If the body occupies the domain Ω 3 then conditions 0) ) ) are satisfied on its faces with x 3 replaced by x 3β in ). On the contact planes x 3 = x 3j x 3 = x 3j is the contact plane of the j-th layer contacting the j+-th layer) the following conditions or T j T j+ = τ j x x ) λ T j j λ T j+ j+ = τ j x x ) u j u j+) = q j x x ) N 3j N 3j+) = Q j x x ) u j u j+) = q j x x ) N 3j N 3j+) = Q j x x ) u 3j u 3j+) = q j3 x x ) N 33j N 33j+) = Q j3 x x ) ω j ω j+) = q j4 x x ) M 3j M 3j+) = Q j4 x x ) ω j ω j+) = q j5 x x ) M 3j M 3j+) = Q j5 x x ) ω 3j ω 3j+) = q j6 x x ) M 33j M 33j+) = Q j6 x x ) 49) T j T j+ = τ j x x ) λ T j j λ T j+ j+ = τ j x x ) N 3j = Q j x x ) N 3j+) = Q j x x ) N 3j = Q j x x ) N 3j+) = Q j x x ) u 3j u 3j+) = q j3 x x ) N 33j N 33j+) = Q j3 x x ) ω j ω j+) = q j4 x x ) M 3j M 3j+) = Q j4 x x ) ω j ω j+) = q j5 x x ) M 3j M 3j+) = Q j5 x x ) M 33j = Q j6 x x ) M 33j+) = Q j6 x x ) 50) are defined. The right-hand sides of equalities 49) and 50) are defined functions and λ j and λ j+ are thermal conductivity coefficients. When we state the problem of thermoelastic equilibrium of a MRP we give expressions for the functions T j) ϕ j) ϕ j) ϕ j) 3 ψ j) ψ j) ψ j) 3 for the j-th layer bearing in mind conditions 0) ) and following the technique of

18 784 N. KHOMASURIDZE the previous section construct a system of β equations in β unknowns and another system of β equations in β 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 j u j 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 σ σ 3 σ 4 and η η naturally using high-quality measuring instruments. References. E. Cosserat and F. Cosserat Theórie des corps déformables. Hermann Paris C. A. Truesdell and R. A. Toupin The classical field theories. Handbuch der Physik Band 3 Teil Springer Berlin W. Nowacki Theory of elasticity. Translated from Polish into Russian) Mir Moscow A. C. Eringer Theory of micropolar elasticity. J. Math. Mech. 5966) W. Nowacki Couple stresses in the theory of thermoelasticity. Bull. Acad. Polon. Sér. Sci. Techn. 4966) No 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. 5 North-Holland Publ. Co. Amsterdam New York Oxford 979; Russian original: Nauka Moscow B. Carbonaro and R. Russo On some classical theorems in micropolar elasticity. Internat. J. Engrg. Sci. 3985) N. Khomasuridze Thermoelastic equilibrium of bodies in generalized cylindrical coordinates. Georgian Math. J. 5998) No Received ; revised.09.00) Author s address: I. Vekua Institute of Applied Mathematics I. Javakhishvili Tbilisi State University University St. Tbilisi Georgia