GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 1, 1998, 25-48

ON THREE-DIMENSIONAL DYNAMIC PROBLEMS OF GENERALIZED THERMOELASTICITY

T. BURCHULADZE AND D. BURCHULADZE

Abstract. Lord–Shulman’s system of partial differential equations of generalized thermoelasticity [1] is considered, in which the finite velocity of heat propagation is taken into account by introducing a relaxation time constant. General aspects of the theory of boundary value and initial-boundary value problems and representation of solutions by series and quadratures are considered using the method of a potential.

As one knows, in the classical theory of thermoelasticity the velocity of heat propagation is assumed to be infinitely large. However, in studying dynamic thermal stresses in deformable solid bodies, when the inertia terms in the equations of motion cannot be neglected, one must take into account that heat propagates not with an infinite but with a finite velocity; a heat flow arises in the body not instantly but is characterized by a finite relaxation time. Presently, there are at least two different generalizations of the classical theory of thermoelasticity: the first of them, Green–Lindsay’s generalization [1] is based on using two heat relaxation time constants; the other one, Lord–Shulman’s generalization [2] admits only one relaxation time constant. Both generalizations were developed as an attempt at explaining the paradox of the classical case that the heat propagation velocity is an infinite value.

In this paper, based on [3], we develop a general theory of solvability, as well as of construction of approximate and effective solutions of dynamic problems for the conjugate system of differential equations of thermoelasticity proposed by Lord and Shulman (L − S theory). Green–Lindsay’s theory (G − L-theory) is developed in [4, 5].

25

1991 Mathematics Subject Classification. 73B30, 73C35.

Key words and phrases. Generalized thermomechanics, dynamic and pseudooscillation boundary value problems, continuation of solution.

1072-947X/98/0100-0025$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Note that the use of relaxation time constants much complicates the basic systems of differential equations and essentially changes the type of equations.

Interesting historical information and bibliography on these issues can be found in [6], [7–11], [12].

1. Notation and Definitions

The following notation is introduced: ℝ^{n} is an n-dimensional Euclidean space; Ox_{1}x_{2}x_{3} is the Cartesian system in ℝ^{3}; x = (x_{k}), y = (y_{k}), … (k = \overline{1, 3}) are the points of R^{3}; |x − y| = [∑_{k=1}^{3}(x_{k} − y_{k})^{2}]^{1/2} denotes the distance between x and y; D ⊂ ℝ^{3} is a finite domain, S ≡ ∂D ∈ L_{1}(α_{0}), α_{0} > 0 (Lyapunov surface) [13]; \overline{D} ≡ D ∪ S; D^{−} ≡ ℝ^{3}\\overline{D} is an infinite domain; Z_{∞} ≡ {(x, t) : x ∈ D, t ∈ [0, ∞[} is a cylinder in ℝ^{4}, t is time, S_{∞} ≡

{(x, t) : x ∈ S, t ∈ [0, ∞[} is the lateral surface of Z_{∞};

n(y) = (n_{k}(y)), k = \overline{1, 3}, is the unit normal vector at the point y ∈ S directed outside D;

dx = dx_{1}dx_{2}dx_{3} is an element of volume, d_{y}S is an element of the area of S at the point y ∈ S;

δ_{jk} is Kronecker’s symbol; i = \sqrt{−1} is the imaginary unit; Δ ≡ Δ(\frac{∂}{∂x}) is a three-dimensional Laplace operator; v(x, t) = (v_{1}, v_{2}, v_{3}) = ∥v_{k}∥_{3×1} is a displacement vector (a one-column

matrix); v_{4}(x, t) is a temperature variation; T as a superscript denotes transposition; σ_{jk}(x, t) are components of elastic stresses; ε_{jk}(x, t) are strain components; A(\frac{∂}{∂x}) ≡ ∥μδ_{jk}Δ (\frac{∂}{∂x})+(λ+μ) \frac{∂^{2}}{∂x_{j}∂x_{k}} ∥_{3×3} is the matrix differential Lamé

operator;

T (\frac{∂}{∂x}, n) ≡ ∥μδ_{jk} \frac{∂}{∂n(x)} + λn_{j}(x) \frac{∂}{∂x_{k}} + μn_{k}(x) \frac{∂}{∂x_{j}} ∥_{3×3} is the matrix operator of elastic stresses;

an m-dimensional vector f = (f_{1}, …, f_{m}) = ∥f_{k}∥_{m×1} (real- or complexvalued) with norm |f| = [∑_{j=1}^{m} |f_{j}|^{2}]^{1/2} is treated as a m × 1 one-column matrix;

the matrix product is obtained by multiplying a row by a column;

if A = ∥A_{kj}∥_{m×m} is a m × m matrix, then A^{k} = ∥A_{jk}∥_{m×1} is the k-th vector-column of the matrix A and we will occasionally write A = ∥A^{1}, …, A^{m}∥ = {A^{1}, …, A^{m}}; obvioulsly, if A and B are m × m matrices and φ is a m × 1 matrix (vector), then

AB = {AB^{1}, …, AB^{m}}, Aφ = ∑_{k=1}^{m} A^{k}φ_{k},

ABφ = ∥ ∑_{(r,j)} A_{kr}B_{rj}φ_{j}∥_{m×1}, (AB^{T})^{T} φ = ∥ ∑_{(r,j)} A_{jr}B_{kr}φ_{j}∥_{m×1};

(f · φ) = f^{T} φ = ∑_{k=1}^{m} f_{k}φ_{k}; I = ∥δ_{jk}∥_{m×m} is the unit matrix.

Lord–Shulman’s fundamental system of equations and relations of a field for homogeneous isotropic linear thermoelasticity proposed is written as follows [1], [12], [15]:

1. A strain and displacement relation

ε_{jk}(x, t) = \frac{1}{2}(\frac{∂v_{j}(x, t)}{∂x_{k}} + \frac{∂v_{k}(x, t)}{∂x_{j}}), j, k = \overline{1, 3},

2. An equation of motion

∑_{j=1}^{3} \frac{∂σ_{jk}(x, t)}{∂x_{j}} + X_{k}(x, t) = ρ\frac{∂^{2}v_{k}(x, t)}{∂t^{2}} , k = \overline{1, 3},

3. An energy equation

− div q(x, t) + r(x, t) = C_{ε} \frac{∂v_{4}}{∂t} + (3λ + 2μ)αΘ_{0} \frac{∂}{∂t} div v,

4. Duhamel–Neuman’s law

σ_{jk}(x, t) = 2με_{jk}(x, t) + [λ div v − (3λ + 2μ)αv_{4}]δ_{jk}, j, k = \overline{1, 3},

5. A generalized heat conductivity equation

q(x, t) + τ_{t} \frac{∂q(x, t)}{∂t} = −k grad v_{4}(x, t),

where τ_{t} > 0 is the relaxation time constant.

In the above formulas q(x, t) = (q_{1}, q_{2}, q_{3}) is the heat flow vector, r(x, t) is the heat source; X_{1}, X_{1}, X_{3} are the given functions; ρ, λ and μ, α, k and C_{ε} denote respectively the density, Lamé moduli, thermal expansion coefficient, conductivity and specific heat capacity for zero deformation; Θ_{0} is the fixed uniform reference temperature (natural state temperature). These constants satisfy the natural restrictions [1], [12], [15]:

ρ > 0, k > 0, C_{ε} > 0, α > 0, 3λ + 2μ > 0, Θ_{0} > 0, τ_{t} > 0 (1)

In the classical case τ_{t} = 0.

For v, v_{4} the above defining relations 1–5 give the basic dynamic system of partial differential equations of generalized thermoelasticity (L–S theory)

A(\frac{∂}{∂x})v(x, t) − γ grad v_{4}(x, t) = ρ \frac{∂^{2}v}{∂t^{2}} − X(x, t),
Δv_{4}(x, t) = \frac{1}{κ} \frac{∂v_{4}}{∂t} + \frac{τ_{t}}{κ} \frac{∂^{2}v_{4}}{∂t^{2}} + η \frac{∂}{∂t} div v +

+ ητ_{t} \frac{∂^{2}}{∂t^{2}} div v − X_{4}(x, t),

(2)

where

γ = (3λ + 2μ)α, η = \frac{γΘ_{0}}{k} , \frac{1}{κ} = \frac{C_{ε}}{k}; ,

X = (X_{1}, X_{2}, X_{3}), X_{4} = \frac{r}{k} .

For the stress components we have

σ_{jk}(x, t) = (λ div v − γv_{4})δ_{jk} + μ(\frac{∂v_{j}}{∂x_{k}} + \frac{∂v_{k}}{∂x_{j}} ), j, k = \overline{1, 3}. (3)

We will consider two possible cases of dependence of v_{k}(x, t), k = \overline{1, 4}, on the time t:

I. v_{k}(x, t) = Re[e^{−ipt}u_{k}(x, p)] are stationary oscillations with frequency p > 0;

II. v_{k}(x, t) = \frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞} e^{ζt}u_{k}(x, ζ)dζ, ζ = σ + iq, σ > 0, is the representation by the Laplace–Mellin’s integral (the general classical case).

One can easily see that in both cases system (2)_{0} (homogeneous) can be reduced to the form (with respect to u_{k}(x, ω))

A(\frac{∂}{∂x})u(x, ω) − γ grad u_{4}(x, ω) + ρω^{2}u(x, ω) = 0,
Δu_{4}(x, ω) + \frac{iω}{κ_{t}} u_{4}(x, ω) + iωη_{t} div u(x, ω) = 0,

(4)

where

\begin{matrix}u = (u_{1}, u_{2}, u_{3}) = ∥u_{k}∥_{3×1}, \frac{1}{κ_{t}} = \frac{1}{κ} (1 − iωτ_{t}), \\ η_{t} = η(1 − iωτ_{t}). \end{matrix}  (5)

Note that ω = p > 0 in case I and ω = iζ = −q + iσ in case II.

2. Energy Identities and Green Formulae

Let V = (v, v_{4}) = ∥v_{k}∥_{4×1} be an arbitrary regular solution of the homogeneous dynamic system (2)_{0}:

V, \frac{∂V}{∂t} ∈ C^{1}(\overline{Z}_{∞}) ∩ C^{2}(Z_{∞}).

We introduce the notation

l_{t} ≡ 1 + τ_{t} \frac{∂}{∂t} , v_{t} ≡ l_{t}v, v_{4t} ≡ l_{t}v_{4}.  (6)

Rewrite system (2)_{0} as

A(\frac{∂}{∂x})v_{t} − γ grad v_{4t} = ρ\frac{∂^{2}v_{t}}{∂t^{2}} ,

Δv_{4} = \frac{1}{κ} \frac{∂v_{4t}}{∂t} + η \frac{∂}{∂t} div v_{t}.

(7)

Multiplying (7)_{1} by \frac{∂v_{t}^{T}}{∂t} gives

\frac{∂v_{t}^{T}}{∂t} [A(\frac{∂}{∂x})v_{t} − γ grad v_{4t}] = ρ \frac{∂v_{t}^{T}}{∂t} \frac{∂^{2}v_{t}}{∂t^{2}} = \frac{ρ}{2} \frac{∂}{∂t}|\frac{∂v_{t}}{∂t} |^{2}.  (8)

The following identity holds [13], [14]:

∫_{D} \{\frac{∂v_{t}^{T}}{∂t} [Av_{t} − γ grad v_{4t}] + E(\frac{∂v_{t}}{∂t} , v_{t}) − γv_{4t} div \frac{∂v_{t}}{∂t} \} dx =

= ∫_{S} \frac{∂v_{t}^{T}}{∂t} (T v_{t} − γnv_{4t}) dS,  (9)

where E(v, u) the well-known bilinear form of the theory of elasticity,

E(v, u) = \frac{3λ + 2μ}{3} div v div u + \frac{μ}{2} ∑_{(p≠q)} (\frac{∂v_{p}}{∂x_{q}} + \frac{∂v_{q}}{∂x_{p}} )(\frac{∂u_{p}}{∂x_{q}} + \frac{∂u_{q}}{∂x_{p}} ) +

+ \frac{μ}{3} ∑_{(p,q)} (\frac{∂v_{p}}{∂x_{p}} − \frac{∂v_{q}}{∂x_{q}} )(\frac{∂u_{p}}{∂x_{p}} − \frac{∂u_{q}}{∂x_{q}} ).

It is easy to verify that E(\frac{∂v_{t}}{∂t} , v_{t}) = \frac{1}{2} \frac{∂}{∂t}E(v_{t}, v_{t}) and by virtue of (8) formula (9) takes the form

\frac{d}{dt} ∫_{D} [\frac{ρ}{2}|\frac{∂v_{t}}{∂t} |^{2} + \frac{1}{2}E(v_{t}, v_{t})] dx − γ ∫_{D} v_{4t} div \frac{∂v_{t}}{∂t} dx =

= ∫_{S} \frac{∂v_{t}^{T}}{∂t} HV_{t} dS,  (10)

where

HV_{t} = T v_{t} − γnv_{4t}.

After multiplying (7)_{2} by v_{4t}, integrating with respect to D and applying the Green formula to the Laplace operator we obtain

−γ ∫_{D} v_{4t} \frac{∂}{∂t} div v_{t} dx = \frac{γ}{ηκ} ∫_{D} v_{4t} \frac{∂v_{4t}}{∂t} dt − \frac{γ}{η} ∫_{D} v_{4t}Δv_{4} dx =

= \frac{γ}{2κη} \frac{d}{dt} ∫_{D} (v_{4t})^{2} dx + \frac{γ}{η} ∫_{D} [ grad v_{4t}]^{T} grad v_{4} dx −

−\frac{γ}{η} ∫_{S} v_{4t} \frac{∂v_{4}}{∂n} dS.  (11)

By taking the identity

[ grad v_{4t}]^{T} grad v_{4} = | grad v_{4}|^{2} + \frac{τ_{t}}{2} \frac{∂}{∂t} | grad v_{4}|^{2}

into account and substituting (11) into (10) we finally have

\frac{d}{dt} ∫_{D} [\frac{ρ}{2} |\frac{∂v_{t}}{∂t} |^{2} + \frac{1}{2} E(v_{t}, v_{t}) + \frac{γ}{2κη} v_{4t}^{2} +

+\frac{γτ_{t}}{2η} | grad v_{4}|^{2}] dx + \frac{γ}{η} ∫_{D} | grad v_{4}|^{2} dx =

= ∫_{S} [\frac{∂v_{t}^{T}}{∂t} HV_{t} + \frac{γ}{η} v_{4t} \frac{∂v_{4}}{∂n} ] dS ≡ M_{S}(V).  (12)

Introducing the notation

J(t) = \frac{1}{2} ∫_{D} [ρ|\frac{∂v_{t}}{∂t} |^{2} + E(v_{t}, v_{t}) + \frac{γ}{κη} v_{4t}^{2} + \frac{γτ_{t}}{η} | grad v_{4}|^{2}] dx. (13)

we rewrite (12) as

\frac{dJ(t)}{dt} + \frac{γ}{η} ∫_{D} | grad v_{4}|^{2} dx = M_{S}(V).  (14)

Note that formula (14) holds for the infinite domain D^{−} ≡ ℝ^{3}\\overline{D} as well provided that the vector V (x, t) satisfies the following condition at infinity:

|\frac{∂^{|α|}V}{∂x_{1}^{α_{1}}∂x_{2}^{α_{2}}∂x_{3}^{α_{3}}∂t^{α_{4}}} | ≤ \frac{const ·e^{σ_{0}t}}{1 + |x|^{1+|α|}} , |α| = \overline{0, 3}, σ_{0} ≥ 0,

where α = (α_{1}, α_{2}, α_{3}, α_{4}) is the multi-index, |α| = ∑_{k=1}^{4} α_{k}.

Let u ∈ C^{1}(\overline{D}) ∩ C^{2}(D), v ∈ C^{1}(\overline{D}), u_{4} ∈ C^{1}(\overline{D}), A(\frac{∂}{∂x})u ∈ L_{1}(D), S ∈ L_{1}(α_{0}), α_{0} > 0. Then the identity [14]

∫_{D} [v^{T} (Au − γ grad u_{4}) + E(v, u) − γu_{4} div v] dx = ∫_{S} v^{T} HU dS (15)

holds. In particular, if U = (u, u_{4}) is a regular solution of system (4) and \overline{U} = (\overline{u}, \overline{u}_{4}) is the comple-conjugate vector, then by simple transformations, from (15) and (14)_{0} we obtain

∫_{D} [ − ρω^{2}|u|^{2} + E(\overline{u}, u) + \frac{γ}{i\overline{ω} \overline{η}_{t}} | grad u_{4}|^{2} + \frac{γ}{\overline{κ}_{t}\overline{η}_{t}} |u_{4}|^{2}] dx =

= ∫_{S} [\overline{u}^{T} HU + \frac{γ}{i\overline{ω} \overline{η}_{t}} u_{4} \frac{∂\overline{u}_{4}}{∂n} ] dS.  (16)

Let ω = p > 0 be the real parameter. In (16) we pass over to the complexconjugate expression and subtract the result from (16); taking into account that Im E(\overline{u}, u) = 0,

\frac{γ}{iω\overline{η}_{t}} − \frac{γ}{−iωη_{t}} = \frac{γ}{iω|η_{t}|^{2}} 2 Re η_{t} = \frac{2γη}{iω|η_{t}|^{2}} ,
\frac{γ}{\overline{κ}_{t}\overline{η}_{t}} − \frac{γ}{κ_{t}η_{t}} = \frac{γ}{|κ_{t}|^{2}|η_{t}|^{2}} 2i Im(κ_{t}η_{t}) = 0,

we obtain

\frac{2γη}{iω|η_{t}|^{2}} ∫_{D} | grad u_{4}|^{2} dx = ∫_{S} \{[\overline{u}^{T} HU − u^{T} H\overline{U}] +

+ \frac{γ}{iω\overline{η}_{t}} u_{4} \frac{∂\overline{u}_{4}}{∂n} + \frac{γ}{iωη_{t}} \overline{u}_{4} \frac{∂u_{4}}{∂n} \} dS.  (17)

Let now ω = iζ = −q + iσ, σ > 0, which means that we are considering the case of pseudo-oscillations. This case is an auxiliary means in studying the general nonstationary problem. Formula (16) takes the form

∫_{D} [ − ρζ^{2}|u|^{2} + E(\overline{u}, u) + \frac{γ}{\overline{ζ} \overline{η}_{t}} | grad u_{4}|^{2} + \frac{γ}{\overline{κ}_{t}\overline{η}_{t}} |u_{4}|^{2}] dx = N_{S}(U) ≡

≡ ∫_{S} [\overline{u}^{T} HU + \frac{γ}{\overline{ζ} \overline{η}_{t}} u_{4} \frac{∂\overline{u}_{4}}{∂n} ] dS,  (18)

where

\frac{1}{κ_{t}} = \frac{1}{κ} (1 + τ_{t}ζ), η_{t} = η(1 + τ_{t}ζ).

Denote by (\overline{18}) the complex-conjugate expression and obtain the sum \overline{ζ}N_{S}(U)+ζ\overline{N_{S}(U)}; since \overline{ζ}ζ^{2}+ζ\overline{ζ}^{2} = 2σ|ζ|^{2}, \overline{ζ}+ζ = 2σ, \frac{γ}{\overline{η}_{t}} + \frac{γ}{η_{t}} = \frac{2γη(1+στ_{t})}{|η_{t}|^{2}},

\frac{\overline{ζ}γ}{\overline{κ}_{t}\overline{η}_{t}} + \frac{ζγ}{κ_{t}η_{t}} = \frac{2γσ}{κη}, we finally have

∫_{D} [2σ|ζ|^{2}ρ|u|^{2} + 2σE(\overline{u}, u) + \frac{2γη(1 + στ_{t})}{|η_{t}|^{2}} | grad u_{4}|^{2} + \frac{2γσ}{κη} |u_{4}|^{2}] dx =

= \overline{ζ}N_{S}(U) + ζ\overline{N_{S}(U)}.  (19)

Another generalized Green formula is helpful in investigating boundary value and initial-boundary problems by the method of a potential. To this end, we introduce the differential (matrix) operators

L(\frac{∂}{∂x}) ≡ ∥ \begin{matrix}A(\frac{∂}{∂x}) _{3×3}, & −γ grad_{x} _{3×1} \\ 0 _{1×3}, & Δ \end{matrix} ∥_{4×4} ,

L^{0}(\frac{∂}{∂x}, \frac{∂}{∂t}) ≡ ∥ \begin{matrix}ρδ_{jk} \frac{∂^{2}}{∂t^{2}} _{3×3}, & 0 _{3×1} \\ η \frac{∂}{∂t}(l_{t} grad_{x}) _{1×3}, & \frac{1}{κ} l_{t} \frac{∂}{∂t} \end{matrix} ∥_{4×4} ,

L(\frac{∂}{∂x}, \frac{∂}{∂t}) ≡ L(\frac{∂}{∂x}) − L^{0}(\frac{∂}{∂x}, \frac{∂}{∂t}).

Obviously, the basic nonstationary homogeneous system (2)_{0} can be rewritten as (V = (v, v_{4}))

L(\frac{∂}{∂x}, \frac{∂}{∂t})V (x, t) = 0,

while system (4) as (U = (u, u_{4}))

L(\frac{∂}{∂x}, −iω)U = 0.

Clearly,

L^{0}(\frac{∂}{∂x}, −iω) ≡ ∥ \begin{matrix}−ρδ_{jk}ω^{2} _{3×3}, & 0 _{3×1} \\ −iωη_{t} grad_{x} _{1×3}, & \frac{iω}{κ_{t}} \end{matrix} ∥_{4×4} ,

L^{0}(\frac{∂}{∂x}, ζ) ≡ ∥ \begin{matrix}ρδ_{jk}ζ^{2} _{3×3}, & 0 _{3×1} \\ ηζl_{ζ} grad_{x} _{1×3}, & \frac{ζ}{κ} l_{ζ} \end{matrix} ∥_{4×4} , l_{ζ} = 1 + τ_{t}ζ.

Let L̃(\frac{∂}{∂x}, −iω) be the (Lagrange-)conjugate operator of L(\frac{∂}{∂x}, −iω),

L̃(\frac{∂}{∂x}, −iω) ≡ ∥ \begin{matrix}A(\frac{∂}{∂x}) + ρδ_{jk}ω^{2} _{3×3}, & −iωη_{t} grad_{x} _{3×1} \\ γ grad_{x} _{1×3}, & Δ + \frac{iω}{κ_{t}} \end{matrix} ∥_{4×4} ,

For the regular in D vectors U = (u, u_{4}) and Ũ = (ũ, ũ_{4}) simple transformations give

U^{T} L̃(\frac{∂}{∂x}, −iω)Ũ − Ũ^{T} L(\frac{∂}{∂x}, −ω)U =

= [u^{T} A(\frac{∂}{∂x})ũ − u^{T} A(\frac{∂}{∂x})u] +

+[u_{4}Δũ_{4} − ũ_{4}Δu_{4}] + γ div(u_{4}u^{T}) − iωη_{t} div(ũ_{4}u^{T}).

Therefore, by the (Gauss–Ostrogradski’s) divergence formula we finally obtain

∫_{D} [U^{T} L̃(\frac{∂}{∂x}, −iω)Ũ − Ũ^{T} L(\frac{∂}{∂x}, −iω)U] dx =

= ∫_{S} [U^{T} R̃Ũ − Ũ^{T} RU] dS,  (20)

where

RU ≡ (HU, \frac{∂u_{4}}{∂n} ), R̃U ≡ (H̃U, \frac{∂u_{4}}{∂n} ),

HU ≡ T u − γnu_{4}, H̃U ≡ T u − iωη_{t}nu_{4}.

Let further

PU ≡ (HU, −u_{4}), P̃U ≡ (H̃U, −u_{4}), QU ≡ (u, \frac{∂u_{4}}{∂n} ).

Now formula (20) can be represented as

∫_{D} [U^{T} L̃(\frac{∂}{∂x}, −iω)Ũ − Ũ^{T} L(\frac{∂}{∂x}, −iω)U] dx =

= ∫_{S} [(QU)^{T} P̃Ũ − (QŨ)^{T} PU] dS.  (21)

3. Decomposition of a Regular Solution. The Properties of Characteristic λ-parameters

In this section we will prove several theorems and lemmas to be essentially used in constructing the general theory of solvability of boundary value problems.

Theorem 1. In the domain D ⊂ ℝ^{3} any solution U = (u, u_{4}) of the homogeneous system (4) of the class U ∈ ℂ^{2}(D) belongs to the class U ∈ ℂ^{∞}(D) and admits a representation

U = (u, u_{4}) = (u^{(1)} + u^{(2)}, u_{4}), rot u^{(1)} = 0, div u^{(2)} = 0,

(Δ + λ_{1}^{2})(Δ + λ_{2}^{2}) (\begin{matrix}u^{(1)} \\ u_{4} \end{matrix}) = 0, (Δ + λ_{3}^{2})u^{(2)} = 0, λ_{3}^{2} = \frac{ρω^{2}}{μ} ,

(22)

while the characteristic constants λ_{k}^{2}(ω), k = 1, 2, are the roots of the quadratic equation

z^{2} − [ρ_{0}ω^{2}(1 + ε_{t}) + \frac{iω}{κ} (1 + ε)]z + [ω^{4} \frac{ρ_{0}}{κ} τ_{t} + \frac{iω}{κ} ρ_{0}ω^{2}] = 0, (23)

where

ρ_{0} = ρ(λ + 2μ)^{−1}, ε_{t} = \frac{τ_{t}}{κρ_{0}} (1 + ε), ε = \frac{κγη}{λ + 2μ} (ε ≪ 1).

Proof. The fact that the vector U belongs to the class ℂ^{∞}(D) is established by means of the formula for a general representation of a regular solution of system (4) using the matrix of fundamental solutions of the considered differential operator of generalized elastothermoelasticity (see [14], [15]).

Since Δ ≡ grad div − rot rot, formula (4) implies

u = u^{(1)} + u^{(2)},
u^{(1)} = grad ( − \frac{λ + 2μ}{ρω^{2}} div u + \frac{γ}{ρω^{2}} u_{4}), u^{(2)} = rot (\frac{μ}{ρω^{2}} rot u). (24)

Obviously, rot u^{(1)} = 0, div u^{(2)} = 0.

Taking into account the value λ_{3}^{2}, we find from (24) that

(Δ + λ_{3}^{2})u^{(2)} = 0.

We have

(λ + 2μ)Δ div u^{(1)} + ρω^{2} div u^{(1)} − γΔu_{4} = 0.  (25)

By putting the value

div u ≡ div u^{(1)} = −\frac{1}{iωη_{t}} Δu_{4} − \frac{1}{κ_{t}η_{t}} u_{4}

from eq. (4)_{2} into (25) we obtain

(Δ + λ_{1}^{2})(Δ + λ_{2}^{2})u_{4} = 0.  (26)

Hence it easily follows that (Δ + λ_{1}^{2})(Δ + λ_{2}^{2}) div u^{(1)} = 0 and by (24) we finally obtain

(Δ + λ_{1}^{2})(Δ + λ_{2}^{2})u^{(1)} = 0.

Theorem 2. If the vector-functions v^{(k)}(x) = ∥v_{j}^{(k)}∥_{3×1}, u^{(2)}(x) = ∥u_{j}^{(2)}∥_{3×1} and the scalar functions v_{k}(x), k = 1, 2, are regular solutions of the equations

(Δ + λ_{k}^{2}) (\begin{matrix}v^{(k)} \\ v_{k} \end{matrix}) = 0, k = 1, 2, (Δ + λ_{3}^{2})u^{(2)} = 0, div u^{(2)} = 0,

and are related through the formula

v^{(k)}(x) = γ[ρω^{2} − λ_{k}^{2}(λ + 2μ)]^{−1} grad v_{k}, k = 1, 2,

then the vector

U(x) = (u^{(1)}(x) + u^{(2)}(x), u_{4}(x)),
u^{(1)}(x) = v^{(1)}(x) + v^{(2)}(x), u_{4}(x) = v_{1}(x) + v_{2}(x),

is a regular solution of system (4).

Proof. By using the conditions of the theorem and performing some simple calculations we obtain

A(\frac{∂}{∂x})u − γ grad u_{4} + ρω^{2}u =

= μΔu + (λ + μ) grad div u − γ grad u_{4} + ρω^{2}u = [μΔu^{(2)} + ρω^{2}u^{(2)}] +

+[(λ + 2μ)Δ(v^{1} + v^{(2)}) + ρω^{2}(v^{(1)} + v^{(2)}) − γ grad(v_{1} + v_{2})] =
= {[ρω^{2} − (λ + 2μ)λ_{1}^{2}]v^{(1)} − γ grad v_{1}} +

+{[ρω^{2} − (λ + 2μ)λ_{2}^{2}]v^{(2)} − γ grad v_{2}} = 0.

Similarly, we have

Δu_{4} + \frac{iω}{κ_{t}} u_{4} + iωη_{t} div u =

= Δ(v_{1} + v_{2}) + \frac{iω}{κ_{t}} (v_{1} + v_{2}) + iωη_{t} div(v^{(1)} + v^{(2)}) =

= [(\frac{iω}{κ_{t}} − λ_{1}^{2})v_{1} + iωη_{t} div v^{(1)}] + [(\frac{iω}{κ_{t}} − λ_{2}^{2})v_{2} + iωη_{t} div v^{(2)}] =

= [\frac{iω}{κ_{t}} − λ_{1}^{2} − \frac{γλ_{1}^{2}iωη_{t}}{ρω^{2} − λ_{1}^{2}(λ + 2μ)}]v_{1} + [\frac{iω}{κ_{t}} − λ_{2}^{2} − \frac{γλ_{2}^{2}iωη_{t}}{ρω^{2} − λ_{2}^{2}(λ + 2μ)}]v_{2} =

= ∑_{k=1}^{2} \frac{v_{k}}{ρω^{2} − λ_{k}^{2}(λ + 2μ)} [\frac{iω}{κ_{t}} ρω^{2} − \frac{iω}{κ_{t}} (λ + 2μ)λ_{k}^{2} −

−λ_{k}^{2}ρω^{2} + λ_{k}^{4}(λ + 2μ) − γλ_{k}^{2}iωη_{t}] =

= ∑_{k=1}^{2} \frac{v_{k}}{ρω^{2} − λ_{k}^{2}(λ + 2μ)} \{\frac{iω}{κ_{t}} ρω^{2} + λ_{k}^{4}(λ + 2μ) −

−λ_{k}^{2}[\frac{iω}{κ_{t}} (λ + 2μ) + ρω^{2} + iωγη_{t}]\} =

= ∑_{k=1}^{2} \frac{v_{k}}{ρω^{2} − λ_{k}^{2}(λ + 2μ)} [\frac{iω}{κ_{t}} ρω^{2} + λ_{k}^{4}(λ + 2μ) − λ_{k}^{2}(λ + 2μ)(λ_{1}^{2} + λ_{2}^{2})] =

= ∑_{k=1}^{2} \frac{v_{k}}{ρω^{2} − λ_{k}^{2}(λ + 2μ)} [\frac{iω}{κ_{t}} ρω^{2} − (λ + 2μ)λ_{1}^{2}λ_{2}^{2}] ≡ 0.

Note that the parameter ε = \frac{κγη}{λ+2μ} is a physical constant and for most real bodies ε ≪ 1 [16]. When ε = 0, the deformation and temperature fields get completely separated and we then have a separate (not conjugate) theory by virtue of which (23) implies

λ_{1}^{2} = \frac{iω}{κ} + \frac{ω^{2}τ_{t}}{κ} , λ_{2}^{2} = \frac{ρω^{2}}{λ + 2μ} , λ_{3}^{2} = \frac{ρω^{2}}{μ} .

Clearly, this case is of no particular interest for us as the investigation of boundary value problems actually reduces to problems of the classical theory of elasticity and thermal conductivity taken separately. In what follows it will therefore be assumed that ε > 0.

In investigating boundary value problems for system (2) and (4) much importance is attached to the parameters λ_{k}(ω) depending on the coefficients of the initial systems of differential equations by formula (23). These properties are described by the following lemmas.

Lemma 1. If ω = p > 0, ε > 0 (stationary oscillation), then λ_{1}^{2} and λ_{2}^{2} are complex numbers.

Proof. If we assume that the opposite is true, then eq. (23) should have the real root z = α, which in turn gives

α = \frac{ρ_{0}ω^{2}}{1 + ε} , α^{2} − ρ_{0}ω^{2}(1 + ε_{t})α + ω^{4} \frac{ρ_{0}}{κ} τ_{t} = 0.

Hence we readily obtain the contradiction ε = 0. Note that if the relations

τ_{t} = \frac{κρ}{λ + 2μ} · \frac{1 − ε}{(1 + ε)^{2}} , ω = \frac{\frac{1}{κ} (1 + ε)}{\sqrt{ρ_{0}^{2}(1 + ε_{t})^{2} − \frac{4ρ_{0}τ_{t}}{κ}}} ,

are not fulfilled simultaneously (which will be assumed in what follows), then λ_{1}^{2} ≠ λ_{2}^{2}.

Thus, for stationary oscillations, λ_{1} and λ_{2} are the complex numbers admitting two signs, and therefore without loss of generality it can always be assumed that Im λ_{k} > 0, k = 1, 2.

Lemma 2. If ε > 0, ω = iζ = −q + iσ, σ > 0, then in the complex half-plane

Π_{σ_{0}^{∗}} = {ζ : Re ζ > σ_{0}^{∗}}

where

σ_{0}^{∗} = max\{σ_{ε}, 0\}, σ_{ε} = \frac{ρ_{0}κ[2 − (1 + ε_{t})(1 + ε)]}{[ρ_{0}κ + τ_{t}(ε − 1)]^{2} + 4ετ_{t}^{2}}

the parameters λ_{k}, k = \overline{1, 3}, possess the properties:

1. Im λ_{k} > 0;

2. λ_{1} ≠ λ_{2} ≠ λ_{3};

3. λ_{k} = λ_{k}(ζ) are the analytic functions ζ which admit estimates λ_{k}(ζ) = O(|ζ|) as ζ → ∞.

Proof. For λ_{3}(ζ) = \sqrt{\frac{ρ}{μ}} · iζ these properties are obvious in the half-plane

Re ζ > 0. We will show that they are valid for λ_{1}(ζ) and λ_{2}(ζ) as well. By (23) we have

z_{1,2} = −\frac{ρ_{0}(1 + ε_{t})ζ + (1 + ε)/κ}{2} ± \sqrt{d},  (∗)

where

z_{k} = \frac{λ_{k}^{2}}{ζ} , d = \frac{1}{4}[ρ_{0}(1 + ε_{t})ζ + (1 + ε)/κ]^{2} − \frac{ρ_{0}}{κ} (1 + τ_{t}ζ)ζ.

One can readily see that the zeros of the discriminant d(ζ) are

ζ_{1,2} = σ_{ε} ± i \frac{\sqrt{|d_{1}|}}{\frac{1}{κ^{2}} [ρ_{0}κ + τ_{t}(ε − 1)]^{2} + 4ετ_{t}^{2}} ,

where

d_{1} = [\frac{ρ_{0}}{κ} (1 + ε_{t})(1 + ε) − \frac{2ρ_{0}}{κ} ] − \frac{(1 + ε)^{2}}{κ^{2}} [ρ_{0}^{2}(1 + ε_{t})^{2} − \frac{4ρ_{0}}{κ} τ_{t}] =

= −\frac{4ρ_{0}^{2}}{κ^{2}} ε < 0.

Hence it is clear that if Re ζ > σ_{0}^{∗}, then the radical in (∗) cannot have branching points (d ≠ 0) and, after making an appropriate choice of branches, one can regard z_{k}(ζ), k = 1, 2, as analytic functions of ζ, z_{1} ≠ z_{2}, z_{k}(ζ) = O(|ζ|). Therefore λ_{k}^{2}(ζ) = z_{k}(ζ)ζ are the analytic functions of ζ and λ_{k}^{2}(ζ) = O(|ζ|^{2}), and, since z_{k} ≠ 0, ζ ≠ 0, it can be assumed that λ_{k}(ζ) are the analytic functions of ζ and λ_{k}(ζ) = O(|ζ|), k = 1, 2. Finally, z_{k}(ζ), k = 1, 2, as analytic functions of ζ in the half-plane Π_{σ_{0}^{∗}}, cannot coincide with expressions of form |a_{k}|^{2}\overline{ζ}, which are not analytic functions of ζ. Therefore in the half-plane Π_{σ_{0}^{∗}} λ_{k}^{2}(ζ) are either complex- or realvalued negative numbers [4] and it can always be assumed that Im_{k}(ζ) > 0, k = 1, 2.

The investigation of boundary value problems rests essentially on the above theorems and lemmas.

4. Uniqueness Theorems. Fundamental Solutions and Principal Potentials

Like in the classical case [13], [14], the constructed mathematical methods enable one to perform a complete mathematical analysis of boundary value problems in all cases: stationary oscillations, pseudo-oscillations and general dynamics.

We will briefly discuss the solution uniqueness. By virtue of Theorem 1 and Lemma 1 we introduce

Definition 1. A regular in the infinite domain D^{−} ≡ ℝ^{3}\\overline{D} solution U = (u^{(1)}+u^{(2)}, u_{4}) of the homogeneous oscillation equation (4) (ω = p > 0) satisfies the thermoelastic radiation condition at infinity if the following asymptotic estimates hold:

\begin{matrix}(\begin{matrix}u^{(1)}(x) \\ u_{4}(x) \end{matrix}) = O(|x|^{−1}), \frac{∂}{∂x_{k}} (\begin{matrix}u^{(1)}(x) \\ u_{4}(x) \end{matrix}) = O(|x|^{−2}), k = \overline{1, 3}, \\ u^{(2)}(x) = O(|x|^{−1}), \frac{∂u^{(2)}(x)}{∂|x|} − iλ_{3}u^{(2)}(x) = o(|x|^{−1}), \end{matrix} (27)

where \frac{∂}{∂|x|} is the derivative along the radius-vector of the point x.

Theorem 3. A regular in D^{−} vector U = (u, u_{4}) ∈ C^{1}(\overline{D}^{−}) ∩ C^{2}(D^{−}), which is a solution of the homogeneous oscillation system (4) (ω = p > 0) satisfying at infinity the radiation condition and admissible homogeneous conditions on S [14], is identically zero in D^{−}.

The proof immediately follows from identity (17) written for the domain D_{R} ≡ D^{−} ∩ {x : ℝ^{3} : |x| < R}, where R is an arbitrary sufficiently large real number (see [14], pp. 242–243). The uniqueness theorems do not hold for the internal oscillation problems (for the finite domain D) because of the emergence of a discrete eigenfrequency spectrum in some boundary value problems of a metaharmonic equation. Such an investigation is carried out similarly to the case of classical thermoelasticity assuming that in all homogeneous internal problems u_{4}(x) ≡ const [13], [14].

Theorem 4. A regular in D vector U ∈ C^{1}(\overline{D}) ∩ C^{2}(D), which is a solution of the homogeneous pseudo-oscillation system (4) (ω = iζ, Re ζ > 0) satisfying one of the admissible homogeneous boundary conditions on S, is identically zero.

The proof immediately follows from identity (19) (see [14], p.245). Note that Theorem 4 remains valid for the infinite domain D^{−} as well (external problems) if the solution U = (u, u_{4}) satisfies at infinity the asymptotic conditions

U = O(|x|^{−1}), \frac{∂U}{∂x_{k}} = O(|x|^{−2}), k = \overline{1, 3}.  (28)

Theorem 5. A regular in the cylinder Z_{∞} vector V = (v, v_{4}) ∈ C^{1}(\overline{Z}_{∞}) ∩ C^{2}(Z_{∞}), which is a solution of the homogeneous nonstationary system (2)_{0} satisfying the homogeneous initial conditions V |_{t=0} = \frac{∂V}{∂t} |_{t=0} = 0 and the corresponding admissible homogeneous boundary conditions on S_{∞}, is identically zero.

Proof. (14) implies M_{S}(V) ≡ 0 by virtute of the boundary condition J′(t) ≤ 0. Therefore J(t) is a nonincreasing function t ≥ 0. On the other hand, one easily obtains J(0) = 0 and therefore J(t) ≤ 0 for t ≥ 0. But by (13) J(t) ≥ 0, i.e., J(t) = 0, which in turn gives v_{t}(x, t) = 0, v_{4t}(x, t) = 0, t ≥ 0, x ∈ D. Thus we finally have ∀x ∈ D, t ≥ 0 : τ_{t} \frac{∂V}{∂t} + V = 0, i.e., V (x, t) = C(x)e^{− \frac{1}{τ_{t}} t}, and by virtue of the initial condition V |_{t=0} = 0 we will have v = 0, v_{4} = 0. The theorem also holds for the external (for the infinite domain D^{−}) initial-boundary problems.

As is well-known, for problems (both stationary and general dynamic) of the conjugate theory of classical thermoelasticity there presently exists a thoroughly developed mathematical theory of solvability which is as general as the available theory for problems of classical elasticity. The final results obtained in this direction are presented in [13], [14], [15].

Further investigations based on the above-presented results showed that the same methods (the method of a potential, theory of multidimensional singular integral equations, Fourier and Laplace–Mellin integral transformation and others) enable one to construct a general mathematical theory of solvability of stationary and nonstationary problems also for problems of the conjugate theory of the nonclassical theory of thermoelasticity by Green–Lindsay (G–L theory) and Lord–Shulman (L–S theory) [3], [4], [5], [15],

In our studies we mostly used the matrix of fundamental solutions of the considered differential operators L(\frac{∂}{∂x}, −iω) and other singular solutions. These solutions are constructed explicitly in terms of elementary functions, which is especially important for obtaining effective and approximate solutions.

The matrix of fundamental solutions has the form [14, 17]:

Φ(x, −iω) = ∥Φ_{jk}(x, −ω)∥_{4×4},

Φ_{kj}(x, −iω) = ∑_{l=1}^{3} \{(1 − δ_{k4})(1 − δ_{j4})(\frac{δ_{kj}}{2πμ} δ_{3l} − α_{l} \frac{∂^{2}}{∂x_{k}∂x_{l}} ) +

+β_{l}[iωη_{t}δ_{k4}(1 − δ_{j4}) \frac{∂}{∂x_{j}} − γδ_{j4}(1 − δ_{k4}) \frac{∂}{∂x_{k}} ] + δ_{k4}δ_{j4}γ_{l}\} \frac{e^{iλ_{l}|x|}}{|x|} ,

k_{1}^{2} = \frac{ρω^{2}}{λ + 2μ} , β_{l} = \frac{(−1)^{l}(δ_{1l} + δ_{2l})}{2π(λ + 2μ)(λ_{2}^{2} − λ_{1}^{2})} , ∑_{l=1}^{3} β_{l} = 0,

γ_{l} = \frac{(−1)^{l}(λ_{l}^{2} − k_{1}^{2})(δ_{1l} + δ_{2l})}{2π(λ_{2}^{2} − λ_{1}^{2})} , ∑_{l=1}^{3} γ_{l} = 0,

α_{l} = \frac{(−1)^{l}(1 − \frac{iω}{κ_{t}} λ_{l}^{−2})(δ_{1l} + δ_{2l})}{2π(λ + 2μ)(λ_{2}^{2} − λ_{1}^{2})} − \frac{δ_{3l}}{2πρω^{2}} , ∑_{l=1}^{3} α_{l} = 0.

The behavior of the fundamental matrix largely depends on the property of the characteristic parameters λ_{k}, k = \overline{1, 3}.

The relation Φ̃(x, −iω) = Φ^{T} (−x, −iω) holds, where Φ̃(x, −iω) is the matrix of fundamental solutions of the conjugated operator L̃(\frac{∂}{∂x}, −iω).

Note that the matrix of fundamental solutions of the static operator L(\frac{∂}{∂x}) (ω = 0) has the form:

Φ(x) = \frac{λ + μ}{4πμ(λ + 2μ)} ×

× ∥ \begin{matrix}\frac{2(λ + 2μ)}{λ + μ} \frac{δ_{jk}}{|x|} − \frac{∂^{2}|x|}{∂x_{j}∂x_{k}} _{3×3}, & \frac{μγ}{λ + μ} grad |x| _{3×1} \\ 0 _{1×3}, & \frac{2(λ+2μ)μ}{λ+μ} · \frac{1}{|x|} \end{matrix} ∥ .

We have

Φ(x, −iω) − Φ(x) = const +O(|x|),
\frac{∂}{∂x_{p}} [Φ(x, −iω) − Φ(x)] = O(1), p = \overline{1, 3},
\frac{∂^{2}}{∂x_{p}∂x_{q}} [Φ_{kj}(x, −iω) − Φ_{kj}(x)] = O(|x|^{−1}), k, j = \overline{1, 3},
k = j = 4, p, q = \overline{1, 3},
\frac{∂^{2}}{∂x_{p}∂x_{q}} [Φ_{kj}(x, −iω) − Φ_{kj}(x)] = O(1), k = \overline{1, 3}, j = 4,
and k = 4, j = \overline{1, 3}.

Fundamental solutions are used in a usual manner to construct new singular solutions of the equations which serve as measures of the generalized potentials of the considered boundary value problems.

Principal potentials for system (4) are written as:

V (x, φ) = ∫_{S} Φ(x − y, −iω)φ(y) d_{y}S (the simple-layer potential);

W (x, φ) = ∫_{S} [R̃(\frac{∂}{∂y} , n)Φ̃(y − x, −iω)]^{T} φ(y) d_{y}S

(the double-layer potential);

M^{(1)}(x, φ) = ∫_{S} [P̃(\frac{∂}{∂y} , n)Φ̃(y − x, −iω)]^{T} φ(y) d_{y}S,

M^{(2)}(x, φ) = ∫_{S} [Q(\frac{∂}{∂y} , n)Φ̃(y − x, −iω)]^{T} φ(y) d_{y}S

the mixed-type potentials;

U(ψ)(x) ≡ U(x, ψ) = ∫_{D} Φ(x − y, −iω)ψ(y) dy (the volume potential).

By virtue of the above constructed mathematical tool the investigation of these potentials, as well as their application in the theory of boundary value problems are accomplished in the same way as in classical thermoelasticity [13], [14]. Therefore all the results obtained in [14] on classical thermoelasticity for oscillations and pseudo-oscillations remain valid in the considered case as well. As for nonstationary problems of general dynamics, they demand a separate investigate, since, for instance, the fourth equation of system (2) is not a parabolic but a hyperbolic one with respect to v_{4}(x, t). The “compatibility” and other conditions will be changed accordingly [14]. However we do not intend to dwell on this topic here.

Remark. By Theorem 1 and Lemma 1 an explicit expression of the fundamental matrix Φ(x−y, −iω) containing expressions of the form exp(iλ_{k}|x− y|), k = \overline{1, 3}, implies that the radiation conditions (27) are satisfied by each column- vector of this matrix, while for pseudo-oscillations condition (28) is satisfied due to Lemma 2. Thus the constructed potentials possess the required properties at infinity. We note further that using (20) and (21) one can derive formulas for general integral representations of an arbitrary regular vector, which can in turn be used as a basis for investigating boundary value problems and as a source of new estimates.

5. Completeness Theorem

We will prove here some completeness theorems for the definite sets of vector-functions generated by the matrix of fundamental solutions Φ(x,−iω). In doing so, a sufficiently general situation will be considered. These theorems underlie the construction of approximate solutions of the corresponding boundary value problems by the Riesz–Fisher–Kupradze method (the method of discrete singularities).

Let: D_{k} ⊂ ℝ^{3} be a finite domain with the boundary S_{k} ∈ L_{2}(α_{0}), α_{0} > 0, (S_{k} ∩ S_{j} = ∅, k ≠ j = \overline{0, m}), S_{0} cover all the other domains but the latter S_{j} not cover one another; \overline{D}_{k} ≡ D_{k} ∪ S_{k}, S ≡ S^{+} ≡ ∪_{k=0}^{m}S_{k}, D^{+} ≡ D_{0}\ ∪_{k=1}^{m} \overline{D}_{k}, D^{−} ≡ ℝ^{3}\ ∪_{k=1}^{m} \overline{D}_{k} be an infinite connected domain with the boundary S^{−} ≡ ∪_{k=1}^{m}S_{k}; D̃_{k} be a domain entirely lying strictly within D_{k}: \overline{D̃}_{k} ⋐ D̃_{0}, k = \overline{1, m}; D̃_{0} be a domain covering D_{0}; \overline{D}_{0} ⋐ D̃_{0}; S̃_{k} = ∂D̃_{k}, k = \overline{0, m}; S̃ ≡ S̃^{+} ≡ ∪_{k=0}^{m}S̃_{k}, S̃^{−} ≡ ∪_{k=0}^{m}S̃_{k}. Clearly, S ∩ S̃ = ∅, S^{−} ∩ S̃ ≡ ∅.

Denote by {x^{k}}_{k=1}^{∞} a dense everywhere countable set of points on the auxiliary surface S̃. We introduce the matrix

M(y − x, −iω) = ∥M^{1}, M^{2}, M^{3}, M^{4} ∥_{4×4},

where

M(y − x, −iω) =

= \begin{cases}Φ(y − x, −iω), & y ∈ ∪^{m_{1}}_{k=0}S_{k}, x ∈ ℝ^{3}, \\ R(\frac{∂}{∂y} , n)Φ(y − x, −iω), & y ∈ ∪^{m_{2}}_{k=m_{1}+1}S_{k}, x ∈ ℝ^{3}, \\ Q(\frac{∂}{∂y} , n)Φ(y − x, −iω), & y ∈ ∪^{m_{3}}_{k=m_{2}+1}S_{k}, x ∈ ℝ^{3}, \\ P(\frac{∂}{∂y} , n)Φ(y − x, −iω), & y ∈ ∪^{m}_{k=m_{3}+1}S_{k}, x ∈ ℝ^{3}, \end{cases}  (29)

m_{j}, j = \overline{1, 3}, m are arbitrary natural numbers; 0 ≤ m_{1} ≤ m_{2} ≤ m_{3} ≤ m.

Theorem 6. The countable set of vectors

{M^{j}(y − x^{k}, ζ)}_{k=1, j=1}^{∞, 4}, y ∈ S′ ≡ ∪^{m}_{k=0}S_{k}  (30)

is linearly independent and complete in the (vector) Hilbert space L_{2}(S).

Proof. a) Linear Independence. By assuming that the opposite is true we have

∑_{j=1}^{4} ∑_{k=1}^{N} c_{jk}M^{j}(y − x^{k}, ζ) ≡ 0, y ∈ S,  (31)

where at least one c_{jk} ≠ 0, N is an arbitrary natural number. Consider the vector

V (x) = ∑_{j=1}^{4} ∑_{k=1}^{N} c_{jk}Φ^{j}(y − x^{k}, ζ).

By (31) V (x) is a regular in D^{+} solution of the homogeneous problem M_{0}^{+}(ζ) : ∀x ∈ D^{+} : L(\frac{∂}{∂x}, ζ)V = 0; ∀z ∈ S_{k}, k = \overline{0, m_{1}} : V^{+}(z) = 0; ∀z ∈ S_{k}, k = m_{1} + 1, …, m_{2} : [RV (z)]^{+} = 0; ∀z ∈ S_{k}, k = m_{2} + 1, …, m_{3} : [QV (z)]^{+} = 0; ∀z ∈ S_{k}, k = m_{3} + 1, …, m : [PV (z)]^{+} = 0. Therefore by virtue of Theorem 4 V (x) ≡ 0, x ∈ D^{+}. Hence, since V is analytic, we have V (x) ≡ 0, x ⊂ D̃_{0}\ ∪_{k=1}^{m} \overline{D̃}_{k}. Further, by making the point x approach x^{k}, k = \overline{1, N}, and repeating the standard reasoning from [13] we obtain the contradiction c_{jk} = 0, j = 1, 4, k = \overline{1, N}.

b) Completeness. Let φ ∈ L_{2}(S) be an arbitrary four-dimensional vector. Since L_{2}(S) is a complete Hilbert space, the necessary and sufficient condition for set (29) to be complete consists in that the orthogonality condition

∫_{S} M^{j}^{T} (y − x^{k}, ζ)\overline{φ}(y) dS = 0, j = \overline{1, 4}, k = \overline{1, ∞},  (32)

would imply the equality φ(y) = 0 almost everywhere on S.

Consider the mixed-type potential

M̃(φ)(x) ≡ M̃(x, φ) = ∑_{k=0}^{m_{1}} ∫_{S_{k}} Φ̃(x − y, ζ)\overline{φ}(y) d_{y}S +

+ ∑_{k=m_{1}+1}^{m_{2}} ∫_{S_{k}} [R(\frac{∂}{∂y} , n)Φ(y − x, ζ)]^{T} \overline{φ}y dS +

+ ∑_{k=m_{2}+1}^{m_{3}} ∫_{S_{k}} [Q(\frac{∂}{∂y} , n)Φ(y − x, ζ)]^{T} \overline{φ}y d_{y}S +

+ ∑_{k=m_{3}+1}^{m} ∫_{S_{k}} [P(\frac{∂}{∂y} , n)Φ(y − x, ζ)]^{T} \overline{φ}y d_{y}S.  (33)

Using (29), (30), (32) and assuming that points x^{k}, k = \overline{1, ∞}, are dense everywhere on S̃, we have M̃(x, φ) ≡ 0, x ∈ S̃. Hence, by the uniqueness theorem for the conjugate problem, we readily obtain

M̃(x, φ) = 0, x ∈ ∪^{m}_{k=1}D̃_{k} ∪ (ℝ^{3}\\overline{D̃}_{0}).  (34)

Since M̃(x, φ) is analytic in the domain ℝ^{3}\ ∪_{k=0}^{m} S_{k}, (34) implies [13]

M̃(x, φ) = 0, x ∈ ∪^{m}_{k=1}D_{k} ∪ (ℝ^{3}\\overline{D}_{0}), \overline{φ} ∈ C^{(1,α)}(S).  (35)

To solve the homogeneous singular integral equations [13] we have used here some theorems of the embedding type. Applying familiar theorems of the Liapunov-Tauber type (see [13], [14], [15]) to (35), we conclude that M̃(x, φ) is a regular solution of the homogeneous boundary value problem M̃_{0}^{+}(ζ):

∀x ∈ D^{+} : L̃(\frac{∂}{∂x}, ζ)M̃(x, φ) = 0,

∀z ∈ ∪^{m_{1}}_{k=0}S_{k} : M̃^{+}(z, φ) = 0, ∀z ∈ ∪^{m_{2}}_{k=m_{1}}S_{k} : [R̃M̃(z, φ)]^{+} = 0,

∀z ∈ ∪^{m_{3}}_{k=m_{2}}S_{k} : [Q̃M̃(z, φ)]^{+} = 0, ∀z ∈ ∪^{m}_{k=m_{3}}S_{k} : [P̃M̃(z, φ)]^{+} = 0,

and therefore, by the uniqueness theorem,

M̃(x, φ) = 0, ∀x ∈ D^{+}.  (36)

Finally, relations (35) and (36) give in turn \overline{φ}(y) = 0, y ∈ S, i.e., φ(y) = 0, which completes the proof of Theorem 6.

Replace, in (29), y ∈ ∪_{k=0}^{m_{1}}S by y ∈ ∪_{k=1}^{m_{1}}S_{k} and denote by M′(y−x, −iω) the matrix M(y − x, −iω) (this means that the surface S_{0} “expands” to infinity).

By a reasoning similar to the above one can prove the following theorems:

Theorem 7. A countable set of vectors

{M^{j}(y − x^{k}, −ip)}_{k=1, j=1}^{∞, 4}, p > 0, y ∈ S ≡ ∪^{m}_{k=0}S_{k},

is linearly independent and complete in the space L_{2}(S) if p^{2} is different from the eigenvalues of the homogeneous problem M_{0}^{+}(−ip).

Theorem 8. A countable set of vectors

{M^{j}′(y − x^{k}, ζ)}_{k=1, j=1}^{∞, 4}, y ∈ S^{−} ≡ ∪^{m}_{k=1}S_{k},

is linearly independent and complete in L_{2}(S^{−}).

Theorem 9. A countable set of vectors

{M^{j}′(y − x^{k}, −ip)}_{k=1, j=1}^{∞, 4}, p > 0, y ∈ S^{−},

is linearly independent and complete in L_{2}(S^{−}).

(In Theorems 8 and 9 {x^{k}} ∈ S̃^{−} ≡ ∪_{k=1}^{m}S̃_{k} is a dense everywhere countable set of points).

6. Solution Continuation Theorems

As is well known, in the three-dimensional domain with the boundary whose some part is plane, a regular solution of the Laplace (also Helmholtz) equation analytically continues across this plane part into an additional domain which is the mirror reflection of the initial domain. In that case the continuable solutions must satisfy additional homogeneous conditions imposed on the plane part of the boundary. This property known by the name of the symmetry principle is important in solving effectively (in quadratures) the boundary value problems for some classes of infinite domains. This principle was generalized by E. I. Obolashvili for equations of classical elasticity [18], [19], by V. D. Kupradze and T. V. Burchuladze for equations of conjugate thermoelasticity, and by T. V. Burchuladze for systems of equations of elastothermodiffusion [15].

Let: Σ be the plane part of the boundary of the domain D given by the equation ∑_{k=1}^{3} b_{k}x_{k} + b_{4} = 0; x̂ = (x̂_{1}, x̂_{2}, x̂_{3}) be the mirror reflection of the point x ∈ D in Σ (it is assumed that D lies on one side of Σ),

Û(x) = (û, û_{4}) = ∥û_{k}(x)∥_{4×1},

û_{4}(x) = b^{−2} ∑_{l=1}^{3} (2b_{k}b_{l} − δ_{lk}b^{2})u(x̂), k = \overline{1, 3},

û_{4}(x) = −u_{4}(x̂), b^{2} = ∑_{k=1}^{3} b_{k}^{2}.

Theorem 10. If U(x) = (u, u_{4}) = ∥u_{k}(x)∥_{4×1} is a regular in D solution of the homogeneous system (4), satisfying, on Σ, the boundary conditions u − n(u · n) = 0, (HU · n) = 0, u_{4} = 0, then the vector V (x) = ∥v_{k}(x)∥_{4×1},

V (x) = \begin{cases}U(x), & x ∈ D, \\ Û(x), & x ∈ D̂ \end{cases}

is a regular solution of system (4) in the domain D ∪ Σ ∪ D̂.

Theorem 11. If the vector U = ∥u_{k}(x)∥_{4×1} is a regular in D solution of the homogeneous system (4), satisfying, on Σ, the boundary conditions (u·n) = 0, HU−n(HU·n) = 0, \frac{∂u_{4}}{∂n} = 0, then the vector V (x) = ∥v_{k}(x)∥_{4×1},

V (x) = \begin{cases}U(x), & x ∈ D, \\ −Û(x), & x ∈ D̂ \end{cases}

is a regular solution of system (4) in the domain D ∪ Σ ∪ D̂.

The proof of these theorems rests on the equalities

Δ(\frac{∂}{∂x}) = Δ(\frac{∂}{∂x̂}), div Û(x) = − div U(x̂)

which are easy to verify, and on general integral representations of the solution of system (4) (see [13], [14]).

Like in [20], [21], [22], [23], one can use these theorems and Theorem 2 to solve effectively (in quadratures) the boundary value problems for system (4) in the case of infinite domains such as a half-space; right, dihedral and trihedral angles (a quarter and eighth part of the space); dihedral and trihedral angles with opening π/2^{m} (m is a natural number), an infinite layer, a half-layer, a nonhomogeneous (piecewise homogeneous) space, a half-space and so on.

References

1. A. E. Green and K. A. Lindsay, Thermoelasticity. J. Elasticity 2(1972), 1–7.

2. H. W. Lord and Y. A. Shulman, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids. 15(1967), No. 5, 299–309.

3. D. T. Burchuladze, To the theory of boundary value problems of generalized thermoelasticity. (Russian) Trudy Gruz. Polytekhn. Inst. 1983, No. 5(262), 22–29.

4. T. V. Burchuladze, Three-dimensional dynamical problems of the nonclassical theory of thermoelasticity. (Russian) Z. Anal. Anwendungen 3(1984), No. 5, 441–455.

5. T. V. Burchuladze, To the theory of dynamical problems of nonclassical thermoelasticity. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 73(1983), 12–24.

6. W. Nowacki, Couopled fields in the mechanics of solids. (Russian) Adv. Mech. 1(1978), No. 1–2, 17–44.

7. J. Ignaczak, On a three-dimensional solution of dynamica thermoelasticity with relaxation times. J. Thermal Stresses 4(1981), No. 3–4, 357–385.

8. J. Ignaczak, Uniqueness in generalized thermoelasticity. J. thermal Stresses 2(1979), No. 2, 171–175.

9. J. Ignaczak, Decomposition theore, for thermoelasticity with finite wave speeds. J. Thermal Stresses 1(1978), No. 1, 41–52.

10. J. Ignaczak, A note on uniqueness in thermoelasticity with one relaxation time. J. Thermal Stresses 5(1982), No. 3–4, 257–263.

11. J. Ignaczak, A uniqueness theorem for stress-temperature equations of dynamic thermoelasticity. J. thermal Stresses 1(1978), 163–170.

12. M. S. Podstrigach and Yu. M. Kolyano, Generalized thermomechanics. (Russian) Naukova Dumka, Kiev, 1976.

13. 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 Publishing Company, Amsterdam–New York–Oxford, 1979; Russian original: Nauka, Moscow, 1976.

14. V. D. Kupradze and T. V. Burchuladze, Dynamical problems of the theory of elasticty and thermoelasticty. (Russian) Current Problems in mathematics, vol. 7, 163–294. Itogi Nauki i Tehniki, Inform., Moscow, 1975. Engl. transl.: Dynamical problems of the theory of elasticity and thermoelasticity. J. of Soviet Mathematics, vol. 7, No. 3, 1977, New York-London.

15. T. V. Burchuladze and T. G. Gegelia, The devlopment of the method of a potentail in the elasticity theory. (Russian) Metsniereba, Tbilisi, 1985.

16. W. Nowacki, Dynamical problems of thermoelasticity. Transleited into Russian, Moscow, Mir, 1970; Polish original, Warszawa, 1966.

17. T. V. Burchuladze, Solution of three-dimensional boundary value problems of nonclassical thermoelasticity in generalized Fourier series. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 81(1986), 3–17.

18. E. I. Obolashvili, Effective solutions of some spatial problems of the elasticticity theory. Rev. Roumaine Math. Pures Appl. 11(1966), No. 8, 965–972.

19. E. I. Obolashvili, Solution of some boundary value problems of the elasticity theory and of the shell theory. (Russian) Dissertation for the Doctor of Science Degree, A. Razmadze Mat. Inst., Tbilisi, 1966.

20. V. D. Kupradze and T. V. Burchuladze, Some problems of thermoelasticity solved in quadratures. (Russian) Differentsial’niye Uravneniya 5(1969), Nos. 10, 11, 1735–1761, 1923–1939.

21. T. V. Burchuladze, On boundary value problems of elastothermodiffusion solved in quadratures. (Russian) Bull. Acad. Sci. Georgian SSR 99(1980), No. 3, 569–572.

22. T. V. Burchuladze, Effective solutions of some boundary value problems of elastothermodiffusion. (Russian) Trudy Gruz. Polytekhnich. Inst. 1982, No. 2(247), 5–11.

23. T. V. Burchuladze, Dynamical problems of the nonclassical theory of thermoelasticity solved in the closed form. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 87(1987), 29–24.

(Received 05.07.1995)

Authors’ address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia