GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 3, 1997, 223-242

ANALOGUES OF THE KOLOSOV–MUSKHELISHVILI GENERAL REPRESENTATION FORMULAS AND

CAUCHY–RIEMANN CONDITIONS IN THE THEORY OF ELASTIC MIXTURES

M. BASHELEISHVILI

Abstract. Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions.

The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.

1. In this section we shall derive analogues of the Kolosov–Muskhelishvili general representation formulas for nonhomogeneous equations of statics in the theory of elastic mixtures. It will be shown that displacement and stress vector components, as well as stress tensor components, are represented in this theory by means of four arbitrary analytic functions.

The representations obtained here will be used in our next papers to investigate two-dimensional boundary value problems for the above-mentioned equations of an elastic mixture.

In the two-dimensional case the basic nonhomogeneous equations of the theory of elastic mixtures have the form (see [1] and [2])

a_{1}Δu′ + b_{1} grad div u′ + cΔu′′ + d grad div u′′ = −ρ_{1}F′ ≡ ψ′,

cΔu′ + d grad div u′ + a_{2}Δu′′ + b_{2} grad div u′′ = −ρ_{2}F′′ ≡ ψ′′, (1.1)

where Δ is the two-dimensional Laplacian, grad and div are the principal operators of the field theory, ρ_{1} and ρ_{2} are the partial densities (positive constants) of the mixture, F′ and F′′ are the mass force, u′ = (u′_{1}, u′_{2}) and

223 u′′ = (u′′_{1}, u′′_{2}) are the displacement vectors, a_{1},, b_{1}, c, d, a_{2}, b_{2} are the known constants characterizing the physical properties of the mixture. We have

1991 Mathematics Subject Classification. 73C02.

Key words and phrases. Generalized Kolosov–Muskhelishvili representation, theory of elastic mixtures, generalized Cauchy–Riemann conditions.

1072-947X/97/0500-0223$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

\begin{matrix}a_{1} = μ_{1} − λ_{5}, b_{1} = μ_{1} + λ_{1} + λ_{5} − ρ^{−1}α_{2}ρ_{2}, a_{2} = μ_{2} − λ_{5}, \\ c = μ_{3} + λ_{5}, b_{2} = μ_{2} + λ_{2} + λ_{5} + ρ^{−1}α_{2}ρ_{1}, \\ d = μ_{3} + λ_{3} − λ_{5} − ρ^{−1}α_{2}ρ_{1} ≡ μ_{3} + λ_{4} − λ_{5} + ρ^{−1}α_{2}ρ_{2}, \\ ρ = ρ_{1} + ρ_{2}, α_{2} = λ_{3} − λ_{4}, \end{matrix} (1.2)

where μ_{1}, μ_{2}, μ_{3}, λ_{1}, λ_{2}, λ_{3}, λ_{4}, λ_{5} are new constants also characterizing the physical properties of the mixture and satisfying the definite conditions (inequalities) [2].

In what follows we shall need the homogeneous equations corresponding to equations (1.1); obviously, they have the form (F′ = F′′ = 0 or ψ′ = ψ′′ = 0)

a_{1}Δu′ + b_{1} grad div u′ + cΔu′′ + d grad div u′′ = 0,
cΔu′ + d grad div u′ + a_{2}Δu′′ + b_{2} grad div u′′ = 0.

(1.3)

In the theory of elastic mixtures the displacement vector is usually denoted by u = (u′, u′′). In this paper u is the four-dimensional vector, i.e., u = (u_{1}, u_{2}, u_{3}, u_{4}) or u_{1} = u′_{1}, u_{2} = u′_{2}, u_{3} = u′′_{1}, u_{4} = u′′_{2}.

The system of basic equations (1.1) can (equivalently) be rewritten as

a_{1}Δu′ + cΔu′′ + b_{1} grad θ′ + d grad θ′′ = ψ′,

cΔu′ + a_{2}Δu′′ + d grad θ′ + b_{2} grad θ′′ = ψ′′, (1.4)

where

θ′ = \frac{∂u′_{1}}{∂x_{1}} + \frac{∂u′_{2}}{∂x_{2}} , θ′′ = \frac{∂u′′_{1}}{∂x_{1}} + \frac{∂u′′_{2}}{∂x_{2}} .  (1.5)

For our further discussion we shall also need the functions

ω′ = \frac{∂u′_{2}}{∂x_{1}} − \frac{∂u′_{1}}{∂x_{2}} , ω′′ = \frac{∂u′′_{2}}{∂x_{1}} − \frac{∂u′′_{1}}{∂x_{2}} .  (1.6)

As mentioned above, here we want to represent the solution (i.e., the displacement vector components) of (1.1) and the stress vector components (calculated by means of the displacement vector) and stress tensor components through analytic functions of a complex variable. To this end, for the basic equations of statics in the theory of elastic mixtures we shall generalize the method developed by Vekua and Muskhelishvili for nonhomogeneous equations of statics of an isotropic elastic body in the two-dimensional case (see [3] or [4]).

THEORY OF ELASTIC MIXTURES 225

We introduce the following variables:

z = x_{1} + ix_{2}, \overline{z} = x_{1} − ix_{2},

i.e.,

x_{1} = \frac{z + \overline{z}}{2} , x_{2} = \frac{z − \overline{z}}{2} ,

where

\begin{matrix}\frac{∂}{∂x_{1}} = \frac{∂}{∂z} + \frac{∂}{∂\overline{z}} , & \frac{∂}{∂x_{2}} = i(\frac{∂}{∂z} − \frac{∂}{∂\overline{z}} ), \\ \frac{∂}{∂z} = \frac{1}{2}(\frac{∂}{∂x_{1}} − i\frac{∂}{∂x_{2}} ), & \frac{∂}{∂\overline{z}} = \frac{1}{2}(\frac{∂}{∂x_{2}} + i\frac{∂}{∂x_{2}} ). \end{matrix}  (1.7)

After performing simple calculations we obtain

\begin{matrix}Δ = 4 \frac{∂^{2}}{∂z∂\overline{z}} , θ′ = \frac{∂w′}{∂z} + \frac{∂\overline{w}′}{∂\overline{z}} , θ′′ = \frac{∂w′′}{∂z} + \frac{∂\overline{w}′′}{∂\overline{z}} , \\ ω′ = −i(\frac{∂w′}{∂z} − \frac{∂\overline{w}′}{∂\overline{z}} ), ω′′ = −i(\frac{∂w′′}{∂z} − \frac{∂\overline{w}′′}{∂\overline{z}} ), \end{matrix}  (1.8)

where

w′ = u′_{1} + iu′_{2}, w′′ = u′′_{1} + iu′′_{2}.  (1.9)

On account of (1.7), (1.8), and (1.9) we can rewrite (1.4) as two complex equations

\begin{matrix}4a_{1} \frac{∂^{2}w′}{∂\overline{z}∂z} + 4c \frac{∂^{2}w′′}{∂\overline{z}∂z} + 2b_{1} \frac{∂θ′}{∂\overline{z}} + 2d \frac{∂θ′′}{∂\overline{z}} = Ψ′, \\ 4c \frac{∂^{2}w′}{∂\overline{z}∂z} + 4a_{2} \frac{∂^{2}w′′}{∂\overline{z}∂z} + 2d \frac{∂θ′}{∂\overline{z}} + 2b_{2} \frac{∂θ′′}{∂\overline{z}} = Ψ′′, \end{matrix}  (1.10)

where

Ψ′ = ψ′_{1} + iψ′_{2}, Ψ′′ = ψ′′_{1} + iψ′′_{2}.

Obviously, (1.10) can be rewritten as

\frac{∂}{∂\overline{z}} (4a_{1} \frac{∂w′}{∂z} + 4c \frac{∂w′′}{∂z} + 2b_{1}θ′ + 2dθ′′) = Ψ′,

\frac{∂}{∂\overline{z}} (4c \frac{∂w′}{∂z} + 4a_{2} \frac{∂w′′}{∂z} + 2dθ′ + 2b_{2}θ′′) = Ψ′′,

which, after applying the Pompeiu formula [4], gives

\begin{matrix}4a_{1} \frac{∂w′}{∂z} +4c \frac{∂w′′}{∂z} +2b_{1}θ′+2dθ′′ =4φ′_{1}(z) + \frac{1}{π} ∫_{D} \frac{Ψ′(y_{1}, y_{2})}{σ} dy_{1}dy_{2}, \\ 4c \frac{∂w′}{∂z} +4a_{2} \frac{∂w′′}{∂z} +2dθ′+2b_{2}θ′′ =4φ′_{2}(z) + \frac{1}{π} ∫_{D} \frac{Ψ′′(y_{1}, y_{2})}{σ} dy_{1}dy_{2}, \end{matrix} (1.11)

where σ = z − ζ, ζ = y_{1} + iy_{2}, φ′_{1}(z) and φ′_{2}(z) are arbitrary analytic functions which we have represented as derivatives of arbitrary analytic functions, while multiplier 4 has been introduced for convenience. In (1.11) D is a finite or infinite two-dimensional domain. In the case of an infinite domain the functions Ψ′ and Ψ′′ satisfy the definite conditions near the point at infinity.

Remark. The integral terms (partial solutions) appear in (1.11) by virtue of the fact that the Pompeiu formula

w(x) = \frac{1}{π} ∫_{D} \frac{F (y_{1}, y_{2})}{σ} dy_{1} dy_{2}

holds (under certain assumptions) for the equation

\frac{∂w}{∂\overline{z}} = F = F_{1} + iF_{2}.

The proof of the Pompeiu formula for both a finite and an infinite domain D is given in [4].

We shall give one more proof of the Pompeiu formula. Let w = u + iv. Then, on separating the real and imaginary parts, the equation for w can be written as two equations:

\frac{∂u}{∂x_{1}} − \frac{∂v}{∂x_{2}} = 2F_{1}, \frac{∂u}{∂x_{2}} + \frac{∂v}{∂x_{1}} = 2F_{2}.

If we now introduce new functions φ and ψ by

u = \frac{∂φ}{∂x_{1}} + \frac{∂ψ}{∂x_{2}} , v = −\frac{∂φ}{∂x_{2}} + \frac{∂ψ}{∂x_{1}} ,

the previous system for φ and ψ can be rewritten as

Δφ = 2F_{1}, Δψ = F_{2}.

By the well-known formula for a partial solution of the Poisson equation, we obtain

φ = \frac{1}{π} ∫_{D} ln rF_{1} dy_{1} dy_{2}, ψ = \frac{1}{π} ∫_{D} ln rF_{2} dy_{1} dy_{2},

where

r = \sqrt{(x_{1} − y_{1})^{2} + (x_{2} − y_{2})^{2}} = |σ|.

By calculating the partial derivatives of first order for φ and ψ we obtain

u = \frac{1}{π} ∫_{D} (\frac{x_{1} − y_{1}}{r^{2}} F_{1} + \frac{x_{2} − y_{2}}{r^{2}} F_{2})dy_{1}dy_{2},

v = \frac{1}{π} ∫_{D} ( − \frac{x_{2} − y_{2}}{r^{2}} F_{1} + \frac{x_{1} − y_{1}}{r^{2}} F_{2})dy_{1}dy_{2},

and hence

w = u + iv = \frac{1}{π} ∫_{D} \frac{F (y_{1}, y_{2})}{σ} dy_{1} dy_{2}.

The latter formula coincides with the Pompeiu formula.

Combining (1.11) with the formulas obtained from (1.11), passing to the conjugate values, and taking (1.8) into account, we obtain, after some transformations for θ′ and θ′′, the system of equations

\begin{matrix}(a_{1} + b_{1})θ′+(c + d)θ′′ =2 Re [φ′_{1}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}], \\ (c + d)θ′+(a_{2} + b_{2})θ′′ =2 Re [φ′_{2}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}], \end{matrix} (1.12)

where the symbol Re denotes the real part.

On subtracting the complex-valued values and again taking into account (1.8), we obtain in a manner similar to the above the following system for ω′ and ω′′:

\begin{matrix}a_{1}ω′ + cω′′ = 2 Im [φ′_{1}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}], \\ cω′ + a_{2}ω′′ = 2 Im [φ′_{2}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}], \end{matrix} (1.13)

where the symbol Im denotes the imaginary part.

228 M. BASHELEISHVILI

By solving system (1.12) for θ′ and θ′′ we obtain

θ′ = \frac{2}{d_{1}} Re \{(a_{2} + b_{2})[φ′_{1}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}] −

− (c + d)[φ′_{2}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}]\},

θ′′ = \frac{2}{d_{1}} Re \{ − (c + d)[φ′_{1}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}] +

+ (a_{1} + b_{1})[φ′_{2}(z) + \frac{1}{4π} ∫_{D} \frac{Ψ′′(y_{1}, y_{2})}{σ} dy_{1} dy_{2}]\},

(1.14)

where d_{1} = (a_{1} + b_{1})(a_{2} + b_{2}) − (c + d)^{2} > 0. For the unknown \frac{∂w′}{∂z} and \frac{∂w′′}{∂z} system (1.11) gives

\begin{aligned}\frac{∂w′}{∂z} & = e_{1}φ′_{1}(z) + e_{2}φ′_{2}(z) + \frac{1}{4π} ∫_{D} (e_{1}Ψ′ + e_{2}Ψ′′)\frac{dy_{1} dy_{2}}{σ} + \\ & + \frac{1}{2d_{2}} [(cd − b_{1}a_{2})θ′ + (cb_{2} − da_{2})θ′′], \\ \frac{∂w′′}{∂z} & = e_{2}φ′_{1}(z) + e_{3}φ′_{2}(z) + \frac{1}{4π} ∫_{D} (e_{2}Ψ′ + e_{3}Ψ′′)\frac{dy_{1} dy_{2}}{σ} + \\ & + \frac{1}{2d_{2}} [(cb_{1} − da_{1})θ′ + (cd − a_{1}b_{2})θ′′], \end{aligned} (1.15)

where

e_{1} = \frac{a_{2}}{d_{2}} , e_{2} = −\frac{c}{d_{2}} , e_{3} = \frac{a_{1}}{d_{2}} , d_{2} = a_{1}a_{2} − c^{2} > 0.  (1.16)

From (1.14) we obtain by elementary calculations

\frac{1}{2d_{2}} [(cd − b_{1}a_{2})θ′ + (cb_{2} − da_{2})θ′′] =

= Re [e_{4}φ′_{1}(z) + e_{5}φ′_{2}(z) + \frac{1}{4π} ∫_{D} (e_{4}Ψ′ + e_{5}Ψ′′)\frac{dy_{1} dy_{2}}{σ} ],

\frac{1}{2d_{2}} [(cb_{1} − da_{1})θ′ + (cd − a_{1}b_{2})θ′′] =

= Re [e_{5}φ′_{1}(z) + e_{6}φ′_{2}(z) + \frac{1}{4π} ∫_{D} (e_{5}Ψ′ + e_{6}Ψ′′)\frac{dy_{1} dy_{2}}{σ} ],

(1.17) where

\begin{aligned}e_{4} & = \frac{(c + d)(da_{2} − cb_{2}) + (a_{2} + b_{2})(cd − b_{1}a_{2})}{d_{1}d_{2}}, \\ e_{5} & = \frac{(a_{1} + b_{1})(cb_{2} − da_{2}) + (c + d)(a_{2}b_{1} − cd)}{d_{1}d_{2}} = \\ & = \frac{(c + d)(a_{1}b_{2} − cd) + (a_{2} + b_{2})(cb_{1} − da_{1})}{d_{1}d_{2}}, \\ e_{6} & = \frac{(a_{1} + b_{1})(cd − a_{1}b_{2}) + (c + d)(da_{1} − cb_{1})}{d_{1}d_{2}}. \end{aligned}  (1.18)

After substituting (1.17) into (1.15), we can rewrite \frac{∂w′}{∂z} and \frac{∂w′′}{∂z} in a simpler form

\frac{∂w′}{∂z} = m_{1}φ′_{1}(z) + m_{2}φ′_{2}(z) + \frac{e_{4}}{2} \overline{φ′_{1}(z)} + \frac{e_{5}}{2} \overline{φ′_{2}(z)} +

+ \frac{1}{4π} ∫_{D} (m_{1}Ψ′ + m_{2}Ψ′′)\frac{dy_{1} dy_{2}}{σ} + \frac{1}{8π} ∫_{D} (e_{4}\overline{Ψ}′ + e_{5}\overline{Ψ}′′)\frac{dy_{1} dy_{2}}{\overline{σ}} ,

\frac{∂w′′}{∂z} = m_{2}φ′_{1}(z) + m_{3}φ′_{2}(z) + \frac{e_{5}}{2} \overline{φ′_{1}(z)} + \frac{e_{6}}{2} \overline{φ′_{2}(z)} +

+ \frac{1}{4π} ∫_{D} (m_{2}Ψ′ + m_{3}Ψ′′)\frac{dy_{1} dy_{2}}{σ} + \frac{1}{8π} ∫_{D} (e_{5}\overline{Ψ}′ + e_{6}\overline{Ψ}′′)\frac{dy_{1} dy_{2}}{\overline{σ}} ,

(1.19)

where

m_{1} = e_{1} + \frac{e_{4}}{2} , m_{2} = e_{2} + \frac{e_{5}}{2} , m_{3} = e_{3} + \frac{e_{6}}{2} .  (1.20)

Since \frac{1}{σ} = \frac{∂}{∂z} (ln σ + ln \overline{σ}) = 2 \frac{∂}{∂z} ln |σ|, we obtain from (1.19) by integration

w′ = m_{1}φ_{1}(z) + m_{2}φ_{2}(z) + \frac{z}{2}[e_{4}\overline{φ′_{1}(z)} + e_{5}\overline{φ′_{2}(z)}] +

+ \overline{ψ_{1}(z)} + \frac{1}{2π} ∫_{D} (m_{1}Ψ′ + m_{2}Ψ′′) ln |σ|dy_{1} dy_{2} +

+ \frac{1}{8π} ∫_{D} \frac{σ}{\overline{σ}} (e_{4}\overline{Ψ}′ + e_{5}\overline{Ψ}′′)dy_{1} dy_{2},

w′′ = m_{2}φ_{1}(z) + m_{3}φ_{2}(z) + \frac{z}{2}[e_{5}\overline{φ′_{1}(z)} + e_{6}\overline{φ′_{2}(z)}] +

+ \overline{ψ_{2}(z)} + \frac{1}{2π} ∫_{D} (m_{2}Ψ′ + m_{3}Ψ′′) ln |σ|dy_{1} dy_{2} +

+ \frac{1}{8π} ∫_{D} \frac{σ}{\overline{σ}} (e_{5}\overline{Ψ}′ + e_{6}\overline{Ψ}′′)dy_{1} dy_{2},

(1.21) where ψ_{1}(z) and ψ_{2}(z) are new arbitrary analytic functions.

In the theory of elastic mixtures, formulas (1.21) obtained for the displacement vector components are analogues of Kolosov–Muskhelishvili general representation formulas.

If the system of equations (1.1) is homogeneous, i.e., ψ′ = ψ′′ = 0 or Ψ′ = Ψ′′ = 0, then the integral terms in (1.21) vanish and we obtain the formulas

\begin{aligned}w′ & = m_{1}φ_{1}(z) + m_{2}φ_{2}(z) + \frac{z}{2}[e_{4}\overline{φ′_{1}(z)} + e_{5}\overline{φ′_{2}(z)}] + \overline{ψ_{1}(z)}, \\ w′′ & = m_{2}φ_{1}(z) + m_{3}φ_{2}(z) + \frac{z}{2}[e_{5}\overline{φ′_{1}(z)} + e_{6}\overline{φ′_{2}(z)}] + \overline{ψ_{2}(z)}, \end{aligned}  (1.22)

which are anlogues of Kolosov–Muskhelishvili formulas for the displacement vector components of equation (1.3).

The integral terms in (1.21) are one particular solution of system (1.1). To rewrite these terms in a different form we introduce the vectors u^{(0)}(x) = (u′_{1}, u′_{2}, u′′_{1}, u′′_{2}) and ψ(x) = (ψ′_{1}, ψ′_{2}, ψ′′_{1}, ψ′′_{2}). Now, after separating the real parts, from (1.21) we have

u^{(0)}(x) = \frac{1}{2π} ∫_{D} ϕ(x − y)ψ(y) dy_{1} dy_{2},  (1.23)

where

ϕ(x − y) = Re Γ(x − y),  (1.24)
Γ(x − y) =

= ∥\begin{matrix}m_{1} ln σ + \frac{e_{4}}{4} \frac{\overline{σ}}{σ} , & \frac{ie_{4}}{4} \frac{\overline{σ}}{σ} , & m_{2} ln σ + \frac{e_{5}}{2} \frac{\overline{σ}}{σ} , & \frac{ie_{5}}{4} \frac{\overline{σ}}{σ} \\ \frac{ie_{4}}{4} \frac{\overline{σ}}{σ} , & m_{1} ln σ − \frac{e_{4}}{4} \frac{\overline{σ}}{σ} , & \frac{ie_{5}}{4} \frac{\overline{σ}}{σ} , & m_{2} ln σ − \frac{e_{5}}{4} \frac{\overline{σ}}{σ} \\ m_{2} ln σ + \frac{e_{5}}{4} \frac{\overline{σ}}{σ} , & \frac{ie_{5}}{4} \frac{\overline{σ}}{σ} , & m_{3} ln σ + \frac{e_{6}}{4} \frac{\overline{σ}}{σ} , & \frac{ie_{6}}{4} \frac{\overline{σ}}{σ} \\ \frac{ie_{5}}{4} \frac{\overline{σ}}{σ} , & m_{2} ln σ − \frac{e_{5}}{4} \frac{\overline{σ}}{σ} , & \frac{ie_{6}}{4} \frac{\overline{σ}}{σ} , & m_{3} ln σ − \frac{e_{6}}{4} \frac{\overline{σ}}{σ} \end{matrix} ∥ .

Here ϕ(x − y) is a fundamental matrix. Each term of matrix (1.24) is a single-valued function on the entire plane and has at most a logarithmic singularity at the point x = y. By direct calculations it can be proved that each column of the matrix ϕ(x − y) (considered as a vector) is a solution of system (1.3) with respect to the cordinates of the point x for x ≠ y. It is obvious from (1.24) that ϕ(x − y) is a symmetric matrix.

Now we shall derive general complex representations for the components of the stress tensor and stress vector in the theory of elastic mixtures. As is known from [2], using the displacement vector u = (u′_{1}, u′′_{2}, u′′_{1}, u′′_{2}) the stress vector components can be written as follows:

\begin{matrix}(T u)_{1} = τ′_{11}n_{1} + τ′_{21}n_{2}, (T u)_{2} = τ′_{12}n_{1} + τ′_{22}n_{2}, \\ (T u)_{3} = τ′′_{11}n_{1} + τ′′_{21}n_{2}, (T u)_{4} = τ′′_{12}n_{1} + τ′′_{22}n_{2}, \end{matrix}  (1.25)

where n = (n_{1}, n_{2}) is an arbitrary unit vector and

\begin{matrix}τ′_{11} & = (λ_{1} − \frac{α_{2}ρ_{2}}{ρ} )θ′ + (λ_{3} − \frac{α_{2}ρ_{1}}{ρ} )θ′′ + \\ & + 2μ_{1} \frac{∂u′_{1}}{∂x_{1}} + 2μ_{3} \frac{∂u′′_{1}}{∂x_{1}} , \\ τ′_{21} & = (μ_{1} − λ_{5})\frac{∂u′_{1}}{∂x_{2}} + (μ_{1} + λ_{5})\frac{∂u′_{2}}{∂x_{1}} + \\ & + (μ_{3} + λ_{5})\frac{∂u′′_{1}}{∂x_{2}} + (μ_{3} − λ_{5})\frac{∂u′′_{2}}{∂x_{1}} , \\ τ′_{12} & = (μ_{1} + λ_{5})\frac{∂u′_{1}}{∂x_{2}} + (μ_{1} − λ_{5})\frac{∂u′_{2}}{∂x_{1}} + \\ & + (μ_{3} − λ_{5})\frac{∂u′′_{1}}{∂x_{2}} + (μ_{3} + λ_{5})\frac{∂u′′_{2}}{∂x_{1}} , \\ τ′_{22} & = (λ_{1} − \frac{α_{2}ρ_{2}}{ρ} )θ′ + (λ_{3} − \frac{α_{2}ρ_{1}}{ρ} )θ′′ + \\ & + 2μ_{1} \frac{∂u′_{2}}{∂x_{2}} + 2μ_{3} \frac{∂u′′_{2}}{∂x_{2}} , \end{matrix} \}  (1.26)

\begin{matrix}τ′′_{11} & = (λ_{4} + \frac{α_{2}ρ_{2}}{ρ} )θ′ + (λ_{2} + \frac{α_{2}ρ_{1}}{ρ} )θ′′ + \\ & + 2μ_{3} \frac{∂u′_{1}}{∂x_{1}} + 2μ_{2} \frac{∂u′′_{1}}{∂x_{1}} , \\ τ′′_{21} & = (μ_{3} + λ_{5})\frac{∂u′_{1}}{∂x_{2}} + (μ_{3} − λ_{5})\frac{∂u′_{2}}{∂x_{1}} + \\ & + (μ_{2} − λ_{5})\frac{∂u′′_{1}}{∂x_{2}} + (μ_{2} + λ_{5})\frac{∂u′′_{2}}{∂x_{1}} , \\ τ′′_{12} & = (μ_{3} − λ_{5})\frac{∂u′_{1}}{∂x_{2}} + (μ_{3} + λ_{5})\frac{∂u′_{2}}{∂x_{1}} + \\ & + (μ_{2} + λ_{5})\frac{∂u′′_{1}}{∂x_{2}} + (μ_{2} − λ_{5})\frac{∂u′′_{2}}{∂x_{1}} , \\ τ′′_{22} & = (λ_{4} + \frac{α_{2}ρ_{2}}{ρ} )θ′ + (λ_{2} + \frac{α_{2}ρ_{1}}{ρ} )θ′′ + \\ & + 2μ_{3} \frac{∂u′_{2}}{∂x_{2}} + 2μ_{2} \frac{∂u′′_{2}}{∂x_{2}} , \end{matrix} \}  (1.27)

where θ′ and θ′′ are defined by (1.5).

Using (1.2), (1.8) and (1.9) and performing some simple transformation, we obtain

τ′_{11} + τ′_{22} = 4 Re [(a_{1} + b_{1} − μ_{1})\frac{∂w′}{∂z} + (c + d − μ_{3})\frac{∂w′′}{∂z} ],

τ′_{11} − τ′_{22} + i(τ′_{21} + τ′_{12}) = 4μ_{1} \frac{∂w′}{∂\overline{z}} + 4μ_{3} \frac{∂w′′}{∂\overline{z}} ,

τ′_{21} − τ′_{12} = 4λ_{5} Im (\frac{∂w′}{∂z} − \frac{∂w′′}{∂z} ),

τ′′_{11} + τ′′_{22} = 4 Re [(c + d − μ_{3})\frac{∂w′}{∂z} + (a_{2} + b_{2} − μ_{2})\frac{∂w′′}{∂z} ],

τ′′_{11} − τ′′_{22} + i(τ′′_{21} + τ′′_{12}) = 4μ_{3} \frac{∂w′}{∂\overline{z}} + 4μ_{2} \frac{∂w′′}{∂\overline{z}} ,

τ′′_{21} − τ′′_{12} = −4λ_{5} Im (\frac{∂w′}{∂z} − \frac{∂w′′}{∂z} ).

After substituting the expressions for w′ and w′′ from (1.21) into the above formulas we have

τ′_{11} + τ′_{22} = 2 Re {(2 − A_{1} − B_{1})φ′_{1}(z) − (A_{2} + B_{2})φ′_{2}(z) +

+ \frac{1}{4π} ∫_{D} [(2 − A_{1} − B_{1})Ψ′ − (A_{2} + B_{2})Ψ′′]\frac{dy_{1} dy_{2}}{σ} \},

τ′_{11} − τ′_{22} − i(τ′_{21} + τ′_{12}) = 2\overline{z}[B_{1}φ′′_{1}(z) + B_{2}φ′′_{2}(z)] + 4μ_{1}ψ′_{1}(z) +

+ 4μ_{3}ψ′_{2}(z) + \frac{1}{2π} ∫_{D} (A_{1}\overline{Ψ}′ + A_{2}\overline{Ψ}′′)\frac{dy_{1} dy_{2}}{σ} −

− \frac{1}{2π} ∫_{D} \frac{\overline{σ}}{σ^{2}} (B_{1}Ψ′ + B_{2}Ψ′′)dy_{1} dy_{2},

τ′_{21} − τ′_{12} = 4λ_{5} Im {(e_{1} − e_{2})φ′_{1}(z) + (e_{2} − e_{3})φ′_{2}(z) +
+ \frac{1}{4π} ∫_{D} [(e_{1} − e_{2})Ψ′ + (e_{2} − e_{3})Ψ′′]\frac{dy_{1} dy_{2}}{σ} \},  (1.28)

τ′′_{11} + τ′′_{22} = 2 Re { − (A_{3} + B_{3})φ′_{1}(z) + (2 − A_{4} − B_{4})φ′_{2}(z) +

+ \frac{1}{4π} ∫_{D} [ − (A_{3} + B_{3})Ψ′ + (2 − A_{4} − B_{4})Ψ′′]\frac{dy_{1} dy_{2}}{σ} \},

τ′′_{11} − τ′′_{22} − i(τ′′_{21} + τ′′_{12}) = 2\overline{z}[B_{3}φ′′_{1}(z) + B_{4}φ′′_{2}(z)] + 4μ_{3}ψ′_{1}(z) +

+ 4μ_{2}ψ′_{2}(z) + \frac{1}{2π} ∫_{D} (A_{3}\overline{Ψ}′ + A_{4}\overline{Ψ}′′)\frac{dy_{1} dy_{2}}{σ} −

− \frac{1}{2π} ∫_{D} \frac{\overline{σ}}{σ^{2}} (B_{3}Ψ′ + B_{4}Ψ′′)dy_{1} dy_{2},

τ′′_{21} − τ′′_{12} = 4λ_{5} Im {(e_{2} − e_{1})φ′_{1}(z) + (e_{3} − e_{2})φ′_{2}(z) +

+ \frac{1}{4π} ∫_{D} [(e_{2} − e_{1})Ψ′ + (e_{3} − e_{2})Ψ′′]\frac{dy_{1} dy_{2}}{σ} \},

where

\begin{matrix}A_{1} = 2(μ_{1}m_{1} + μ_{3}m_{2}), & A_{2} = 2(μ_{1}m_{2} + μ_{3}m_{3}), \\ A_{3} = 2(μ_{3}m_{1} + μ_{2}m_{2}), & A_{4} = 2(μ_{3}m_{2} + μ_{2}m_{3}), \\ B_{1} = μ_{1}e_{4} + μ_{3}e_{5}, & B_{2} = μ_{1}e_{5} + μ_{3}e_{6}, \\ B_{3} = μ_{3}e_{4} + μ_{2}e_{5}, & B_{4} = μ_{3}e_{5} + μ_{2}e_{6}, \end{matrix}  (1.29)

and the constants e_{1}, e_{2}, e_{3} are defined by (1.16).

It is easy to calculate the stress tensor components τ′_{11}, τ′_{22}, τ′_{21}, τ′_{12}, τ′′_{11}, τ′′_{22}, τ′′_{21}, τ′′_{12} by (1.28). They are expressed through four arbitrary analyic functions and their derivatives. Since for the time being we do not need the specific values of the stress tensor components, we shall not write them out.

For the stress tensor components formulas (1.28) are the generalized Kolosov–Muskhelishvili formulas in the theory of elastic mixtures.

As said above, the four-dimensional vector u^{(0)}(x) defined by (1.23) is a particular solution of system (1.1). By using this solution one can always reduce, without loss of generaliy, the nonhomogeneous equation (1.1) to the homogeneous equation (1.3). Hence in what follows we shall consider only equation (1.3).

Now the expressions for the stress vector components from (1.25) can be rewritten in a simpler form. By virtue of (1.2), (1.5) and (1.6), we rewrite (1.26) and (1.27) as

\begin{matrix}τ′_{11} = (a_{1} + b_{1})θ′ + (c + d)θ′′ − 2μ_{1} \frac{∂u′_{2}}{∂x_{2}} − 2μ_{3} \frac{∂u′′_{2}}{∂x_{2}} , \\ τ′_{21} = −a_{1}ω′ − cω′′ + 2μ_{1} \frac{∂u′_{2}}{∂x_{1}} + 2μ_{3} \frac{∂u′′_{2}}{∂x_{1}} , \\ τ′_{12} = a_{1}ω′ + cω′′ + 2μ_{1} \frac{∂u′_{1}}{∂x_{2}} + 2μ_{3} \frac{∂u′′_{1}}{∂x_{2}} , \\ τ′_{22} = (a_{1} + b_{1})θ′ + (c + d)θ′′ − 2μ_{1} \frac{∂u′_{1}}{∂x_{1}} − 2μ_{3} \frac{∂u′′_{1}}{∂x_{1}} , \end{matrix} \}  (1.30)

\begin{matrix}τ′′_{11} = (c + d)θ′ + (a_{2} + b_{2})θ′′ − 2μ_{3} \frac{∂u′_{2}}{∂x_{2}} − 2μ_{2} \frac{∂u′′_{2}}{∂x_{2}} , \\ τ′′_{21} = −cω′ − a_{2}ω′′ + 2μ_{3} \frac{∂u′_{2}}{∂x_{1}} + 2μ_{2} \frac{∂u′′_{2}}{∂x_{1}} , \\ τ′′_{12} = cω′ + a_{2}ω′′ + 2μ_{3} \frac{∂u′_{1}}{∂x_{2}} + 2μ_{2} \frac{∂u′′_{1}}{∂x_{2}} , \\ τ′′_{22} = (c + d)θ′ + (a_{2} + b_{2})θ′′ − 2μ_{3} \frac{∂u′_{1}}{∂x_{1}} − 2μ_{2} \frac{∂u′′_{1}}{∂x_{1}} , \end{matrix} \}  (1.31)

Now, applying (1.12) and (1.13) and introducing the notation

\frac{∂}{∂s(x)} = n_{1} \frac{∂}{∂x_{2}} − n_{2} \frac{∂}{∂x_{1}} ,  (1.32)

which expresses the derivative with respect to the tangent when the point x(x_{1}, x_{2}) is on the boundary, we obtain from (1.25) with (1.30) and (1.31) taken into account

(T u)_{1} = 2 Re φ′_{1}(z)n_{1} − 2 Im φ′_{1}(z)n_{2} − 2μ_{1} \frac{∂u′_{2}}{∂s(x)} − 2μ_{3} \frac{∂u′′_{2}}{∂s(x)},

(T u)_{2} = 2 Im φ′_{1}(z)n_{1} + 2 Re φ′_{1}(z)n_{2} + 2μ_{1} \frac{∂u′_{1}}{∂s(x)} + 2μ_{3} \frac{∂u′′_{1}}{∂s(x)},

(T u)_{3} = 2 Re φ′_{2}(z)n_{1} − 2 Im φ′_{2}(z)n_{2} − 2μ_{3} \frac{∂u′_{2}}{∂s(x)} − 2μ_{2} \frac{∂u′′_{2}}{∂s(x)},

(T u)_{4} = 2 Im φ′_{2}(z)n_{1} + 2 Re φ′_{2}(z)n_{2} + 2μ_{3} \frac{∂u′_{1}}{∂s(x)} + 2μ_{2} \frac{∂u′′_{1}}{∂s(x)}.

Hence, using notation (1.9) and (1.32) and performing some simple transformations, we can write

i(T u)_{1} − (T u)_{2} = \frac{∂}{∂s(x)}(2φ_{1}(z) − 2μ_{1}w′ − 2μ_{3}w′′),

i(T u)_{3} − (T u)_{4} = \frac{∂}{∂s(x)}(2φ_{2}(z) − 2μ_{3}w′ − 2μ_{2}w′′).

After substituting the expressions for w′ and w′′ from (1.22) into the above formulas we easily obtain

\begin{aligned}i(T u)_{1} & − (T u)_{2} = \frac{∂}{∂s(x)}\{(2 − A_{1})φ_{1}(z) − A_{2}φ_{2}(z) − \\ & − z[B_{1}\overline{φ′_{1}(z)} + B_{2}\overline{φ′_{1}(z)}] − 2μ_{1}\overline{ψ_{1}(z)} − 2μ_{3}\overline{ψ_{2}(z)}\}, \\ i(T u)_{3} & − (T u)_{4} = \frac{∂}{∂s(x)}\{ − A_{3}φ_{1}(z) + (2 − A_{4})φ_{2}(z) − \\ & − z[B_{3}\overline{φ′_{1}(z)} + B_{4}\overline{φ′_{2}(z)}] − 2μ_{3}\overline{ψ_{1}(z)} − 2μ_{2}\overline{ψ_{2}(z)}\}, \end{aligned} (1.33)

where A_{1}, A_{2}, A_{3}, A_{4} and B_{1}, B_{2}, B_{3}, B_{4} are defined by (1.29).

Thus for the stress vector components we have obtained a general representation expressed in terms of analytic functions. For the stress vector components in the theory of elastic mixtures these formulas are the generalization of the Kolosov–Muskhelishvili formulas.

Formulas (1.33) imply that M. Levy’s theorem does not hold in the theory of elastic mixtures, i.e., the stress vector components depend on constants characterizing the physical properties of an elastic mixture.

In the theory of elastic mixtures, in addition to the stress vector, much importance is also attached to the so-called generalized stress vector

T^{κ}u = T u + κ \frac{∂u}{∂s},  (1.34)

where T u is the stress vector, \frac{∂}{∂s} is defined by (1.32), u is the four-dimensional displacement vector, and κ is the constant matrix:

κ = ∥\begin{matrix}0 & κ_{1} & 0 & κ_{3} \\ −κ_{1} & 0 & −κ_{3} & 0 \\ 0 & κ_{3} & 0 & κ_{2} \\ −κ_{3} & 0 & −κ_{2} & 0 \end{matrix}∥ ,  (1.35)

where κ_{1}, κ_{2}, κ_{3} take arbitrary real values. We have written an arbitrary matrix κ in form (1.35) (some of its term are zero) due to system (1.3); this is the highest arbitrariness that system (1.3) can provide. Note that analogous generalized stress vectors were introduced by us for equations of statics of isotropic and anisotropic elastic bodies in our earlier studies.

Let us consider some particular values of the constant matrix κ. In

(1.35) we write κ_{1} = κ_{2} = κ_{3} = 0, i.e., κ = 0. In that case T^{∘} ≡ T and the generalized stress vector coincides with the stress vector. Assuming now that κ_{1} = 2μ_{1}, κ_{2} = 2μ_{2}, κ_{3} = 2μ_{3}, we obtain κ = κ_{L} and denote the generalized stress vector by L. In view of the above calculations we have

(Lu)_{1} = [(a_{1} + b_{1})θ′ + (c + d)θ′′]n_{1} − (a_{1}ω′ + cω′′)n_{2} =
= 2 Re φ′_{1}(z)n_{1} − 2 Im φ′_{1}(z)n_{2},
(Lu)_{2} = (a_{1}ω′ + cω′′)n_{1} + [(a_{1} + b_{1})θ′ + (c + d)θ′′]n_{2} =
= 2 Im φ′_{1}(z)n_{1} + Re φ′_{1}(z)n_{2},
(Lu)_{3} = [(c + d)θ′ + (a_{2} + b_{2})θ′′]n_{1} − (cω′ + a_{2}ω′′)n_{2} =
= 2 Re φ′_{2}(z)n_{1} − 2 Im φ′_{2}(z)n_{2},
(Lu)_{4} = (cω′ + a_{2}ω′′)n_{1} + [(c + d)θ′ + (a_{2} + b_{2})θ′′]n_{2} =
= 2 Im φ′_{2}(z)n_{1} + Re φ′_{2}(z)n_{2},

where

i(Lu)_{1} − (Lu)_{2} = 2\frac{∂φ_{1}(z)}{∂s(x)}, i(Lu)_{3} − (Lu)_{4} = 2\frac{∂φ_{2}(z)}{∂s(x)}. (1.36)

Since (1.34) implies

Lu = T u + κ_{L} \frac{∂u}{∂s},

the generalized stress vector can be rewritten as

T^{κ}u = Lu + (κ − κ_{L})\frac{∂u}{∂s},

which by virtue of (1.36) yields

\begin{matrix}i(T^{κ}u)_{1} − (T^{κ}u)_{2} = \frac{∂}{∂s(x)}[2φ_{1}(z) + (κ_{1} − 2μ_{1})w′ + (κ_{3} − 2μ_{3})w′′], \\ i(T^{κ}u)_{3} − (T^{κ}u)_{4} = \frac{∂}{∂s(x)}[2φ_{2}(z) + (κ_{3} − 2μ_{3})w′ + (κ_{2} − 2μ_{2})w′′]. \end{matrix} (1.37)

Let us consider one more specific value of the constant matrix κ which is important for studying the first boundary value problem in the theory of elastic mixtures. Assume that in (1.37)

\begin{matrix}κ_{1} − 2μ_{1} = −\frac{m_{3}}{Δ_{0}} , κ_{2} − 2μ_{2} = −\frac{m_{1}}{Δ_{0}} , \\ κ_{3} − 2μ_{3} = \frac{m_{2}}{Δ_{0}} , Δ_{0} = m_{1}m_{3} − m_{2}^{2}. \end{matrix}  (1.38)

Then we have κ ≡ κ_{N}, and we denote the generalized stress vector by N. Performing simple calculations we obtain from (1.37)

\begin{aligned}i(Nu)_{1} − (Nu)_{2} & = \frac{∂}{∂s(x)}\{φ_{1}(z) + z[ε_{1}\overline{φ′_{1}(z)} + ε_{2}\overline{φ′_{2}(z)}] − \\ & − \frac{m_{3}}{Δ_{0}} \overline{ψ_{1}(z)} + \frac{m_{2}}{Δ_{0}} \overline{ψ_{2}(z)}), \\ i(Nu)_{3} − (Nu)_{4} & = \frac{∂}{∂s(x)}\{φ_{2}(z) + z[ε_{3}\overline{φ′_{1}(z)} + ε_{4}\overline{φ′_{2}(z)}] + \\ & + \frac{m_{2}}{Δ_{0}} \overline{ψ_{1}(z)} − \frac{m_{1}}{Δ_{0}} \overline{ψ_{2}(z)}), \end{aligned} (1.39)

where

\begin{matrix}ε_{1} = \frac{e_{5}m_{2} − e_{4}m_{3}}{2Δ_{0}} , ε_{3} = \frac{e_{4}m_{2} − e_{5}m_{1}}{2Δ_{0}} , \\ ε_{2} = \frac{e_{6}m_{2} − e_{5}m_{3}}{2Δ_{0}} , ε_{4} = \frac{e_{5}m_{2} − e_{6}m_{1}}{2Δ_{0}} , \end{matrix}  (1.40)

and e_{4}, e_{5}, e_{6} and m_{1}, m_{2}, m_{3} are defined by (1.18) and (1.20), respectively.

Using the values m_{1}, m_{2}, m_{3} and e_{4}, e_{5}, e_{6}, we obtain, after obvious calculations, the following new expressions for the coefficients ε_{k} (k = \overline{1, 4}):

δ_{0}ε_{1} = b_{1}(2a_{2} + b_{2}) − d(2c + d), δ_{0}ε_{3} = 2(da_{2} − cb_{2}),
δ_{0}ε_{2} = 2(da_{1} − cb_{1}), δ_{0}ε_{4} = b_{2}(2a_{1} + b_{1}) − d(2c + d),
δ_{0} = (2a_{1} + b_{1})(2a_{2} + b_{2}) − (2c + d)^{2}.

The application of the generalized stress vector we have introduced here will be discussed in our future works.

2. In this section the usual Cauchy–Riemann conditions will be generalized for homogeneous equations of statics in the theory of elastic mixtures.

First we introduce the vectors

φ = (φ_{1}, φ_{2}, φ_{3}, φ_{4}, φ_{5}, φ_{6}, φ_{7}, φ_{8})

and

ψ = (ψ_{1}, ψ_{2}, ψ_{3}, ψ_{4}, ψ_{5}, ψ_{6}, ψ_{7}, ψ_{8}),

where

\begin{matrix}φ_{1} = \frac{∂u_{1}}{∂x_{1}} , φ_{2} = \frac{∂u_{2}}{∂x_{2}} , φ_{3} = \frac{∂u_{3}}{∂x_{1}} , φ_{4} = \frac{∂u_{4}}{∂x_{2}} , \\ φ_{5} = \frac{∂u_{1}}{∂x_{2}} , φ_{6} = \frac{∂u_{2}}{∂x_{1}} , φ_{7} = \frac{∂u_{3}}{∂x_{2}} , φ_{8} = \frac{∂u_{4}}{∂x_{1}} , \\ ψ_{1} = \frac{∂v_{1}}{∂x_{1}} , ψ_{2} = \frac{∂v_{2}}{∂x_{2}} , ψ_{3} = \frac{∂v_{3}}{∂x_{1}} , ψ_{4} = \frac{∂v_{4}}{∂x_{2}} , \\ ψ_{5} = \frac{∂v_{1}}{∂x_{2}} , ψ_{6} = \frac{∂v_{2}}{∂x_{1}} , ψ_{7} = \frac{∂v_{3}}{∂x_{2}} , ψ_{8} = \frac{∂v_{4}}{∂x_{1}} , \end{matrix}  (2.1)

where u = (u_{1}, u_{2}, u_{3}, u_{4}) and v = (v_{1}, v_{2}, v_{3}, v_{4}) are arbitrary differentiable vectors.

Definition. The vectors u and v will be said to be conjugate vectors or to satisfy the generalized Cauchy–Riemann conditions if the following conditions are fulfilled:

aφ = bψ,  (2.2)

where

a = (\begin{matrix}A, & 0 \\ 0, & B \end{matrix}) , b = (\begin{matrix}0, & C \\ D, & 0 \end{matrix}) ,  (2.3)

A = ∥\begin{matrix}a_{1}+b_{1}, & a_{1}+b_{1}− \frac{m_{3}}{Δ_{0}} , & c + d, & c+d+ \frac{m_{2}}{Δ_{0}} \\ a_{1} + b_{1} − \frac{m_{3}}{Δ_{0}} , & a_{1} + b_{1}, & c+d+ \frac{m_{2}}{Δ_{0}} , & c + d \\ c + d, & c+d+ \frac{m_{2}}{Δ_{0}} , & a_{2} + b_{2}, & a_{2}+b_{2}− \frac{m_{1}}{Δ_{0}} \\ c+d+ \frac{m_{2}}{Δ_{0}} , & c + d, & a_{2}+b_{2}− \frac{m_{1}}{Δ_{0}} , & a_{2} + b_{2} \end{matrix} ∥ ,

B = ∥\begin{matrix}−a_{1}, & a_{1} − \frac{m_{3}}{Δ_{0}} , & −c, & c + \frac{m_{2}}{Δ_{0}} \\ −a_{1} + \frac{m_{3}}{Δ_{0}} , & −a_{1}, & −c − \frac{m_{2}}{Δ_{0}} , & c \\ −c, & c + \frac{m_{2}}{Δ_{0}} , & −a_{2}, & a_{2} − \frac{m_{1}}{Δ_{0}} \\ −c − \frac{m_{2}}{Δ_{0}} , & c, & −a_{2} + \frac{m_{1}}{Δ_{0}} , & a_{2} \end{matrix} ∥ ,

C = \frac{1}{Δ_{0}} ∥\begin{matrix}m_{3}, & 0, & −m_{2}, & 0 \\ 0, & −m_{3}, & 0, & m_{2} \\ −m_{2}, & 0, & m_{1}, & 0 \\ 0, & m_{2}, & 0, & −m_{1} \end{matrix}∥ ,

D = \frac{1}{Δ_{0}} ∥\begin{matrix}m_{3}, & 0, & −m_{2}, & 0 \\ 0, & m_{3}, & 0, & −m_{2} \\ −m_{2}, & 0, & m_{1}, & 0 \\ 0, & −m_{2}, & 0, & m_{1} \end{matrix}∥ .

In (2.3) the symbol 0 denotes a four-dimensional matrix all of whose terms are zero.

The matrix equation (2.2) immediately implies that if v a twice continuously differentiable vector, then the vector u satisfies the homogeneous equations (1.3).

Let us now prove

Theorem. Conditions (2.2) remain valid if the vectors φ is replaced by ψ, and the vector ψ by (−φ). This means that the vector u must be replaced by v, and the vector v by (−u).

Proof. By (2.2) we have

ψ = b^{−1}aφ,

where

b^{−1} = (\begin{matrix}0, & m \\ E, & 0 \end{matrix}) ,

\begin{matrix}m = ∥\begin{matrix}m_{1}, & 0, & m_{2}, & 0 \\ 0, & m_{1}, & 0, & m_{2} \\ m_{2}, & 0, & m_{3}, & 0 \\ 0, & m_{2}, & 0, & m_{3} \end{matrix}∥ , \\ E = ∥\begin{matrix}m_{1}, & 0, & m_{2}, & 0 \\ 0, & −m_{1}, & 0, & −m_{2} \\ m_{2}, & 0, & m_{3}, & 0 \\ 0, & −m_{2}, & 0, & −m_{3} \end{matrix}∥ . \end{matrix}  (2.4)

After long but simple calculations we obtain

ab^{−1}a = −b,

which implies that

aψ = −bφ.

From the proven theorem it follows that the vector v is also a solution of system (1.3).

Now multiply the first equality of (2.2) by n_{1}, the fifth equality by (−n_{2}), and combine them. Then, recalling the definition of the operator N by (1.38), we obtain

(Nu)_{1} = \frac{m_{3}}{Δ_{0}} \frac{∂v_{1}}{∂s} − \frac{m_{2}}{Δ_{0}} \frac{∂v_{3}}{∂s} .  (2.5)

Quite similarly, with the operator N taken into account, we find from (2.2) that

\begin{matrix}(Nu)_{2} = \frac{m_{3}}{Δ_{0}} \frac{∂v_{2}}{∂s} − \frac{m_{2}}{Δ_{0}} \frac{∂v_{4}}{∂s} , \\ (Nu)_{3} = −\frac{m_{2}}{Δ_{0}} \frac{∂v_{1}}{∂s} + \frac{m_{1}}{Δ_{0}} \frac{∂v_{3}}{∂s} , \\ (Nu)_{4} = −\frac{m_{2}}{Δ_{0}} \frac{∂v_{2}}{∂s} + \frac{m_{1}}{Δ_{0}} \frac{∂v_{4}}{∂s} . \end{matrix}  (2.6)

Formulas (2.5) and (2.6) can be rewritten as

Nu = m^{−1} \frac{∂v}{∂s} ,  (2.7)

where m is defined by (2.4) and

\begin{matrix}m^{−1} = \frac{1}{Δ_{0}} ∥\begin{matrix}m_{3}, & 0, & −m_{2}, & 0 \\ 0, & m_{3}, & 0, & −m_{2} \\ −m_{2}, & 0, & m_{1}, & 0 \\ 0, & −m_{2}, & 0, & m_{1} \end{matrix}∥ , \\ Δ_{0} = m_{1}m_{3} − m_{2}^{2}. \end{matrix}  (2.8)

By virtue of the above-proven theorem one can easily verify that, along with (2.7), the formula

Nv = −m^{−1} \frac{∂u}{∂s}  (2.9)

is also valid.

Therefore the conjugate vectors u and v satisfy conditions (2.7) and (2.9). We introduce the notation

w = u + iv;  (2.10)

then (2.8) and (2.9) can be written as

Nw = −im^{−1} \frac{∂w}{∂s} ,

which enables us to rewrite (1.34) as

T^{κ}w = [ − im^{−1} + (κ − κ_{N})]\frac{∂w}{∂s} .

For the stress vector we have

T w = (−im^{−1} − κ_{N})\frac{∂w}{∂s} .

The generalized Cauchy–Riemann conditions written as (2.2) are rather cumbersome as they contain eight scalar equations. Let us rewrite (2.2) in a more convenient form. For this we solve the first, third, sixth, and eighth equations of (2.2) with respect to the vector \frac{∂u}{∂x_{1}} and obtain

\frac{∂u}{∂x_{1}} = ∥\begin{matrix}0 & A_{11} − 1, & 0, & −A_{12} \\ 1 − A_{21}, & 0, & A_{22}, & 0 \\ 0 & −A_{31}, & 0, & A_{32} − 1 \\ A_{41}, & 0, & 1 − A_{42}, & 0 \end{matrix}∥ \frac{∂u}{∂x_{2}} +

+ ∥\begin{matrix}A_{11}, & 0, & −A_{12} & 0 \\ 0, & A_{21}, & 0, & −A_{12} \\ −A_{31}, & 0, & A_{32}, & 0 \\ 0, & −A_{41}, & 0, & A_{42} \end{matrix}∥ \frac{∂v}{∂x_{1}} ,  (2.11)

where

A_{11} = \frac{m_{2}(c + d) + m_{3}(a_{2} + b_{2})}{Δ_{0}d_{1}}, A_{21} = \frac{m_{3}a_{2} + m_{2}c}{Δ_{0}d_{2}},

A_{12} = \frac{m_{2}(a_{2} + b_{2}) + m_{1}(c + d)}{Δ_{0}d_{1}}, A_{22} = \frac{m_{2}a_{2} + m_{1}c}{Δ_{0}d_{2}},

A_{31} = \frac{m_{2}(a_{1} + b_{1}) + m_{3}(c + d)}{Δ_{0}d_{1}}, A_{41} = \frac{m_{2}a_{1} + m_{3}c}{Δ_{0}d_{2}},

A_{32} = \frac{m_{1}(a_{1} + b_{1}) + m_{2}(c + d)}{Δ_{0}d_{1}}, A_{42} = \frac{m_{1}a_{1} + m_{2}c}{Δ_{0}d_{2}}.

In quite a similar manner, from system (2.11), which is valid by the aboveproven theorem, we obtain

\frac{∂v}{∂x_{1}} = ∥\begin{matrix}0 & A_{11} − 1, & 0, & −A_{12} \\ 1 − A_{21}, & 0, & A_{22}, & 0 \\ 0 & −A_{31}, & 0, & A_{32} \\ A_{41}, & 0, & 1 − A_{42}, & 0 \end{matrix}∥ \frac{∂v}{∂x_{2}} +

+ ∥\begin{matrix}−A_{11}, & 0, & A_{12} & 0 \\ 0, & −A_{21}, & 0, & A_{22} \\ A_{31}, & 0, & −A_{32}, & 0 \\ 0, & A_{41}, & 0, & −A_{42} \end{matrix}∥ \frac{∂u}{∂x_{1}} .  (2.12)

Using (1.2), (1.18) and the formulas

e_{1} + e_{4} = \frac{a_{2} + b_{2}}{d_{1}} , e_{2} + e_{5} = −\frac{c + d}{d_{1}}, e_{3} + e_{6} = \frac{a_{1} + b_{1}}{d_{1}} ,

obtained from (1.16) and (1.18), we have with obvious transformations

A_{11} + A_{21} = 2, A_{12} + A_{22} = 0,
A_{31} + A_{41} = 0, A_{32} + A_{42} = 2.

We introduce the notation

A_{11} = 1 − ε_{1}, A_{12} = ε_{3}, A_{31} = ε_{2}, A_{32} = 1 − ε_{4},

where ε_{k} (k = \overline{1, 4}) is defined by (1.40). Now, by virtue of (2.11), (2.12) with notation (2.10) taken into account we obtain

\frac{∂w}{∂x_{1}} = P \frac{∂w}{∂x_{2}} ,  (2.13)

where

P = ∥\begin{matrix}i(ε_{1} − 1), & −ε_{1}, & iε_{3}, & −ε_{3} \\ −ε_{1}, & −i(ε_{1} + 1), & −ε_{3}, & −iε_{3} \\ iε_{2}, & −ε_{2}, & i(ε_{4} − 1), & −ε_{4} \\ −ε_{2}, & −iε_{2}, & −ε_{4}, & −i(ε_{4} + 1) \end{matrix}∥ .

The solution of system (2.13) gives

\frac{∂w}{∂x_{2}} = Q \frac{∂w}{∂x_{1}} ,  (2.14)

where

Q = ∥\begin{matrix}i(ε_{1} + 1), & −ε_{1}, & iε_{3}, & −ε_{3} \\ −ε_{1}, & i(1 − ε_{1}), & −ε_{3}, & −iε_{3} \\ iε_{2}, & −ε_{2}, & i(1 + ε_{4}), & −ε_{4} \\ −ε_{2}, & −iε_{2}, & −ε_{4}, & i(1 − ε_{4}) \end{matrix}∥ .

Direct calculations prove that

PQ = QP = E, det P = 1,

where E is the unit four-dimensional matrix.

In what follows by the generalized Cauchy–Riemann condition we mean (2.13) or (2.14).

Finally, note that for the vector u = (u′_{1}, u′_{2}, u′′_{1}, u′′_{2}) we have written a general representation through four analytic functions in for (1.22). To write a general representation for the conjugate vector v = (v_{1}, v_{2}, v_{3}, v_{4}) it is sufficient to compare (1.39) and (2.7). We obtain

\begin{matrix}iv_{1} − v_{2} = m_{1}φ_{1}(z) + m_{2}φ_{2}(z) − \frac{z}{2}[e_{4}\overline{φ′_{1}(z)} + e_{5}\overline{φ′_{2}(z)}] − \overline{ψ_{1}(z)}, \\ iv_{3} − v_{4} = m_{2}φ_{1}(z) + m_{3}φ_{2}(z) − \frac{z}{2}[e_{5}\overline{φ′_{1}(z)} + e_{6}\overline{φ′_{2}(z)}] − \overline{ψ_{2}(z)}. \end{matrix} (2.15)

Now, using (1.22) and (2.15), for the components of the vector w = u + iv we have the following expressions:

w_{1} = m_{1}φ_{1}(z) + m_{2}φ_{2}(z) + \frac{\overline{z}}{2}[e_{4}φ′_{1}(z) + e_{5}φ′_{2}(z)] + ψ_{1}(z),
w_{2} = (−i)\{m_{1}φ_{1}(z) + m_{2}φ_{2}(z) − \frac{\overline{z}}{2}[e_{4}φ′_{1}(z) + e_{5}φ′_{2}(z)] − ψ_{1}(z)\},
w_{3} = m_{2}φ_{1}(z) + m_{3}φ_{2}(z) + \frac{\overline{z}}{2}[e_{5}φ′_{1}(z) + e_{6}φ′_{2}(z)] + ψ_{2}(z)
w_{4} = (−i)\{m_{2}φ_{1}(z) + m_{3}φ_{2}(z) − \frac{\overline{z}}{2}[e_{5}φ′_{1}(z) + e_{5}φ′_{2}(z)] − ψ_{2}(z)\}.

Hence we immediately conclude that

w_{1} + iw_{2} = 2[m_{1}φ_{1}(z) + m_{2}φ_{2}(z)], w_{3} + iw_{4} = 2[m_{2}φ_{1}(z) + m_{3}φ_{2}(z)]

are analytic functions.

The application of the formulas derived in this section will be given in our next works.

References

1. T. R. Steel, Applications of a theory of interacting continua. Q. J. Mech. Appl. Math. 20(1967), part 1, 57–72.

2. D. G. Natroshvili, A. Ya. Jagmaidze, and M. Zh. Svanadze, Some problems of the linear theory of elastic mixtures. (Russian) Tbilisi University Press, 1986.

3. I. N. Vekua and N. I. Muskhelishvili, Methods of the analytic function theory in elasticity. (Russian) Trans. USSR Meeting Theor. Appl. Mekh. (Russian), USSR Acad. Sci. Publ. House, 1962, 310–338.

4. N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. 5th edition. (Russian) Nauka, Moscow, 1966.

(Received 4.12.1995)

Author’s address:

I. Vekua Institute of Applied Mathematics

Tbilisi State University

2, University St., Tbilisi 380043

Georgia