GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 3, 1999, 251-258

FIRST BOUNDARY VALUE PROBLEM OF ELECTROELASTICITY FOR A TRANSVERSALLY ISOTROPIC PLANE WITH CURVILINEAR CUTS

L. BITSADZE

Abstract. The first boundary value problem of electroelasticity for a transversally isotropic plane with curvilinear cuts is investigated. The solvability of a system of singular integral equations is proved by using the potential method and the theory of singular integral equations.

In this paper the first boundary value problem of electroelasticity is investigated for a transversally isotropic plane with curvilinear cuts. The second boundary value problem of electroelasticity for this kind medium was solved in [1] using the theory of analytic functions and singular integral equations. The boundary value problems of electroelasticity for transversally isotropic media were studied in [2] (Ch. VI), while the boundary value problems of elasticity for anisotropic media with cuts were considered in [3], [4]. The uniqueness theorems for boundary value problems of electroelasticity are given in [5], [6]. Here we shall be concerned with the plane problem of electroelasticity (it is assumed that the second component of the threedimensional displacement vector is equal to zero, while the components u_{1}, u_{3} and the electrostatic potential u_{4} depend only on the variables x_{1} and x_{3} [2]).

Let an electroelastic transversally isotropic plane be weakened by curvilinear cuts l_{j} = a_{j}b_{j}, j = 1, …, p. Assume that l_{j}, j = 1, …, p, are simple nonintersecting open Lyapunov arcs. The direction from a_{j} to b_{j} is taken as the positive one on l_{j}. The normal to l_{j} will be drawn to the right relative to motion in the positive direction. Denote by D^{−} the infinite plane with curvilinear cuts l_{j}, j = 1, …, p, l = ∪_{j=1}^{p}l_{j}. Suppose that D^{−} if filled with

some material.

251

1991 Mathematics Subject Classification. 73R05.

Key words and phrases. Electroelasticity, curvilinear cuts, potential method, integral equations.

1072-947X/99/0500-0251$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

We introduce the notation z = x_{1} + ix_{3}, ζ_{k} = y_{1} + α_{k}y_{3}, τ_{k} = t_{1} + α_{k}t_{3}, z_{k} = x_{1} + α_{k}x_{3}, τ = t_{1} + it_{3} and σ_{k} = z_{k} − ζ_{k}.

The basic homogeneous equations of the plane theory of electroelasticity have the form [2]

C(∂_{x})U = 0,  (1)

where C(∂_{x}) = ∥C_{ij}∥_{3×3},

C_{11} = c_{11} \frac{∂^{2}}{∂x_{1}^{2}} + c_{44} \frac{∂^{2}}{∂x_{3}^{2}} , C_{12} = (c_{13} + c_{44})\frac{∂^{2}}{∂x_{1}∂x_{3}} ,
C_{13} = (e_{13}+e_{15})\frac{∂^{2}}{∂x_{1}∂x_{3}} , C_{22} = c_{44} \frac{∂^{2}}{∂x_{1}^{2}} + c_{33} \frac{∂^{2}}{∂x_{3}^{2}} ,
C_{23} = e_{15} \frac{∂^{2}}{∂x_{1}^{2}} + e_{33} \frac{∂^{2}}{∂x_{3}^{2}} , C_{33} = −ε_{11} \frac{∂^{2}}{∂x_{1}^{2}} − ε_{33} \frac{∂^{2}}{∂x_{3}^{2}} ,
U = U(u_{1}, u_{3}, u_{4}),

u_{1}, u_{3} are the components of the displacement vector, u_{4} is the electrostatic potential, c_{11}, c_{44}, … are the constants characterizing the medium under consideration.

The first boundary value problem is formulated as follows: let the boundary values of the displacement vector and electrostatic potential be given on both edges of the arc l_{j}. Further, assume that at infinity we have the principal vector of external forces acting on l, stress, the electrostatic potential and the induction vector. It is required to define the deformed state of the plane.

If we denote by u^{+} (u^{−}) the limit of u on l from the left (right), then the boundary conditions of the problem will take the form

u^{+} = f^{+}, u^{−} = f^{−},  (2)

where f^{+} and f^{−} are the known vector-functions on l of the Hölder class H which have derivatives in the class H^{∗} (for the definitions of the classes H and H^{∗} see [7]) and satisfying, at the ends a_{j} and b_{j} of l_{j}, the conditions

f^{+}(a_{j}) = f^{−}(a_{j}), f^{+}(b_{j}) = f^{−}(b_{j}).

It is obvious that displacement vector discontinuities along the cut l_{j} generate a singular stress field in the medium. Hence it is of interest for us to study the solution behavior in the neighborhood of the cuts.

We seek for a solution of the problem in the form a double-layer potential [8], [9]

U(z) = \frac{1}{π} Im ∑_{k=1}^{3} E_{(k)} ∫_{l} \frac{∂ ln σ_{k}}{∂s} [g(s) + ih(s)]ds +

FIRST BOUNDARY VALUE PROBLEM OF ELECTROELASTICITY 253

+ ∑_{j=1}^{p} V_{j}(z) + A^{∞} Re z + B^{∞} Im z + M^{p+1},  (3)

where s is the arc coordinate,

E_{(k)} = −∥A_{pq}^{(k)}∥_{3×3} ∥h_{pq}∥_{3×3} \frac{ib_{3}}{A_{1}h_{11}}

and Im ∑_{k=1}^{3} E_{(k)} \frac{∂}{∂s} ln σ_{k} denotes a special fundamental matrix that reduces

the first boundary value problem to a Fredholm integral equation of sec-

ond order. It is assumed here that Γ = Im ∑_{k=1}^{3} ∥A_{pq}^{(k)}∥_{3×3} ln σ_{k} is the basic

fundamental matrix for equation (1), Γ is the symmetric matrix whose all elements are one-valued functions on the entire plane and possess a logarithmic singularity. It is easy to show that every element of the matrix Γ is a solution of equation (1) at any x ≠ y. For the expression of the matrix Γ see [8]. The elements of the real symmetric matrix h are defined as follows:

h_{11} = c_{11}c_{44} \frac{1}{\sqrt{a_{1}a_{2}a_{3}}} , α_{k} = i\sqrt{a_{k}}, k = 1, 2, 3,

h_{22} = c_{11}c_{44}C − αB + c_{44}c_{33}A,

−h_{33} = c_{11}ε_{11}C + B[c_{11}ε_{33} + c_{44}ε_{11} + (e_{13} + e_{15})^{2}] + c_{44}ε_{33}A,

h_{23} = c_{11}e_{15}C − B[e_{13}(c_{13} + c_{44}) + c_{13}e_{15} − c_{11}e_{33}] + c_{44}e_{33}A,

α = c_{13}^{2} − c_{11}c_{33} + 2c_{13}c_{44}, A_{1} = \frac{b_{0}}{c_{11}} C + α_{0}B + \frac{b_{3}}{c_{44}} A,

b_{0} = c_{11}(c_{44}ε_{11} + e_{15}^{2}), b_{3} = c_{44}(c_{33}ε_{33} + e_{33}^{2}), h_{12} = h_{13} = 0

(the values of A, B, C are given in [8]).

A^{∞} and B^{∞} are the preassigned column vectors whose components are:

A_{1}^{∞} = a_{11}τ_{11}^{∞} + a_{13}τ_{33}^{∞} + a_{14}τ_{43}^{∞},

A_{3}^{∞} = \frac{1}{2}(b_{11}τ_{13}^{∞} + (b_{33} − a_{14})τ_{41}^{∞}) + ε^{∞},

A_{4}^{∞} = (b_{33} − a_{14})τ_{13}^{∞} − b_{44}τ_{41}^{∞},

B_{1}^{∞} = A_{3}^{∞} − 2ε^{∞}, B_{3}^{∞} = a_{13}τ_{11}^{∞} + a_{33}τ_{33}^{∞} + a_{34}τ_{43}^{∞},

B_{4}^{∞} = a_{14}τ_{11}^{∞} + a_{34}τ_{33}^{∞} − a_{44}τ_{43}^{∞}.

Here τ_{11}, τ_{33}, …, τ_{43} denote the components of the electromechanical stress tensor (see [8]); a_{11}, …, a_{44} are the real constant values which are the combinations of the known piezoelastic, elastic and dielectric constants

a_{11} = \frac{b_{3}}{c_{44}Δ}, a_{44} = \frac{c_{11}c_{33} − c_{13}^{2}}{Δ} , a_{34} = \frac{c_{11}e_{33} − c_{13}e_{13}}{Δ} ,

a_{33} = \frac{c_{11}ε_{33} + e_{13}^{2}}{Δ} , a_{13} = −\frac{ε_{33}c_{13} + e_{33}e_{13}}{Δ} , a_{14} = \frac{c_{33}e_{13} − e_{33}c_{13}}{Δ} ,

Δ = \frac{c_{11}c_{33} − c_{13}^{2}}{c_{33}c_{44}} b_{3} + \frac{1}{c_{33}} (c_{33}e_{13} − e_{33}c_{13})^{2},

b_{11} = \frac{c_{11}ε_{11}}{b_{0}} , b_{44} = \frac{c_{11}c_{44}}{b_{0}} , b_{33} − a_{14} = \frac{c_{11}e_{15}}{b_{0}} ;

α_{k}, k = 1, …, 6, are the roots of the characteristic equation

b_{3}α^{6} + b_{2}α^{4} + b_{1}α^{2} + b_{0} = 0,
b_{1} = c_{11}α_{0} − ε_{11}(c_{13} + c_{44})^{2} + c_{44}(e_{13} + e_{15})^{2} −

− 2e_{15}(e_{13} + e_{15})(c_{13} + c_{44}) + c_{44} \frac{b_{0}}{c_{11}} ,

b_{2} = c_{11} \frac{b_{3}}{c_{44}} + c_{44}α_{0} − ε_{33}(c_{13} + c_{44})^{2} + c_{33}(e_{13} + e_{15})^{2} −

− 2e_{33}(c_{13} + c_{44})(e_{13} + e_{15}).

M^{(j)}, j = 1, …, p + 1, are the unknown real constant vectors to be defined later on; g and h are the unknown real vectors from the Hölder class that have derivatives in the class H^{∗}.

We write

V_{j}(z)= \frac{1}{π} Im ∑_{k=1}^{3} A_{(k)} \frac{(z_{k} − b_{j}^{(k)}) ln(z_{k} − b_{j}^{(k)})−(z_{k} − a_{j}^{(k)}) ln(z_{k} − a_{j}^{(k)})}{b_{j}^{(k)} − a_{j}^{(k)}}M^{j},

a_{j}^{(k)} = Re a_{j} + α_{k} Im a_{j}, b_{j}^{(k)} = Re b_{j} + α_{k} Im b_{j}.

The vector V_{j}(z) satisfies the following conditions:

1. V_{j} has the logarithmic singularity at infinity

V_{j} = \frac{1}{π} Im ∑_{k=1}^{3} A_{(k)}(− ln z_{k} + 1)M^{j} + O(z_{k}^{−1}).

2. By V_{j} is meant a branch which is uniquely defined on the plane cut along l_{j}.

3. V_{j} is continuously extendable on l_{j} from the left and the right, the end points a_{j} and b_{j} inclusive, i.e., we have the equalities

V_{j}^{+}(a_{j}) = V_{j}^{−}(a_{j}), V_{j}^{+}(b_{j}) = V_{j}^{−}(b_{j}),

V_{j}^{+} − V_{j}^{−} = 2 Re ∑_{k=1}^{3} A_{(k)} \frac{τ_{k} − a_{j}^{(k)}}{b_{j}^{(k)} − a_{j}^{(k)}} M^{j}, j = 1, …, p.

To define the unknown density, we obtain, by virtue of (3)–(2), a system of singular integral equations of the normal type

±g(τ) + \frac{1}{π} Im ∑_{k=1}^{3} E_{(k}) ∫_{l} \frac{∂ ln(τ_{k} − ζ_{k})}{∂s} (g + ih) ds +

+ ∑_{j=1}^{p} V_{j}^{±} + A^{∞} Re τ + B^{∞} Im τ + M^{p+1} = f^{±}.

This formula implies

2g(τ) = f^{+} − f^{−} − Re ∑_{j=1}^{p} ∑_{k=1}^{3} A_{(k)} \frac{τ_{k} − a_{j}^{(k)}}{b_{j}^{k} − a_{j}^{(k)}} M^{j},  (4)

\frac{1}{π} ∫_{l} \frac{h(ζ)ds}{ζ − τ} + \frac{1}{π} ∫_{l} K(τ, ζ)h(ζ) ds = Ω(τ),  (5)

where

K(τ, ζ) = −i \frac{∂Θ}{∂s} E + Re ∑_{k=1}^{3} E_{(k}) \frac{∂}{∂s} ln (1 + λ_{k} \frac{\overline{τ} − \overline{ζ}}{τ − ζ} ),

λ_{k} = \frac{1 + iα_{k}}{1 − iα_{k}} , Θ = arg(τ − ζ), Ω(τ) = \frac{1}{2} (f^{+} + f^{−}) −

−\frac{1}{2} ∑_{j=1}^{p}(V_{j}^{+} + V_{j}^{−}) − A^{∞} Re τ − B^{∞} Im τ − M^{p+1} −

−\frac{1}{π} Im ∑_{k=1}^{3} E_{(k}) ∫_{l} \frac{∂}{∂s} ln(τ_{k} − ζ_{k})g(ζ) ds.

Thus we have defined the vector g on l. It is not difficult to verify that g ∈ H, g′ ∈ H^{∗}, g(a_{j}) = g(b_{j}) = 0, Ω ∈ H, Ω′ ∈ H^{∗}. Formula (5) is a system of singular integral equations of the normal type with respect to the vector h. The points a_{j} and b_{j} are nonsingular, while the total index of the class h_{2p} is equal to −3p (for the definition of the class h_{2p} see [7]).

A solution of system (5), if it exists, will be expressed by a vector of the Hölder class that vanishes at the end points a_{j}, b_{j}, and has derivatives in the class H^{∗}.

Next we shall prove that the homogeneous system of equations corresponding to (5) admits on a trivial solution in the class h_{2p}. Let the contrary be true. Assume h^{(0)} to be a nontrivial solution of the homogeneous system corresponding to (5) in the class h_{2p} and construct the potential

U_{0}(z) = \frac{1}{π} Re ∑_{k=1}^{3} E_{(k)} ∫_{l} \frac{∂}{∂s} ln(z_{k} − ζ_{k}) h^{0}(ζ)ds.

Clearly, U_{0}^{+} = U_{0}^{−} = 0 and by the uniqueness theorem we have U_{0}(z) = 0, z ∈ D^{−}. Then T U_{0} = 0, z ∈ D^{−} and

(T U_{0})^{+} − (T U_{0})^{−} = 2∥h_{ij}∥\frac{b_{3}}{A_{1}h_{11}} \frac{∂h^{(0)}}{∂s} = 0.

Therefore, since h^{0}(a_{j}) = 0, we obtain h^{0} = 0, which completes the proof. Thus the homogeneous system adjoint to (5) will have 3p linearly independent solutions σ_{j}, j = 1, …, 3p, in the adjoint class and the condition for system (5) to be solvable will be written as

∫_{l} Ωσ_{j} ds = 0, j = 1, …, 3p.  (6)

Taking into account the latter condition and that

∫_{l} [(T U)^{+} − (T U)^{−}] ds = −2 ∑_{j=1}^{p} M^{j},  (7)

we obtain a system of 3p + 3 algebraic equations with the same number of unknowns with respect to the components of the unknown vector M^{j}.

We shall show that system (6)–(7) is solvable. Assume that the homogeneous system obtained from (6)–(7) has a nontrivial solution M_{j}^{0} = (M_{1j}^{0}, M_{3j}^{0}, M_{4j}^{0}), j = 1, …, p + 1, and construct the potential

U_{0} = \frac{1}{π} Im ∑_{k=1}^{3} E_{(k)} ∫_{l} \frac{∂}{∂s} ln(z_{k} − ζ_{k}) (g_{0} + ih_{0}) ds + ∑_{j=1}^{p} V_{j}^{0} + M_{P +1}^{0},

where

g_{0} = − Re ∑_{k=1}^{3} A_{(k)} \frac{τ_{k} − a_{p}^{(k)}}{b_{p}^{(k)} − a_{p}^{(k)}} M_{p}^{(0)}, τ ∈ l_{p},

V_{j}^{(0)} = \frac{1}{π} Im ∑_{k=1}^{3} A_{(k)} \frac{(z_{k} − b_{j}^{(k)}) ln(z_{k} − b_{j}^{(k)}) − (z_{k} − a_{j}^{(k)} ln(z_{k} − a_{j}^{(k)})}{b_{j}^{(k)} − a_{j}^{(k)}} M_{j}^{(0)}.

It is obvious that U_{0}^{+} = U_{0}^{−} = 0. Applying the formulas ∑_{j=1}^{P} M_{j}^{0} = 0 and

∫_{l_{j}} ((T U_{0})^{+} − (T U_{0})^{−})ds = −2M_{j}^{0} = 0, j = 1, …, p, U_{0}(∞) = M_{0}^{p+1} = 0,

we obtain M_{j}^{0} = 0, which contradicts the assumption. Therefore system (6)–(7) has a unique solution.

For M_{0}^{(j)} system (5) is solvable in the class h_{2p}. The solution of the problem posed is given by potential (3) constructed using the solution h of system (5) and the vector g.

Let us consider a particular case with a rectilinear cut on the segment ab of the real axis. Assuming that the principal vector of external forces, displacement, electrostatic potential and induction vector vanish at infinity, we obtain

U(z) = \frac{1}{2π} Im ∑_{k=1}^{3} E_{(k)} ∫_{a}^{b} \frac{u^{+}(t) − u^{−}(t)}{t − z_{k}} dt +

+ \frac{1}{2π} Im ∑_{k=1}^{3} E_{(k)}X(z_{k}) ∫_{a}^{b} \frac{u^{+}(t) − u^{−}(t)}{X^{+}(t)(t − z_{k})} dt,

where X(z_{k}) = \sqrt{(z_{k} − a)(b − z_{k})}.

References

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9. L. P. Bitsadze, Generalized Cauchy–Riemann conditions in the twodimensional theory of electroelasticity. (Russian) Reports Enlarged Sessions Seminar I. Vekua Inst. Appl. Math. (Dokl. Rassh. Zosed. Sem. Inst. Prikl. Mat. Tbiliss. Gos. Univ.) 8(1993), No. 2, 21–25.

(Received 21.04.1997)

Author’s address:

I. Vekua Institute of Applied Mathematics

Tbilisi State University

2, University St., Tbilisi 3800043

Georgia