Georgian Mathematical Journal 1(94), No. 5, 459-467

BOUNDARY VALUE PROBLEMS OF ELECTROELASTICITY WITH CONCENTRATED SINGULARITIES

T. BUCHUKURI AND D. YANAKIDI

Abstract. We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighbourhood of infinity or in the neighbourhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.

The basic equations of the static state of an electroelastic medium are written in terms of displacement and electric potential components as follows [1, 2]:

c_{ijkl} \frac{∂^{2}u_{k}}{∂x_{j}∂x_{l}} + e_{kij} \frac{∂^{2}φ}{∂x_{j}∂x_{k}} + F_{i} = 0,  (1)
−e_{ikl} \frac{∂^{2}u_{k}}{∂x_{i}∂x_{l}} + E_{ik} \frac{∂^{2}φ}{∂x_{i}∂x_{k}} = 0, i = 1, 2, 3,  (2)

where u = (u_{1}, u_{2}, u_{3}) is a displacement vector, φ is an electric field potential, c_{ijkl}, e_{kij}, E_{ik}, are constants, F = (F_{1}, F_{2}, F_{3}) is mass force. It is assumed that the constants c_{ijkl}, e_{kij} , E_{ik} satisfy the conditions

c_{ijkl} = c_{jikl} = c_{klij}, e_{kij} = e_{kji}, E_{ij} = E_{ji}, i, j, k, l = 1, 2, 3. (3)

System (1), (2) can be written in the matrix form. We introduce the operator

A(∂x) = ∥A_{ij}(∂x)∥_{4×4}, A_{ik}(∂x) = c_{ijkl} \frac{∂^{2}}{∂x_{j}∂x_{l}} , i, k = 1, 2, 3,

A_{i4}(∂x) = e_{kij} \frac{∂^{2}}{∂x_{k}∂x_{j}} , i = 1, 2, 3,  (4)

459

1991 Mathematics Subject Classification. 73C30, 73C15, 73C35.

1072-947X/94/0900-0459$07.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

A_{4k}(∂x) = −e_{ikl} \frac{∂^{2}}{∂x_{i}∂x_{l}} , k = 1, 2, 3, A_{44}(∂x) = E_{ik} \frac{∂^{2}}{∂x_{i}∂x_{k}} .

Introducing the four-component vectors U = (U_{1}, U_{2}, U_{3}, U_{4}) = (u_{1}, u_{2}, u_{3}, φ) and χ = (F_{1}, F_{2}, F_{3}, 0), system (1), (2) is rewritten as

A(∂x)U + χ = 0.  (5)

It is easy to show that the operator A(∂x) is a second order homogeneous operator of the elliptic type.

Assume that an electroelastic medium occupies a bounded domain Ω^{+} of the three-dimensional space ℝ^{3}. Let Ω^{−} = ℝ^{3}\Ω̄^{+}, S = ∂Ω^{+} = ∂Ω^{−}.

Assume that the surface S is partitioned into four parts: S_{11}, S_{12}, S_{13}, S_{14}, where S_{1i} ∩ S_{1j} = ∅, i ≠ j and ∪_{i=1}^{4}S_{1i} = S. Also assume that we have another partitioning of S into two parts: S_{21} and S_{22}; S_{21} ∩ S_{22} = ∅, S_{21} ∪ S_{22} = S.

We shall consider a boundary value problem for system (5) when the following conditions are given: displacements on the part S_{11} of the boundary S, boundary mechanical stresses on the part S_{12}, normal components of the displacement vector and tangential components of the mechanical stress vector on the part S_{13}, and normal components of the boundary stress vector and tangential components of the displacement vector on the part S_{14}. These conditions can be written in the form

u_{i}|_{S_{11}}(y) = f_{i}(y), i = 1, 2, 3;  (6)
τ_{ji}n_{j}|_{S_{12}}(y) = g_{i}(y), i = 1, 2, 3;  (7)

u_{i}n_{i}|_{S_{13}}(y) = f^{(n)}(y), (τ_{ji}n_{j} − n_{k}τ_{jk}n_{j})|_{S_{13}}(y) = g_{i}^{(τ)}(y), i = 1, 2, 3,(8)

n_{i}τ_{ji}n_{j}|_{S_{14}}(y) = g^{(n)}(y), (u_{i} − n_{k}u_{k}n_{i})|_{S_{14}}(y) = f_{i}^{(τ)}(y), i = 1, 2, 3.(9)

Here f_{i}, g_{i}, f^{(n)}, g^{(n)}, f_{i}^{(τ)}, g_{i}^{(τ)} are the known functions.

In additition to the above ”mechanical” boundary conditions we should also be given ”electric” conditions

φ|_{S_{21}}(y) = ψ(y),  (10)
D_{i}n_{i}|_{S_{22}}(y) = h(y).  (11)

In the above formulas τ_{ji} denotes the mechanical stress tensor, D_{i} the electric induction vector. These values are related with the unknown displacement vector U and the electric potential φ by the relations

τ_{ij} = \frac{1}{2}c_{ijkl}(\frac{∂u_{k}}{∂x_{l}} + \frac{∂u_{l}}{∂x_{k}} ) + e_{kij} \frac{∂φ}{∂x_{k}} ,  (12)

D_{i} = \frac{1}{2}e_{ikl}(\frac{∂u_{k}}{∂x_{l}} + \frac{∂u_{l}}{∂x_{k}} ) − E_{ik} \frac{∂φ}{∂x_{k}} .  (13)

We shall apply the term ”the basic internal regular boundary value problem of electrostatics” to the following problem:

Find in the domain Ω^{+} the four-component vector U = (u, φ) of the class C^{2}(Ω^{+}) ∩ C^{1}(Ω̄^{+}) which is a solution of system (5) and satisfies conditions (6)-(11). Denote this problem by (E)^{+}.

The external boundary value problem (E)^{−} is formulated absolutely in the same manner. In that case the vector U is sought for in the domain Ω^{−}.

From the basic problem (E)^{±} one can obtain, as the particular case, various problems. Denote by (p, q)^{±} the problem (E)^{±} when S_{1p} = S (S_{1i} = ∅, if i ≠ p) and S_{2q} = S (S_{2i} = ∅, if i ≠ q).

Denote by (E)_{0}^{±} the problem (E)^{±} with the homogeneous boundary conditions f_{i} = 0, g_{i} = 0, f_{i}^{(τ)} = 0, g_{i}^{(τ)} = 0, f^{(n)} = 0, g^{(n)} = 0, ψ = 0, h = 0, when χ = 0. The notation (p, q)_{0}^{±} has the same meaning as above.

The following uniqueness theorem is valid:

Theorem 1. If U = (u, φ) is a solution of the problem (E)_{0}^{±}, then it has the form

U_{i}(x) = ε_{ijk}a_{i}x_{k} + b_{i}, φ = φ_{0}, i = 1, 2, 3,

where φ_{0}, a_{i}, b_{i} are arbitrary constants and ε_{ijk} is the Levy-Civita symbol. In that case if S_{11} ≠ ∅ and is not a subset of some plane, then a_{i} = 0, b_{i} = 0, (i = 1, 2, 3); if S_{21} ≠ ∅, then φ_{0} = 0.

The proof is based on the Green formula

∑_{i,k=1}^{4} ∫_{Ω^{+}} U_{i}(x)A_{ik}(∂x)V_{k}(x) dx = ∑_{i,k=1}^{4} ∫_{S} U_{i}(y)T_{ik}(∂y, n)V_{k}(y) d_{y}S −

− ∫_{Ω^{+}} E(U, V)(x) dx,  (14)

where T_{ik}(∂y, n) are the components of the boundary stress operator

T_{ik}(∂y, n) = c_{ijkl}n_{j}(y)\frac{∂}{∂y_{l}} , i, k = 1, 2, 3,

T_{i4}(∂y, n) = e_{kij}n_{j}(y)\frac{∂}{∂y_{k}} , i = 1, 2, 3,  (15)

T_{4k}(∂y, n) = −e_{ikl}n_{i}(y)\frac{∂}{∂y_{l}} , k = 1, 2, 3, T_{44}(∂y, n) = E_{ik}n_{i}(y)\frac{∂}{∂y_{k}};

\begin{matrix}E(U, V)=c_{ijkl} \frac{∂u_{i}}{∂x_{j}} \frac{∂v_{k}}{∂x_{k}} + e_{kij} \frac{∂u_{i}}{∂x_{j}} \frac{∂ψ}{∂x_{k}} − e_{ikl} \frac{∂φ}{∂x_{i}} \frac{∂v_{k}}{∂x_{l}} + E_{ik} \frac{∂φ}{∂x_{i}} \frac{∂ψ}{∂x_{k}} , \\ U = (u, φ), V = (v, φ). \end{matrix}(16)

If U = (u, φ) is a solution of the problem (E)_{0}^{+}, then

∀y ∈ S : ∑_{i,k=1}^{4} U_{i}(y)T_{ik}(∂y, n)U_{k}(y) = 0.  (17)

Applying (17), from the formula (14) we conclude that

∀x ∈ Ω^{+} : E(U, U) = 0,  (18)

where U is a solution of the problem (E)_{0}^{+}. From (18) we obtain

∀x ∈ Ω^{+} : \frac{∂u_{i}(x)}{∂x_{j}} + \frac{∂u_{j}(x)}{∂x_{i}} = 0, \frac{∂φ(x)}{∂x_{i}} = 0, i, j = 1, 2, 3. (19)

This formula immediately implies that all the statements of Theorem 1 are valid.

Theorem 2. If U is a solution of the problem (E)_{0}^{−} and satisfies the conditions

U_{i}(x)=O(|x|^{−1}), i = 1, 2, 3, 4, \frac{∂U_{i}(x)}{∂x_{j}} =o(|x|^{−1}), j = 1, 2, 3, (20)

in the neighbourhood of a point at infinity, then ∀x ∈ Ω^{−} : U(x) = 0.

The theorem is proved by the reasoning used in proving Theorem 1 for the domain Ω_{r} ≡ Ω^{−}\\overline{B(0, r)} where B(0, r) is the ball with centre at the point 0 and radius r and with passage to the limit as r → ∞.

We can formulate Theorem 2 more precisely, since it turns out that conditions (20) can be considerably weakened.

The results of [3] imply, as the particular case,

Lemma 1. Let U be a solution of the system

A(∂x)U = 0  (5_{0})

of the class C^{2}(Ω^{−}) in the domain Ω^{−} and one of the conditions below

lim_{r→∞} r^{−(p+4)}  ∫_{B(0,r)\B(0,r/2)} |U(y)| dy = 0,  (21)

U(y) = o(|y|^{p+1}), |y| → ∞,  (22)
∫_{Ω^{−}} \frac{|U(y)|}{1 + |y|^{p+4}} dy < +∞  (23)

be fulfilled for some nonnegative integer number p. Then for any nonnegative integer q we have the representation

\begin{matrix}∀x ∈ Ω^{−} : U_{j}(x) = ∑_{|α|≤p} c_{j}^{(α)}x^{α} + ∑_{|β|≤q} d_{k}^{(β)}∂^{β}Φ_{jk}(x) + ψ_{j}(x), \\ j = 1, 2, 3, 4, \end{matrix} (24)

c_{j}^{(α)} =const, d_{k}^{(β)} =const, ψ_{j} ∈C^{2}(Ω^{−}), and in the neighbourhood of infinity

∂^{γ}ψ_{j}(x) = O(|x|^{−2−|γ|−q}).  (25)

Here Φ = ∥Φ_{jk}∥_{4×4} is the matrix of fundamental solutions of equations (5_{0}).

Theorems 2 and 3 imply directly the following uniqueness theorem:

Theorem 3. Let U be a solution of the problem (E)_{0}^{−} and satisfy at infinity the condition

U(x) = o(1).  (26)

Then ∀x ∈ Ω^{−} : U(x) = 0.

Consider now the boundary value problem: Find in the domain Ω^{−} the solution U of the Problem (E)^{−}, satisfying at infinity the condition

U(x) = o(|x|^{p+1}).  (27)

This problem will be denoted by (E_{p})^{−} and the corresponding homogeneous problem by (E_{p})_{0}^{−}.

Let K(p) be the number of linearly independent polynomial solutions of system (5_{0}) with a degree not higher that p. Repeating the reasoning given in [4] for an equation of classical elasticity we can readily prove that

K(p) = 4 [(\begin{matrix}p + 2 \\ 2 \end{matrix}) + (\begin{matrix}p + 1 \\ 2 \end{matrix})] = 4(p + 1)^{2}.  (28)

Now it is easy to prove the following

Lemma 2. Let the homogeneous problem (E)_{0}^{−} has a solution satisfying condition (26) (it will be trivial by virtue of Theorem 3). Then the homogeneous problem (E_{p})_{0}^{−} has at most K(p) = 4(p + 1)^{2} linearly independent solutions.

Proof. Let U^{(1)}, …, U^{(r)} (r > 4(p + 1)^{2}) be solutions of the homogeneous problem (E_{p})_{0}^{−}. By virtue of Lemma 1 U^{(i)} = P^{(i)} + V^{(i)} where P^{(i)} is a polynomial solution of system (5_{0}) of a degree not higher than p and V^{(i)} is a solution of system (5_{0}) satisfying condition (26). Then by the condition of the lemma there exist numbers c_{i} not all equal to zero such that ∑_{i=1}^{r} c_{i}P^{(i)} = 0. Consider the vector W ≡ ∑ c_{i}U^{(i)} = ∑ c_{i}V^{(i)}. W is a linear combination of U^{(i)} and hence will be a solution of the homogeneous problem (E_{p})_{0}^{−}, but at the same time W is a linear combination of solutions V^{(i)}, therefore satisfying condition (26), and hence, on account of Theorem 3, W = 0. Thus solutions U^{(i)} are linearly dependent. ■

Lemma 2 immediately yields

Corollary 1. If the nonhomogeneous problem (E)^{−} has the unique solution for any f_{i}, g_{i}, f^{(n)}, g^{(n)}, f_{i}^{(τ)}, g_{i}^{(τ)}, ψ, and h belonging to the class C^{∞}, then the homogeneous problem (E_{p})_{0}^{−} has exactly K(p) = 4(p + 1)^{2} linearly independent solutions, while the nonhomogeneous problem (E_{p})^{−} has the solution U for arbitrary boundary data and this solution is represented as U = U_{0} + U^{(p)}, where U_{0} is the solution of the problem (E)^{−} satisfying condition (26) and U^{(p)} is an arbitrary solution of the problem (E_{p})_{0}^{−}.

The problem (E)^{+} is treated with sufficient completeness in [5]. This paper also contains the proof of the existence of a generalized solution in Sobolev spaces. Using the well-known regularization theorems [6], from these results we easily obtain the existence of classical solutions for sufficiently smooth S and boundary data. In particular, we have

Lemma 3. Let the boundary S of the domain Ω^{+} and the boundary data belong to the class C^{∞}(Ω̄^{+}). Then:

problems (1.1)^{+} and (4.1)^{+} have the unique solution of the class C^{∞}(Ω̄^{+}); the problem (3.1)^{+} has the unique solution of the class C^{∞}(Ω̄^{+}) if S is

not the rotation surface;

the problem (2.1)^{+} has a solution of the class C^{∞}(Ω̄^{+}) if and only if the conditions

∫_{S} g_{i}(y) d_{y}S = 0, i = 1, 2, 3,  (29)

∫_{S} ε_{ijk}y_{j}g_{k}(y) d_{y}S = 0, i = 1, 2, 3,  (30)

are fulfilled to within a term of the form U = (u_{1}^{(0)}, u_{2}^{(0)}, u_{3}^{(0)}, 0) where

U_{i}^{(0)}(x) = ε_{ijk}x_{j}a_{k} + b_{i},  (31)

a_{i} and b_{i} are arbitrary constants (i = 1, 2, 3);

problems (1.2)^{+} and (4.2)^{+}, and also problem (3.2)^{+} if S is not the rotation surface, have a solution if and only if

∫_{S} h(y) d_{y}S = 0;  (32)

the solution is defined to within a term of the form U = (0, 0, 0, φ_{0}) where φ_{0} is an arbitrary number;

problem (2.2)^{+} has a solution if and only if conditions (29), (30), (32) are fulfilled; the solution is defined to within a term of the form U = (u, φ_{0}) where u is written as (31) and φ_{0} is an arbitrary number.

We are interested in investigating not smooth solutions of the problems (E)^{+}, but such solutions that at some given points have singularities not higher than given power orders.

Let x^{(1)}, …, x^{(r)} be points lying in the domain Ω^{+}, M_{r} ≡ {x^{(1)}, …, x^{(r)}}. Consider the problem with concentrated singularities: Find the solution

U of equation (5) which belongs to the class C^{2}(Ω^{+}\M_{r}) ∩ C^{1}(Ω̄^{+}\M_{r}), satisfies the boundary conditions (6)-(11) and, in the neighbourhood of the point x^{(i)}, the condition |U(x)| ≤ \frac{c}{|x−x^{(i)}|^{p_{i}}} , i = 1, …, r, where p_{i} are given nonnegative numbers. Denote this problem by (E)_{cs}^{p}.

The investigation of this problem is largely based on one proposition following from the theorem proved in a more general situation in [3].

Lemma 4. Let Ω ⊂ ℝ^{3}, y ∈ Ω, U be a solution of (5_{0}) of the class C^{2}(Ω\{y}) in the domain Ω\{y} and, for some c > 0 and p ≥ 0, |U(x)| ≤ \frac{c}{|x−y|^{p}} . Then

U_{j}(x) = U_{j}^{(0)}(x) + ∑_{k=1}^{4} ∑_{|α|≤[p]−1} a_{k}^{(α)}∂^{α}Φ_{jk}(x − y), j = 1, …, 4,

where U^{(0)} is the solution of system (5_{0}) of the class C^{2}(Ω), α = (α_{1}, α_{2}, α_{3}) is the multiindex, [p] is the integer part of the number p, a_{k}^{(α)} = const, Φ = ∥Φ_{jk}∥_{4×4} is the matrix of fundamental solutions of equation (5).

Using this lemma, by a reasoning analogous to that from [4], we prove

Lemma 5. Let F_{p} be a finite-dimensional space stretched onto the system of vectors {∂^{α}Φ^{(k)}(· − x^{(i)}); k = 1, 2, 3, 4; |α| ≤ [p_{i}] − 1}, where Φ^{(k)} = (Φ_{1k}, Φ_{2k}, Φ_{3k}, Φ_{4k}). Then dim F_{p} = 4 ∑_{i=1}^{r}[p_{i}]^{2}.

Proof. We assume V = (v, φ) and introduce the notation

ϵ_{ij}^{(V)} = \frac{1}{2}(\frac{∂v_{i}}{∂x_{j}} + \frac{∂v_{j}}{∂x_{i}} ), E^{(V)} = −\frac{∂φ}{∂x_{k}} ,

τ_{ij}^{(V)} = c_{ijkl}ϵ_{kl}^{(V)} − e_{kij}E_{k}^{(V)}, D_{i}^{(V)} = e_{ikl}ϵ_{kl}^{(V)} + E_{ik}E_{k}^{(V)}.

Let {ω^{(k)}; k = 1, …, dim F_{p}} be base spaces F. Denote by U^{(k)} a solution of system (5_{0}) satisfying the boundary conditions (6)-(11) for

f_{i} = ω_{i}^{(k)}, g_{i} = τ_{ji}^{(ω^{(k)})}n_{j}, f^{(n)} = ω_{i}^{(k)}n_{i},
g_{i}^{(τ)} = τ_{ji}^{(ω^{(k)})}n_{j} − n_{l}τ_{jl}^{(ω^{(k)})}n_{j}n_{i}, g^{(n)} = n_{i}τ_{ji}^{(ω^{(k)})}n_{j},
f_{i}^{(τ)} = τ_{ji}^{(ω^{(k)})}n_{j} − n_{l}τ_{jl}^{(ω^{(k)})}n_{j}n_{i}, ψ = ω_{4}^{(k)}, h = D_{i}^{(ω^{(k)})}n_{i}.

Consider the vectors V^{(k)} = U^{(k)} − ω^{(k)}. Obviously, the vector V^{(k)} is a solution of the homogeneous problem (E_{p})_{0}^{+}. We shall prove that the system of vectors {V^{(k)}, ψ^{(i)}; k = 1, …, dim F_{p}; i = 1, …, q}, where {ψ^{(i)}} a linearly independent system of solutions of the homogeneous problem (E)_{0}^{+}, is linearly independent. Indeed, if

∑_{k=1}^{dim F_{p}} c_{k}V^{(k)} + ∑_{i=1}^{q} d_{i}ψ^{(i)} = 0,

then

∑_{k=1}^{dim F_{p}} c_{k}ω^{(k)} = ∑_{k=1}^{dim F_{p}} c_{k}U^{(k)} + ∑_{i=1}^{q} d_{i}ψ^{(i)}.

By virtue of (28) ∑ c_{k}ω^{(k)} = 0 and therefore c_{k} = 0 and ud_{i} = 0. Now from Lemma 2 we obtain the proof of Theorem 4. ■

Theorem 4. If the problem (E)^{+} has a solution for any boundary data of class C^{∞}, then the homogeneous problem (E_{0})_{cs}^{p} has exactly 4 ∑_{i=1}^{r}[p_{i}]^{2} + q linearly independent solutions, where q is the number of linearly independent solutions of the problem (E_{0})^{+} and [p_{i}] is the integer part of the number p_{i}.

This theorem readily implies

Corollary 2. The homogeneous problems (1.1)_{cs}^{p} and (4.1)_{cs}^{p} have exactly 4 ∑_{i=1}^{r}[p_{i}]^{2} linearly independent solutions and if the boundary S is not the surface of rotation, then Problem (3.1)_{cs}^{p}, too, has the same number of linearly independent solutions.

Similar theorems hold for the other problems as well.

References

1. R. Holland and E,P. EerNisse, Design of resonant piezoelectric devices. M.I.T. Press, Cambridge–London, 1969.

2. A.F. Ulitko, To the theory of oscillation of piezoceramic bodies. (Russian) Teplov. napryazheniya v elementakh konstruktsii. Respubl. mezhvedomstv. sbornik, Kiev 15(1975), 110-114.

3. T.V. Buchukuri and T.G. Gegelia, Qualitative properties of solutions of the basic equations of elasticity near singular points. (Russian) Trudy Tbil. Mat. Inst. Razmadze, 90(1988), 40-67.

4. —–, Boundary value problems of the elasticity theory with concentrated singularities. (Russian) Differentsial’nye Uravneniya 25 (1989), No.10, 38-50.

5. A.V. Belokon and I.I. Vorovich, Some mathematical questions of the theory of electroelastic bodies. Topical problems of mechanics of deformable bodies. (Russian) Dnepropetrovsk University Press, 1979.

6. G. Fichera, Existence theorems in elasticity. Handb. d. Physik, Bd. VI/2, No.3 Springer–Verlag, Heidelberg, 1973.

(Received 30.09.1993)

Authors’ address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z. Rukhadze St., Tbilisi 380093 Republic of Georgia