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Latin American Journal of Solids and Structures

On-line version ISSN 1679-7825

Lat. Am. j. solids struct. (Online) vol.8 no.1 Rio de Janeiro  2011 

On asymptotic analysis of spectral problems in elasticity



S.A. NazarovI; J. SokolowskiII,*

IInstitute of Mechanical Engineering Problems, V.O., Bolshoi pr., 61, 199178, St. Petersburg - Russia
IIUniversité H.Poincaré, Nancy 1, Départ. de Mathematiques, B.P. n. 239, 54506 Vandoeuvre les Nancy Cédex - France




The three-dimensional spectral elasticity problem is studied in an anisotropic and inhomogeneous solid with small defects, i.e., inclusions, voids, and microcracks. Asymptotics of eigenfrequencies and the corresponding elastic eigenmodes are constructed and justified. New technicalities of the asymptotic analysis are related to variable coefficients of differential operators, vectorial setting of the problem, and usage of intrinsic integral characteristics of defects. The asymptotic formulae are developed in a form convenient for application in shape optimization and inverse problems.

Keywords: singular perturbations; spectral problem; asymptotics of eigenfunctions and eignevalues; elasticity boundary value problem.




1.1 Asymptotic analysis of eigenvalues in singularly perturbed domains

In the paper asymptotic analysis of eigenvalues and eigenfunctions is performed with respect to singular perturbations of geometrical domains (see Fig. 1). By singular perturbations of the boundary it is understood e.g., the creation of the new parts of the boundary due to the nucleation of small voids.

The case of low frequencies is considered for elasticity spectral problems in three spatial dimensions. The results established here can be directly used in some applications, for example in inverse problems of identification of small defects in the body based on the observation of elastic eigenmodes. Compared to the existing results in the literature, the technical difficulties of the present paper mainly concern vectorial setting of boundary value problems, anisotropy of physical properties, and variable coefficients of differential operators, i.e., inhomogeneity of elastic materials. Exisiting results on elasticity problems with singular perturbations of boundaries (see monographs [36,368] and [13]) deal with homogeneous, mainly isotropic elastic bodies. For a system of differential equations, an asymptotic analysis is required to be much more elaborated and direct adopting of the methods proper for scalar equations may lead to an unfortunate mistake (cf. [18] and corrections in [1]).

The known results are given in particular for singular perturbations of isolated points of the boundary (small holes in the domain, see [14], [15], [5], [1], [13], [15] and others), perturbations of straight boundaries including perturbations by changing the type of boundary conditions (cf. [2]-[3]), and the dependence of the obtained results in more general geometrical domains on the curvature is clarified in [8,22,23] in the case of scalar equations. The most of attention is paid in the present paper to derivation of explicit formulae for solutions and extraction of principal characteristics of elastic fields and defects which influence these formulae. To this end, we employ matrix/column notation, use the notion of elastic polarization matrix (tensor), and perform certain additional technical calculations which are not needed in the case of homogeneous, isotropic elastic materials.

Small defects can be regarded as singular perturbations of the interior piece of the boundary of the body. In this way we can consider e.g., the finite number of isolated points which approximate small cavities. More generally, by means of asymptotic analysis we can model the creation of caverns, i.e., some piece of material is taken off from the elastic body. We can also fill the cavern with some other elastic material and model such a phenomenon by formation of one or more inclusions in the body.

Roughly speaking, the influence of a substantial change of local properties of the elastic body cannot be analysed by the classical tools of the shape sensitivity analysis or any other type of sensitivity analysis, but it requires the application of asymptotic methods. Especially, such methods turn out to be of importance for the microcracks, since the microcrack implies the creation of a new portion of internal boundary in the body, which cannot be taken into account in the framework of classical sensitivity analysis based on regular perturbations of the coefficients and of the boundary. The asymptotic methods seem to be the only avalaible tool to perform the efficient analysis of solutions, eigenvalues and eigenfunctions, and of shape functionals, in general setting.

We leave aside an important and still not completed topic related to the so-called concentrated masses. Since the pioneering work [37] of E. Sanchez-Palencia, a lot of attention has been paid to mathematical analysis of vibrations of elastic bodies, with small parts wich are very heavy (e.g., pellets in an aspic or in a meat-jelly); see papers [4,9,12,25,30,33,39], as well as the monographs [34,38] in an incomplete list. Such problems are the best examples of the topping role of the boundary layer effect. Although we analyse the boundary layers in details, the purposes of the present paper is essentially different so that we cannot mutually serve for an analysis of concentrated masses.

1.2 Preliminaries, anisotropic inhomogeneous elastic body

Let us consider in three spatial dimensions the elasticity problem for an elastic body Ω, written in the matrix/column notation, see e.g., [10], [19] for more details,

where () is a symmetric positive definite matrix function in of size 6×6, with measurable or smooth elements, consisting of the elastic material moduli (the Hooke's or stiffness matrix) and (x) is (6×3)-matrix of the first order differential operators,

u = (u1, u2, u3)T is displacement column, n = (n1,n2,n3)T is the unit outward normal vector on ∂Ω and T stands for transposition. In this notation the strain ε(u;x) and stress σ(u;x) (x) = (x) u(x) columns are given respectively by

The factors 2-1/2 and imply that the norms of strain and stress tensors coincide with the norms of columns (4) and (5), respectively. From the latter property in the matrix/column notation, any orthogonal transformation of coordinates in 3 gives rise to orthogonal transformations of columns (4) and (5) in 6 (cf. [[18];Ch. 2]).

Remark 1.1 The strains and the stresses degenerate on the space of rigid motions,


This subspace plays a critical role in many questions in the elasticity theory, it appears also in the so-called polynomial property [26,27] (see also [21]).

The following equalities can be veried by a direct computation,

where N and N are the unit and null (N × N)-matrices, respectively.

The boundary load gΩ is supposed to be self equilibrated in order to assure the existence of a solution to the elasticity problem,



Consider inhomogenuous anisotropic elastic bodyΩ3 with the Lipschitz boundary ∂Ω. Spectral problems for the body are formulated in a fixed Cartesian coordinate system x = (x1,x2,x3)T, and in the matrix notation.

We assume that the matrix of elastic moduli is a matrix function of the spatial variable x 3, symmetric and positive definite for x ∈ Ω∂Ω. The problem on eigenvibrations of the bodyΩ takes the form

where γ > 0 is the material density, λ is an eigenvalue, the square of eigenfreguency. The part Γ of the surface ∂Ω is clamped, and the first boundary condition is prescribed on the traction free remaining part Σ = \ ∂Ω\ of the surface. We denote by (Ω;Γ)3 the energy space, i.e., the subspace of the Sobolev space H1(Ω)3 with null traces on the subset Γ. The variational formulation of problem (10)-(11) reads:

Find a non trivial function u (Ω;Γ)3 and a number λ such that for all test functions v (Ω;Γ)3 the following integral identity is verified

where (,)Ω is the scalar product in the Lebesgue space L2(Ω).

If the stiffness matrix and the density γ are measurable functions of the spatial variables x, and in addition uniformly positive definite and bounded, then the variational problem (12) admits normal positive egenvalues λp, which form the sequence

taking into account its multiplicities, and the corresponding eigenfunctions u(p), the elastic vibration modes, are subject to the orthogonality and normalization conditions

where δp,q is the Kronecker symbol.

In the sequel it is assumed that elements of the matrix and the density γ are smooth functions in Ω, continuous up to the boundary. In such the case Ω is called a smooth inhomogenuous body. For such a body the elastic modes u(p) are smooth functions in the interior of Ω, and up to the boundary in the case of the smooth surface ∂Ω. We have also the equivalence between the variational form and the differential form (10)-(11) of the spectral problem. We require only the interior regularity of elastic modes in the sequel, in any case the elastic modes have singularities on the collision line and therefore, are excluded from the Sobolev space H2(Ω)3.

Along with the smooth inhomogenuous bodyΩ let us consider a bodyΩh with defects; here h > 0 stands for a small dimensionless geometrical parameter, which describes the relative size of defects. Actually, we select in the interior of Ω the points P1,...,PJ and denote by ω1,...,ωj elastic bodies bounded by the Lipschitz surfaces ω1,...,ωj , furthemore, for the sake of simplicity we assume that the origin belongs to ωj, j = 1,...,J. The body with defects is defined by


The stiffness matrix and the density of the composite body take the form

The matrices and (j) as well as the scalars γ and γ(j) are different from each other, i.e., are inhomogenuous inclusions of small diameters. We assume that (j) and γ(j) are measurable, bounded and positive definite uniformly on ωj. In particular, for almost all ξ ∈ ωj the eigenvalues of the matrix (j)ξ are bounded from below by a constant cj > 0. There is no special assumption on the relation between the properties of the inclusions and of the matrix (body without inclusions), we assume only that the densities γ, γ(j), and entries of the matrices , (j) are of similar orders, respectively. We point out that in the framework of our asymptotic analysis, in section there are performed the limit pasages (j) 0 and γ(j) 0 (a hole) as well as (j)→ ∞ and γ(j)→ ∞ (an absolutely rigid inclusion). However, the passage γ(j)→ ∞ with the fixed matrix function (j) (heavy concentrated masses) can be analysed with some other ansätze, cf. [4,36,39].

In the fracture mechanics, the most intereting case is the weakening of elastic material due to the crack formation. The cracks are modelled by two-sided, two dimensional surfaces, with the first boundary conditions from (11) prescribed on the both crack lips, i.e. the surface is traction free from both sides. The case of a microcrack is not formally included in our problem statement, since we assume that the defect ωj is of positive volume and with the Lipschitz boundary ωj. However, the asymptotic procedure works also for the cracks. Small changes which are required in the justification part, are given separately (see the end of section , proof of Proposition and Remark ). The polarization matrices for the cracks can be found in [41], [24].

The exchange of γ and (IMG) by γh and h from (17), respectively, transforms in the integral identity for the body weakened by the defects ,...,, this integral identity is further denoted by (12)h. We observe also, that for smooth stiffness matrix (IMG) and the density γ the differential problem for vibrations of a composite body does not consist only of the system of equations, denoted in our notation by (10)h, restricted to the union of domains (15), along with the boundary conditions (11)h, but in addition it contains transmition conditions on the surface where the ideal contact is assumed. Since we use only the variational formulations of the spectral problems, the transmission conditions are not explicitely written. In a similar way as for problem (13), there is the sequence of eigenvalues for the problem (12)h

and the corresponding eigenfunctions meet the orthogonality and normalization conditions


3 Formal construction of asymptotics

We introduce the following asymptotic ansätze for eigenvalues and eigenfunctions in problem (12)h

where χj (Ω), j = 1,..., J, are cut-off functions, with non overlaping supports in Ω, and for each j, χj(x) = 1 for x and χi(Pj) = δi,j.

First, we assume that the egenvalue λ = λp in problem (12) is simple, and for brevity the subscript p is omitted. The corresponding eigenfunction u = u(p) (Ω; Γ), normalized by condition (14), is smooth in the interior of the domain Ω.

Columns of the matrices d(x) and (x)T form a basis in twelve dimensional space of linear vector functions in 3. In this way, the Taylor formula takes the form

and, by equalities(4), (5) and (8), the columns

aj = d(x)T u(Pj), εj = (x) u(Pj),

represent the column of rigid motions, and of strains, at the point Pj. Since in the vicinity of the inclusion we have

ε(u;x) = εj + O(x) = εj + O(h),

the main terms of discrepancies, left by the field u in the problem (12)h for the composite bodyΩh, appear in the system of equations in and in the transmition conditions on . For the compensation of the discrepancies are used the special solutions of the elasticity problem in a homogenuous space with the inclusion ωj of unit size

Here ν is the unit vector of the exterior normal on the boundary ωj of the bodyωj, ek = (δ1,k, ...,δ6,k)T is a orthant in the space 6, W+ and W- are limit values of the function W on the surface ωj evaluated from outside and from inside of the inclusion ωj, respectively.

We denote by Φ j the fundamental (3×3)-matrix of the operator L0j(ξ) in 3. This matrix is infinitely differentiable in 3\ and enjoys the following positive homogeneity property

It is known (see, e.g., [[20], Ch. 6]), that the solutions Wjk of problem (23) admit the expansion

where p = , , is a line of the matrix (see (3)), Φj1, Φj2, Φj3 are columns of the matrix Φj, and the radius R of the ball R = { ξ : | ξ | < R} is chosen such that R. The coefficients in (25) form the (6×6)-matrix Mj which is called the polarization matrix of the elastic inclusion ωj (see[28,41] and also [[20]; Ch. 6], [5], [21]). Some properties of the polarization matrix, and some comments on the solvability of problem (23) are given in section .

The columnes Wj1,..., Wj6 compose the (3×6)-matrix Wj and we set

In section 5 it is verified, that the right choice of boundary layer is given by formula (26), since it compensates the main terms of discrepancies. From (25) and (26) it follows that

Relation (27) can be differentiated term by term on the set 3\R under the rule ξO(|ξ|-p) = O(|ξ|-p-1) for the remainder.

In view (24) of the detached asymptotics term equals

It produces discrepancies of order h3 (we point out that there is the factor h on w1j in (21)), which should be taken into account when constructing the regular type term h3u. On the other hand, discrepancies of the same order h3 are left in the problem for u by the subsequent term h2w(h-1(x-Pj)), which solves the transmission problem analoguous to (23)

and with the decay rate O(|ξ|-1) at |ξ| , smaller compared to the decay rate of w1j.

Now, we evaluate the right-hand sides of the problems (29),(30) . First, by the representation of the stiffness matrix

and the corresponding splitting of differential operator with the variable coefficients 0(x,x) from (10), we find that the right-hand side of system (29) is the main term of the expression

We note that L0j(x) w1j(h-1(x-Pj)) = 0 in (32), and the dots... stand for the terms of lower order, which are unimportant for our asymptotic analysis. The following discrepancy appears in the second transmission condition (30):

The second term comes out from the elaborated Taylor formula (31)

and involves the quadratic vector function

Finally, the right-hand side of system (29) takes the form

Besides the term obtained from the quadratic vector function (35) in the Taylor formula (34), the expression (36) contains the discrepancy λγj u(Pj) which originates from the inertial term λh γjuh in accordance to the asymptotic ansätze (34) and (35).

In order to establish properties of solutions to the problem (29), (30), we need some complementary results.

Lemm 3.1 Assume that Z(ξ) = (ξ)T Y(ξ) and

where (ρ,θ) are spherical coordinates and C(2)6, C(2)3 are smooth vector functions on the unit sphere. The model problem

admits a solution X(ξ) = ρ-1(θ), which is defined up to the term Φj(ξ)c with c 3, and becomes unique under the orthogonality condition

Proof After separating variables and rewriting the operator L0j(ξ) = r-2 (θ,θ, r / r) in the spherical coordinates, the system (38) takes the form

Since (θ,θ, 0 ) is the formally adjoint operator for j(θ,θ, -1) (see, for example, [[20]; Lemma 3.5.9]), the compability condition for the system of differential equations (40) implies the equality

The equality represents the orthogonality condition in the space L2(2) of the right-hand side of system (40) to the solutions of the system

which are nothing but constant columns. Indeed, after transformation to the Cartesian coordinate system ξ equations(42) take the form L0j(ξ) V(ξ) = 0, ξ ∈ 3\, and any solution V(ξ) = ρ0(θ) is constant. Let b > a > 0 be some numbers, and let Ξ be the annulus {ξ : a < r < b}. We have

We have used here the Green formula and the fact that the integrands on the spheres of radii a and b are equal to b-2(θ)T and a-2(θ)T , respectively, i.e., the integrals cancel one another.

Therefore, the compability condition (41) is verified and the system (40) has a solution C(2)3. The solution is determined up to a linear combination of traces on 2 of columns of the fundamental matrix Φ(ξ); recall that the columns of matrix Φ(ξ) are the only homogenuous solutions of degree -1 of the homogenuous model problem (38).

According to the definition and utility the columns Φq verify the relations

where ξ is the unit outer normal to the sphere 2 = 1, 1 = {ξ : ρ < 1}, δ is the Dirac mass, eq = ( δ1 ,q, δ 2,q,δ 3,q)T is the basis vector of the axis xq, and the last integral over 1 is understood in the sense of the theory of distributions. Thus, owing to (43), the orthogonality condition (39) can be satisfied that implies the uniqueness of the solution to the problem (38), (39). ■

In view of (32) and (27), (28), the right-hand side (38) of takes the form

General results of [7] (see also[[20]; §3.5, §6.1, §6.4]) show that there exists a unique decaying solution of problem (29), (30), which admits the expansion

In the same way as in relation (27), the relation (45) can be differentiated term by term under the rule ξO(|ρ|-p(1+|ln ρ|)) = O(|ρ|-p-1(1+|ln ρ|)).

The method [16] is applied in order to evaluate the column Cj.

Lemma 3.2 The equality is valid

where |ωj| is the volume, and = |ωj|-1 ωj γj (ξ)dξ the mean scaled density of the inclusion ωj, i.e., its mass is |ωj|, and

Proof In the ball R we apply the Gauss formula and obtain, that for R → ∞,

We have also taken into accout equalities (39) and (43). On the other hand, in view of formulae (36) and (32) it follows that

We turn back to the decomposition (27), and taking into account the homogeneity degree of the integrand, we see that the integral over the sphere = R equals

The integrals over the surfaces ωj in the right-hand sides of (49) cancel with two integrals, which according (33) to appear in the formula

Finally, by the equality

(– x)T 0(Pj)(– x)Uj(ξ) +(– x)T (xT x0(Pj))εj = λγ0(Pj)u(Pj) ,

resulting from equation (33) at the point x = Pj, the sum of the pair of two last integrals in (51) takes the form

ωj ((– ξ)T 0(Pj)(ξ)Uj(ξ) + (– ξ)T (ξT x0(Pj))εj)dξ = λγ0(Pj)|ωj|u(Pj) .

It remains to pass to the limit R + .

Now, we are in position to determine the terms ν and µ in the ansätze (21) and (20), which are given by solutions of the problem

The weak formulation of (52)-(53) is given below by (59) in the subspace (Ω;Γ)3 of the Sobolev space H1(Ω). The right-hand side ƒ includes the discrepancies, which results from the terms of boundary layer type and of the order h3. By decompositions (27) and (45) we obtain

The terms in the curly braces enjoy the singularities O(|x-Pj|-2) and O(|x-Pj|-1), respectively, therefore, it should be clarified in what sense the differential problem (52), (53) is considered. Equation (52) is posed in the punctured domain Ω , thus the Dirac mass and its derivatives, which are obtained by the action of the operator on the fundamental matrix, are not taken into account. Beside that, by virtue of the definition of the term Xj implying a solution to the model problem (38)with the right-hand side (44), and according to the estimates of remainders in the expansions (27), (45), the following relations are valid

which accept the differentation according to the standard rule

xO((1 + | ln rj|))) = O((1 + |ln rj|))) .

In other words, expression (54) should be written in the combersome way

Here, [A,B] = AB-BA is the commutator of operators A and B, and Sj1, Sj2 = Sj1 +Xj + Φj Cj are expressions in curly braces in (54).

Lemma 3.3 Let λ be a simple eigenvalue in the problem (10), (11), and u the corresponding vector eigenfunction normalized by the condition (14). Problem (52), (53) admits a solution ν H1(Ω)3 if and only if

where Ωδ = Ω \ ( ... ) and = {x :rj < δ}.

Proof The variant of the one dimensional Hardy's inequality

provides the estimate

In this way, the last term in the integral identity serving for problem (52), (53)

is a continuous functional over the Sobolev space H1(Ω)3, owing to the inequalities

|(f, V)Ω| < c < cV ;H1(Ω)∥,

Thus, Lemma follows from the Riesz representation theorem and Fredholm alternative, in addition, formula (57) is valid because the integrand is a smooth function in Ω \ { P1,...,PJ}, with absolutely integrable singularities at the points P1,...,PJ. ■

Remark 3.1 If the points Pj are considered as tips of the complete cones 3\Pj, the elliptic theory in domains with conical points (see the fundamental contributions [7,16,17] and also e.g., monograph [20]) provides estimates in weighted norms of the solution v to problem (52), (53). Indeed, owing to relation (55) for any τ > 1/2 the inclusions L2(j)3 are valid, where j stands for a neighbourhood of the point Pj, in addition jk = for j k, therefore, the terms -2u, -1xv and are square integrable in j. ■

We evaluate the limit in the right-hand side of (57) for δ + 0. By the Green formula and representation (54), the limit is equal to the sum of the surface integrals

We apply the Taylor formulae (31) and (32) to the matrix and the vector u, and take into account relations (8) for the matrices d and . We also introduce the stretched coordinates ξ = δ-1(x - Pj). As a result, up to an infinitesimal term as δ → +0, integral equals to

The integrals I0 and I1 vanish. Indeed, due to the second equality in (8) we have

These equalities are understood in the sense of distributions. By formula (47), we obtain

I2 = -u(Pj)Ij.

Relations (39) and (43) yield

I3 = u(Pj)Cj.

Finally, in the same way as in (62), we obtain

Now, we could apply the derived formulae. We insert the obtained expressions for Iq into (61) (60) (57) and in view of equation (46) for the column Cj, we conclude that

If equality (64) holds, then problem (52), (53) admits a solution υ H1(Ω)3. The construction of the detached terms in the asymptotic ansätze (20) and (21) is completed.

In the forthcoming sections the formal asymptotic analysis is confirmed and generalized into the following result.

Theorem 3.2 Let λp be an eigenvalue in problem (12) with multiplicity , i.e., in the sequence (13)

There exist hp > 0 and cp > 0 such that for h (0, hp] the eigenvalues , . . . , of the singularly perturbed problem (12)h, and only the listed eigenvalues, verify the estimates

where cp(α) is a multiplier depending on the number p and the exponent α (0,1/2) but independent of h (0,hp], while , . . . , stand for eigenvalues of symmetric ( × )-matrix p with the entries

Mj is the polarization matrix of the scaled inclusion (see (25) and (27)), u(p), . . . , are vector eigenfunctions in the problem (12) corresponding to the eigenvalue λp and orthonormalized by condition (14), finally the quantities and |ωj| are defined in Lemma 3.2.

We explain which changes are necessary in the asymptotic ansätze (20), (21) and in the asymptotic procedure in order to construct asymptotics in the case of a multiple eigenvalue λp. First, for µp and u(p) in and should be selected unknown number and the linear combination

of vector eigenfunctions; the column b(q) = (, . . . , )T is of the unit norm. After the indicated changes the formulae for the boundary layers w1jq and w2jq remain unchanged. The same applies to problem (52), (53) for the correction term of regular type. However, the compability conditions are modified, and turn into the relations

The left-hand side of (69) equals to by (14) and (68). It can be evaluated by the same method as for formula (57), that (69) becomes the system of algebraic equations

with coefficients from (67). In this way, the eigenvalues of the matrix (p) and its eigenvectors b(q) furnish the explicit values for the terms of the asymptotic ansätze (20) and (21). We emphasise that by the orthogonality and normalization conditions (b(q))Tb(k) = δq,k for the eigenvectors of the symmetric matrix (p), it follows that the vector eigenfunctions u(p) = (, . . . , ), p = 1, . . . , , in problem (12), which are given by formulae (68), are as well orthonormalized by the conditions (14).

If we have good luck, and from the beginning the eigenvectors u(p), . . . , have the required form (68), then the matrix (p) is diagonal and the system of equations (70) is decomposed into a collection of independent relations, fully analoguous to relations (64) in the case of a simple eigenvalue. Such an observation is the key ingredient of the algorithm of defects identification which will be described in a forthcoming paper, and it makes the identification method insensitive to the multiplicity of eigenvalues in the limit problem.



The results presented in this section are borrowed from [28], and paper [24].

Variational formulation of problem (23) for the special fields Wjk, which define the elements of the polarization matrix Mj in decomposition (25), are of the form

where (3) is the Kondratiev space [7], which is the completion of the linear space (3) (infinitely differentiable functions with compact supports) in the weighted norm

The following result, established in [24,28] can be shown by using transformations analoguous to (62) and (63) operating with the fields Wjk and jm = ξTek + Wjm.

Pronposition 4.1 The equalities hold true

From the above representation it is clear that the matrix Mj is symmetric, the property follows by the symmetry of the stiffness matrices A0, Aj and of the energy quadratic form Ej. In addition, the representation allows us to deduce if the matrix Mj is negative or positive definite. We write M1 < M2 for the symmetric matrices M1 and M2 provided all eigenvalues of M2- M1 are positive.

Pronposition 4.2 (see [24]) 1° If (j)ξ < (Pj) for ξ ∈ ωj (the inclusion is softer compared to the matrix material), then Mj is a negative definite matrix.

2° If the matrix(j) is constant and < (Pj)-1 (the homogenuous inclusion is rigid compared to the matrix), then Mj is a positive definite matrix.

It is also possible to consider the limit cases, either of a cavity with j = 0, or of an absolutely stiff inclusion with (j) = . For the case of a cavity the diifferential problem takes the form

For an absolutely rigid inclusion the integral-differential equations occur as follows

where the matrices and d are introduced in (3) and (7), respectively.

The Dirichlet conditions (75) in contains an arbitrary column cjk 6, which permits for rigid motion of ωj and can be determined by the integral conditions which annulate the principal vector and moment of forces applied to the bodyωj. The variational formulation of problems (74) and (75) can be established in the Kondratiev space (see [7], and e.g., [20]) normed by the weighted norm (72)(cf. the right-hand side of (76)) and in its linear subspace {W :W|∂ωj }, respectively, where is the linear space of rigid motions (6). The asymptotic procedures of derivation of problems (74) and (75) from problems (23) and (71) can be found in [11,36]. The required estimates can be extracted from these references as well.

In accordance with Proposition 4.2 the polarization matrix for a cavity is always negative definite, and that for an absolutely rigid inclusion, it is always positive definite. Theorem 3.2 gives an asymptotic formula, which can be combined with the indicated facts and the information from Proposition 4.2, and it makes possible to deduce the sign of the variation of a given eigenvalue in terms of the defect properties. For example, in the case of a defect-crack, with the null volume and negative polarization matrix, the eigenvalues of the weakened body are smaller compared to the initial body. Such an observation is already employed in the bone China porcelane shops by the qualified personel.



We proceed with the following statements, which are fairly known for the entire body (see [6,32]) but should be verified for a body with small cavities (see (16)). We emphasize that a body with small inclusions is to be regarded in some sense as an intermediate case. In this way, some of given below axiliary results for the intact body are fit for the body with foreign inclusions, however, in some situations it is much simpler to compare the latter with the body with small voids. On the other hand, the whole justification procedure works for any sort of defects.

Proposition 5.1 For a vector function u (Ω; Γ)3 the inequality

holds true. The above inequality remains valid with a constant independent of h (0,h0], if the domain Ω is replaced by the domain Ω(h) with defects.

Proof For analysis of displacement fields in the domain Ω(h) with cavities (in particular, with cracks) we apply the method described in review papier [[31]; §2.3] - in this framework the body with elastic inclusions is considered as intact or entire. Let us consider the restriction û of u to the set and radius hR of the balls is selected in such a way that . We construct an extension to Ω of the field û. To this end, we introduce the annulae = and perform the stretching of coordinates x ξ j = h-1(x - Pj). The vector functions û and u written in the ξ j-coordinates are denoted by ûj and uj, respectively. It is evident that

where Ξ = . Let

where d is the matrix (7), and the column aj 6 is selected in such a way that

where the matrix d is given by (7). By the orthogonality condition (79), the Korn inequality is valid

(see, e.g., [6], [[31]; §2] and [[19]; Thm 2.3.3]), and the last equality follows from the second formula (8) since the rigid motion daj generates null strains (4). Let denote an extension in the Sobolev class H1 of the vector function from Ξ onto R, such that

Now, the required extension of the field u onto the entire domain Ω is given by the formula

In addition, according to (78) and (77), (80), (81) we have

Applying the Korn's inequality in Ω, we obtain

We turn back to the function ûj and find

The other variant of the Korn's inequality

(see e.g., [6], [[31]; §2] or [[19]; §3.1]), after returning to the x-coordinates leads to the relations

By virtue of Ch > rj > ch > 0 for x 2hR\ , the multiplier h-1 can be inserted into the norm, and transformed to rj-1, but the norm || rj-1 u; L2()|| is already estimated in (84), owing to = u on . Estimates (87), j = 1, . . . , J, modified in the indicated way along with relation (84) imply the Korn inequality in the domain Ω(h). ■

Remark 5.1 If ωj is a domain, then in the proof of Proposition 5.1 we do not need to restrict û to Ωh, but operate directly with the sets Ω(h) and 2R\ ωj since there is a bounded extension operator in the class H1 over the Lipschitz boundary ∂ωj with the estimate of type (81). The presence of cracks makes the existence of such an extension impossible. However, the Korn's inequality (87) is still valid in this case, since to maintain the validility the union of Lipschitz domains is only required (see [6]).

The bilinear form

can be taken as a scalar product in the Hilbert space (Ω; Γ)3. In this way, the integral identity (12)h can be rewritten as the abstract spectral equation

where h = (λh)-1 is the new spectral parameter, and h is a compact, symmetric, and continuous operator, thus selfadjoint,

Eigenvalues of the operator h constitute the sequence

with the elements related to the sequence in (18) by the first formula in (90).

The following statement is known as Lemma on almost eigenvalues and eigenvectors (see, e.g., [40]).

Proposition 5.2 Let and be such that

Then there exists an eigenvalue of the operator h, which satisfies the inequality

Moreover, for any δ• > δ the following inequality holds

where is a linear combination of eigenfunctions of the operator h, associated to the eigenvalues from the segment [ - δ, + δ], and ||•|| = 1.

For the asymptotic approximations and of solutions to the abstract equation (89) we take

where U stands for the sum of terms separated in the asymptotic ansatz (21). Let us evaluate the quantity δ from formula (92). By virtue of λp > 0, for h (0,hp] and hp > 0 small enough, we have

where = { V : ||V; || = 1} is the unit sphere in the space . In addition, to estimate the norm ||U; || the following relations are used

where the first relation follows from the continuity at the points Pj of the second order derivatives of the vector function u(p) combined with the integral identity (12) and the normalization condition (19). We transform the expression under the sign sup in (96). Substituting into the expression the sum of terms in ansatz (21), we have

In (98) we used that u(p) and λp verify the integral identity (12). Furthermore, by the Taylor formulae (34) and (31), we obtain

Let explain the derivation of above formulae. The following substitutions are performed

with pointwise estimates for remainders of orders h2, h2, and h, respectively. These gave rise to the following multipliers in the majorants

Note that the factor h3/2 is proportional to (mes3)1/2, and h-1rj does not exceed a constant on the inclusion . Beside that, the Poincaré inequality

is employed together with the relation

Here stands for the mean value of V over . Finally, all the norms of the test function V are estimated by Proposition 5.1.

In similar but much simpler way, by virtue of Remark 3.1, the term from (100) satisfies

where τ > 1/2 is arbitrary. It is clear that h6|| < Ch6. The integral cancels the integral in (98) and some parts of the integrals from (99), which we are going to consider.

In the notation of formula (56) we have

Furthermore, the integrals and cancel each other according to the integral identities

The latter formulae are provided by (71), (26) and (29), (30), (32), (33), (36). We point out that the test functionξ χj (hξ + Pj)(hξ + Pj) in (106) has a compact support, i.e., the function belongs to the Kondratiev space (3), and in the analysed integrals the stretching of coordinates x ξ = h-1(x - Pj) has to be performed.

The expressions including asymptotic terms (h-1(x - Pj)) = h3-i(x - Pj) are detached from the integrals and ,

and the remainders are estimated by virtue of the decompositions (27) and (45), namely,

Inequalities for the integrals from (99) are obtained in a similar way and look as follows:

According to formula (56) for the right-hand side f of the problem (52), (53) and the associated integral identity (59), the sum of the expressions from (100) and from (107), (109) (the latter is summed over j = 1, . . . , J and q = 0,1,2) turns out to vanish. As a result, collecting the obtained estimates, we conclude that the quantity δ from formula (95) (see also (92)) satisfies the estimate

for any α (0,1/2).

Now we are in position to prove the main theorem on asymptotics of solutions of singularly perturbed problem.

Proof of Theorem 3.2 From the columns b(1), . . . , of matrix (p) with elements (67) can be constructed linear combinations (68) of vector eigenfunctions u(p), . . . , as well as the subsequent terms of asymptotic ansatz (21). As a result, for q = p, . . . , p + - 1 the approximate solutions {(λp + h3µp)-1, } of the abstract equation (89) are obtained, such that the quantity δ from relations (92) verifies the inequality (110). We apply the second part of Proposition 5.2 with

Let the list

include all eigenvalues of the operator h, located in the segment

for sufficiently small h• > 0, such that (λp + h3µp)-1 with h (0, h•] belongs to segment (113). Our immediate objective becomes to show that

The quantities for m > n + N - 1 are uniformly bounded in h (0, h•]. By Proposition 5.1, the same assumptions provide the uniform boundedness of the norm ||; (Ω; Γ)|| of the vector functions h constructed for the vector eigenfunctions in (12)h according to (86). Hence, there exists an infinitesimal sequence {hi}, such that the limit passage hi +0 leads to the convergences

We substitute into the integral identity (12)h the test function v (\(Γ∪{P1, . . ., PJ}))3. According to definition (17) and for sufficiently small h > 0, the stiffness matrix h and the density γh coincide on the support of v with and γ , respectively. Therefore, the limit passage hi +0 in the integral identity (12)h leads to the equality

Since (\(Γ∪{P1, . . ., PJ}))3 is dense in ((Ω; Γ)3, the integral identity (116) holds true for all test functions v (Ω; Γ)3. We observe that the weighted norms ; L2(Ω)|| are uniformly bounded by virtue of inequality , thus

In this way, taking into account formulae (19) and (115), we find out that

Hence, is an eigenvalue, and is a normalized vector eigenfunction of the limit problem (12). This implies that p + > n + N. Considering consenquently the eigenvalues λp, . . . , λ1, we conclude that

In order to establish the inequalities p < n and < N we select the factor c• in (111) such that for the number is excluded from the segment

Let be the multiplicity of the eigenvalue of matrix M(p). By Proposition 5.1 and estimate (120) there are, not necessarily distinct, eigenvalues ,..., of the operator h such that

In addition, Proposition 5.1 furnishes the normalized columns , such that

where ,..., + N-1 are normalized in vector eigenfunctions of the operator h corresponding to all eigenvalues from segment (119). By formulae (97), and (12), (14),

|λpδk,l| = o(1) for h +0 .

Furthermore, owing to formula (121), we have

| - λp((k)) T (l)| = o(1) for h +0 .

Thus, for sufficiently small h the number N cannot be smaller than . Hence, there are eigenvalues ,. . ., which verify inequality (120)with the majorant (since the exponent α ∈ (0,1/2) is arbitrary, we can choose α < α without loosing of the precision in the final estimate (66)). Selecting all eigenvalues of the matrix M(p), and subsequently the numbers λp-1,...,λ1, it turns out that necesserily the equality in occurs, and also N = .

The proof of Theorem 3.2 is completed. ■

Remark 5.2 Theorem 3 provides inequality (121), which allows for derivation of some asymptotic formulae for vector eigenfunctions of the problem (12)h. We emphasise that, first, the estimates of remainder are not as good as in the case of eigenvalues, and, second, for multiple eigenvalues of matrix M(p) even the initial approximation for is not available. And this is not a lack of the obtained estimates but just the matter of asymptotic procedures; we refer the reader to the chapter 7 of book [19] and to papers [11,29], where is discussed the notion of individual and collective asymptotics of solutions to spectral problems. We present one variant of the estimates proved above.

If is a simple eigenvalue of the matrix M(p) (for example, λp is a simple eigenvalue of problem (12)) and b(q) the corresponding normalized eigenvector, then there is an eigenvalue in problem (12)) (if λp is simple than p = q), which is simple, and together with the corresponding vector eigenfunction verifies the estimates

where α (0,1/2) is arbitrary, and the factors cp(α), Cp(α) are independent of parameter h (0,hp].

Acknoledgements The research of S.A.N. was partially supported by the grant Russian Foundation of Basic Research-09-01-00759. The research of J.S. was partially supported by the Projet ANR GAOS Geometrical analysis of optimal shapes.



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Received 5 Sep 2010
In revised form 13 Sep 2010



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