Journal of Applied Mathematics and Mechanics 74 (2010) 633–636

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Journal of Applied Mathematics and Mechanics

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The oscillations of layered elastic media with a wavy boundary�

V.A. Babeshko ∗, O.M. Babeshko, O.V. YevdokimovaKrasnodar, Russia

a r t i c l e i n f o

Article history:Received 25 March 2010

a b s t r a c t

Block element methods are used to investigate the oscillations of an elastic layer with a rough surface.For simplification, the investigation is carried out separately for the rotational and potential componentsof the boundary-value problem, which enables the solution to be obtained by analysing Helmholtz’sequations.

© 2011 Elsevier Ltd. All rights reserved.

The problem in question was initiated by Vorovich,1,2 who investigated, for the first time, the spectral and resonance properties of aninhomogeneous elastic strip, where the inhomogeneity was formed, including waviness of the boundary of the strip. His papers are thebasis of further research on boundary value problems in a three-dimensional formulation and the development of new methods for solvingthem. In this paper we suggest that the waviness can be described by a set of block elements, introduced previously in Refs 3–6.

Differential and integral methods of factorization, which have a topological basis,7–9 are the foundation of the theory of block structuresand a block element. Their application in boundary value problems of the mechanics of a deformable solid in static and dynamic cases10

involves the factorization of matrix functions, which complicates the discussion. In order to eliminate a lengthy investigation of systemsof partial differential equations and the factorization of the matrix functions that arise, below we use an expansion of the solutions ofthe boundary value problems into potential and rotational components, which are employed in many boundary value problems of themechanics of a deformable solid.11,12

Using this approach, for the equation of the theory of elasticity in vector form

(in the standard notation), we will seek a solution of the boundary value problem in the form u = grad� + rot�. The function � gives thepotential component of the solution while the components �n(n = 1, 2, 3) of the vector � give the rotational component. Substituting thisrepresentation for u into the equation of the theory of elasticity, in the special case, we obtain

(1)

The parameters k1 and k2 represent the phase velocities of the longitudinal waves �k−11 and the transverse waves �k−1

2 in the elasticmedium.

Block elements for equations that are more general than (1) have been constructed3–6, for example[A11 ∂ 2x1 + A22 ∂ 2x2 + A33 ∂ 2x3 + A]�(x1, x2, x3) = 0, in regions which have the form of a rectangular parallelepiped and various formsof pyramid. The Dirichlet or Neumann conditions are imposed at the boundaries of the regions. Solutions of the boundary value problemsare sought in the spaces of slowly increasing generalized functions. The problem of finding the classical component of the solution isdescribed in detail in previous papers.3–9

We recall that, unlike a finite element, a block element, more correctly, albeit in a more complex way, describes the local properties of thesolutions of boundary value problems.3 Using block elements and combining them, one can form complex structures or, conversely, complexstructures can be separated into fragments in the form of block elements. When using block elements, accompanying expressions, obtained

� Prikl. Mat. Mekh. Vol. 74, No. 6, pp. 890–894, 2010.∗ Corresponding author.

E-mail address: babeshko@kubsu.ru (V.A. Babeshko).

0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.jappmathmech.2011.01.002

634 V.A. Babeshko et al. / Journal of Applied Mathematics and Mechanics 74 (2010) 633–636

using external forms, are derived, without which a block element is not completely described: functional equations, pseudodifferentialequations and formulae for representing the solutions in integral form.3–9

Using the above, we will consider the case when we have a boundary value problem for an elastic unbounded layer of thicknessh with a wavy upper boundary. We will describe the waviness by an elastic rectangular parallelepiped lying on the upper boundaryof the layer and rigidly connected to it. Obviously, this construction, split along the upper boundary of the layer, can be decom-posed into two blocks – a rectangular parallelepiped (a hexagonal block) and a layer (the simplest dihedral block). From the pointof view of the differential method of factorization, this structure is no more than a double-layer object, but with a larger number offaces.

We will present the main characteristics of these objects. Some of the characteristics of a block element in the form of a rectangularparallelepiped for the boundary value problem considered were presented previously in Ref. 3, where a tangential stratification of themanifold was carried out and local systems of coordinates were introduced.

We will present an external analysis using one expression from the functional and pseudodifferential equations and the repre-sentations of the solutions in one of the local systems of coordinates, characteristic for the topological approach. The remainingsimilar expressions can be obtained using the relations presented in the above-mentioned publications. It is worth noting thata block element is an element of a differentiable topological manifold, and its topology is induced by the topology of theneighbourhood-carriers of the block elements. A set of block elements is an Abelian group with a group operation in the form ofaddition.

We will represent the functional equations for the block in the form

(2)

In the case of a layer we have

(3)

We have used the notation employed previously in Ref. 3: �′m3 denotes a derivative in the local system of coordinates xm

1 , xm2 , xm

3 withrespect to the variable xm

3 , and �m3 is the value of the function in the system xm1 , xm

2 , xm3 .

The functional equations give the equivalent formulation of boundary-value problems, but in terms of integral transformations. Theyidentify the functions or matrix functions, which require factorization.

V.A. Babeshko et al. / Journal of Applied Mathematics and Mechanics 74 (2010) 633–636 635

We will introduce the Fourier transformation operators

The pseudodifferential equations, which are the result of a calculation of a Leray residue form, describe all possible forms of boundaryconditions which can be formulated for this system of differential equations of the boundary value problem. After selecting the boundaryconditions we change to integral or integro-differential equations for solving the boundary value problem. In the case considered we havethe following pseudodifferential equations

(4)

(5)

The dots in formulae (4) and (5) denote the expressions on the right-hand sides of formulae (2) and (3) respectively, with �13 replaced

by ˛13− (see below).

The functions on the left-hand side of the functional equations, which require factorization, give the characteristic equations in the localsystems of coordinates. They have the form

The root sets which are of interest are described by the relations

We have taken those branches of the analytic functions which ensure that the roots belong to the lower half-plane for real parametersof the Fourier transform of sufficiently large modulus.

We will obtain representations of the solution in each of the local systems of coordinates introduced, i.e., we will obtain representationsof the block elements. Note that the presence of several formulae for representing the solutions in each block element is a consequence ofthe use of the external analysis method, which enables us to carry out the investigation. We have

(6)

The dots denote the expression on the right-hand side of Eq. (2) in the case of a block and in Eq. (3) in the case of a layer.The above relations give a description of the block structure, corresponding to all the boundary-value problems possible for the differen-

tial form introduced above. When selecting specific boundary conditions, for example, Dirichlet, Neumann or mixed boundary conditions,the functions specified on the boundary are introduced into the pseudodifferential equations. Functions not specified on the boundary,but also occurring in the pseudodifferential equations, are regarded as unknown; to determine them the pseudodifferential equations arereduced to integral or integro-differential equations. Corresponding boundary conditions10 are specified on the contact surface betweentwo blocks, depending on the contact conditions: rigid bonding, partial bonding or the presence of a crack or inclusion. For example, in thecase of rigid bonding of the block with the layer we must take �3 = �10, �′

33 = �′130 in the whole group of equations of the block. The

system of equations in the unknowns becomes closed.The following theorem holds.Theorem. The first operators of the pseudodifferential equations, which are the principal ones, are convolution operators. The next

operators are subordinate operators, completely continuous in the space of slowly increasing generalized functions, if their carriers arebounded.

This theorem enables different formulations of the boundary-value problems which are to be investigated to be set up very simply.Inverting the principal operators and introducing the known functions on to the right-hand side, we obtain equations for determining theunknowns. By solving the equations and introducing the required functions into the representation of the solutions of the boundary valueproblem, we obtain the spectral representation of the solution in explicit form, which is the expression following the Fourier transformation

636 V.A. Babeshko et al. / Journal of Applied Mathematics and Mechanics 74 (2010) 633–636

operator in the previous formulae. This approach is particularly convenient in problems of seismology when a whole series of boundaryconditions for the block strucutres is unknown and the most acceptable ones are found by varying the formulations of the boundaryconditions in accordance with experiment.

Problems of combining the assumed boundary conditions with the boundary conditions of the stress-strain state of the block structure,and estimates of the behaviour of the solutions in the neighbourhood of edges, corner points and at infinity, are solved by additionalinvestigations, which, in a number of cases, are fairly simple.

In the case of more complex waviness or lamination of the structure, in order to describe it it is necessary to make use of the necessarynumber of block elements, while preserving the investigation scheme described.

The equations were investigated by a numerical-analytical method. However, for certain sets of values of the parameters one cananalytically obtain an approximate discrete spectrum for a layer with a wavy surface. As an example, we will take the wavy surface formedby the block considered above, assuming it to be of unlimited extent, obtained by taking the limit as c → ∞. We will assume that this blockstructure occurs under antiplane deformation conditions.

A similar problem arises in seismology (B B Golitsyn assumed that it was the shear motions of lithospheric plates that were responsiblefor destructive earthquakes). We will seek the natural oscillations of this block structure assuming rigid bonding between the lower baseof the layer and the stress-free remaining boundary of the wavy surface. The corresponding pseudodifferential equations are simplified,and one can obtain, using the integral method of factorization,10 equations describing the discrete values of the block-structure spectrum.This analysis is fairly simple but lengthy.

The stages of the investigation include the combining of the blocks in the block structure by replacing the boundary conditions for alayer with the boundary conditions for a block, the approximate solution of the integral equation obtained and a determination, from theserelations, of the parameters which ensure that the correct solution exists for zero boundary conditions. As a result, for a relatively thinlayer (h « a), discrete frequencies of the block structure, with an upper bound, are obtained

It can be seen that for any numbers n and s one can always find the width a and the height b of a block, for which the layer with thewavy boundary will have a finite number of discrete resonance frequencies. The oscillations of this structure differ in this feature from theoscillations of a bounded body having a denumerable frequency spectrum.

It is these discrete values of the spectrum which were predicted by Vorovich in his outstanding papers.1,2

Acknowledgements

Individual parts of this investigation were supported by the Russian Foundation for Basic Research (09-08-00170, 08-08-00468, 09-08-00171, 08-08-00669), the “South Russia” Programme (09-01-96500, 09-01-96503, 09-08-96522, 09-08-96527, 09-08-00294), the StateProgramme for the Support of the Leading Scientific Schools (NSh-3765.2010.1), the Federal Special Purpose Programme (2009-1.5-503-004-006), the Programme for the Support of Young Russian Scientists – Doctors of Science (ND-1554.2009.1), and the programmes ofthe Department of Power Engineering, Machine Building, Mechanics and Control Processes and the programmes of the Praesidium of theRussian Academy of Sciences, carried out by the Southern Scientific Centre of the Russian Academy of Sciences.

References

1. Vorovich II. Spectral properties of the boundary-value problem of the theory of elasticity for an inhomogeneous strip. Dokl Akad Nauk SSSR 1979;245(4):817–20.2. Vorovich II. Resonance properties of an inhomogeneous elastic strip. Dokl Akad Nauk SSSR 1979;245(5):1076–9.3. Babeshko VA, Babeshko OM, Yevdokimova OV. The problem of the block structures of Academician M. A Sadovskii Dokl Ross Akad Nauk 2009;427(4):480–5.4. Babeshko VA, Babeshko OM, Yevdokimova OV. The theory of the block element. Dokl Ross Akad Nauk 2009;427(2):183–6.5. Babeshko VA, Babeshko OM, Yevdokimova OV. The pyramidal block element. Dokl Ross Akad Nauk 2009;428(1):30–4.6. Babeshko VA, Babeshko OM, Yevdokimova OV. A block element in the form of an arbitrary triangular pyramid. Dokl Ross Akad Nauk 2009;429(6):758–61.7. Babeshko VA, Babeshko OM, Yevdokimova OV. An integral method of factorization in mixed problems for anisotropic media. Dokl Ross Akad Nauk 2009;426(4):471–5.8. Babeshko VA, Yevdokimova OV, Babeshko OM. A differential method of factorization in block structures and nanostructures. Dokl Ross Akad Nauk 2007;415(5):596–9.9. Yevdokimova OV, Babeshko OM, Babeshko VA. A differential method of factorization in inhomogeneous problems. Dokl Ross Akad Nauk 2008;418(3):321–3.

10. Babeshko VA, Yevdokimova OV, Babeshko OM, Zaretskaya MV, Pavlova AV. A differential method of factorization for a block structure. Dokl Ross Akad Nauk2009;424(1):36–9.

11. Nowacki W. Dynamic Problems of Thermoelasticity. Warsaw: PWN; 1966.12. Kochin N. Ye. Vector Calculus and the Principles of Tensor Calculus. Moscow: Nauka; 1965.

Translated by R.C.G.