Plane problem of diffraction of elastic harmonic waves on periodic curvilinear inserts
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PLANE PROBLEM OF DIFFRACTION OF ELASTIC HARMONIC WAVES ON PERIODIC CURVILINEAR INSERTS O. M. Nazarenko and O. M. Lozhkin
UDC 539.3
Using the method of integral equations, we solved the plane problem of diffraction of elastic waves on a periodic system of rigid tunnel inserts. The obtained system of singular integral equations was solved numerically by the method of mechanical quadratures. We present the diagrams of stress distribution near the insert tips.
In studying the stress-strain state of bodies with inhomogeneities within the framework of plane and threedimensional dynamic problems, the method of integral equations works efficiently. With its use [1], we investigated the behavior of a medium with a system of radial thin rigid inclusions [2] and the diffraction of elastic waves in a space with a rigid insert. The asymptotic approach to the solution of three-dimensional dynamic problems of the scattering of elastic waves on thin inclusions of different types was proposed earlier [3, 4]. In the present paper, we develop the approach [2] for the case where plane harmonic waves interact with a periodic system of rigid curvilinear inserts. Formulation of the Problem Let an isotropic elastic space contain a 2d-periodic system of tunnel rigid inserts, whose cross section in the plane x1 O x2 is situated along curvilinear contours Lj = L ( mod 2d ), which do not intersect. The thickness of these inserts is small as compared with their length, and, hence, we may neglect their masses. Suppose that a harmonic dilatation-compression wave ( P-case) U1(0) = 0,
U2(0) = τ1e − iγ 1x2 ,
γ1 =
ω , c1
c1 =
λ + 2μ , ρ
τ1 = const
(1)
μ , ρ
τ2 = const
(2)
or a transverse wave ( S V-case) U1(0) = τ2e − iγ 2 x2 = 0,
U2(0) = 0,
γ2 =
ω , c2
c2 =
runs against this system of inserts from infinity. Here, λ and μ are the Lamé coefficients, ν is Poisson’s ratio, c1 and c2 are the velocities of longitudinal and transverse waves, respectively, and ω is the angular frequency of oscillations. Since the strain is stationary, we solve the problem in amplitude values, omitting the time coordinate t, the dependence on which is expressed by the multiplier e − iωt ( i2 = – 1 ). Sumy State University, Sumy. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 43, No. 2, pp. 94 – 99, March – April, 2007. Original article submitted May 24, 2006. 1068–820X/07/4302–0249
© 2007
Springer Science+Business Media, Inc.
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250
O. M. NAZARENKO
AND
O. M. LOZHKIN
The interaction of the incident wave with the system of inserts generates reflected longitudinal and transverse waves. We represent the general field of displacement amplitudes Uk and components of the tensor of stress amplitudes σmn as sums Uk = Uk(0) + Uk(1) ,
0) 1) σmn = σ(mn + σ(mn
( k, m, n = 1, 2 ),
(3)
0) 1) and Uk(1) , σ(mn are the amplitudes of components of the displacement vector and the tensor where Uk(0) , σ(mn of stress amplitudes of the incident and reflected fields, respectively. In the case of plane strain, the displacement amplitudes of the reflecte
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