Diffraction of waves in an elastic space is investigated, when a shear wave is incident in an arbitrary direction on a. semi-infinite inclusion in the form of elastic layer of a small thickness. The problem is to determine the wave field both in the contact region and in the space. In the both regions of shade and reflected wave and, naturally, in the contact region a wave part i s discovered ~ localized (surface) wave of Love (if only the. shear wave speed in the space is greater than that in the inclusion)+ As an extreme case solutions of problems of space with a semi-infinite crack and also a semi-infinite rigid inclusion are derived. Asymptotic presentations of displacements and st#resses in the far field and asymptotics of stresses near the inclusion's edge are dcrived.1. Consider an elastic space? in Cartesian coordinate system Ozyz, containing an elastic semi-infinite inclrision in the form of a semi-infinite strip of a sma.11 thickness 2 h , which occupies the region Ro (-m < 5 5 0, IyI 5 h, Irl < m).A plane shear wave with time harmonic factor e-iwt and amplitude(1) u,im) (z, y) = e -i k r c o s P -i h~s i n P is incident form infinite at an angle j ? (0 < , O < 7~/2), where k. = w/c is the wave number, c = is the shear watm speed, fi, p and D O , po are the elastic moduli and densities of the spacc and the inclusion. The space is in the state of an anti-plane deformamtion.The task is to determine the diffracted wave field both in the contact region and in t,he space.One can present uzm) ( (5, y) in tlie form of it sum of its even part and odd part Then the displacement; will dso bc represented i n t,he form 21, (w, = w (w) + w 2 (G?l) where w1 (x, y) is the even part and iu2 (2: y) is thc odd part of thc unknown displacement amplitude, which is i o he determincd.
Аннотация. Рассмотрена задача об установившихся гармонических колебаниях слоистой предвари-тельно напряженной среды, представляющей собой электромагнитоупругий слой, жестко сцепленный с электромагнитоупругим полупространством. Начальные напряжения в среде определяются величиной деформации слоя или полупространства. В частности, исследованы случаи одновременного сжатия или растяжения слоя и полупространства. Для описания динамического процесса в среде использовали ли-неаризованные приближения уравнений движения и квазистатических уравнений Максвелла. Постро-ена функция Грина среды с учетом непрерывности механических, электрических и магнитных полей. В качестве исследуемой среды был выбран композит, выполненный из широко используемых материа-лов: верхний слой композита выполнен из пьезоэлектрика BaTiO 3 , а подстилающее полупространство из пьезомагнетика CoFe 2 O 4 .Рассчитаны фазовые скорости среды при различных величинах начальных деформаций пьезоэлек-трической и пьезомагнитной компонент среды. В качестве вариативных условий для задания начальных напряжений были использованы различные режимы начальной деформации: одноосный, двухосный и трехосный (гидростатический). Полученные результаты могут быть полезны при проектировании вы-соконагруженных устройств на базе электромагнитоупругих материалов.Ключевые слова: начальные напряжения, электромагнитоупругость, гетероструктура, функция Грина. Abstract. The problem of steady harmonic oscillations of a layered prestressed medium, which is an electromagnetoelastic layer rigidly coupled to an electromagnetoelastic half-space, is considered. The initial stresses in the medium are determined by the amount of deformation of the layer or half-space. In particular, the cases of simultaneous compression or extension of a layer and a half-space are investigated. The dynamic process in the medium was described by linearized approximations of the equations of motion and quasistatic Maxwell equations. The Green's function of the medium is constructed with allowance for the continuity of mechanical, electric, and magnetic fields. As the studied medium, a composite made of widely used materials was chosen: the upper layer of the composite is made of the piezoelectric BaTiO 3 , and the underlying halfspace from the piezomagnetic CoFe 2 O 4 . PECULIARITIES OF DYNAMICS OFThe phase velocities of the medium are calculated for different values of the initial deformations of the piezoelectric and piezomagnetic components of the medium. As various conditions for setting initial stresses, different initial strain modes were used: uniaxial, biaxial and triaxial (hydrostatic). The obtained results can be useful in the design of highly loaded devices based on electromagnetoelastic materials.
The plane contact problem for compound elastic half-plane consisting from endless strip and half-plane from different orthotropic materials. The compound half-plane weakened by crack of final length on the junction line of the strip and half-plane. Normal and shearing stresses are given on the free boundary of the strip and at the edges of the crack. There are conditions of full contact on junction line of the strip and at the edges of crack. On junction line of the strip and half-plane, outside of the crack there are conditions of full contact. The discontinuous solution of the elasticity theory for the compound orthotropic plane is obtained by the method of Airy stress function. The closed solution of problem is built by means of the last, bringing to the singular integral equation of the second kind./Рассматривается плоская контактная задача для составной упругой полуплоскости, состоящей из бесконечной полосы и полуплоскости из различных ортотропных материалов. На линии контакта полосы и полуплоскости составная полуплоскость ослаблена трещиной конечной длины. На свободной границе полосы и на краях трещины заданы нормальные и касательные напряжения. На линии контакта, вне трещины, имеют место условия полного контакта. Методом функции напряжений Эйри получено разрывное решение теории упругости для составной ортотропной плоскости. При помощи последнего построено замкнутое решение задачи, сведя к сингулярному интегральному уравнению второго рода.
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