A method for determining the stress-strain state of an elastic half-strip under a nonstationary load applied to its boundary is proposed. The corresponding initial-boundary-value problem is formulated. The Laplace transform is used and the solution is expanded into a Fourier series. The variation in the stress and displacement with time and space coordinates is studied Introduction. Studies of nonstationary processes in elastic bodies have an extensive bibliography. Many of these studies address the deformation of an elastic half-space or a half-plane under a nonstationary load. Some results of such studies are reported in [3,[6][7][8][9][10][11][13][14][15][16][17]. The problem of the deformation of an elastic half-plane under a nonstationary load applied to its boundary was addressed in [5]. Its solution was found by applying the Laplace transform with respect to time and the Fourier transform with respect to the linear coordinate running along the boundary. The joint transforms will be inverted using the Cagniard technique [12]. This technique, which is primarily applicable to transforms homogeneous in the transformation parameters, makes it possible to derive the exact expressions for the normal stress and displacement as functions of time and distance to the boundary for some types of external load.Here we develop a combined numerical/analytical approach to solving the problem under consideration for certain constraints on the time interval and for loads of quite general form. We will use the analytic solution from [5] to validate our results.The essence of the approach is as follows. Instead of an elastic half-plane, we will consider an elastic half-strip of certain width with such boundary conditions that the general solution for the wave potentials expanded into a Fourier series satisfy these conditions. The solution of the thus modified problem coincides with the solution of the original problem (for a half-plane) up to the instant the waves are reflected from the lateral edges. We will apply the Laplace transform with respect to the time variable and Fourier series expansion in the width coordinate of the strip.These transformations make it possible to separate out jump functions in the solution. For the remaining smooth portion of the solution, the inversion of the Laplace transform is reduced to the solution of a sequence of Volterra equations of the second kind. Our approach is similar to that applied in [13] to impact problems for a blunt body.1. Problem Formulation. Basic Equations. Let us address the plane problem for an elastic half-space under a nonstationary load applied to its surface (plane strain).We introduce Cartesian coordinates x, y, z such that the wave process occurs in the half-plane xz, and we will use the following dimensionless notation:

A two-layer (metal-piezoceramic) beam with hinged ends under a nonstationary distributed load is considered. A method for solving the problem of active damping of the nonstationary flexural vibrations of this structure is proposed. The shape of the electric signal applied to the electrode of the electroelastic layer to keep the beam almost undeformed is identified. The solution is obtained using the Laplace transform with respect to time. Numerical results are presented and analyzed Keywords: electroelasticity, nonstationary processes, two-layer (metal-piezoceramic) beam, Laplace transformIntroduction. Electroelasticity is an important and intensively developed area of the mechanics of coupled fields. Most publications on the subject dealt with stationary dynamic processes in piezoelectric bodies. There are also relevant monographs [4,7].Nonstationary problems of electroelasticity and hydroelectroelasticity are of current interest. Results on transient processes accompanying the interaction of piezoceramic shells with a fluid are reviewed in [9]. Of the recent publications on nonstationary electroelasticity, [8,11,12,13] are noteworthy.Among many piezoelectric transducers, bimorph plates and beams (consisting of elastic and electroelastic layers) are of wide use. The nonstationary vibrations of such structural elements under mechanical or electric loads were studied in [2,3].It is of interest to find the shape of an electric signal that, depending on the type of external shock load, could be used for controlling the stress-strain state of bimorph piezoelectric transducers, including active damping. This is the subject of the present study.

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