Nonstationary contact of a rigid body with an elastic medium: plane problem (review)

2016

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An approach to studying nonstationary wave processes in an elastic half-plane with mixed boundary conditions of the fourth boundary-value problem of elasticity is proposed. The Laplace and Fourier transforms are used. The sequential inversion of these transforms or the inversion of the joint transform by the Cagniard method allows obtaining the required solution (stresses, displacements) in a closed analytic form. With this approach, the problem can be solved for various types of loads Keywords: elastic half-plane, stress state, nonstationary waves, integral transforms, mixed boundary conditionsIntroduction. There are four types of boundary-value problems in elasticity (see, for example, [9]): (i) the stress vector is given (first boundary-value problem); (ii) the displacement vector is given (second boundary-value problem); (iii) the normal component of the displacement vector and the tangential components of the stress vector are given (third boundary-value problem); (iv) the normal component of the stress vector and the tangential components of the displacement vector are given (fourth boundary-value problem). The first two types are main conditions, and the last two conditions are mixed. Whether the analytic solution of nonstationary problems can be found depends on the type of boundary conditions. Nonstationary wave processes in an isotropic elastic half-space (half-plane) under concentrated or distributed surface loads were subject of many studies (see, for example, [3,6,7,11] and the references therein). To solve corresponding boundary-value problems, use is made of effective analytic and combined numerical/analytical methods that employ the Laplace transform with respect to the time coordinate and the Fourier (Hankel) transform with respect to the linear coordinate. The chief difficulty is the recovery of the original functions, which depends, primarily, on the space-time distribution of load over the boundary and the type of boundary conditions. Studies usually focus on separate aspects of wave processes: asymptotic construction of the displacement and stress fields near wavefronts [26,27]; the displacements of particles of the surface of the half-space and the symmetry axis [15,25] or at a considerable distance from it [15]. The joint transform is inverted using the Cagniard technique [8,12] and taking into account the homogeneity of the joint transform in the transform parameters. Solutions for the displacements and stresses inside a half-space under a uniformly distributed step load were found in [14,[21][22][23] using the residue theory to invert the Laplace transform. The expressions derived in [10] are the superposition of the analytic solutions of the Lamb problem and the convolution with respect to the radial coordinate of the load distribution function and the fundamental solution. Two types of spatial distribution of load were considered and the elastic displacements resulting from a load varying with time as a delta function were calculated. This resulted in discontinuity of the calculated dis...

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confidence: 99%

“…A combined numerical/analytical method to solve the first boundary-value problem for a half-plane without such a constraint was used in [17,18]. Formulations of the third boundary-value problem for shock processes were reviewed in [6].…”

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confidence: 99%

Nonsteady Problem for an Elastic Half-Plane with Mixed Boundary Conditions

2016

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“…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].…”

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confidence: 99%

Nonstationary Load on the Surface of an Elastic Half-Strip

Yanchevskii

2015

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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:

“…Particular attention is usually given to problems for half-spaces (axisymmetric problem) or half-planes (plane problem) that are most often subject to a shock load or a normal stress suddenly applied to the boundary. The major results achieved in this field can be found in [3,6,13]. Some approaches and features of studies are described in [8-10, 12, 14-17].…”

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confidence: 99%

Stress State of an Elastic Half-Plane under Nonstationary Loading

2015

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A technique for determining the stress-strain state of an elastic half-plane under a nonstationary load applied to its boundary is developed. The corresponding boundary-value problem with initial conditions is formulated. Laplace and Fourier transforms are used. The inversion of the joint transform enables obtaining the exact analytical expressions for the stress and displacement as functions of time and the distance to the boundary for some types of loads Keywords: elastic plane problem, nonstationary process, stress-strain state, Laplace transform, Fourier transform, Cagniard techniqueIntroduction. The nonstationary problem of elasticity has a long history and has intensively been developed due to practical importance, specific features of the physical processes, and interesting aspects of solving the corresponding boundary-value problems. Particular attention is usually given to problems for half-spaces (axisymmetric problem) or half-planes (plane problem) that are most often subject to a shock load or a normal stress suddenly applied to the boundary. The major results achieved in this field can be found in [3,6,13]. Some approaches and features of studies are described in [8-10, 12, 14-17].We will find the analytic solution to the nonstationary plane problem of elasticity for a half-plane under a distributed nonstationary load (normal stress) suddenly applied to its surface. The general solution of the problem will be found using the Laplace and Fourier transforms. The transforms will be jointly inverted using the Cagniard technique [9]. Doing so will lead to the exact analytic expression for the normal stress on the axis of symmetry of the problem as a function of time and depth for two types of load (stress linearly or parabolically distributed over the half-plane surface).1. Let us address the plane problem for an elastic half-space under a nonstationary load applied to its surface. We introduce Cartesian coordinates x y z , , such that the wave process occurs in the half-plane xz, the x-axis is directed along the boundary of the half-plane, and the z-axis is directed deep into it (Fig. 1).It is assumed that the normal stress symmetric about the z-axis is induced at some initial time t = 0 and, in the general case, is a function of time and the coordinate x.We introduce the following dimensionless notation: