Nonstationary Load on the Surface of an Elastic Half-Strip

Kubenko, V. D.; Yanchevskii, I. V.

doi:10.1007/s10778-015-0690-x

Cited by 8 publications

(12 citation statements)

References 12 publications

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“…This can be ascertained by comparing the given solutions. Figures a , 6 b show stress computed for a load expanding with constant velocity k (see solution ) in the statement of this paper (dashed line) and in the statement of the first boundary problem (solid line). In the quoted paper expression of the

σ_{z z} (t, z, x = 0)

is obtained in the next form:

\begin{matrix} true \\ left {σ_{z z} (() t, z)|}_{x = 0} = \frac{2}{π} Q_{0} k [H (t - \frac{z}{α}) \int_{0}^{\frac{\sqrt{α^{2} t^{2} - z^{2}}}{α z}} R_{α} (η) d η - 4 \frac{β^{3}}{α} H (t - \frac{z}{β}) \int_{0}^{\frac{\sqrt{β^{2} t^{2} - z^{2}}}{β z}} R_{β} (η) d η] \\ left R_{α} ((), η) = \frac{1}{(1 + k^{2} η^{2})} \frac{{(() 1 + 2 β^{2} η^{2})}^{2}}{{(1 + 2 β^{2} η^{2})}^{2} - 4 \frac{β^{3}}{α} η^{2} \sqrt{1 + α^{2} η^{2}} \sqrt{1 + β^{2} η^{2}}}, \\ left R_{β} ((), η) = \frac{1}{(1 + k^{2} η^{2})} \frac{η^{2} \sqrt{1 + α^{2} η^{2}} \sqrt{1 + β^{2} η^{2}}}{{(1 + 2 β^{2} η^{2})}^{2} - 4 \frac{β^{3}}{α} η^{2}} \end{matrix}

…”

Section: Comparison Of Quantitative Results' Of the First And Fourth mentioning

confidence: 99%

“…This can be ascertained by comparing the given solutions. Figures 6a, 6b show stress computed for a load expanding with constant velocity k (see solution (29)) in the statement of this paper (dashed line) and in the statement of the first boundary problem [16] (solid line). In the quoted paper expression of the σ zz (t, z, x = 0) is obtained in the next form:…”

Section: Comparison Of Quantitative Results' Of the First And Fourth mentioning

confidence: 99%

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On a non‐stationary load on the surface of a semiplane with mixed boundary conditions

Kubenko¹

2015

Z Angew Math Mech

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An exact analytical solution has been constructed for the plane problem on action of a non‐stationary load on the surface of an elastic semiplane for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on the boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non‐stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the object at an arbitrary point of time. Some variants of non‐stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. For a particular case, computed numerical results are compared with the solution of the first boundary problem. Constructing exact analytical solutions, even if infrequently used in practice, besides being significant on their own, can also help refine various numerical and approximate approaches, for which the types of boundary conditions are not critical.

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σ_{z z} (t, z, x = 0)

is obtained in the next form:

\begin{matrix} true \\ left {σ_{z z} (() t, z)|}_{x = 0} = \frac{2}{π} Q_{0} k [H (t - \frac{z}{α}) \int_{0}^{\frac{\sqrt{α^{2} t^{2} - z^{2}}}{α z}} R_{α} (η) d η - 4 \frac{β^{3}}{α} H (t - \frac{z}{β}) \int_{0}^{\frac{\sqrt{β^{2} t^{2} - z^{2}}}{β z}} R_{β} (η) d η] \\ left R_{α} ((), η) = \frac{1}{(1 + k^{2} η^{2})} \frac{{(() 1 + 2 β^{2} η^{2})}^{2}}{{(1 + 2 β^{2} η^{2})}^{2} - 4 \frac{β^{3}}{α} η^{2} \sqrt{1 + α^{2} η^{2}} \sqrt{1 + β^{2} η^{2}}}, \\ left R_{β} ((), η) = \frac{1}{(1 + k^{2} η^{2})} \frac{η^{2} \sqrt{1 + α^{2} η^{2}} \sqrt{1 + β^{2} η^{2}}}{{(1 + 2 β^{2} η^{2})}^{2} - 4 \frac{β^{3}}{α} η^{2}} \end{matrix}

…”

Section: Comparison Of Quantitative Results' Of the First And Fourth mentioning

confidence: 99%

Section: Comparison Of Quantitative Results' Of the First And Fourth mentioning

confidence: 99%

On a non‐stationary load on the surface of a semiplane with mixed boundary conditions

Kubenko¹

2015

Z Angew Math Mech

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“…The original of expression ξ 2 s −3 e − zn β √ s 2 +β 2 ξ 2 and expressions which containz n are obtained by replacement in (19) α by β and z n byz n .…”

Section: The General Casementioning

confidence: 99%

“…For a more general loading, ref. [19] offers a numerical-analytical solution of the first boundary problem for a semi-plane. Ref.…”

mentioning

confidence: 99%

Non-stationary stress state of an elastic layer at the mixed boundary conditions

Kubenko¹

2016

Z. Angew. Math. Mech.

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An exact analytical solution has been constructed for the plane problem on action of a non-stationary load on the surface of an elastic layer for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on one boundary and normal displacement and shear stress (the second boundary problem) for another boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non-stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the layer at an arbitrary moment of time. Some variants of non-stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. Distinctive features of wave processes are analyzed.The boundary conditions in problems of elasticity theory belong to one of four types; see e.g. [1]. The principal types are the first and second, wherein the stress and displacement vectors, respectively, are specified. The third and fourth types are called mixed boundary conditions because each employs both displacement and stress, i.e. type three specifies the normal component of the displacement vector and the tangential components of the stress vector while type four species the normal stress and tangential displacement components. The mixed boundary conditions are physically less realistic and sometimes considered approximations of types one and two. The choice of which boundary condition to use could significantly inform the likelihood of obtaining an analytic solution for non-stationary problems.The effectively analytical and numerically analytical methods, grounded mainly in the application of the Laplace transform in the time domain and Fourier (Hankel) transform for the linear coordinate, are developed for a solution of relevant boundary value problems. The problem thus is the build-up of originals which primarily depends on the character of the space-time distribution of the load on the boundary and the type thereof. Interested readers are directed to refs.[2-5] for studies of non-stationary wave processes at an operation of the concentrated or distributed surface actions in an isotropic elastic half-space (half-plane).Refs.[6] and [7] give the solutions of many non-stationary problems. The displacements of points on the surface of the semi-space as well as on the axis of symmetry were studied in refs.[8] and [9]. Thereat, for joint inversion of transforms, the authors used the Cagniard technique with account of transform uniformity with respect to transform parameters [10]. The saddle-point was used in ref. [9] for an approximate search for displacements of points remote from the axis of symmetry. Similarly, with the inverse Laplace transform, using the theory of residues, refs. [11][12][13] solved for the displacements and stress...

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“…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

Kubenko

2016

Int Appl Mech

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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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Nonstationary Load on the Surface of an Elastic Half-Strip

Cited by 8 publications

References 12 publications

On a non‐stationary load on the surface of a semiplane with mixed boundary conditions

On a non‐stationary load on the surface of a semiplane with mixed boundary conditions

Non-stationary stress state of an elastic layer at the mixed boundary conditions

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

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