Stress State of an Elastic Half-Plane under Nonstationary Loading

Kubenko, V. D.

doi:10.1007/s10778-015-0678-6

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…This can be proved by comparing the solutions. Figures 6a and 6b show the stress induced by a load extending at a constant rate k and calculated from solution (4.5) (dotted line) and from the solution of the first boundary-value problem [16] (solid line). Figure 6a illustrates the distribution of the stress along the z-axis for t = 15 and different values of k. Figure 6b shows the stress as a function of time for z = 5, 15.…”

Section: Comparison Of the Solutions Of The First And Fourth Boundarymentioning

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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Section: Comparison Of the Solutions Of The First And Fourth Boundarymentioning

confidence: 99%

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

Kubenko

2016

Int Appl Mech

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“…In particular, the variations of elastic modulus and Poisson's ratio with depth are considered. e numerical method used for analysis is developed by applying the fundamental solution of layered elastic solids [10][11][12][13][14][15][16][17] and integrating it over the loading area. e adaptive numerical quadrature and parallel computation techniques are used to evaluate the integrals over the elements on the discretized area.…”

Section: Introductionmentioning

confidence: 99%

Elastic Fields of a Nonhomogeneous Half-Space Subject to an Inclined Circular Load

Xie

Xiao

Yue

2020

Advances in Materials Science and Engineering

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The paper examines the elastic fields of displacements and stresses for a nonhomogeneous elastic half-space where the elastic parameters have a linear variation over a finite depth beyond which it is constant. The circular loading area is subjected to a uniform inclined load. The numerical method is developed by applying the fundamental solution of a layered elastic solid and integrating numerically it over the loading area. As a result, only the loading area needs to be discretized in using the proposed numerical method. Numerical examples of calculation of displacements are conducted, and excellent agreement with the existing closed-form solutions is obtained. The results obtained are used to understand the elastic fields induced by different types of loads in a nonhomogeneous medium.

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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 was developed in [8]. The corresponding boundary-value problem with initial conditions was formulated, and Laplace and Fourier transforms were used there.…”

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Non-stationary problem of elasticity for a quarter-plane

Vaysfeld

Zhuravlova

2021

BKNUPhM

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The plane problem for an elastic quarter-plane under the non-stationary loading is solved in the article. The method for solving was proposed in the previous authors’ papers, but it was used for the stationary case of the problem there. The initial problem is reduced to the one-dimensional problem by using the Laplace and Fourier integral transforms. The one-dimensional problem in transform space is written in vector form. Its solution is constructed as the superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is derived using the matrix differential calculations. The partial solution is found through Green’s matrix-function. The derived expressions for displacements and stresses are inverted by using of mutual inversion of Laplace-Fourier transforms. The solving of the initial problem is reduced to the solving of the singular integral equation regarding the displacement function at the one of the boundary of the quarter-plane. The time discretization is used, and the singular integral equation is solved using the orthogonal polynomials method at the fixed time moments. Based on numerical research some important mechanical characteristics depending on the time and loading types were derived.

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Stress State of an Elastic Half-Plane under Nonstationary Loading

Cited by 7 publications

References 10 publications

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

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

Elastic Fields of a Nonhomogeneous Half-Space Subject to an Inclined Circular Load

Non-stationary problem of elasticity for a quarter-plane

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