Fundamental solutions in an elastic space in the case of moving loads

Alekseyeva, L.A.

doi:10.1016/0021-8928(91)90119-f

Cited by 4 publications

(5 citation statements)

References 2 publications

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“…Differential Equations -Theory and Current Research as u ½ F ¼ 0: For the gaps of these functions, the theorem has been proved on the basis of classic methods (see [4,5]). For full proof of this theorem, see [6].…”

Section: Shock Waves As Generalized Solutions Of Transport Lame Equatmentioning

confidence: 99%

“…(17) and solving the corresponding system of linear algebraic equations for the Fourier transforms U(ξ 1 ,ξ 2 ,ξ 3 ). It is reduced to the form (see [4]).…”

Section: ð21þmentioning

confidence: 99%

“…(14), as noted earlier, since the solutions contain waves of two types. Green's tensor for the Lame transport equation was constructed by Alekseyeva [4] by applying fundamental solutions of the Laplace and wave equations and regularization functions f¯k m , which depends on the speed of transport load. Green's tensor has the regular form:…”

Section: ð23þmentioning

confidence: 99%

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General Functions Method in Transport Boundary Value Problems of Elasticity Theory

Alexeyeva¹

2018

Differential Equations - Theory and Current Research

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The Lame system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, and supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Transport boundary value problem for an elastic medium bounded by a cylindrical surface of arbitrary cross section and subjected to transport loads is considered in the subsonic and supersonic case with regard to shock waves. To solve problems, the generalized functions method is developed. In the space of generalized functions, generalized solutions are constructed and their regular integral presentations are obtained. Singular boundary equations solving the boundary value problems are presented.

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Section: Shock Waves As Generalized Solutions Of Transport Lame Equatmentioning

confidence: 99%

“…(17) and solving the corresponding system of linear algebraic equations for the Fourier transforms U(ξ 1 ,ξ 2 ,ξ 3 ). It is reduced to the form (see [4]).…”

Section: ð21þmentioning

confidence: 99%

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General Functions Method in Transport Boundary Value Problems of Elasticity Theory

Alexeyeva¹

2018

Differential Equations - Theory and Current Research

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“…This tensor was constructed in Ref. 4, and its components have the form where and (z) is Heaviside's function. Note that, at supersonic velocities, the carrier U is the interior of a cone with angle arctg (1/m 1 ) at the vertex; supp U = {(x, z): z > m 1 ||x||}.…”

Section: Green's Tensor For An Elastic Half-space With a Free Boundarmentioning

confidence: 99%

“…To do this, we employ the complete Fourier transform of this tensor, which has previously been constructed in Ref. 4. Its incomplete transform with respect to two of the Fourier variables corresponding to x 1 and x 2 enables us to construct the Fourier transform of the tensor U with respect to z:for subsonic speeds and, for transonic and supersonic speeds, where F() = 2i −1 K 0 () for transonic speeds and F() = H 0 () for supersonic speeds, K 0 is MacDonald's function, H 0 is the Hankel function of the second kind (H (2) 0 when > 0) and of the first kind (H (1) 0 when < 0) and r = x 2 1 + x 2 2 .…”

Section: Fundamental Spatially-periodic Solutionsmentioning

confidence: 99%