The plane problem of the theory of elasticity

Lurie, A. I.; Belyaev, Аlexander K.

doi:10.1007/978-3-540-26455-2_7

Cited by 25 publications

(32 citation statements)

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“…Following the well-known approach in, 49 we will seek a solution of Eq. (6.1) in the form of expansions in Legendre polynomials P n (summation from n = 2 to = ∞).…”

Section: The Local Stability Of Centrisymmetric Deformationsmentioning

confidence: 99%

The stability of the equilibrium of two-phase elastic solids

Yeremeyev¹,

Freidin²,

Sharipova³

2007

Journal of Applied Mathematics and Mechanics

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“…Following the well-known approach in, 49 we will seek a solution of Eq. (6.1) in the form of expansions in Legendre polynomials P n (summation from n = 2 to = ∞).…”

Section: The Local Stability Of Centrisymmetric Deformationsmentioning

confidence: 99%

The stability of the equilibrium of two-phase elastic solids

Yeremeyev¹,

Freidin²,

Sharipova³

2007

Journal of Applied Mathematics and Mechanics

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“…This observation suggests to apply the theory of thin elastic rods (see [11,12] for engineering theories and [13], [14,Ch. 6], [15,Ch.…”

Section: Preamblementioning

confidence: 99%

Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Kozlov

Назаров

2017

J Elast

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Elastic bodies with cuspidal singularities at the surface are known to support wave processes in a finite volume which lead to absorption of elastic and acoustic oscillations (this effect is recognized in the engineering literature as Vibration Black Holes). With a simple argument we will give the complete description of the phenomenon of wave propagation towards the tip of a three-dimensional cusp and provide a formulation of the Mandelstam (energy) radiation conditions based on the calculation of the Umov-Poynting vector of energy transfer. Outside thresholds, in particular, above the lower bound of the continuous spectrum, these conditions coincide with ones due to the traditional Sommerfeld principle but also work at the threshold frequencies where the latter principle cannot indicate the direction of wave propagation. The energy radiation conditions supply the problem with a Fredholm operator of index zero so that a solution is determined up to a trapped mode with a finite elastic energy and exists provided the external loading is orthogonal to these modes. An example of a symmetric cuspidal body is presented which supports an infinite sequence of eigenfrequencies embedded into the continuous spectrum and the corresponding trapped modes with the exponential decay at the cusp tip. We also determine (unitary and symmetric) scattering matrices of two types and derive a criterion for the existence of a trapped mode with the power-low behavior near the tip.

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“…9 In the General case the tensor of elastic properties Ê is considered constant by in the equation (5), moreover, the start timing is the moment occurrence of dynamic programming process, а but age of concrete has its start timing; the tensor of kernels of relaxation is calculated at a certain age of the concrete from wave or other dynamic experiments. To determine the dynamic states of an object, we use the Lagrange-d'Alembert variational equation [12]:…”

Section: Modal Approachmentioning

confidence: 99%

“…Here def(u) denotes a tensor of the II rank calculated by the Cauchy formula [12] for small deformation.…”

Section: R-s-p Seminar 2017 Theoretical Foundation Of Civil Engmentioning

confidence: 99%

“…The solution of equation (10) is easily obtained by using the integral Laplace transform with respect to time: (12) In the last formula the first two terms are the General solution of (10) defined by the initial conditions, the last term (integral Duhamel) is a particular solution corresponding to given loads. Then the function ) W(t is the normalized matrix of fundamental solutions having the obvious properties of (12) (11) Thus, if the known oscillation modes Y and normalized matrix of fundamental solutions W( )…”

Section: Diagmentioning

confidence: 99%

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Modal approach in structural-dynamic equations

Kovalchuk

2017

MATEC Web Conf.

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Abstract. Some aspects of the solution of inhomogeneous problems of rods and plates are considered in the article. Dynamic effects from the effects of short-term loads are revealed with the help of modal decomposition under the assumption of the constancy of the elastic properties of concrete corresponding to its age during the application of the perturbing dynamic effect. Superposition of reactions to long-and-shortterm effects are possible due to the inadmissibility of finite deformations in building structures.

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The plane problem of the theory of elasticity

Cited by 25 publications

References 0 publications

The stability of the equilibrium of two-phase elastic solids

The stability of the equilibrium of two-phase elastic solids

Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Modal approach in structural-dynamic equations

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