Summary
The model problem of scattering of a sound wave by an infinite plane structure formed by a semi-infinite acoustically hard screen and a semi-infinite sandwich panel perforated from one side ...and covered by a membrane from the other is exactly solved. The model is governed by two Helmholtz equations for the velocity potentials in the upper and lower half-planes coupled by the Leppington effective boundary condition and the equation of vibration of a membrane in a fluid. Two methods of solution are proposed and discussed. Both methods reduce the problem to an order-2 vector Riemann–Hilbert problem. The matrix coefficients have different entries, have the Chebotarev–Khrapkov structure and share the same order-4 characteristic polynomial. Exact Wiener–Hopf matrix factorization requires solving a scalar Riemann–Hilbert on an elliptic surface and the associated genus-1 Jacobi inversion problem solved in terms of the associated Riemann θ-function. Numerical results for the absolute value of the total velocity potentials are reported and discussed.
Nonlinear bending models for beams and plates Antipov, Y. A.
Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences,
10/2014, Letnik:
470, Številka:
2170
Journal Article
Recenzirano
Odprti dostop
A new nonlinear model for large deflections of a beam is proposed. It comprises the Euler-Bernoulli boundary value problem for the deflection and a nonlinear integral condition. When bending does not ...alter the beam length, this condition guarantees that the deflected beam has the original length and fixes the horizontal displacement of the free end. The numerical results are in good agreement with the ones provided by the elastica model. Dynamic and two-dimensional generalizations of this nonlinear one-dimensional static model are also discussed. The model problem for an inextensible rectangular Kirchhoff plate, when one side is clamped, the opposite one is subjected to a shear force, and the others are free of moments and forces, is reduced to a singular integral equation with two fixed singularities. The singularities of the unknown function are examined, and a series-form solution is derived by the collocation method in terms of the associated Jacobi polynomials. The procedure requires solving an infinite system of linear algebraic equations for the expansion coefficients subject to the inextensibility condition.
Analytical solutions to two axisymmetric problems of a penny-shaped crack when an annulus-shaped (model 1) or a disc-shaped (model 2) rigid inclusion of arbitrary profile are embedded into the crack ...are derived. The problems are governed by integral equations with the Weber–Sonine kernel on two segments. By the Mellin convolution theorem, the integral equations associated with models 1 and 2 reduce to vector Riemann–Hilbert problems with 3 × 3 and 2 × 2 triangular matrix coefficients whose entries consist of meromorphic and plus or minus infinite indices exponential functions. Canonical matrices of factorization are derived and the partial indices are computed. Exact representation formulae for the normal stress, the stress intensity factors (SIFs) at the crack and inclusion edges, and the normal displacement are obtained and the results of numerical tests are reported. In addition, simple asymptotic formulae for the SIFs are derived.
Summary
This article analyzes the axisymmetric contact problem of two elastic inhomogeneous bodies whose Young moduli are power functions of depth and the exponents are not necessarily the same. It ...is shown that the model problem is equivalent to an integral equation with respect to the pressure distribution whose kernel is a linear combination of two Weber–Schafheitlin integrals. The pressure is expanded in terms of the Jacobi polynomials, and the expansion coefficients are recovered by solving an infinite system of linear algebraic equations of the second kind. The coefficients of the system are represented through Mellin convolution integrals and computed explicitly. The Hertzian and Johnson–Kendall–Robertson adhesive models are employed to determine the contact radius, the displacement of distant points of the contacting bodies, the pressure distribution and the elastic normal displacement of surface points outside the contact circular zone. The effects of the exponents of the Young moduli and the surface energy density on the pressure distribution and the displacements are numerically analyzed.
Summary
Two model problems of plane elasticity on subsonic steady-state motion of a thin rigid body in an elastic medium are analyzed. Both models concern a finite body symmetric with respect to the ...plane of motion and assume that the body contacts with the surrounding medium according to the Coulomb friction law. The body, while moves, leaves a trailing semi-infinite crack-like cavity moving at the body speed. The first model also assumes that ahead of the body a finite crack-like cavity is formed, and it is moving at the same speed. The second model does not admit the existence of this finite cavity. Both problems reduce to two sequently solved Riemann–Hilbert problems with piece-wise constant coefficients. Analysis of the solution to these problems obtained by quadratures reveals that the normal and tangential traction components and the normal velocity are continuous for any point of separation of the medium from the body. A criterion for the separation point based on the analysis of the sign of the normal traction component is proposed. Numerical results for the length of the fore crack (the first model), the normal traction and the resistance force for some ogive-nose penetrators are reported.
An exact formula for the conformal map from the exterior of two slits onto the doubly connected flow domain is obtained when a fluid flows in a wedge about a vortex. The map is employed to determine ...the potential flow outside the vortex and the vortex domain boundary provided the circulation around the vortex and constant speed on the vortex boundary are prescribed, and there are no stagnation points on the walls. The map is expressed in terms of a rational function on an elliptic surface topologically equivalent to a torus, and the solution to a symmetric Riemann-Hilbert problem on a finite and a semi-infinite segments on the same genus-1 Riemann surface. Owing to its special features, the Riemann-Hilbert problem requires a novel analogue of the Cauchy kernel on an elliptic surface. Such a kernel is proposed and employed to derive a closed-form solution to the Riemann-Hilbert problem and the associated Jacobi inversion problem. The final formula for the conformal map possesses a free geometric parameter and two model parameters. It is shown that the solution exists and the vortex has two cusps, while the solution does not exist when the wedge angle exceeds π.
Summary
A convolution integral equation of the first kind and integro-differential equation of the second kind with the kernel $e^{-\gamma |y-\eta|}$ on a finite and semi-infinite interval are ...analyzed. For the former equation necessary and sufficient conditions for the existence and uniqueness of the solution are obtained, and when the solution exists, a closed-form representation for the solution is derived. On the basis of these results new integral relations for the spheroidal functions and Laguerre polynomials are obtained. The integro-differential equations on a finite and semi-infinite interval are transformed into a vector and scalar Riemann–Hilbert problem, respectively. Both problems are solved in closed-form. An application of these solutions to bending problems of a strip-shaped and a half-plane-shaped plate contacting with an elastic linearly deformable three-dimensional foundation characterized by the Korenev kernel $AK_0(\delta r)$ ($A$ and $\delta$ are parameters, $K_0(\cdot)$ is the modified Bessel function, and $r=\sqrt{(x-\xi)^2+(y-\eta)^2}$) is considered.
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