Half-plane Problem Research Articles

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

Lamb’s problem for a non-homogeneous half-plane

Lamb’s problem for a non-homogeneous half-plane

Theoretical analysis of the surrounding rock stress field and energy field in shallow rectangular chambers during excavation

The mechanical analysis of surrounding rock after chamber excavation is of great significance for revealing the essence of stratum disturbances. Based on the Schwarz alternating method, a shallow rectangular chamber model was decomposed into a rectangular hole problem in an infinite plane and a half-plane problem. The general iterative expression and the explicit stress solution in shallow rectangular chambers when the mapping coefficient takes three terms were established by processing the boundary conditions with a mapping function and Fourier series. Then the analytical solution of the energy density accumulation of the surrounding rock was deduced, and the reliability of the theoretical solutions of the stress field and energy field were verified by numerical simulations. The disturbance laws of the lateral pressure coefficient, burial depth and chamber shape on the two fields were also analysed. The results show that the horizontal stress at the sidewall increases with increasing lateral pressure coefficient and burial depth. The peak of the vertical stress moves away from the wall with increasing lateral pressure coefficient. The energy accumulation at the corner increases with increasing chamber height and decreases with increasing burial depth and chamber width, forming a high stress area at the sidewall. With increasing Poisson's ratio, the corner energy distribution gradually flattens, and the energy decays more rapidly with distance.

Conversion of the UTD Method to an Exact Method at All Frequencies for 2-D Conducting Half-Plane Problems

In addition to two recently discovered series, we discover a new asymptotic series for the exact solution of 2-D conducting half-plane problems. These series are compared with the ones used in the conventional UTD method solutions of the problem. By attaching some coefficients, which are not complex, to the terms of those series we convert the UTD method to an exact method at all frequencies, including low ones, regarding the given problem type. Schemes for calculations of these series with 7D accuracy (7 digits of accuracy) or 2D accuracy for high-speed calculations are described.

The Flamant problem for an orthotropic half-plane

The Flamant problem for an orthotropic half-plane

Analytical solution of an orthotropic elasticity for a frictional rigid flat punch on a half plane with an oblique edge crack using a mapping function

A solution of an orthotropic elastic plane problem is derived for a frictional rigid flat punch on a half plane with an oblique edge crack. A mapping function is used for arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped half plane problems can be solved by exchanging the mapping function. Therefore, once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, the stress distributions are shown for a frictional coefficient and an oblique edge crack angle. Four elastic constants are expressed by two independent parameters.

RESEARCH OF THE STRESS-STRAIN STATE OF THE ORTHOTROPIC HALF-PLANE UNDER THE PLANAR DEFORMATION CONDITIONS

The first basic boundary problem of the elasticity theory connected with the determination of the stress-strain state of the orthotropic half-plane under the planar deformation conditions has been considered. The stresses at the boundary y = 0 are known. The stresses tend to zero at infinity. It is necessary to determine the stress and displacement at any point of the half-plane. A brief overview of the scientific works which highlight the methods and approaches to solving the problems of the theory of elasticity, the strength as for the determination of stresses and deformations in orthotropic bodies, in particular plates, slabs, and beams is provided. The solution of the given boundary problem for the orthotropic half-plane is sought in the transformant space of the one-dimensional integral Fourier transformation. At the same time, all the main equations of the problem and the boundary conditions are subjected to direct one-dimensional integral Fourier transformation. The solution of the formulated planar problem is based on the construction of the Fourier transform of the stress function, which satisfies the corresponding analogue of the biharmonic differential equation in the transformant space for the case of an orthotropic material. The form of the transformant of the stress function depends on the values of the orthotropic material elastic constants, namely on the values of the roots of the characteristic equation obtained in the transformant space. One of three possible cases have been considered. The relations between the transformant of the stress function and the transformants of stresses and displacements are established. The transformants of the stress function are expressed in terms of four auxiliary functions, which are related to the loads on the surface of the half-plane. We find two auxiliary functions from the boundary condition ( y = 0 ). The infinity conditions make it possible to establish the connection between two found auxiliary functions and the other two functions. After substituting the found expressions into transformants of stresses and displacements and applying the inverse integral Fourier transformation, we obtain the true values of stresses and displacements at the points of the orthotropic half-plane. The solutions for the specific cases have been obtained and the numerical results have been analyzed. The obtained calculations demonstrate the adequacy of the results and the logic of using the chosen method for the solution of the given problem.

УЧЕТ ТРЕНИЯ НА НАЧАЛЬНОМ ЭТАПЕ ВЕРТИКАЛЬНОГО ВНЕДРЕНИЯ ВЫПУКЛОГО УДАРНИКА В УПРУГУЮ ПОЛУПЛОСКОСТЬ

The initial stage of indentation of a convex rigid punch into isotropic elastic half-plane with friction is considered. A closed mathematical formulation in Cartesian coordinates is presented by equations of motion in the absence of a body force for homogeneous isotropic elastic medium in terms of displacement potentials, Cauchy equations for deformations, Hooke law and translational equation for the punch. Initial conditions are homogeneous. Outside of contact region surface is free from stresses. Inside contact region normal displacements of the surface of a half-space and the surface of a punch are taken to be equal and the relation between tangential and normal stresses in a form of Coulomb friction law is given. Due to the short duration of the supersonic stage and smallness of the contact region radius the tangential stress direction is considered to be point independent. Resolving functional equations are given in a form of convolutions with influence functions. The latter is the solution the original problem for half-plane with a special boundary condition in a form of the Dirac delta function. This solution is obtained in the domain of Laplace transform by time and Fourier transform by spatial coordinate. The original is constructed with the combined Fourier–Laplace inversion method. Solution is given with consideration to the initial stage characteristic property of supersonic velocity of contact region expansion (no smaller than propagation velocity of expansion-compression waves). It is shown that kinematic parameters of the punch, resulting force and discontinuities of the first kind on boundaries of the contact region are independent of friction unlike contact pressure. Calculation examples for different values of friction coefficient are presented.

Oblique water wave scattering by vertical elastic porous barriers

The mixed boundary value problem associated with scattering of obliquely incident water waves by a flexible porous barrier of different barrier configurations is considered here. The originally defined half-plane problem is converted into a quarter-plane problem. Then a novel connection is established between the solution potential of the converted problem and a resolvable potential in the quarter-plane. Further Galerkin approximation is used to obtain close bounds for the reflection and the transmission coefficients. Numerical results are presented for scattering quantities against various parameters in each of the problems under consideration.

A reasonable solution to the elastic problem of half-plane

A reasonable solution to the elastic problem of half-plane

Solution of three basic problems of elasticity for half-plane

Solution of elastic problem for half-plane is of the highest concern in almost every field of engineering. However, there are no relations binding the values of all components of the stress-strain state at the half-plane boundary This paper shows that arbitrary formulation of the boundary conditions considerably simplifies integral equations and transfers them to the class of Fredholm’s first kind equations. Considering symmetry, the problem for half-plane with mathematical cut is reduced to the problem on half-plane boundary. Regardless of the method, any solution is reduced to ill-posed problem.

An elastic half-plane with a fixed cycloid wavy boundary

We present a rigorous analysis of the plane problem of an elastic half-plane with a fixed cycloid wavy boundary under biaxial tension using Muskhelishvili’s complex variable method. A closed-form expression for the pair of analytic functions characterizing the stresses and displacements in the elastic body is derived by means of analytic continuation. Furthermore, elementary expressions for the interfacial and hoop stresses along the cycloid interface and stress distributions along the right of an interface valley are obtained. For a cusped cycloid interface, the stresses exhibit the square root singularity at the cusp tips. The mode I stress intensity factor at a cusp tip is expediently extracted from the resulting analytic functions. The interfacial normal stress and hoop stress are constant along a cusped cycloid interface except at the isolated cusp tips.

Investigation of Riemann Boundary Value Problem for Half Plane in Weighted Spaces of Analytic Functions

The boundary value problems that deal with the piecewise continuous solution of an elliptic system that satisfies a certain jump condition on the curves for a given closed curve or a set of finite non intersecting curves are Riemann boundary value problems. In the first part of this study, the literature summary on the Riemann boundary value problem, the Riemann boundary value problem for the half plane and the Riemann boundary value problem for the half plane in the weighted spaces are given. In the second chapter, the Riemann boundary value problem for the half plane in the weighted spaces is established, in the third chapter, Lemmas, which are the results obtained for the solution of the problem, are given together with their proofs, and finally, in the fourth chapter, two theorems and their proofs that indicate the necessary and sufficient conditions for the solution of the problem are given. Keywords: Riemann boundary value problem, Riemann boundary value problem for half plane, Riemann boundary value problem for half plane in weighted spaces DOI: 10.7176/JSTR/7-01-06

Half-plane differential-difference elliptic problems with general-kind nonlocal potentials

In the half-plane, the Dirichlet problem is considered for elliptic differential-difference equations with nonlocal general-kind potentials, which are linear combinations of translations of the desired function, not bounded by commensurability conditions. We find a condition for the symbol of the corresponding differential-difference operator, providing the classical solvability of the specified problem for each continuous and bounded boundary-value function. The representation of the specified classical solution by a Poisson-type integral is constructed.

Representation of incomplete contact problems by half-planes

Methodsfor finding the optimal choices of the applied remote loads – the applied normal force, moment, shear force and remote bulk stresses – needed to solve frictional contact problems in partial-slip using half-plane theory are derived by using data from contacts analysed by the finite element method. While the normal and shear forces and moment are readily found from equilibrium considerations, in order to determine the bulk stresses we must exploit details of the traction ratio and the direct strain within the contact, both of which are readily extracted from simulations. These contact loads enable the formulation of an equivalent half-plane problem for the contact, which can be used to determine much more precise estimates of the slip-zone sizes than are obtainable from direct use of frictional finite element analysis, as aggregated data from the finite element output is employed, and the half-plane analysis will add precision in terms of satisfaction of the laws of frictional slip and stick.

New Equations for the Solution of Line Source Illuminated 2-D Conducting Half-Plane Problems

A slowly convergent series that arises in the solution of 2-D conducting half-plane problems with line source (cylindrical wave) illumination is transformed into a new series by attaching an arbitrary parameter α to the terms of series. The effect of α on the convergence behavior of the new series is analyzed theoretically and also illustrated by figures. For some values of α , the new series reduces to some rapidly convergent and accurate forms as demonstrated by calculations.

О применении метода гиперсингулярных интегральных уравнений к решению задач для упругой плоскости с коллинеарной системой трещин

In the present paper, using the method of hypersingular integral equations, based on the formulas of the inversion of the corresponding singular integral equations, the exact quadrature solution of the classical problems of the mechanics of an elastic plane with a collinear system of cracks is constructed. The elastic plane is in a state of antiplane deformation or plane deformation; in case of antiplane deformation, crack edges are symmetrically loaded by tangential forces, while in case of plane deformation, they are again loaded symmetrically but by normal forces. Mixed boundary-value problems for an elastic half-plane equivalent to these problems are formulated. Under plane deformation, the mixed boundary-value problem for an elastic half-plane is discussed as well when the plane boundary is reinforced by two similar and symmetrically located semi-infinite stringers between which a system of an arbitrarily final number of stringers is situated. It is considered that the stringers are absolutely rigid for expansion and compression and absolutely flexible for bending. A particular case of two similar symmetrically located cracks is considered more in detail. In this case, the exact solution to the problem is also constructed by the method of Chebyshev orthogonal polynomials.

BEM-FORM Model for the Probabilistic Response of Circular Tunnels in Elastic Media

Problems involving cavities or excavations are widely addressed in geomechanics, in both analytical and numerical approaches. The boundary element method (BEM) is well-known as an interesting choice for half plane problems, providing accurate results at a low computational cost. This work deals with the probabilistic analysis of circular tunnels embedded in elastic media, coupling a BEM formulation to a structural reliability model. The gravitational loading and material parameters are treated as random variables, whose statistical description is taken from the literature. The loadings considered include the vertical overburden stress and the lateral earth pressure. Regarding the reliability evaluation, first order reliability method (FORM) and Monte Carlo simulation technique are employed, being compared in terms of accuracy. Regarding the BEM model, the Multiple Reciprocity Method (MRM) is used in the evaluation of domain integrals, and the subregion technique is employed for the analysis of the tunnel lining. Some analyses are presented, in order to validate the coupled BEM-FORM model and apply it to the estimation of failure probability, evaluating the influence of the random variables taken into account in the probabilistic response.

On contact problem for the microperiodic composite half-plane with slant layering

This paper deals with the plane frictional contact problem for the microperiodic composite half-plane with slant layering. Problem was considered as a contact of a non-deformable punch pressed into the composite half-plane with normal and tangential forces at the state of limit equilibrium. Microperiodic composite consists of small periodic layers made of two components, differing in their stiffness’ modules. To describe macro-properties of the considered material the homogenized model with microlocal parameters, presented by [1,2] was used. A mathematical problem was formulated and solved using solutions obtained by [3]. Solution to presented problem was received in a general form for any load parameters and material properties. The paper shows contact pressures and stress distributions considered separately for both components of the composite material and results are presented in the form of graphs.

Autoproducts in and near acoustic shadow zones created by barriers.

The autoproducts are nonlinear mathematical constructs developed from acoustic fields with non-zero bandwidth. When averaged through the field's bandwidth, the autoproducts may mimic a genuine acoustic field at frequencies that are lower or higher than the original field's bandwidth. The resulting opportunity to extend signal processing to user-selectable below- or above-band frequencies is intriguing for many signal processing algorithms. Based on prior work, the limitations of the autoproducts' mimicry of out-of-band fields are understood when the in-band acoustic field is well-represented by ray acoustics. Thus, the focus in this study is on autoproducts in acoustic shadow zones behind barriers containing only diffracted acoustic fields where a sum of ray-path contributions is not an adequate field description. Diffraction is expected to be a detriment to autoproduct techniques due to its sensitivity to frequency. Two ideal shadow-zone environments with exact analytic Helmholtz-equation solutions are considered: Sommerfeld's half-plane problem, also known as knife-edge diffraction, and Mie scattering from a sphere with ka = 40, where k is the wavenumber and a is the sphere's radius. With the exception of the shadow regions, autoproducts experience only mild degradation in field-mimicry performance when compared to what the ray-based theory would predict.