Boundary Of Half-plane Research Articles

Interaction of elastic waves with a curvilinear crack in a half-plane

We consider the problem of the interaction of monochromatic displacement waves with a curvilinear crack-cut in a half-plane. We find integral representations of the solution. The boundary-value problem is reduced to a system of singular integral equations. A parametric investigation is carried out for the effect of the form of the load, the fastening conditions on the boundary of the half-plane, and the curvature of the crack on the dynamic coefficients of the stress intensity.

Dynamic stress intensity factors of an inclined surface-breaking crack

The interaction of time-harmonic elastic waves with a surface-breaking crack, which is inclined under an arbitrary angle with the free boundary of an elastic half-plane, is investigated. Particular attention is devoted to the near-tip stress intensities induced by the incidence of elastic waves. By using the elastodynamic representation integral for the scattered displacement field in conjunction with Hooke's law, a system of traction boundary integral equations for the crack opening displacements is obtained. A special regularization technique is applied for solving these hyper singular integral equations. Numerical calculations have been carried out for various crack orientations and for different types of incident elastic waves. Numerical results for the elastodynamic stress intensity factors are presented as functions of the crack inclination angle, the angle of incidence and the dimensionless frequency of incident waves.

On the subsonic stationary motion of stamps and flexible cover-plates on the boundary of an elastic half-plane and a composite plane

A mixed dynamic problem for an elastic half-plane on different sections of whose boundary shear and normal stresses and displacements are given simultaneously in four fundamental combinations is considered. It is assumed that all the sections move at an identical constant subsonic velocity along the half-plane boundary and their number and mutual arrangement are arbitrary. An analogous problem on the interaction of two half-planes of different materials (a composite plane) is examined under the formulation of six kinds of contact conditions simultaneously in two modifications. The solutions are constructed in quadratures on the basis of new representations of the complex Galin potentials /1/. The first problem is reduced to a scalar combined Hilbert-Riemann boundary value problem /2/ for a plane with slots, and the second to unrelated Hilbert-Riemann and Hubert problems for the same domain. Both problems of the theory of analytic functions are solved by a new method different from /2/. The problem of the wedging of a composite plane by a finite stamp moving at a sub-Rayleigh velocity /3/, and the problem of the motion of a stamp and a flexible cover plate over a half-plane boundary at subsonic velocity are examined as examples. The exact solutions of stationary contact problems for a half-plane with two kinds of boundary conditions were first obtained by Galin /1/. The problem was formulated for a composite plane with three kinds of boundary conditions, whose solution is obtained in quadratures in the case of one slipping section /4/. However, as shown in /3, 5/, the method described in /4/ does not result in an exact solution for a large number of sections.

Non-stationary dynamic contact problem for a periodic system of stamps under arbitrary loading

The contact stresses are determined under a periodic system of stamps on the boundary of a homogeneous plastic half-plane and moving under the action of a load that is arbitrary in time. Unlike /1/, a more general problem is solved when the system of stamps is firstly invariant not only with respect to shear (translations) along the half-plane boundary by vectors that are multiples of a certain vector a 0, but also with respect to reflections in a periodic system of parallel planes π k ( k = 0, ± 1, ± 2, …) perpendicular to the vector a 0 and separated from each other by a distance l = | a 0|/2 (these transformations form a group C 1 h /2/, and secondly, the load is not assumed to be identical for all stamps (Fig.1).

Certain problems of an arbitrarily oriented stringer in a composite an isotropic plane

The fundamental solution for a composite anisotropic plane is constructed taking defects into account on the line separating the materials. On this basis, a mathematical formulation is given of the contact problem for an arbitrarily oriented stringer located in one of the half-planes, in the form of a singular integral equation with a fixed singularity. The solvability of the equation obtained is proved. The power nature of the behaviour of the solution is clarified (by using the asymptotic properties of Cauchy-type integrals), and the conditions are established for the satisfaction of which the mentioned asymptotic form is strengthened by a logarithmic polynomial. An exact solution is constructed for the problem of a semi-infinite stringer leaving the line of material separation at an arbitrary angle, and an analogous problem for a finite inextensible stringer. It is hown that the asymptotic form of the contact shear stresses at the point of departure does not depend on the elastic properties of the stringer. Solutions of analogous problems for anisotropic and orthotropic half-planes can be obtained by a passage to the limit. Such problems were examined earlier for a finite stringer in an anisotropic half-plane /1/ and for a semi-infinite stringer perpendicular to the boundary of an orthotropic half-plane /2, 3/. An incorrect assumption was made here about the power-logarithmic nature of the behaviour of the solution at the point of stringer departure at the boundary (the authors incorrectly utilized the asymptotic properties of Cauchy-type integrals). The error of the result /2/ is mentioned in /3/ where an exact solution of the problem is constructed. However, the true reason for the error is not given here, in which connection a false deduction is made about the fact that the asymptotic form of Cauchy-type integrals does not permit a unique solution of the question of the nature of the singularity if it is assumed to be power-logarithmic. The error of this deduction is shown below.

Interaction of curvilinear cracks with the boundary of an elastic half-plane

Interaction of curvilinear cracks with the boundary of an elastic half-plane

The Sliding With Coulomb Friction of a Rigid Indentor Over a Power-Law Inhomogeneous Linearly Viscoelastic Half-Plane

In a previous paper, the title problem was solved for a homogeneous power-law linearly viscoelastic half-plane. Such material has a constant Poisson’s ratio and a shear modulus with a power-law dependence on time. In this paper, the shear modulus is assumed also to have a power-law dependence on depth from the half-plane boundary. As in the earlier paper, only a quasi-static analysis is presented, that is, the enertial terms in the equations of motion are not retained and the indentor is assumed to slide with constant speed. The resulting boundary value problem is reduced to a generalized Abel integral equation. A simple closed-form solution is obtained from which all relevant physical parameters are easily computed.

Two problems with mixed boundary conditions for an incompressible isotropic hyperelastic material

Within the framework of nonlinear elasticity theory there is considered the equilibrium of a layer of incompressible Isotropie hyperelastic material under plane strain under the effect of gravity and forces P applied at infinity. The linearized equations generated by this state of stress and strain are investigated. It is shown that under for relationships between the material parameters, the layer thickness and the force P the equilibrium position can become unstable. Two problems are considered: the contact problem for a strip and the problem of a vertical crack of finite length emerging on the half-plane boundary. The action of the stamp and the crack is considered a small perturbation of the state of stress and strain caused by the action of the intrinsic weight and the force P.

A note on the solvability of simultaneous Wiener–Hopf equations

Sufficient conditions are given for the solvability of simultaneous Wiener–Hopf equations by means of the Wiener–Hopf–Hilbert technique, for the case when the kernel matrix of the system has the special form appropriate to half-plane boundary value problems.

Plastic deformation at the tip of an edge crack: PMM vol. 43, no. 1, 1979, pp. 160–166

An exact analytic solution of the problem of a crack emerging at the boundary of a free half-plane, is given for the case when the plastic zone is concentrated at the extension of the crack (a plastic zone appears in precisely such a form in such materials as low- C steel [1–3]. In particular, when the plastic zone has zero length, we arrive at the problem of stretching of an elastic half-plane with an edge crack. A general formula is obtained for the stress intensity co-efficient K I, and the formula yields the results of [4–6] as particular cases. An approximate solution of the problem discussed below was constructed in [7] for a particular case of a constant normal stress σ y at infinity.

Dynamic contact problem for a visco-elastic half-plane: PMM vol. 41, n≗ 5, 1977, pp. 926–934

Steady oscillations of a rigid stamp at the boundary of a visco-elastic half-plane are considered. No tractions exist outside the region of contact, and the absence of friction, Coulomb friction or coupling of the stamp to the half-plane all prevail within the region. A system of integral equations of the first kind obtained with the help of a Fourier transformation is reduced, by differentiating with respect to the coordinate and separating the kernal singularities, to a system of singular integral equations. At zero oscillation frequency the latter system coincides with the equations of the analogous static problem of the theory of elasticity. The exact solutions of the static problem are utilized for regularization of the system according to Carleman-Vekua when the oscillation frequency is not zero. The low frequency asymptotic of the system kernals is investigated using the contour integration, and the asymptotic properties of the Laplace transformation. The solution of the system is constructed in first approximation for low frequency oscillations. Oscillations of a stamp on an elastic half-plane were studied in [1, 2], while the results of the problem with coupling were presented at the All- Union Winter School (∗) (∗) Zlatina I.N. and Zil'bergleit A.S. Use of dual integral equations in the dynamic contact problem of oscillations of a rigid stamp on an elastic half-space. In “ Contact Problems of the Mechanics of Deformable Solids ”. Erevan, 1976. .

On two-dimensional electro-gasdynamic flows with allowance for the inertia of charged particles: PMM vol. 40, n≗1, 1976, pp. 65–73

Electro-gasdynamic flows in inertia-free approximation and those with allowance for inertia forces are investigated. Conditions under which inertia effects are considerable, are determined. Simple analytical solutions are derived for systems of electro-gasdynamic equations that describe the motion of particles in a uniform external electric field in the presence of tangential discontinuity of gasdynamic velocity at the half-plane boundary. The possibility of reverse current generation, i.e. of the return of particles to the emitter is demonstrated. Obtained results are compared with data related to inertia-free approximation. A numerical method is developed for solving the complete system of equations of electro-gasdynamics with allowance for particle inertia. The proposed method is used for investigating the expansion of electro-gasdynamic streams in channels. Results of numerical calculations for various values of controlling parameters are presented. Effects of inertia are set appart.

Jet Flap Diffuser: A New Thrust Amplifying Device

A jet sheet is proposed as an alternative to a rigid diffuser for a momentum propulsor. This appears attractive technically since the diffuser shape can be tailored by modulating jet momentum and angle and can be switched off in forward flight, its main function being to increase thrust-power ratio at static speeds. Theoretical global analysis for a steady inviscid incompressible flow predicts impressive thrust amplifications, showing effects of blowing coefficient, angle and jet thickness. Taking into account the energy required to feed the jet sheet, the propulsor thrust/power ratio is generally increased. The device can be applied to ducted fans, jet engines and seems particularly attractive for ejector thrust systems. To determine flow details, a linearized solution of the planar problem is given. A mapping transforms it into a half-plane boundary value problem of the Riemann-Hilbert-Poincare type. It is solved by combining Hilbert Transforms, asymptotic expansion, and a digital computer program.

Motion of a rigid stamp on the boundary of a viscoelastic half-plane: PMM vol. 32, no. 3, 1968, pp. 445–453

Motion of a rigid stamp on the boundary of a viscoelastic half-plane: PMM vol. 32, no. 3, 1968, pp. 445–453

The Stress on the boundary of an elastic half-plane in which body forces are acting

In this note we consider the problem of determining the stress on the boundary y = 0 of the elastic half-plane y ≡ 0 when there are prescribed body forces acting in the interior and the boundary is free from applied stress. Expressions for the components of stress at a general point of the half-plane when the imposed body force is concentrated at a single point have been derived by Melan [1], Sneddon [2] and Green [3], each author making use of a different method.

Flax growing within the empire : Journal of Royal Society of Arts, November 7, 1919.)

Steady oscillations of a rigid stamp at the boundary of a visco-elastic half-plane are considered. No tractions exist outside the region of contact, and the absence of friction, Coulomb friction or coupling of the stamp to the half-plane all prevail within the region. A system of integral equations of the first kind obtained with the help of a Fourier transformation is reduced, by differentiating with respect to the coordinate and separating the kernal singularities, to a system of singular integral equations. At zero oscillation frequency the latter system coincides with the equations of the analogous static problem of the theory of elasticity. The exact solutions of the static problem are utilized for regularization of the system according to Carleman-Vekua when the oscillation frequency is not zero.The low frequency asymptotic of the system kernals is investigated using the contour integration, and the asymptotic properties of the Laplace transformation. The solution of the system is constructed in first approximation for low frequency oscillations. Oscillations of a stamp on an elastic half-plane were studied in [1, 2], while the results of the problem with coupling were presented at the All- Union Winter School (∗).