Piecewise-homogeneous Plane Research Articles

Limit Equilibrium of a Piecewise Homogeneous Plane with Small-Scale Interfacial Shear Cracks at a Corner Point in the Presence of a Loaded Internal Semi-Infinite Crack

Limit Equilibrium of a Piecewise Homogeneous Plane with Small-Scale Interfacial Shear Cracks at a Corner Point in the Presence of a Loaded Internal Semi-Infinite Crack

Contact problem for a piecewise-homogeneous plane with an interfacial crack under dry friction

The article considers the contact problem of indentation of an absolutely rigid stamp with dry friction into one of the edges of an interfacial finite crack located in a piecewise homogeneous elastic plane. The governing system of equations for the problem is derived in the form of a system of three singular integral equations with respect to the normal contact pressure under the stamp and dislocation components of the displacements of the crack edge points. The exact solution is constructed. The behavior of contact stresses near the end points of a crack is studied depending on the ratio of the elastic characteristics of heterogeneous materials of half-planes that formed a piecewise-homogeneous plane.

Two-phase piecewise homogeneous plane deformations of a fibre-reinforced neo-Hookean material with application to fibre kinking and splitting

Two-phase piecewise homogeneous plane deformations are examined in respect of a neo-Hookean matrix material reinforced with embedded aligned fibres characterized by a single stiffness parameter. The deformations are interpreted in terms of fibre kinking and fibre splitting. Previous work has shown that such a transversely isotropic material can lose ellipticity if the reinforcing stiffness is sufficiently large and the fibre direction is sufficiently compressed. In particular, it was shown that the associated failure modes are characterised by the emergence of weak surfaces of discontinuity that are normal to the fibre direction (the onset of fibre kinking) or parallel to the fibre direction (the onset of fibre splitting). Here, the analysis of strong surfaces of discontinuity, developing from weak ones, is studied. The considered model can give rise to piecewise smooth plane deformations separated by a plane stationary surface of discontinuity, interpreted as either kinking or splitting. Attention is restricted to (plane) deformations in which, on one side of the surface of discontinuity, the load axis is aligned with the fibre axis. Then the fibre stretch on this side of the discontinuity is a natural load parameter. The ellipticity status of the two-phase piecewise homogeneous plane deformations is shown to span all four possible ellipticity/non-ellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be non-elliptic. Moreover, it is shown that the mechanism is dissipative, and maximally dissipative quasi-static failure motion is examined in respect of both kinking and splitting. It follows that, firstly, surfaces of discontinuity perpendicular to the fibre direction, associated with fibre kinking, are nucleated followed by surfaces of discontinuity parallel to the fibre direction, associated with fibre splitting. With respect to kinking, such maximally dissipative kinks nucleate only in compression as weak surfaces of discontinuity, with the subsequent motion converting non-elliptic deformation to elliptic deformation.

On the Stress State of a Heterogeneous Plate Made of a Material with a Step Change in Elastic Characteristics in the Presence of an Inclusion System and Temperature Influence

In the formulation of thermoelasticity and in the framework of the conventional theory of thermal stresses, the problem on the stress state of an elastic piecewise-homogeneous plane or an infinite plate at non-uniform steady-state heating is considered. On the interface of dissimilar materials, the compound plane is reinforced by a collinear system of absolutely rigid thin inclusions and is subjected to mechanical and thermal influences. First, to determine the temperature distribution in a piecewise-homogeneous plane the corresponding boundary value problem of the theory of steady-state heat conduction is solved using the integral Fourier transform. Solving this problem is reduced to solving a singular integral equation (SIE) that allows an exact solution. Further, the elastic displacements of points of the compound plane, caused by mechanical and temperature influence, are determined by the known methods of thermoselasticity. Based on these results, solving the problem of the contact interaction between the system of inclusions and a compound plane is again reduced to solving SIE, which also allows an exact solution. A special case is considered.

Stress state of a piecewise uniform layered space with doubly periodic internal cracks

The present paper deals with the stress state of a piecewise homogeneous plane formed by alternation junction of two distinct strips of equal height manufactured of different materials. There is a doubly periodic system of cracks on the plane. The governing system of singular integral equations of the first kind for the density of the crack dislocation is derived. The solution of the problem in the case where only one of the repeated strips contains one doubly-periodic crack is obtained by the method of mechanical quadratures.

О решении уравнения Пуассона на кусочно-однородной плоскости с круговым включением

О решении уравнения Пуассона на кусочно-однородной плоскости с круговым включением

Discontinuous Solutions of a Doubly Periodic Problem for a Piecewise Homogeneous Plane with Interphase Defects

Discontinuous solutions of equations of a plane elasticity problem for a piecewise homogeneous plane obtained by joining two dissimilar strips are found using the Airy stress function. In the general case, doubly periodic interphase defects are on their connection lines, on which displacements and stresses are discontinuous. Based on the solutions obtained, a system of governing singular integral equations is derived for a particular problem where the defect is a finite crack. This system is solved by the method of mechanical quadratures. Numerical calculations are also carried out.

Об упругом равновесии кусочно-однородной плоскости с трещиной в угловой точке границы раздела

Об упругом равновесии кусочно-однородной плоскости с трещиной в угловой точке границы раздела

Stress Concentration Near the Tip of an Internal Semi-Infinite Crack in a Piecewise-Homogeneous Plane with a Nonsmooth Interface

The exact solution to the symmetric problem of the elastic equilibrium of a piecewise-homogeneous isotropic plane with a V-shaped interface and a loaded internal semi-infinite crack is found using the Wiener–Hopf method

Reflection and transmission coefficients for radio waves incident obliquely on N identical plane plasma layers and zebra patterns in the spectrum of solar radio emission

The reflection and transmission coefficients for quasi-monochromatic radio waves incident at an arbitrary angle on an arbitrary number of identical piecewise-homogeneous plane plasma layers are calculated analytically and numerically. It is shown that alternating transparency and opacity stripes in the spectrum of radio waves passing through such a plasma structure (the zebra pattern effect) can be observed at any angle of incidence. The opacity stripes for ordinary waves are wider than those for extraordinary waves. For the zebra pattern to be well pronounced, the radio wave flux in the Sun’s atmosphere should be narrowly directed, which is possible during bursts.

The contact problem for a piecewise-homogeneous plane with a finite inclusion

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite inclusion, which meets the interface at a right angle and is loaded with shear forces, is considered. The contact stresses along the contact line are determined, and the behaviour of the contact stresses in the neighbourhood of singular points is established. By using the methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite section. Using an integral transformation, a Riemann problem is obtained, the solution of which is presented in explicit form.

Solution of boundary value problems for Laplace’s equation in a piecewise homogeneous plane with a parabolic crack (screen)

Boundary value problems for Laplace’s equation are considered in a piecewise homogeneous plane divided into two zones by a strongly permeable crack or a weakly permeable screen in the form of a parabola. The desired potentials have prescribed singular points (sources, sinks, etc.). Formulas are derived expressing the potentials in terms of harmonic functions that have the given singular points and describe similar processes in a homogeneous plane.

The contact problem for a piecewise-homogeneous plane with a semi-infinite inclusion

A piecewise-homogenous elastic plate, reinforced with a semi-infinite inclusion, which intersects the interface at a right angle and is loaded with shear forces is considered. The contact stresses along the contact line are determined and the behaviour of the contact stresses in the neighbourhood of singular points is established. Using methods of the theory of analytical functions and integral transformations the problem is reduced to a system of singular integro-differential equations on the semi-axis. The solution is presented in explicit form.

The steady subsonic motion of cracks and narrow notches along the interface in a combined anisotropic plane

Solutions of mixed dynamic problems are constructed in the case of a piecewise-homogeneous plane coinciding with the plane of symmetry of the elastic properties of the medium, that is, the most common case of anisotropy when the plane and the antiplane problems separate. Conditions for the stress and displacements vectors are specified in three combinations and of certain equivalent versions on the straight interface between the two materials. For example, the tractions on the open intervals are given, at the slippage, the shear resistance and the discontinuity in the normal displacement are specified and, in domains of cohesion, the discontinuities in the stresses and displacements are specified. By introducing new complex potentials, the common representations of solutions in steady-state dynamics [1, 2] can be successfully transformed to a type of boundary-value problems which has previously been studied [3–5]. The following problems are investigated as examples: the problem of a notch with an unknown section of contact of the edges due to an external stress field, and the problem of the motion of a semi-infinite cut under the action of forces applied to the cut surfaces, with alternative conditions of their contact (opening or slipping).

Interface Crack Problem for Electroelastic Body

Abstract A two-dimensional crack problem in the Comninou formulation is investigated for a piecewise homogeneous plane. Applying a special integral representation formula for the displacement vector the problem is reduced to a system of singular integral equations. The system is analysed and its solvability is proved using the potential method and the theory of singular integral equations.

Interaction between a circular opening and fractures originating from its boundary in a piecewise homogeneous plane

The complex variables boundary element method (CVBEM) is used to study interaction between a circular opening and fractures originating from its boundary in a piecewise homogeneous plane. A new complex hypersingular equation for piecewise homogeneous media with a circular opening is obtained. The equation is solved using the CVBEM technique with circular and straight boundary elements and polynomial approximations (with square root asymptotics for crack tip elements) for the unknown functions. The algorithm is verified through comparison with known semi-analytical and numerical solutions that involve interaction between a circular opening and specific systems of cracks or other openings. New numerical results concerning the interaction of the circular opening with the cracks and circular inclusions are presented. The method is applied to an important problem in the petroleum industry : modelling propagation of hydraulic fractures in the vicinity of a borehole.

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.

A Contact Problem of the Interaction of a Semi-Finite Inclusion with a Plate

Abstract A piecewise-homogeneous plane made up of twodifferent materials and reinforced by an elastic unclusion is considered on a semi-finite section where the different materials join. Vertical and horizontal forces are applied to the inclusion which haz a variable thichness and a variable elasticity modulus. Under certain conditions the problem is reduced to integrodifferential equations of third order. The solution is constructed effectively by applying the methods of theory of analytic functions to a boundary value problem of the Carleman type for a strip. Asymptotic estimates of normal contact stress are obtained.

On Some Contact Problems for Bodies with Elastic Inclusions

The paper deals with the contact problems of the theory of elasticity. The problems are reduced to Prandtl-type integral dif- ferential equations with a coecient at the singular operator which has higher-order zeros at the ends of the integration interval. In some concrete cases the solution is constructed eciently. Asymptotic rep- resentations are obtained. Investigation of the problem of stress concentration when contact be- tween dierent elastic media takes place remains one of the most topical tasks. Inclusions such as punches and cuts are stress concentrators and hence the study of the insuence of inclusions on the stress-strained state of solid deformable bodies, as well as the elaboration of methods allowing one to reduce stress concentration, are of great theoretical and practical importance. The problems of contact interaction between rigid inclusions of various geometrical forms and elastic bodies are considered in (1-5). In (6) some problems of contact interaction between a piecewise-homogeneous plane and a rigid inclusion, and also the anti-plane problem of an elastic half-space and problems of bending of plates with thin-shelled inclusions, are reduced to integral equations by means of the generalized method of integral transforms. As distinct from the previous papers, this paper deals with the problems of contact interaction between a thin elastic inclusion of varying rigidity and (a) an elastic half-space which is in the state of anti-plane deformation, and (b) an elastic piecewise-homogeneous plane.

A mixed dynamic problem of the theory of elasticity for a piecewise homogeneous plane

We study the process of splitting of a compressed piecewise homogeneous medium under high-speed motion of a wedge with the formation of a crack of unknown extent ahead of the wedge. The motion of the wedge occurs along the interface between physico-mechanical properties of the piecewise homogeneous plane. We carry out numerical studies. We obtain asymptotic representations of the elastic potentials at the limiting velocities of motion of the wedge. Two figures. One table. Bibliography: 6 titles.