Surface Of Half-plane Research Articles

A semi-analytical approach for two-dimensional frictional contact of anisotropic magneto-electro-elastic solids

In this work, we present a semi-analytical modeling approach for solving the frictional contact of two-dimensional anisotropic magneto-electro-elastic (MEE) solids. The method is established in a broad framework in which the contact body is made of generally anisotropic MEE materials, the punch head can be arbitrary, and the punch can be subjected to generalized loading conditions including mechanical forces and electric and magnetic charges. An analytical solution of generalized distributed load along the half-plane surface is derived and utilized to formulate the semi-analytical equations governing the relation between the generalized displacements and tractions. Based upon the semi-analytical equations, the semi-analytical model in which the influence matrices can be calculated explicitly and its effectiveness is established. An iterative procedure is then used for determining the contact region, contact status, generalized surface deformation and the distribution of contact tractions, as well as the punch’s indentation depth and electric and magnetic potentials. The present approach is verified by comparing its numerical results with those obtained by the existing analytical solutions and boundary element method. The influences of different factors, such as punch profile, mechanical load, electric and magnetic charges, friction coefficient and material properties, are also analyzed. The capability of this method to handle complicated geometry, such as rough surfaces, is finally discussed, showing the potential performance of the developed method for practical engineering applications.

Dynamic response of a shallow lined circular tunnel by incident P1 and SV waves in an unsaturated half-plane

The dynamic response of a shallow lined circular tunnel in the unsaturated half-plane during earthquakes is investigated in this paper using the complex function method. The wave fields of the unsaturated medium with unknown coefficients are solved by Fredlund’s theory and Helmholtz’s decomposition based on the momentum and mass balance equations. General expressions for the displacements and total stresses of the solid skeleton, pore pressures, and seepage are derived using the complex function method and conformal mapping technique. The half-plane surface and the tunnel in the physical domain are mapped onto two bonded ring regions in the image plane using two conformal mapping functions. The boundary value problem yields a series of algebraic equations by the boundary conditions and the continuity conditions along the liner-medium interface. The unknown coefficients in the infinite series of algebraic equations can be solved numerically by truncating the series number. A parametric study is performed to investigate the dynamic responses of the medium and the liner by the incident P1 waves. Numerical results reveal that the embedded depth of the tunnel, the saturation degree of the medium, the liner-medium rigidity ratio, the frequency and angle of the incident P1 waves significantly affect the dynamic responses of the medium and the liner. The shielding effect on the tunnel to the incident P1 waves is obvious. The response of the liner-medium system to the incident waves is independent of the saturation degree of the unsaturated medium when its value is high. The effect of the liner-medium rigidity ratio is great on the dynamic stress concentrations of the liner-medium system and weak on the displacements along the half surface. The frequency of the incident excitation has a great influence on the displacements of the half surface.

The Problem of Wedge Indenter with Flat-Rounded Bottom Indenting Half-Plane Elastic Body

In this paper, the problem of a wedge indenter with flat-rounded bottom indenting an infinite half-plane elastic body is solved. It can be used to study the penetration force or the surface stress distribution of the cutter when the tool intrudes into an elastic body. First, a first-order continuity method is proposed to eliminate the stress singularity caused by the sharp corners of the wedge indenter with flat bottom. The two-dimensional elastic contact problem is transformed into a Riemann–Hilbert boundary value problem with discontinuous coefficients expressed in the form of complex variables. For the frictionless contact problem, the explicit analytical solution is obtained for contact stress on the half-plane surface. Furthermore, the expression of the stress potential function is also derived to solve the stress field in the half-plane. Verification indicates that the wedge indenter with flat-rounded bottom can degenerate to several well-studied shaped indenters, including the Hertz model and flat-rounded model. The model in this paper is also validated with its numerical counterpart built in the finite element program LS-DYNA. Finally, the sensitivity of different configuration parameters on the normal contact stress and interior stress field in numerical model was investigated.

Mathematical modeling of stress waves under concentrated vertical action in the form of a triangular pulse: Lamb’s problem

The aim of the work. The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. Methods. To solve the non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements was used. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasiregular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the area under study is proposed. The method is based on the schemes: point, line and plane. The study area is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the study area is divided into linear end elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is given about numerical modeling of elastic stress waves in an elastic half-plane with a concentrated wave action in the form of a Delta function. The estimated area under study has 12 008 001 nodal points. A system of equations consisting of 48 032 004 unknowns is solved. The change of elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of Rayleigh surface waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves with a concentrated vertical action in the form of a triangular pulse on the surface of an elastic half-plane. After surface Rayleigh waves, a dynamic process is observed in the form of standing waves.

Analytical Solution of Stress in a Transversely Isotropic Floor Rock Mass under Distributed Loading in an Arbitrary Direction

Rock masses with a distinct structure may present a transversely isotropic character; thus, the stress state in a transversely isotropic elastic half-plane surface is an important way to assess the behavior of the interaction between the distributed loading and the surroundings. Most previous theoretical analyses have considered a loading direction that is either vertical or horizontal, and the stress distribution that results from the effect of different loading directions remains unclear. In this paper, based on the transversely isotropic elastic half-plane surface theory, a stress solution that is applicable to distributed loading in any direction is proposed to further examine the loading effect. The consistency between the analytical solution and numerical simulations showed the effectiveness of the proposal that was introduced. Then, it was utilized to analyze the stress distribution rule by changing the Poisson’s ratio and Young’s modulus of the model. The effects of the formation dip angle on the stress state are also examined. The stress distribution, depending on the physical property parameters and relative angle, is predicted using an analytical solution, and the mechanisms associated with the transversely isotropic elastic half-plane surface subjected to the loading in any direction are clarified. Additionally, extensive analyses regarding this case study, with respect to the mechanical behavior associated with changes in the stress boundary, is available. Hence, the proposed analytical solution can more realistically account for the loading problem in many engineering practices.

A finite crack in a half-plane under uniform heat flux or surface heat source

A novel and effective method is proposed to determine the temperature and thermal stresses in the case of a finite Griffith crack lying perpendicular to the surface of an isotropic half-plane under uniform remote heat flux or a surface heat source. The surface of the half-plane and the two faces of the crack are otherwise thermally insulating and traction-free. For the heat conduction problem, the thermally insulating crack is simulated by a continuous distribution of heat dipoles, and the resulting Cauchy singular integral equation (SIE) is solved numerically to arrive at the associated density function. The original thermoelastic problem can be treated equivalently as a mode II Zener-Stroh crack (ZSC) under isothermal conditions. The net dislocation of ZSC is determined by the aforementioned density function and the crack is modeled by a pileup of edge dislocations. The resulting SIE is solved numerically leading to the mode II stress intensity factors at the two crack tips induced by uniform remote heat flux or surface heat source. The net dislocation of an actual Zener-Stroh crack can be designed in such a way that the crack will become neutral to the uniform heat flux or surface heat source.

A loaded beam in full frictionless contact with a couple stress elastic half-plane: Effects of non-standard contact conditions

The plane problem of a loaded Euler-Bernoulli beam of finite length in frictionless bilateral contact with a microstructured half-plane modelled by the couple stress theory of elasticity is considered here. The study is aimed to investigate the size effects induced on the beam internal forces and moments by the contact pressure and couple stress tractions transmitted across the contact region. Use is made of the Green’s functions for point force and point couple applied at the surface of the couple stress elastic half-plane. The problem is formulated by imposing compatibility of strain between the beam and the half-plane along the contact region and three alternative types of microstructural contact conditions, namely vanishing of couple stress tractions, vanishing of microrotations and compatibility between rotations of the beam cross sections and microrotations of the half-plane surface. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, without any sound motivation, although the third boundary condition seems the most correct one. The problem is thus reduced to one or two (singular) integral equations for the unknown distributions of contact pressure and couple stress tractions, which are expanded in series of Chebyshev orthogonal polynomials displaying square-root singularity at the beam ends. By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the series. The contact pressure and couple stress along the contact region and the shear force and bending moment along the beam are then calculated under various loading conditions applied to the beam, varying the flexural stiffness of the beam and the characteristic length of the elastic half-plane. The size effects due to the characteristic length of the half-plane and the implications of the generalized contact conditions are illustrated and discussed.

Oblique surface cracking and crack closure in an orthotropic medium under contact loading

This article presents analytical methods for analysis of an inclined surface crack in an orthotropic medium subjected to contact loading. Governing partial differential equations are derived in accordance with plane theory of orthotropic elasticity. Contact and crack problems are formulated separately utilizing Fourier transformation techniques. The contact problem consists of a rigid flat punch pressing upon an orthotropic half-plane, and is reduced to a singular integral equation of the second kind. The surface crack is assumed to be tilted at an arbitrary angle with respect to the normal of the half-plane surface. Three different crack configurations are considered, which are completely closed, partially-closed, and fully-open cracks. Frictional contact between the crack faces is implemented in the cases of completely closed, and partially-closed cracks. A single singular integral equation is derived for the completely closed crack, whereas each of the problems of partially- and fully-open cracks requires derivation of a set of two singular integral equations. The contact problem and each of the crack problems are solved numerically by means of the expansion-collocation technique. A flowchart is developed and implemented to detect the type of crack closure configuration under a given contact loading. The parametric analyses carried out illustrate the influences of factors such as crack inclination angle, crack face and contact surface friction coefficients, material orthotropy, and relative punch location on crack face contact stresses, degree of crack closure, and mode I and II stress intensity factors.

On stress analysis of load for microperiodic composite half-plane with slant lamination

The paper deals with the plane problem of normal and tangential load for microperiodic composite with slant lamination. The boundary condition was considered as a normal and tangential load given on the surface of the composite half-plane. Microperiodic composite consisted of two components, differing in their Young’s modules. The homogenized model with microlocal parameters, which is presented by Woźniak (Int J Eng Sci 25:483–498, 1987. https://doi.org/10.1016/0020-7225(87)90102-9), Jędrysiak (Compos Struct 202:1253–1262, 2018. https://doi.org/10.1016/J.COMPSTRUCT.2018.05.155) was used to solve the considered problem. The problem was formulated and solved using the averaged boundary condition given by Perkowski et al. (Compos Sci Technol 67:2683–2690, 2007. https://doi.org/10.1016/J.COMPSCITECH.2007.02.013). The considered boundary value problem will be solved using the Fourier transform method. An analytical solution was obtained in a general form for any form of the function describing normal and tangential loads on the surface of a half-plane. An analytical solution was determined for the special cases of an elliptical, normal and tangential loads and for the special case of the concentrated force. The results are shown in the form of figures.

Урахування зношування при плоскому контакті прямокутного штампа з пружною півплощиною

The plane contact problem on wear of elastic half-plane by a rigid punch has been considered. The punch moves with constant velocity. Arising thermal effects are neglected because the problem is investigated in stationary statement. In this case the crumpling of the nonhomogeneities of the surfaces and abrasion of half-plane take place. Out of the punch the surface of half-plane is free of load. The solution for problem of theory of elasticity is constructed by means of Fourier integral transformation. Contact stresses are found in Fourier series which coefficients satisfy the dual integral equations. It leads to the system of nonlinear algebraical equations for unknown coefficients by a method of collocations. This system is reduced to linear system in the partial most interesting cases for computing of maximum and minimum wear. The iterative scheme is considered for investigation of other nonlinear cases, for initial approximation the mean value of boundary cases is used. The evolutions of contact stresses, wear and abrasion in the time are given. For both last cases increase or invariable of vertical displacement correspondently is obtained. In the boundary cases coincidence of results with known is obtained.

The elastic field distributions in damaged composite half-planes created by various surface loadings

A method of solution is presented for the prediction of the elastic field distributions in composite half-planes possessing periodic microstructure and embedded internal damage. The elastic field is created by the application of a compressive loading on a portion of the half-plane surface or by its indentation by rectangular, circular or wedge-shaped rigid punches. The method is based on a two scale (micro-to-macro) analysis. In the microscale analysis, a micromechanical model is employed for the determination of the effective stiffness tensor of the undamaged composite. This is followed by a macroscale analysis where the partially loaded composite half-plane which consists of distinct phases and an embedded damage is considered. The macroscale analysis consists of the solution of an auxiliary elasticity problem, followed by the application of the discrete Fourier transform in the domain of which the damaged composite problem is solved in conjunction with an iterative procedure and the transform inversion. Applications and verifications are given for a half-plane which consists of a porous alumina that includes a damage in the form of a short crack, a semi-infinite (long) crack or a rectangular hole.

Explicit formulation for the Rayleigh wave field induced by surface stresses in an orthorhombic half-plane

Abstract We develop an explicit asymptotic model for the Rayleigh wave field arising in case of stresses prescribed on the surface of an orthorhombic elastic half-plane. The model consists of an elliptic equation governing the behaviour within the half-plane, with boundary values given on the half-plane surface by a wave equation. Consequently, propagation along the surface is entirely accounted for by the hyperbolic equation, which, besides, may be immediately recast in terms of the associated surface displacement. The model readily solves otherwise involved dynamic problems for prescribed surface stresses, and its effectiveness is demonstrated for the classical Lamb's problem, as well as for the steady-state moving load problem. The latter example shows that the proposed model is really obtained by perturbation around the steady-state solution for a load moving at the Rayleigh speed.

Consideration of spatial variation of the friction coefficient in contact mechanics analysis of laterally graded materials

This paper presents a new analytical approach for sliding contact analysis of laterally graded materials, which allows taking into account the spatial variation of the friction coefficient. The method is developed by considering a sliding frictional contact problem between a laterally graded elastic medium and a rigid flat punch. Governing partial differential equations entailing the displacement components are derived in accordance with the theory of plane elasticity. General solutions are determined and boundary conditions are implemented by the use of Fourier transformation; and the problem is reduced to a singular integral equation of the second kind. Both the shear modulus and the coefficient of friction are assumed to be a functions of the lateral coordinate in the derivations. The singular integral equation is solved numerically by means of an expansion-collocation technique in which the primary unknown is represented as a series in terms of Jacobi polynomials. Outlined procedures yield the stresses at the half-plane surface and the tangential contact force required for sliding. Proposed techniques are verified by making comparisons to the contact stresses computed for constant-friction type sliding contact problems involving homogeneous and laterally graded materials. Parametric analyses are presented so as to demonstrate the influences of the variations in the friction coefficient and the shear modulus upon the contact stresses and the tangential contact force.

О нестационарной нагрузке на поверхности полуплоскости при смешанных граничных условиях

О нестационарной нагрузке на поверхности полуплоскости при смешанных граничных условиях

Off-critical parafermions and the winding angle distribution of the O(n) model

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents γ1 and γ11 as well as wedge critical exponents γ2(α) and γ21(α) and the exponent characterizing the winding angle distribution of the O(n) model in the half-plane, or more generally in a wedge of wedge-angle α. We assume only the existence of these exponents and, for some values of n, the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality γ1 − γ11 ⩾ 1, and its extension γ2(α) − γ21(α) ⩾ 1 for the edge exponents. We provide conjectured values for all exponents for n ∈ [ − 2, 2).

The contact problem for a wheel having a ‘flat’

The problem of quasi-static contact between a half-plane and a wheel having a small flat portion on its rim is considered, for the case where the ‘flat’ is parallel with the half-plane surface. The contact pressure distribution and corresponding internal state of stress are found and particular attention is given to characterising the nature of the singularity arising.

On the possibility of local buckling of the surface of an elastic half-space under compression

The possibility of local buckling of the free surface of the lower half-plane under compression is studied in a static formulation within the framework of plane deformation. It is shown that in some media small subcritical strains can lead to local buckling of the half-plane surface. It is found that two forms of local surface buckling correspond to one critical compression load.

Thermoelastodynamic disturbances in a half-space under the action of a buried thermal/mechanical line source

The transient dynamic coupled-thermoelasticity problem of a half-space under the action of a buried thermal/mechanical source is analyzed here. This situation aims primarily at modeling underground explosions and impulsively applied heat loadings near a boundary. Also, the present basic analysis may yield the necessary field quantities required to apply the Boundary Element Method in more complicated thermoelastodynamic problems involving half-plane domains. A material response for the half-space predicted by Biots thermoelasticity theory is assumed in an effort to give a formulation of the problem as general as possible (within the confines of a linear theory) . The loading consists of a concentrated thermal source and a concentrated force (mechanical source) having arbitrary direction with respect to the half-plane surface. Both thermal and mechanical line sources are situated at the same location in a fixed distance from the surface. Plane-strain conditions are assumed to prevail. Our problem can be viewed as a generalization of the classical Nakano–Lapwood–Garvin problem and its recent versions due to Payton (1968) and Tsai and Ma (1991) . The initial/boundary value problem is attacked with one- and two-sided Laplace transforms to suppress, respectively, the time variable and the horizontal space variable. A 9×9 system of linear equations arises in the double transformed domain and its exact solution is obtained by employing a program of symbolic manipulations. From this solution the two-sided Laplace transform inversion is then obtained exactly through contour integration. The one-sided Laplace transform inversion for the vertical displacement at the surface is obtained here asymptotically for long times and numerically for short times.

Dynamic indentation of an elastic half-plane by a rigid wedge: Frictional and tangential-displacement effects

A dynamical contact problem is studied in this paper. It involves an elastic half-plane which is indented by a rigid wedge-shaped body. In an effort to depart from the classical formulation of this problem, we consider frictional and tangential-displacement effects. More specifically, it is assumed that Coulomb friction develops between the contacting bodies and also that the tangential surface displacements are not negligible and should thus be coupled with the normal surface displacements in imposing the contact-zone boundary conditions. Certainly, the foregoing considerations model the dynamic indentation of an elastic half-plane in a more realistic way than the usual frictionless and uncoupled formulation. The contact region is assumed to extend at a constant sub-Rayleigh speed (this situation can be achieved by conveniently specifying the indentor kinetics), whereas, due to symmetry, friction may act in opposing directions on opposite sides of the indentor. The study exploits the problem's self-similarity by utilizing homogeneous-function techniques along with the Riemann-Hilbert problem analysis. As the present exact analysis shows, both the sign reversal of the tangential traction and the coupling of the displacement components along the contact length strongly influence the contact-stress behavior at the wedge-apex station. In particular, friction tends to create a power-type singularity at the changeover point of boundary conditions (due to symmetry, this point here is the point where the wedge apex makes contact with the half-plane surface), whereas the tangential-displacement effect tends to eliminate singular behavior there. Representative numerical results are given for the normal stress and tangential displacement along the contact zone, and the relation between the contact-zone expansion velocity and the indentor velocity.

Dynamic frictional indentation of an elastic half-plane by a rigid punch

Dynamic rigid indentation of a linearly elastic half-plane in the presence of Coulomb friction is studied in this paper. A rigid punch, which is either wedge- or parabolic-shaped, is rapidly driven into the deformable body so that stress waves are generated. The contact region is assumed to extend at a constant sub-Rayleigh speed (this situation can be achieved by conveniently specifying the kinetic and geometric characteristics of indentor), whereas, due to symmetry, friction acts in opposing directions on opposite sides of the indentor. As the present exact analysis shows, this sign reversal of the tangential traction along the half-plane surface creates an extra stress-singularity at the changeover point of the boundary conditions (due to symmetry, this point here coincides with the point where the indentor apex makes contact with the half-plane surface). The study exploits the problem's self-similarity by utilizing homogeneous-function techniques previously used by L.M. Brock, along with the Riemann-Hilbert problem analysis. Representative numerical results are given for the wedge indentation case.