Unknown Contact Stresses Research Articles

Контактная задача о взаимодействии штампа в форме параболоида вращения и пороупругого слоя, закрепленного на упругом основании

The contact problem of the introduction of a rigid punch in the form of a paraboloid of rotation into a poroelastic layer fixed on an elastic half-space is considered on the basis of the equations of the theory of poroelastic bodies of Cowin-Nunziato. It is assumed that there is no friction in the contact zone. With the help of the Hankel integral transformation, the problem posed is reduced to solving the integral equation for an unknown contact stress. The collocation method is used to solve the integral equation. The area of contact, the values of contact stresses are found, and the deformation of the surface outside the punch is investigated. The relationship between the force acting on the punch and its displacement has also been investigated. A comparative analysis of the studied quantities for different values of the parameters of the poroelastic layer and the elastic foundation has been carried out. Numerical results are presented in the form of graphs.

INVESTIGATION OF CONTACT INTERACTION WITH STRIPE WITH INITIAL STRESSES IF SHE REINFORCED BY THE ELASTIC THIN FINITE STRINGER

In this paper, within the framework of the linearized theory of elasticity, the contact problem of reinforcement of a prestressed elastic band by a finite elastic stringer is considered. Elastic strip, free from reinforcement and load face, rigidly clamped to the base. The problem is solved for the cases of equal and unequal roots of the defining equation in the case of elastic potentials of an arbitrary structure. The research is carried out in general for the theory of large initial deformations and two variants of the theory of small initial deformations within the linearized theory of elasticity at arbitrary structure of elastic potential. It is assumed that the initial stress state of the elastic stringer and the band in the contact area is homogeneous. The research is carried out in the coordinates of the initial deformed state, which are related to the Lagrangian coordinates (natural state). In addition, the external concentrated load causes small perturbations of the corresponding values of the ground stress-strain state. The general solutions of the basic differential equations of the linearized theory of elasticity are given. Assuming that the stringer is loaded with both vertical and horizontal forces, it should be noted that the elastic stringer bends in the vertical direction like a normal beam, and in the horizontal direction is compressed or stretched like a normal rod with finite stiffness, which is in uniaxial stress. deformed state, the problem is mathematically formulated as a system of integro-differential equations with respect to unknown contact stresses. Using the Fourier transform, the system is solved in a closed form. Ultimately, the expressions for the contact voltages are represented as Fourier integrals. In the case of uneven roots for chemically active rubber SKU-6 and Treloir potential (neogukov type material), the results of numerical analysis are presented, presented in the form of a graph illustrating a fairly significant effect of initial stresses. Therefore, the effect of initial stresses on the stressstrain state of the elastic band is that the initial stresses in the strip lead to a decrease in stress in the contact area in the case of compression, and in the case of tension to increase them. As for the movements in the area of contact of the stringer with the strip, the opposite is true.

Contact Interaction of an Axisymmetric Stamp and an Elastic Layer Fixed on a Poroelastic Base

The work presented is devoted to the axisymmetric contact problems on the interaction of a rigid stamp and an elastic layer fixed on a porous elastic half-space by using the Cowin–Nunziato model. It is assumed that the stamp basis has a flat or parabolic shape and friction is absent in the contact zone. With the help of the Hankel integral transform, the problems stated are reduced to solving integral equations for the unknown contact stress by using the collocation method. The values of contact stresses and the contact area in the case of a parabolic stamp are found. The normal stresses on the interface between the elastic layer and the poroelastic half-space are investigated. The relationship between the force acting on the stamp and the corresponding displacement, which is one of the main characteristics in determining the mechanical parameters of the material by the indentation method, is investigated. A comparative analysis of the values found for various parameters of the elastic layer and the poroelastic base is carried out. The numerical results are presented in a table and in the form of graphs.

RELATIVE PORE VOLUME UNDER INDENTATION OF POROUS MATERIALS

Investigation of the function of the relative volume of pores under the load action is carried out on the base of the solution of the static contact problem of the indentation of a layer made of a material with voids or unfilled pores. A rigid strip indenter with a flat base is pressed into a porous layer that is adhered to a non-deformable base along the lower boundary. The formulated 3D problem of the indentation of a porous layer is reduced to solving the plane contact problem of the indentation of a porous strip. The plane contact problem is reduced to solving an integral equation for unknown contact stresses, the solution of which is constructed by the method of successive approximations in the form of an asymptotic expansion in the dimensionless parameter of the problem. The obtained contact stresses and the force acting on the indenter made it possible to study the influence of the nonclassical moduli of the layer porous material (the connectivity modulus and pore rigidity modulus) on the main contact characteristics and on the distribution of the function of the relative pore volume. The connectivity modulus increase leads to an increase in the compliance of the layer porous material, the pore rigidity modulus increase leads to an increase in the rigidity of the layer porous material. The maximum value of the distribution function of the relative pore volume in the porous material of the layer is achieved under the indenter base centre, regardless of the change in the porous material non-classical moduli.

Hertz-type frictional contact problem of a bidirectionally graded half-plane indented by a sliding rounded punch

The quasi-static frictional contact problem of a rigid rounded punch sliding over a bidirectionally graded half-plane is treated within the framework of plane-strain elasticity. The material heterogeneity in the half-plane is assumed to entail independent shear modulus variations along the depth and longitudinal directions in terms of exponential functions. In-plane field variables are converted through the Fourier transform method and subsequently reduced to a Cauchy singular integral equation (SIE) as a function of the unknown contact stress. The SIE is solved using a collocation approach on the roots of the Chebyshev polynomial of the first kind. A finite element approach (FEA) is also implemented for the same contact problem employing the homogeneous finite element method and the augmented Lagrange algorithm in the environment of a structural analysis software. Besides the verification results generated by the foregoing techniques, the effects of such factors as the friction coefficient and the degree of material heterogeneities are also addressed on the contact stress, the in-plane tensile stress, the normal contact force and on the centerline position of the punch. The bidirectional gradation is evidently shown superior to the conventional one-directional gradation, as far as their potentials are considered for the alleviation of the contact-induced failure risks.

Контактна задача про взаємодію смуги з початковими напруженнями з періодично розміщеними пружними накладками

The present article considers the formulation and solution of the problem on contact interaction of a pre-stressed band with regularly placed  elastic cover plates. The study has been carried out within the framework of the linearized elasticity theory in general form for the theory of large (finite) initial deformations and two variants of the theory of small initial deformations with an arbitrary structure of the elastic potential [1]. The solution of the problem comes down to a singular integro-differential equation for unknown contact stresses with a kernel expressed in the form of a sum of the kernel and some regular kernel under certain boundary conditions. The solution sought is given in the form of a series of Jacobi polynomials. A quasiregular infinite system of linear equations is obtained in order to determine the unknown coefficients of the series [2].

Контактна взаємодія пружної смуги з початковими напруженнями з нескінченим і скінченним стрингерами

The present article the problems of the contact interaction of infinite and finite elastic stringer with a prestressed elastic strip. The study is performed in the framework of the linearezed theory of elasticity in common form for the theory of large (finite) initial deformations and two variants of the theory of small initial deformations with elastic potential of arbitrary structure. Based on the assume that the stringers load at the same time the vertical and horizontal forces, fair model of bending beams in combination with horizontall elongation. This problem is mathematically formulated as a system of integro-differential equations of a relatively unknown contact stresses. The system is solved in a closed forms using transformation of Fourier. Expressions of stresses are represented by Fourier integrals with a simple enough structure. Influence of initial stress on the distribution of contact stresses is study and discovered the mechanical effects under the influence of concentrated loads.

Modeling of partial closure of slots system in perforated isotropic medium reinforced by stringers

On the basis of the methods of the theory of elasticity, a mathematical description of the model of partial closure of a system of slits in a perforated isotropic medium with foreign transverse inclusions is given. Such a medium can be considered as a perforated unrestricted plate, reinforced by a system of stringers of a very narrow cross section. It is believed that the medium is weakened by a periodic system of circular holes and rectilinear variable width slits. The variable width of the slits is comparable to elastic deformations. A method of solving the periodic elastic problem and an explicit method of constructing complex potentials corresponding to the unknown normal displacements along rectilinear slits are applied. General representations of solutions are constructed, that describe a class of problems with a periodic distribution of stresses outside circular holes and slits with contact zones. To determine the unknown contact stresses and sizes of contact zones, a singular integral equation is obtained, that reduces to a system of nonlinear algebraic equations. The system of algebraic equations can be solved by the method of successive approximations. As a result, the contact stresses and sizes of contact zones have been found.

The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

In this article, we study the axisymmetric torsional contact problem of a half-space coated with functionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thickness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral transform technique and reduce the problem to a singular integral equation, and then numerically determine the unknown contact stress and electric displacement at the contact surface. The results show that the surface contact stress, surface azimuthal displacement, surface electric displacement, and inner electromechanical field are obviously dependent on the gradient index of the FGPM coating. It is found that we can adjust the gradient index of the FGPM coating to modify the distributions of the electric displacement and contact stress.

On surface wave fields arising in soil-structure interaction problems

The paper aims at generalization of the specialized formulation, originally developed for the surface wave fields induced by prescribed surface stresses. We extend this formulation to soil-structure interaction problems with unknown contact stresses and internal sources. The problem for an internal source embedded in an elastic half-plane is reduced to that for prescribed surface stresses by considering the point source solution for an unbounded medium. In this case the sub-problems corresponding to normal and tangential stresses assume a separate treatment. Then, we analyze interaction of an elastic half-plane with a one degree of freedom mass-spring system. The focus is on a near-resonant regime investigated by a perturbation technique.

Analytical solution of the frictional contact problem of asemi-circular punch sliding over a homogeneous orthotropic half-plane

An analytical solution to the frictional sliding contact problem for homogeneous orthotropic materials indented by a semi-circular punch is developed.The principal axes of orthotropy are assumed to be parallel and perpendicular to the contact. Coulomb friction assumption is used to model the friction between the punch and the orthotropic medium. The mixed boundary value problem is reduced into a Fredholm integral equation of the second kind by using Fourier transform technique. The singular integral equation is solved analytically using Jacobi Polynomials for the unknown surface contact stresses. Numerical results show the effect of the orthotropic material parameters, coefficient of friction on the contact stress distribution and load vs. contact length behavior. Keywords: Contact mechanics; Friction; Orthotropic materials; Singular integral equation; Semi-circular punch DOI: 10.17350/HJSE19030000033 Full Text:

The effective solution of two‐dimensional integro‐differential equations and their applications in the theory of viscoelasticity

The effective solutions for integro‐differential equations related to problems of interaction of an elastic thin finite inclusion with a plate, when the inclusion and plate materials possess the creep property are constructed. If the geometric parameter of the inclusion is measured along its length according to the parabolic and linear law we have managed to investigate the obtained boundary value problems of the theory of analytic functions and to get exact solutions and establish behavior of unknown contact stresses at the ends of an elastic inclusion.

Frequency-dependent impedance of a strip foundation group and its representation in time domain

A semi-analytical procedure is presented to study the dynamic cross-interference among a group of long strip foundations on an elastic half-space, which is simplified as a two-dimensional problem. In addition to analytical solutions in terms of Green functions, a discretization method is applied to determine the lateral/rocking dynamic impedances of a foundation group. To determine the unknown contact stresses between the foundations and the supporting medium, the soil-foundation interfaces are discretized into a series of strip elements which are only subjected to constant lateral and vertical tractions. Multiple foundations with different widths and separation distances are considered in the formulations. For the time history analysis of those strongly frequency-dependent impedances of a foundation group through a substructure approach, complex Chebyshev polynomials are used to fit the lumped-parameter (L-P) model in the domain-transformation. Numerical results over a wide range of vibration frequencies for impedance calculation are verified with the existing methods. The effects of cross-interference on contact stress distributions and impedances for a group of two and three rigid foundations are discussed in detail. Finally, the accuracy and the validity of the L-P model are rigorously studied by simulating adjacent foundations on an elastic half-space within a close distance.

Partial closure of a curved crack in a plane using an induced thermoelastic stress field

On the basis of the methods of thermoelasticity, a mathematical description of the model of partial closure of a curved crack in a plane with the help of an induced temperature field was carried out. The determination of unknown contact stresses and the sizes of the contact zones in each approximation is reduced to the solution of a singular integral equation. Normal and tangential contact stresses and the values of the size of the tip zones where the faces of a curved crack close are found.

Frictional contact of anisotropic piezoelectric materials indented by flat and semi-parabolic stamps

An exact analysis of frictional contact of anisotropic piezoelectric materials indented by a rigid flat or semi-parabolic stamp is conducted. Fundamental solutions that can lead to the real values of physical quantities are detailed for each eigenvalue distribution. The complicated mixed boundary value problems are reduced into a singular integral equation of the second kind in terms of the unknown surface contact stress beneath the stamp. Employing excellent properties of Jacobi Polynomials, the exact solution of the reduced second kind singular integral equation can be obtained. Exact and explicit expressions of various surface stresses and electric displacement are given in terms of elementary functions. Relationships between the applied load and contact area are derived, and stress intensity factors at stamp edges are given. Numerical results are presented to show the effects of the friction coefficient on various surface stresses and electric displacement. The present investigation could provide a scientific basis for the theoretical and experimental test of contact behaviors of anisotropic piezoelectric materials.

The solution of system of integral differential equations and its application in the theory of elasticity

AbstractA contact problem of the theory of elasticity of infinite compound and infinite orthotropic plates with an elastic semi‐infinite or finite inclusion of variable rigidity is considered. The problem is reduced to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, using the methods of analytical functions to get exact solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion. In some case using the method of orthogonal polynomials we obtain the dual infinite system of linear algebraic equations. We can manage to investigate the obtained system on the quasi‐regularity and the method of reduction for approximate solution is developed.

Modeling of partial closure of cracks in a perforated isotropic medium reinforced by a regular system of stringers

A mathematical model of partial closure of a crack in a perforated isotropic medium with a system of rectilinear foreign inclusions is constructed. Such a medium can be interpreted as an unbounded plate reinforced by a regular system of ribs whose cross sections are shaped as narrow rectangles. The medium is assumed to be attenuated by a periodic system of circular holes and straight-line cracks. Determination of unknown contact stresses and contact zone sizes is reduced to solving a singular integral equation, which is transformed by an algebraization procedure to a system of nonlinear algebraic equations solved by the method of consecutive approximations.

The contact problem for a double-layer cylindrical base with account of friction forces

The contact problem of a double-layer cylindrical base is discussed with account of friction forces. The problem is reduced to an integral equation of the first kind in relation to unknown contact stresses. A direct method of collocation is used to solve it, enabling us to obtain sufficiently accurate solutions for virtually any values of the parameters. The contact stresses and the contact-zone area are calculated for various material and geometrical parameters of the base layers.

The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

The contact problems of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integro-differential equations or to systems of integro-differential equations with variable coefficient and singular operator. If such coefficient varies according to power law, we investigate the obtained equations and get exact or approximate solutions and study behavior of unknown contact stresses at the ends of the elastic inclusion.

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied