The viscoelastic deformation of an orthotropic body (half-plane) by a rigid punch

The method of operator continued fractions is used to solve the problem on the stress state in a viscoelastic orthotropic half-plane loaded by a punch at the instantt=0. The pressure in the half-plane is determined on the basis of the Volterra principle and by solving the corresponding elastic problem. The influence of the rheological parameters on the stress state of the half-plane is shown by an example for a composite material. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 81–91, July, 2000.     

Rolling of a Rigid Cylinder over a Half-Plane with Initial Stresses (Harmonic and Bartenev–Khazanovich Potentials)

An asymmetric quasistationary problem for a prestressed half-plane with harmonic and Bartenev–Khazanovich potentials is solved based of the linearized theory of elasticity. The Mehler–Fock integral transform is used to solve the differential equations that describe the stress–strain state of the half-plane. The dependences of the normal and tangential stresses and stress intensity factors on the elongation are plotted     

Electro-mechanical frictionless contact behavior of a functionally graded piezoelectric layered half-plane under a rigid punch

The frictionless contact problem of a functionally graded piezoelectric layered half-plane in-plane strain state under the action of a rigid flat or cylindrical punch is investigated in this paper. It is assumed that the punch is a perfect electrical conductor with a constant potential. The electro-elastic properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. The problem is reduced to a pair of coupled Cauchy singular integral equations by using the Fourier integral transform technique and then is numerically solved to determine the contact pressure, surface electric charge distribution, normal stress and electric displacement fields. For a flat punch, the normal stress intensity factor and electric displacement intensity factor are also given to quantitatively characterize the singularity behavior at the punch ends. Numerical results show that both material property gradient of the FGPM layer and punch geometry have a significant influence on the contact performance of the FGPM layered half-plane.     

The tilted shallow wedge problem

The problem of indentation of a half-plane by a titled, shallow wedge, under frictionless conditions, is studied, using half-plane theory. The contact law itself, together with the pressure distribution and internal state of stress are found. Particular attention is then given to the nature of the pressure and state of stress at the apex, and an asymptotic form appropriate to all angular features within contacts, and therefore of widespread application, is introduced.     

Subsonic semi-infinite crack with a finite friction zone in a bimaterial

Propagation of a semi-infinite crack along the interface between an elastic half-plane and a rigid half-plane is analyzed. The crack advances at constant subsonic speed. It is assumed that, ahead of the crack, there is a finite segment where the conditions of Coulomb friction law are satisfied. The contact zone of unknown a priori length propagates with the same speed as the crack. The problem reduces to a vector Riemann–Hilbert problem with a piece-wise constant matrix coefficient discontinuous at three points, 0, 1, and ∞. The problem is solved exactly in terms of Kummer's solutions of the associated hypergeometric differential equation. Numerical results are reported for the length of the contact friction zone, the stress singularity factor, the normal displacement u₂, and the dynamic energy release rate G. It is found that in the case of frictionless contact for both the sub-Rayleigh and super-Rayleigh regimes, G is positive and the stress intensity factor K_II does not vanish. In the sub-Rayleigh case, the normal displacement is positive everywhere in the opening zone. In the super-Rayleigh regime, there is a small neighborhood of the ending point of the open zone where the normal displacement is negative.     

Mixed Boundary Value Problem in a Non-homogeneous Medium Under Steady Distribution of Temperature

This paper is concerned with a mixed boundary value problem of a non-homogeneous medium under steady distribution of temperature whose elastic constants are exponential functions of y. By using Fourier cosine transforms the mixed boundary value problem of heat conduction is reduced to a Fredholm integral equation of the second kind. Then the elastic problem of the non-homogeneous semi-infinite half-plane under distribution of load over a plane face is solved. The influence of the non-homogeneity of the material on the thermal stress distribution is illustrated graphically.     

Dynamics of a system comprising a pre-stressed orthotropic layer and pre-stressed orthotropic half-plane under the action of a moving load

This paper investigates the dynamic response to a moving load of a system comprising an initially stressed covering layer and initially stressed half-plane, within the scope of the piecewise-homogeneous body model utilizing three-dimensional linearized wave propagation theory in the initially stressed body. It was assumed that the materials of the layer and half-plane are anisotropic (orthotropic), and that the velocity of the line-located moving load is constant as it acts on the free face of the covering layer. The investigations were made for a two-dimensional problem (plane-strain state) under subsonic velocity of the moving load for complete and incomplete contact conditions. Corresponding numerical results were obtained for the stiffer layer and soft half-plane system in which the modulus of elasticity of the covering layer material (for the moving direction of the load) is greater than that of the half-plane material, which was assumed to be isotropic. Numerical results are presented and discussed for the critical velocity and stress distribution for various values of the problem parameters. In particular, it was established that, the critical velocity of the moving load is controlled mainly with a Rayleigh wave speed of a half-plane material and the initial stretching of the covering layer causes to increase these values.     

Motion of a rigid punch at a low velocity along the boundary of a viscoelastic half-plane

The contact problem of the interaction of a rigid punch with a viscoelastic half-plane is considered. The dependence of the displacement of the boundary of half-plane on the normal load applied to it is determined, and the integral equation for determining the contact pressure is derived and solved by the method of “small λ”. Distributions of contact pressures under the punch are graphically represented.     

Vertical vibrations of a rigid edge inclusion under a harmonic load

The paper considers the problem of vibrations of a rigid edge inclusion, which lies in an elastic half-plane and emerges on the surface perpendicular to that half-plane. The vibrations are initiated by a harmonic force acting on the end of the inclusion, which emerges on the surface. The field of translations in the half-plane is shown to be represented by the superposition of two discontinuous solutions with discontinuities at the boundary between the half-plane and the line of the inclusion. The unknown discontinuities are determined from the boundary conditions and the conditions of the inclusion-medium interaction. The problem is thus reduced to one of solving a singular integral equation with an immobile singularity for the jump in shear stresses on the line of the inclusion. The equation obtained is solved numerically by the method of mechanical quadratures. The amplitudes of the inclusion vibrations and the stressed state of the medium near it are studied.Odessa State Marine Academy, Odessa, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 46–55, July, 1995.     

Dynamics of a system comprising an orthotropic layer and orthotropic half-plane under the action of an oscillating moving load

This paper investigates the dynamic response to a time-harmonic oscillating moving load of a system comprising a covering layer and half-plane, within the scope of the piecewise-homogeneous body model utilizing of the exact equations of the linear theory of elastodynamics. It is assumed that the materials of the layer and half-plane are anisotropic (orthotropic), and that the velocity of the line-located time-harmonic oscillating moving load is constant as it acts on the free face of the covering layer. Our investigations were carried out for a two-dimensional problem (plane-strain state) under subsonic velocity for a moving load in complete and incomplete contact conditions. The corresponding numerical results were obtained for the stiffer layer and soft half-plane system in which the modulus of elasticity of the covering layer material (for the moving direction of the load) is greater than that of the half-plane material. Numerical results are presented and discussed for the critical velocity, displacement and stress distribution for various values of the problem parameters. In particular, it is established that the critical velocity of the moving load is controlled mainly with a Rayleigh wave speed of a half-plane material and the existence of the oscillation of the moving load causes two types of critical velocity to appear: one of which is less, but the other one is greater than that attained for the case where the mentioned oscillation is absent.     

Stresses in a weighable half-plane with a semicircular notch

The influence of the surface roughness on the stress state of a rock is studied. For an elastic half-plane in the gravity field that contains a notch shaped like a semicircle, the stress distribution is constructed. It is shown that depending on the Poisson ratio, the notch bottom can be in a state of tension or compression. The polynomial dependence of pressure on depth is given on the axis of symmetry. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 152–161, January–February, 2000.     

Subinterface crack in an anisotropic piezoelectric bimaterial

In this paper, the problem of a subinterface crack in an anisotropic piezoelectric bimaterial is analyzed. A system of singular integral equations is formulated for general anisotropic piezoelectric bimaterial with kernel functions expressed in complex form. For commonly used transversely isotropic piezoelectric materials, the kernel functions are given in real forms. By considering special properties of one of the bimaterial, various real kernel functions for half-plane problems with mechanical traction-free or displacement-fixed boundary conditions combined with different electric boundary conditions are obtained. Investigations of half-plane piezoelectric solids show that, particularly for the mechanical traction-free problem, the evaluations of the mechanical stress intensity factors (electric displacement intensity factor) under mechanical loadings (electric displacement loading) for coupled mechanical and electric problems may be evaluated directly by considering the corresponding decoupled elastic (electric) problem irrespective of what electric boundary condition is applied on the boundary. However, for the piezoelectric bimaterial problem, purely elastic bimaterial analysis or purely electric bimaterial analysis is inadequate for the determination of the generalized stress intensity factors. Instead, both elastic and electric properties of the bimaterial’s constants should be simultaneously taken into account for better accuracy of the generalized stress intensity factors.     

Elastoplastic equilibrium of a weighable isotropic half-plane with a circular hole

The elastoplastic state of a weighable isotropic half-plane with a circular hole is studied. Complex Kolosov-Muskhelishvili functions which describe the elastic state of the half-plane are constructed. The unknown interface between the plastic and elastic regions is studied with allowance for the single-valuedness of the elastic displacements. The problem is also solved by the small-parameter method, and the two solutions are compared. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 93–98, March, 1999.     

A generalised stress intensity approach to characterising the process zone in complete fretting contacts

The known analytical contact solution for the stress field induced by a rigid, square-ended punch, sliding on an elastic half-plane defines the stress state everywhere in the half-plane. An asymptotic approach is then used to determine the characteristic stress field at the edge of the contact, which is matched with the contact solution. Hence, the regions over which the asymptotic solution is valid are found. Using a method analogous to the crack-tip stress field, a generalised stress intensity factor is defined, with the aim of providing a single variable characterisation of the stress state at the punch corner. The crack initiation process zone for a fretting fatigue crack is therefore captured, and the conditions for small scale yielding explicitly found.     

A combined approach of the Laplace transform and Padé approximation solving viscoelasticity problems

In this paper the Laplace transform method is combined with Padé approximations to solve linear viscoelastic problems. This approach allows to avoid the usual difficulties of original function determination. An algorithm is given to find solution with arbitrary precision. As an example the solution for problem of viscoelastic orthotropic half-plane stress state under concentrated normal force is given.     

Comprehensive bounded asymptotic solutions for incomplete contacts in partial slip

Solutions for the traction distributions and corresponding sub-surface state of stress adjacent to the edge of an incomplete contact suffering partial slip are found. The effects of frictional shakedown and a synchronously varying in-plane tension on the solution are found in closed form. The value of the asymptote, and its characterisation by just three independent parameters is illustrated by applying it to the finite problem of a rigid, tilted punch pressed onto a half-plane, and suffering partial slip induced by the application of in-plane tension.     

The mixed mode crack problem in an FGM layer bonded to a homogeneous half-plane

In this paper, the plane elasticity problem of an arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane is considered. The problem is modeled by assuming that the elastic properties of the FGM layer are exponential functions of the thickness coordinate and are continuous at the interface of the FGM layer and the half-plane.The Fourier transform technique is used to reduce the problem to the solution of a system of Cauchy-type singular integral equations, which are solved numerically. The stress intensity factors are computed for various crack orientations, crack locations and material parameters. The results show that crack length, crack orientation and the non-homogeneity parameter of the strip material have significant effect on the fracture of the FGM layer.     

Surface instability of a semi-infinite anisotropic elastic body under surface van der Waals forces

We investigate the surface instability of an anisotropic elastic half-plane subjected to surface van der Waals forces due to the influence of another rigid contactor by means of the Stroh formalism. It is observed that the surface of a generally anisotropic elastic half-plane subjected to van der Waals forces from another rigid flat is always unstable. The wave number of the surface wrinkling is only reliant on the positive M₂₂ component of the 3 × 3 surface admittance tensor M, the van der Waals interaction coefficient β and the surface energy γ of the elastic half-plane. The decay rate of surface perturbation along the direction normal to the surface of the anisotropic half-plane is different from the wave number, a phenomenon different from that observed for an isotropic half-plane.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Linear viscoelastic analysis of a semi-infinite porous medium

This paper considers the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, non-overlapping circular holes. The sizes and the locations of the holes are arbitrary. Constant or time dependent far-field stress acts parallel to the boundary of the half-plane and the boundaries of the holes are subjected to uniform pressure. Three types of loading conditions are assumed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, a force uniformly distributed over the whole boundary of the half-plane. The solution of the problem is based on the use of the correspondence principle. The direct boundary integral method is applied to obtain the governing equation in the Laplace domain. The unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex Fourier series. A system of linear equations is obtained by using a Taylor series expansion. The viscoelastic stresses and displacements at any point of the half-plane are found by using the viscoelastic analogs of Kolosov–Muskhelishvili’s potentials. The solution in the time domain is obtained by the application of the inverse Laplace transform. All the operations of space integration, the Laplace transform and its inversion are performed analytically. The method described in the paper allows one to adopt a variety of viscoelastic models. For the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered. The accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ANSYS.