Impact of a plane die on an elastic orthotropic half-plane

To study the process of impact of a rigid body on the surface of an elastic body made of a composite material, we consider a nonstationary dynamic contact problem about the impact of a plane rigid die on an elastic orthotropic half-plane. The problem is reduced to solving an integral equation of the first kind for the Laplace transform of the contact stresses under the die base. An approximate solution of the integral equation is constructed with the use of a special approximation to the symbol of the kernel of the integral equation in the complex plane. The inverse Laplace transform of the solution results in determining the scalar contact stress field on the die base, the force exerted by the die on the elastic medium, and the vertical displacement field of the free surface of the orthotropic medium out side the die. The solutions thus obtained permit studying specific features of the process of die penetration into an orthotropic medium and the strain properties of the medium.     

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.     

Pressure pulse on the boundary of an elastic inhomogeneous half-plane

The problem about the motion of a pressure pulse at constant velocity along the boundary of an elastic homogeneous half-plane has been examined in [1–3]. The problem was considered as stationary in [1, 2], while in [3] it was solved by using a Laplace time transformation. An analogous problem is considered in this paper for an elastic half-plane with variable Lamé parameters and density of the medium.     

Plane contact problem for an elastic rectangular punch and a half-plane with initial stresses

International Applied Mechanics -     

Characteristics of local compliance of an elastic body under a small punch indented into the plane part of its boundary

An asymptotic solution of the contact problem of an elastic body indented (without friction) by a circular punch with a flat base is obtained under the assumption of a small relative size of the contact zone. The resulting formulas involve integral characteristics of the elastic body, which depend on its shape, dimensions, fixing conditions, Poisson's ratio, and location of the punch center. These quantities have the mechanical meaning of the coefficients of local compliance of the elastic body. Relations that, generally, reduce the number of independent coefficients in the asymptotic expansion are obtained on the basis of the reciprocal theorem. Some coefficients of local compliance at the center of an elastic hemisphere are calculated numerically. The asymptotic model of an elastic body loaded by a point force is discussed.     

Uniform indentation of the elastic half-space by a rigid rectangular punch

The problem considered is that of a rigid flat-ended punch with rectangular contact area pressed into a linear elastic half-space to a uniform depth. Both the lubricated and adhesive cases are treated. The problem reduces to solving an integral equation (or equations) for the contact stresses. These stresses have a singular nature which is dealt with explicitly by a singularity-incorporating finite-element method. Values for the stiffness of the lubricated punch and the adhesive punch are determined: the effect of adhesion on the stiffness is found to be small, producing an increase of the order of 3%.     

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.     

Edge crack opening in an elastic plane with a wedge-like cut in contact with a punch

The equilibrium of an elastic plane with a wedge-like cut and an internal or edge crack on the symmetry axis was studied in [1] in the case of punch indentation in the lateral faces of the cut at a distance from the cut tip. In [1], the systemof singular integral equations of the problemwas solved numerically by the mechanical quadrature method. In this paper, the generalized Wiener-Hopf method [2] is used to obtain the analytic solution of a similar problem in the case of an edge crack under punch pressure on parts of the cut lateral faces adjacent to the cut tip. Some special cases of this problem were considered earlier without a crack [3, 4] or a punch [5, 6].     

Effect of interface stresses on the elastic deformation of an elastic half-plane containing an elastic inclusion

The effect of the interface stresses is studied upon the size-dependent elastic deformation of an elastic half-plane having a cylindrical inclusion with distinct elastic properties. The elastic half-plane is subjected to either a uniaxial loading at infinity or a uniform non-shear eigenstrain in the inclusion. The straight edge of the half-plane is either traction-free, or rigid-slip, or motionless, which represents three practical situations of mechanical structures. Using two-dimensional Papkovich–Neuber potentials and the theory of surface/interface elasticity, the elastic field in the elastic half-plane is obtained. Comparable with classical result, the new formulation renders the significant effect of the interface stresses on the stress distribution in the half-plane when the radius of the inclusion is reduced to the nanometer scale. Numerical results show that the intensity of the influence depends on the surface/interface moduli, the stiffness ratio of the inclusion to the surrounding material, the boundary condition on the edge of the half-plane and the proximity of the inclusion to the edge.     

Diffraction of acoustic and elastic waves on a half-plane for boundary conditions of various types

The classical problem of wave diffraction on a half-plane with boundary conditions of different types and its generalizations to elastic media are considered. As a solution method it is proposed to combine the Fourier method of separation of variables and the series summation technique based on the use of integral representations of Bessel functions. The analytic solutions thus obtained are equally efficient in the near- and far-field diffraction regions. The two-term singularity at a corner point (in stresses for elastic media and in the velocity for acoustic media) was discovered for the first time. The knowledge of singularities in the scalar problem allowed one to construct the solution of the vector problem of elastic longitudinal wave diffraction. It is investigated how different types of boundary conditions on both sides of the half-plane affect the solution behavior in the far-field region. Possible physical interpretations of the obtained results are given.     

Phase boundary motion in an elastic bar of finite length

Using the continuum mechanical model of solid-solid phase transitions of Abeyaratne and Knowles, this paper examines the large time dynamical behavior of a phase boundary. The problem studied concerns a finite elastic bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. Interaction between the moving phase boundary and the elastic waves generated by an impact at the end of the bar and subsequent reflections is studied in detail by using a finite difference scheme. The numerical results show that the phase boundary in a finite bar returns to an equilibrium state after a disturbance of finite duration, whether the two-phase material is trilinear or not.     

Asymptotic analysis of stresses in an isotropic linear elastic plane or half-plane weakened by a finite number of holes

The problem of an isotropic linear elastic plane or half-plane weakened by a finite number of small holes is considered. The analysis is based on the complex potential method of Muskhelishvili as well as on the theory of compound asymptotic expansions by Maz’ya. An asymptotic expansion of the solution in terms of the relative hole radii is constructed. This expansion is asymptotically valid in the whole domain, i.e. both in the vicinity of the holes and in the far-field. The approach leads to closed-form approximations of the field variables and does not require any numerical approximation. Several examples of the interaction between holes or holes and an edge are presented.     

Quasiperiodic contact problem for an elastic half-plane

Chuvash State University, Cheboksary. Translated from Prikladnaya Mekhanika, Vol. 28, No. 6, pp. 22–28, June, 1992.