Indentation of a Rigid Body into an Elastic Plate

The problem of a punch shaped like an elliptic paraboloid pressed into an elastic plate is studied under the assumption that the contact region is small. The action of the punch on the plate is modeled by point forces and moments. The method of joined asymptotic expansions is used to formulate the problem of one–sided contact for the internal asymptotic representation; the problem is solved with the use of the results obtained by L. A. Galin. The coordinates of the center of the elliptic contact region, its dimensions, and the angle of rotation are determined. The moments which ensure translational indentation of the punch are calculated and an equation that relates displacements of the punch to the force acting on it is given.     

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch. We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular punch, and a concave punch.     

Indentation of a Punch with a Fine-Grained Base into an Elastic Foundation

The linear contact problem for a system of small punches located periodically on a part of the boundary of an elastic foundation is studied. An averaged contact problem is derived using the Marchenko–Khruslov averaging theory. An asymptotic formula is obtained for the translational capacity of a smooth punch with a fine-grained flat base.     

On the Material Symmetry of Elastic Rods

This paper presents a treatment of material symmetry for hyperelastic rods. The rod theory of interest is based on a Cosserat (or directed) curve with two director fields, and was developed in a series of works by Green, Naghdi and several of their co-workers. The treatment is based on Murdoch and Cohen's work on material symmetry of Cosserat surfaces. Two material symmetry groups are discussed: one pertains to the strain-energy function, while the other pertains to the response functions. The paper closes by showing how the treatment relates to the form-invariant approach used in Green and Naghdi's papers and a treatment proposed recently by Cohen. This revised version was published online in July 2006 with corrections to the Cover Date.     

Instabilities in Dynamic Anti-plane Sliding of an Elastic Layer on a Dissimilar Elastic Half-Space

The stability of dynamic anti-plane sliding at an interface between an elastic layer and an elastic half-space with dissimilar elastic properties is studied. Friction at the interface is assumed to follow a rate- and state-dependent law, with a positive instantaneous dependence on slip velocity and a rate weakening behavior in the steady state. The perturbations at the interface are of the form exp(ikx ₁+pt), where k is the wavenumber, x ₁ is the coordinate along the interface, p is the time response to the perturbation and t is time. A key feature of the problem is that interfacial waves both in freely slipping contact as well as in bonded contact exist for the problem. Attention is focused on the role of the interfacial waves on slip stability. Instabilities are plotted in the $\operatorname{Re} (pL/V_{o})$ versus $\operatorname{Im} (p/|k|c_{s})$ plane, where L is a length scale in the friction law, V _o is the unperturbed slip velocity and c _s is the shear wave speed of the layer. Stability of both rapid and slow slip is studied. The results show one mechanism by which instabilities occur is the destabilization by friction of the interfacial waves in freely slipping contact/bonded contact. This occurs even in slow sliding, thus confirming that the quasi-static approximation is not valid for slow sliding. The effect of material properties and layer thickness on the stability results is studied.     

Pressure of a Narrow Band-Like Punch Upon an Elastic Half-Space

An asymptotic solution is obtained to the contact problem of a band-like punch acting upon an elastic half-space. The method of joined asymptotic expansions is used. The results of numerical calculations are presented. The efficiency of the approach is tested by comparing it with another method     

Numerical Simulation of the Stress Concentration in an Elastic Half-Space with a Two-Layer Inclusion

The stress concentration around a two-layered spherical inclusion located near the surface of an elastic half-space under uniform compression is considered. A numerical solution algorithm is elaborated, application software is created, and numerical experiments demonstrating the capabilities of the approach proposed are conducted.     

Indentation of an Elastic Half-space by a Rigid Flat Punch as a Model Problem for Analysing Contact Problems with Coulomb Friction

One important problem which still remains to be solved today is the uniqueness of the solution of contact problems in linearized elastostatics with small Coulomb friction. This difficult question is addressed here in the case of the indentation of a two-dimensional elastic half-space by a rigid flat punch of finite width, which has been previously studied by Spence in Proc. Camb. Philos. Soc. 73, 249–268 (1973). It is proved that all the solutions have the same simple structure, involving active contact everywhere below the punch and a sticking interval surrounded by two inward slipping intervals. All these solutions show the same local asymptotics for surface traction and displacement at a border between a sticking and a slipping zone. These asymptotics describe (soft) singularities, which are universal (they hold with any geometry) and are explicitly given. It is also proved that in cases where the friction coefficient is small enough, the sticking intervals present in two distinct solutions, if two distinct solutions exist, cannot overlap.     

Study into the Interaction of Cracks in an Elastic Half-Space under a Shock Load by Means of Boundary Integral Equations

The effect of a shock load on the interaction of circular cracks in an elastic half-space is studied. In the space of Fourier time transforms, the problem is reduced to a system of two-dimensional boundary integral equations in the form of the Helmholtz potential with unknown densities characterizing the discontinuities in the displacements of the opposite crack faces. Discrete analogs of those equations are constructed. As an example, two cracks are considered whose faces are under the action of shock tensile loads varying in time as the Heaviside function. The time dependences of the dynamic stress intensity factors are obtained. Their dependence on the relative position of the cracks in the half-space is analyzed.     

Propagation of Waves in an Elastic Half-Space Interacting with a Viscous Liquid Layer

International Applied Mechanics - The problem of the propagation of acoustic waves in a layer of a compressible viscous fluid that interacts with an elastic half-space is solved using the...     

On Indentation of a System of Irregularly Shaped Rigid Punches into a Coated Foundation

We consider the contact problem of interaction between a coated viscoelastic foundation and a system of rigid punches in the case where the punch shape is described by rapidly varying functions. A system of integral equations is derived, and possible versions of the statement of the problem are given. The analytic solution of the problem is constructed for one of the versions.     

On the Number of Invariants in the Strain Energy Density of an Anisotropic Nonlinear Elastic Material with Two Material Symmetry Directions

We determine the minimum number of independent invariants that are needed to characterize completely the strain energy density of a compressible hyperelastic solid having two distinct material symmetry directions. We use a theory of representation of isotropic functions to express this energy density in terms of eighteen invariants, from which we extract ten invariants to analyze two cases of material symmetry. In the case of orthogonal directions, we recover the classical result of seven invariants and offer a justification for the choice of invariants found in the literature. If the directions are not orthogonal, we find that the minimum number is also seven and correct a mistake in a formula found in the literature. An energy density of this type is used to model, on the macroscopic scale, engineering materials, such as fiber-reinforced composites, and biological tissues, such as bones.     

Love Waves in a Piezoelectric Semiconductor Thin Film on an Elastic Dielectric Half-Space

In this paper,Love waves propagating in a piezoelectric semiconductor(PSC)layered structure are investigated,where a PSC thin film is perfectly bonded on an elastic dielectric half-space.The dispersion equations are derived analytically.The influence of semiconducting properties on the propagation characteristics is examined in detail.Numerical results show that the semiconducting effect reduces the propagation speed,and that the Love waves can propagate with a speed slightly higher than the shear wave speed of the elastic dielectric half-space.The wave speed and attenuation significantly depend on the steady-state carrier density and the thickness of the PSC thin film.It is also found that when the horizontal biasing electric field is larger than the critical value(corresponding to the zero attenuation),the wave amplitude is increased.These findings are useful for the analysis and design of various surface wave devices made of PSC.     

Propagation of Quasi-Lamb Waves in an Elastic Layer Interacting with a Viscous Liquid Half-Space

International Applied Mechanics - The propagating of quasi-Lamb waves in an elastic layer that interacts with a half-space of a viscous compressible fluid is studied. The three-dimensional...     

Elastic Displacement in a Half-Space Under the Action of a Tensor Force. General Solution for the Half-Space with Point Forces

The elastic displacement in an isotropic elastic half-space with free surface is calculated for a point tensor force which may arise from the seismic moment of seismic sources concentrated at an inner point of the half-space. The starting point of the calculation is the decomposition of the displacement by means of the Helmholtz potentials and a simplified version of the Grodskii-Neuber-Papkovitch procedure. The calculations are carried out by using generalized Poisson equations and in-plane Fourier transforms, which are convenient for treating boundary conditions. As a general result we compute the displacement in the isotropic elastic half-space with free surface caused by point forces with arbitrary structure and orientation, localized either beneath the surface (generalized Mindlin problem) or on the surface (generalized Boussinesq-Cerruti problems). The inverse Fourier transforms are carried out by means of Sommerfeld-type integrals. For forces buried in the half-space explicit results are given for the surface displacement, which may exhibit finite values at the origin, or at distances on the surface of the order of the depth of the source. The problem presented here may be viewed as an addition to the well-known static problems of elastic equilibrium of a half-space under the action of concentrated loads. The application of the method to similar problems and another approach to the starting point of the general solution are discussed.