Solution to certain contact problems without axial symmetry in the case of a punch with an elliptical cross section

A solution is obtained to two new contact problems in the class of problems considered earlier [3]. The analytically established relations are suitable for use in several mechanical and engineering applications as, for instance, in the problem of heat and electric current conduction through a contact between solid deformable bodies. The most essential parameters have been computed and the results are shown here in the form of graphs.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 2, pp. 334–342, February, 1973.     

Plane strain problem of a punch indenting an initially stressed Neo-Hookean anisotropic elastic half plane

Summary The plane strain problem of an initially stressed neo-Hookean orthotropic incompressible elastic half plane indented by a smooth rigid punch is formulated through a generalised Fourier transform and is reduced to the solution of a well-known singular integral equation of first kind. The solution of the integral equation directly gives the surface pressure under the punch. The normal displacement outside the region of contact of the punch is also determined. Three different shapes of punches are considered. Results are deduced for an sotropic incompressible medium free of initial stress.     

Fracture behaviours of brittle ceramics under elliptical ultrasonic vibration: near-to-limit contact analysis of an elastic flat punch

International Journal of Mechanics and Materials in Design - Conform and non-conformal contact problem in the face of piezoelectric motors can be hard to grasp, especially both the friction drive...     

Frictionless elastic contact problem for a curved rigid punch of arbitrary shape

Summary A new method is proposed for the analysis of elastic contact problems for a curved punch of non-elliptic planform under the action of a normal force. The punch base is assumed to be a quadratic surface. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some general relationships are established between the applied force and the punch settlement. Specific formulae are derived for a punch whose planform has a shape of a polygon, a rectangle, a rhombus and a cross. An example of a finite rigid cylinder lying on its generator and pressed against an elastic half-space is considered in detail. The method allows to have singular stresses at the cylinder edges and zero stresses at the rest of the boundary of the contact domain. The last condition serves for defining the width of the domain of contact. All the formulae are checked against the solutions known in the literature, and a good accuracy is confirmed in a sufficiently wide range of the aspect ratio.With 3 Figures     

Propagation properties of an elliptical anisotropic metamaterial cylinder

The propagation characteristics were investigated of an open elliptical cylinder comprising a uniaxial anisotropic metamaterial. The dispersion relation for hybrid modes in this cylinder was examined theoretically. It is shown that forward and backward waves can be supported by such a metamaterial cylinder. Even and odd modes have different losses in a realistic anisotropic elliptical cylinder. The forward wave can propagate longer distances than the backward wave because of its lower loss. The light can propagate backward on the cylinder without the limit of structure size. The odd modes have higher operating wavelengths than the even modes. The extraordinary properties in this structure are identified as multilayer plasmons. The validity of the effective medium approximation of this structure is examined.     

An improved initial value method for a contact problem with a nonplane punch

Solution of an axisymmetric three dimensional contact problem of a nonplane punch pressing into a finitely thick elastic layer is presented. The method consists in deriving a Cauchy system for two auxiliary functions. This initial value problem is readily solved by modern computers. Complete displacement and stress fields as well as the penetration of the center of the punch, and the force required to push it are determined. This is a direct method, and no successive approximations or series expansions are used. The results are obtained for all radii of contact of the punch with the elastic layer.     

On the modelling of complex anisotropic frictional contact laws

In this paper, the formulation of complex anisotropic frictional models with orthotropic friction condition and non-associated sliding rule is discussed. The friction law is described by a superellipse, which allow to consider a wide range of convex friction condition by simply varying the roundness factor affecting the shape of the limit surface. The sliding potential is also a superellipse but with a different semi-axis ratio, which lead to a non-associated sliding rule. For bodies in contact, the Signorini conditions can be formulated in terms of velocities and combined with the sliding rule to give the frictional contact law describing interfacial interactions. Its is shown that the frictional contact law as well as its inverse can be derived from the same scalar valued function called bi-potential. Due to the non-associated nature of the law, this relation is implicit. The advantage of the present formulation lies in the existence of stationary points of a functional, called bi-functional, that depends on both the displacements and the stresses.     

Hertz contact problems for an anisotropic physically nonlinear elastic medium

Conditions under which Hertz contact problems exhibit the property of self-similarity are determined. Qualitative conclusions concerning the character of selfsimilitude solutions from which, among other things, an equation similar to the familiar Mayer equation follows, are drawn. The problem of the collision between nonlinearly elastic bodies is also examined.Moscow Institute of Radio Engineering, Electronics, and-Automated Equipment. Translated from Problemy Prochnosti, No. 12, pp. 47–53, December, 1989.     

On the parameter identification problem for failure criteria in anisotropic bodies

Summary The elliptic paraboloid failure surface criterion (EPFS) is adopted in this paper to describe the failure behaviour of anisotropic bodies. A method is described, based on an inequality-constrained least square problem for the determination of the parameters of the EPFS criterion. After the discussion of the influence of the strength differential effect on the failure behaviour of the material, a neural network learning approach is introduced to the problem of extrapolating the given experimental results beyond the given range of experimental data by establishing an appropriate law of evolution of the failure surface valid for the material up to fracture.     

A special case of contact between an asymmetric punch and an elastic half-space

A solution is obtained to a group of problems concerning the contact between an asymmetric punch and an elastic half-space. The special case is considered where the punch penetrates from a contact point along a rectilinear segment.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 1, pp. 138–146, January, 1973.     

A dislocation and a point force in an anisotropic medium with an elliptical hole

Summary The solutions of a dislocation and a point force in an infinite medium with a crack of elliptical cross-section (or an elliptical hole) are given for plane problems of anisotropic solids with one plane of symmetry.With 1 Figure     

Comparative solutions for a punch problem of an elastic sphere encapsulated in an elastic shell

One of the major causes of mechanical damage incurred in agricultural commodities is attributed to the frequent impacts they receive in harvesting and handling. In agricultural operations, in general, only the local contact phenomenon is considered and the effect of wave propagation is ignored. The punch problem is a special case in the class of contact problems that is of particular practical interest in the impact loading encountered in fruit handling and harvesting.Two potential methods are proposed for analysing the deformation of an elastic sphere encapsulated in an elastic shell and subjected to punch loading. Numerical evaluation of both models showed that the results are comparable. Although more examples related to different geometries and loadings would be required to substantiate the data, the results provide a useful tool for the selected specific geometry which is very common in agriculture. Because good agreement was obtained between the two methods, the choice of which to implement should be made according to the specific problem in question. The Boussinesq method is a more rigorous mathematical approach and, as such, offers a better insight into the actual behaviour of the domain under given boundary conditions. Its utilization is limited to well defined geometrics because of the complexity involved. The finite element method is capable of handling irregular shapes but requires large computer memory for an exact solution. Both methods have been successfully implemented in a practical agricultural problem in an attempt to decrease the mechanical damage encountered during mechanized fruit harvesting.     

A plane contact problem for an elastic orthotropic strip

The contact of a punch with an elastic orthotropic strip is considered. A singular integral equation is derived for the contact pressure. The analytic expression of the associated kernel is unique for all types of orthotropy. An iterative solution method is developed to investigate a thick strip. A direct asymptotic procedure proposed for a thin strip leads to simple explicit formulae. Numerical examples are presented for various values of the relative strip thickness.     

Plane problem for an anisotropic body containing an inclusion surrounded with an elastic layer

We construct an algorithm for the solution of the plane problem of the theory of elasticity for a three-component composition in which the boundary between the matrix and inclusion is regarded as an elastic layer. We give a quantitative estimate of the dependence of the generalized stress concentration (intensity) factors near the tip of a curvilinear notch and a perfectly rigid inclusion on the loading mode, elastic characteristics, the width of the indicated elastic layer, and the (small) radius of curvature at the tip of the concentrator. Translated from Fizyko-Khimichna Mekhanika Materialiv. Vol. 36, No. 3. pp. 27–34. May-June, 2000.     

Axisymmetric contact problem for an elastic layer and an elastic foundation

This paper is concerned with the axisymmetric problem of an elastic layer lying on a semi-infinite foundation. The layer is pressed against the foundation by a uniform clamping pressure applied over its entire surface and a uniform vertical body force due to the effect of gravity. In addition, an axisymmetric vertical line load is applied to the layer. It is assumed that the contact between the layer and the foundation is frictionless and that only compresive normal tractions can be transmitted through the interface. The contact along the interface will be continuous if the value of the line load is less than a critical value. However, interface separation takes place if it exceeds this critical value. The problem is formulated and solved for the cases of tensile and compressive line loads. Numerical results for contact stress distributions are given for different material combinations.     

Numerical solution of the internal inverse problem for an anisotropic system with sources

The authors have developed and implemented an approach to the problem of modeling the temperature fields of anisotropic systems with sources on the basis of the equivalence principle.Institute of Engineering Thermophysics, Kiev, Ukraine. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 65, No. 6, pp. 715–717, December, 1993.