On a three-dimensional Kelvin problem for an elastic nonlocal medium

Summary Making use of the Eringen-Kroener form of the nonlocal constitutive equations and the exponential Fourier transformation, a system of two coupled differential equations of the second order describing the equilibrium of the body is derived. By appeal to the Helmholtz representation, the system is reduced to a single differential equation of the fourth order for the Love function, reminding a Bessel type transform of the biharmonic equation. A solution of this equation is found, and inverse transforms of the stress components using the convolution theorem established. A recourse to the formula of de la Vallée Poussin shows that, in contrast to the classical result, the stress singularity at the point of application of a concentrated force fails to appear, though the stress concentration at that point is extremely high.     

On the three-dimensional Boussinesq problem for an elastic nonlocal medium

After determining the nonlocal elastic moduli and the constitutive equations used, a brief review of the Kelvin problem in nonlocal setting is given. The Westergaard procedure of transition from the classical Kelvin problem to the classical Boussinesq problem is discussed, and applied to the nonlocal case using Fourier's exponential transformation. An example illustrating the application of the method to calculate the stress system in a nonlocal half-space is given.     

The contact problem of an orthotropic non-homogeneous elastic half space

The problem of a rigid punch on an elastic half plane with orthotropic and non-homogeneous material is considered. The axes of orthotropy are chosen to coincide with the Cartesian coordinate system in which one axis is parallel to the edge of the half plane and the other is perpendicular to it. Non-homogeneity is introduced in both directions of orthotropy as continuous functions along these directions. Using the Fourier Transform Technique, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically. The formulation of the problem is obtained for a rigid punch with arbitrary shape.     

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.     

Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation

In this paper, we propose a modified nonlocal boundary value problem method for an homogeneous biharmonic equation in a rectangular domain. We show that the considered problem is ill-posed in the sense of Hadamard, i.e. the solution does not depend continuously on the given data. Convergence estimates for the regularized solution are obtained under a priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method.     

The axisymmetric double contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the smooth receding contact between an elastic layer and a half space when the layer is compressed by a frictionless semi-infinite elastic cylinder. Upon loading, the contact along the layer-subspace interface shrinks to a circular area, radius of which is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of contact and the stress intensity factor round the edge of the cylinder are calculated for various material combinations.     

The problem of a rigid punch on a non-homogeneous elastic half space

The elastostatic problem of a rigid punch on an elastic half space is considered. The medium is assumed to exhibit a non-homogeneity varying with depth. Using the Fourier Transform Technique, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically. The effect of non-homogeneity on the stress distribution under the punch and on the stress singularity is studied. The influence of Poisson's ratio on the results is also considered.     

A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space

The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.     

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 nonlocal continuum theory for diatomic elastic solids

In this study, taking the effects of long range intermolecular forces into account, a mechanical model for diatomic solids is presented. The model we use is based upon the assumption that a diatomic solid may be considered to consist of two simple elastically interacting media which are initially overlapping but may have relative motions at a later time $\text{t}$. Based on this assumption, the kinematics and balance laws, i.e. conservation of mass, balance of momenta, conservation of energy and equation of entropy production are presented. The nonlinear and linear constitutive equations and the discussion of some special cases are introduced in Sections 3 and 4, respectively. To illustrate the theory, in the final part, the propagation of time harmonic waves in such a medium is studied, and by comparing present result with those of lattice dynamics, the explicit form of influence functions is obtained.     

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.     

A griffith crack problem for an inhomogeneous elastic material

Summary This paper examines the problem of a Mode I crack in a nonhomogeneous elastic medium. It is assumed that the shear modulus varies exponentially with the coordinate perpendicular to the plane of the crack. The problem is reduced to a Fredholm integral equation and in terms of its solution the normal components of stress and displacement are described. Expressions are also derived for the stress intensity factor and the crack energy. The effect of the inhomogeneity is examined and comparisons made with the corresponding results for the homogeneous material.     

Axisymmetric contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the elastostatic contact problem of a semi-infinite cylinder compressed against a layer lying on a rigid foundation. It is assumed that all the contacting surfaces are frictionless and that only compressive normal tractions can be transmitted through the interfaces. Upon loading the contact along the layer-foundation interface shrinks to a circular area whose radius is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of the contact area between the layer and the foundation, and the stress intensity factor round the edge of the cylinder are calculated for various material pairs.     

Axially symmetric contact problem of pressing of an absolutely rigid ball into an elastic half space with inhomogeneous coating

We consider an axially symmetric contact problem of pressing of an absolutely rigid ball into an inhomogeneous half space formed by a homogeneous base and an inhomogeneous surface layer. The Poisson’s ratio of the layer is constant and its Young modulus is an exponential function of the distance from the surface of the half space. The solution of the problem of the theory of elasticity with continuous dependence of the Young modulus on the coordinate is compared with the solution of the problem in which the inhomogeneous layer is replaced with a package of homogeneous layers.     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

On a boundary value problem for an elastic dielectric half-plane

Summary Using Papkovitchtype representations for the displacement and polarization vectors and Fourier transforms, a general solution to boundary value problems of a half plane subjected to an arbitrary charge distribution is constructed within Mindlin's theory of elastic dielectrics. Explicit expressions for various mechanical and electric potentials are obtained for a point charge located on the boundary of the elastic dielectric half-space.

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Über ein Randwertproblem für eine elastische, dielektrische Halbebene

Zusammenfassung Auf der Basis von Mindlins Theorie des elastischen Dielektrikums und mit Benutzung der Papkovitch-Darstellungen für Verschiebungs- und Polarisationsvektoren wird mit Hilfe von Fourier-Transformationen eine allgemeine Lösung für Randwertprobleme einer Halbebene mit einer beliebigen Ladungsverteilung gefunden. Es werden explizite Ausdrücke für verschiedene mechanische und elektrische Potentiale für eine Punktladung an der Berandung des elastischen, dielektrischen Halbraums erhalten.

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With 2 Figures     

On a nonlocal theory of longitudinal waves in an elastic circular bar

After determining the values of the nonlocal moduli for longitudinal waves in an infinite space, Fourier transforms of the equations of axially symmetric longitudinal waves in an infinite circularly cylindrical rod are established and decoupled according to the Pochhammer procedure. Dispersion equation is obtained from the conditions of traction free surface of the rod, and compared with its classical counterpart. While the velocity of long waves coincides, as required, with that derived in the classical case, the velocity of short waves turns out to be about 36% less.Notation a interatomic spacing - a ₁,a ₂,a ₃ coefficients defined by (2.12.5) - c wave phase velocity - d rod diameter - h, l defined by (2.15) - k wave number - overbar denotes Fourier transform - R rod radius - r, r vector of the point of observation and of generic point, respectively - u displacement in thex ₁-direction - u, w displacements in ther- andz-direction, respectively - double Fourier transform ofu(r, z, t) - (r–r) Dirac delta function - , Lamé constants - , nonlocal moduli - Fourier transforms of 03BC; and - A wave length - mass density - ₁₁ normal stress in thex ₁-direction - _rr , _rz , _zz stress components in polar coordinates,r, ,z - dilatation - , ^* defined by (2.20.2) - wave frequency - notation defined by (2.13.2) With 1 FigurePrepared with partial support of the University of Delaware.