The indentation of a half-space of hexagonal elastic material by a circular punch of arbitrary end-profile

Summary The problem is considered of the indentation by a smooth rigid punch of a half-space composed of linear elastic material of hexagonal symmetry whose plane boundary is parallel to the basal planes. The case is considered in which the area of contact between the punch and the half-space is circular, the end of the punch with is in contact with the half-space having an arbitrary profile. An integral equation is formulated and solved for the boundary value of the normal displacement in the half-space, and an expression is derived for the distribution of pressure under the punch.     

An indenter moving on an anisotropic elastic half-space

Summary The problem is considered of a semi-infinite homogeneous anisotropic elastic solid deformed in plane strain by an indenter moving steadily over its surface. General expressions are derived for the components of stress throughout the medium for arbitrary anisotropy of the elastic modulus tensor. It is shown that the distribution of normal traction under the indenter and the normal displacement of the free surface outside the region of contact depend on the elastic modulus components and the velocity of the indenter through a single constant, denoted by . A simple, practical method of calculating is presented.The solutions are examined in detail for the case of orthotropic materials and analytical expressions are given for and for the complete stress distribution throughout the medium in this case. Representative numerical results are presented graphically to illustrate the angular variations of the scaled radial shear and tangential stress components near the edge of the punch.

---

Ein bewegter Stempel auf einem anisotropen elastischen Halbraum

Zusammenfassung Das Problem eines homogenen, anisotropen, elastischen Halbraumes, auf dessen Oberfläche sich ein Stempel stetig bewegt, wird betrachtet. Allgemeine Ausdrücke werden für die Spannungskomponenten im Körper für beliebige Anisotropie des Elastizitätsmodultensors abgeleitet. Es wird gezeigt, daß die Normaldruckverteilung unter dem Stempel und die Normalverschiebung der freien Oberfläche außerhalb des Kontaktbereiches von den Elastizitätsmodulkomponenten und der Stempelgeschwindigkeit durch eine einzige Konstante, bezeichnet mit , abhängen. Eine einfache praktische Methode zur Bestimmung von wird angegeben.Die Lösungen werden für den Fall eines orthotropen Materials bestimmt, und die analytischen Ausdrücke für und für die Spannungsverteilung im Körper für diesen Fall angegeben. Numerische Ergebnisse werden graphisch angegeben zur Darstellung der Radialschub-und Tangentialspannungskomponenten in der Nähe der Stempelspitze.

---

---

With 2 Figures     

A crack within a half-space of orthotropic elastic material

Summary The dislocation layer method combined with a technique of images is used to study a mode III loaded Griffith-type elastic strip crack situated parallel to the free-surface of a semi-infinite orthotropic crystal. The density function of the proposed distribution of dislocations is shown to satisfy a complex singular integral equation. Its closed-form solution provides a compact expression for an appropriate combination of the resulting stress field components. Some representative numerical results are presented in tabular form and discussed.     

Bending of long, rectangular-section bonded rubber blocks

Readily-calculable explicit closed-form representations are determined for the bending stiffness of, and stresses within, generally loaded bonded rubber blocks of long, rectangular cross-section. They satisfy exactly the stated governing equations and conditions based upon the classical theory of elasticity. From these, more detailed investigations are given for the especially interesting loading cases corresponding to simple bending, cantilever loading and apparent shear. It is observed that the deformed profiles of the side surfaces are not in general parabolic as has been usually assumed previously. In simple bending an improved approximate elementary expression is deduced for the couple needed to create a specified rotation of the loaded end of the block. Similarly, in apparent shear a more realistic simple estimate is derived from the exact expression for the ratio of the true to the apparent shear modulus.     

Elastoplastic cracks in orthotropic crystals using dislocation layers

Summary The method of dislocation layers is used to study the stress-field created around an infinite row of collinear Griffith-type elastoplastic strip cracks in an orthotropic crystal loaded at infinity. Formal solutions are obtained in detail for the mode III antiplane shear case, leading to explicit expressions for the length of the plastic zones and the total plastic displacement at the crack-tips. Some representative numerical results are given. It is observed that the problems of a single, elastoplastic crack within a finite orthotropic plate and a finite plate containing a surface crack have solutions which are actually also provided by this analysis. The mode I and mode II analogous situations are briefly discussed.     

An analogy between radially-loaded rubber bush mountings and axially-loaded bonded rubber blocks

The validity of the widely-quoted experimental data for the reduced radial stiffness of a cylindrical rubber bush mounting of finite length is discussed by proposing an analogy between such a bush and two tandem cylindrical rubber blocks of circular cross-section which are axially loaded. Previously-derived expressions for the radial stiffness of a finite-length cylindrical bush and the apparent Young’s modulus of an axially-loaded bonded rubber block of circular cross-section are exploited. This leads to an adaptation of the data that incorporates the possible failure of the bushes due to internal rupture. The analytically-derived values of this stiffness are demonstrated to have improved agreement with these appropriately modified data. Representative numerical results are presented for a range of bushes in tabular and graphical forms.     

Dislocation layers applied to moving crack problems

In this paper we consider a moving Griffith-type strip crack subjected to three fundamental modes of surface tractions in an infinite, homogeneous, isotropic, elastic solid. Closed form solutions are obtained in detail for the mode II case for the components of the resulting stress fields using a method of dislocation layers and some numerical results are given. Some implications of the results for brittle and ductile crack growth are discussed.     

A study of cracks in orthotropic crystals using dislocation layers

Summary The method of dislocation layers is used to study the stress field of a loaded Griffith-type elastic strip crack in an orthotropic crystal. Three fundamental modes of tractions applied to the crack faces are discussed. Formal solutions are derived for the stress components and the distribution of stress near a crack tip is deduced, before representative numerical results are presented graphically. Finally, the method is applied to the BCS model of elastoplastic cracks. In particular, the results are convenient for studying certain orientations of elastic and elastoplastic cracks in hexagonal and cubic crystals.     

Propagation of waves from a spherical cavity with and without a shell embedment

Summary A spherical cavity in an infinite, elastic medium with and without a shell embedment is subjected to axisymmetric, non-torsional surface loads in the radial and meridional directions. The so-called Residual Variable Method (RVM) is used to obtain exact, closed-form solutions of the wave propagation problems. Some representative numerical results are presented graphically for the stresses created in two realistic loading situations.     

Moving antiplane shear crack in transversely isotropic magnetoelectroelastic media

An antiplane shear strip crack moving uniformly in transversely isotopic magnetoelectroelastic media when subjected to representative non-constant crack-face loading conditions is studied. Readily calculable explicit closed-form representations are determined and discussed for the components of the stress, electric and magnetic fields created throughout the material. Representative numerical data are presented. Alternative boundary conditions for which corresponding analyses can be derived analogously are listed.