Ian N. Sneddon

The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile

Abstract A solution of the axisymmetric Boussinesq problem is derived from which are deduced simple formulae for the depth of penetration of the tip of a punch of arbitrary profile and for the total load which must be applied to the punch to achieve this penetration. Simple expressions are also derived for the distribution of pressure under the punch and for the shape of the deformed surface. The results are illustrated by the evaluation of the expressions for several simple punch shapes.

The Distribution of Stress in the Neighbourhood of a Crack in an Elastic Solid

The distribution of stress produced in the interior of an elastic solid by the opening of an internal crack under the action of pressure applied to its surface is considered. The analysis is given for ‘Griffith’ cracks (§2) and for circular cracks (§3), it being assumed in the latter case that the applied pressure varies over the surface of the crack. For both types of crack the case in which the pressure is constant over the entire crack surface is considered in some detail, the stress components being tabulated and the distribution of stress shown graphically. The effect of a crack (of either type) on the stress produced in an elastic body by a uniform tensile stress is considered and the conditions for rupture deduced.

Boussinesq's problem for a rigid cone

1. The problem of determining the distribution of stress in a semi-infinite solid medium when its plane boundary is deformed by the pressure against it of a perfectly rigid cone is of considerable importance in various branches of applied mechanics. It arises in soil mechanics where the cone is the base of a conical-headed cylindrical pillar and the semi-infinite medium is the soil upon which it rests ( 1 ). In this instance the distribution of stress in the soil is known to be more or less similar to that calculated on the assumption that the soil is a perfectly elastic, isotropic and homogeneous medium, at least if the factor of safety of a mass of soil with respect to failure by plastic flow exceeds a value of about three ( 2 ). The same problem occurs in the theory of indentation tests in which a ductile material is indected by cylindrical punches with conical heads ( 3 ).

The Classical Theory of Elasticity

The theory of elasticity is concerned with the mechanics of deformable bodies which recover their original shape upon the removal of the forces causing the deformation. The first discussions of elastic phenomena occur in the writings of Hooke (1676) but the first real attempts to construct a theory of elasticity using the continuum approach, in which speculations on the molecular structure of the body are avoided and macroscopic phenomena are described in terms of field variables, date from the first half of the eighteen century1. Since that time a tremendous amount of scientific effort has been devoted to the study of the mathematical theory of elasticity and its applications to physics and engineering. The sheer volume of the published work in the subject makes it quite impossible for an author to cover the entire subject at all adequately within the compass of a single book. The present article has a much more modest aim than that: It tries to give a brief survey of certain parts of the basic theory of elasticity with sufficient discussion of special problems to give some indication of the mathematical techniques available for the solution of such problems. Even within that limited framework there are notable omissions; for example, nothing is said about such an important technological topic as the theory of elastic stability or about such a basic topic as the calculation of the elastic constants of a crystal by the theory of crystal lattices.

The effect of a penny-shaped crack on the distribution of stress in a long circular cylinder

Abstract This paper contains an analysis of the distribution of stress in a long circular cylinder of elastic material when it is deformed by the application of pressure to the inner surfaces of a penny-shaped crack situated with its centre on the axis of the cylinder and its plane perpendicular to that axis. It is assumed that the cylindrical surface is free from shear and is supported in such a way that the radial component of the displacement vector vanishes on the surface. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a Fredholm integral equation of the second kind. Expressions for the various quantities of physical interest are derived for small values of the ratio of the radius of the crack to that of the cylinder by finding an iterative solution of this equation. For values of this ratio near unity the integral equation has been solved numerically using a high-speed computer and the relevant quantities calculated.

The stress field in the vicinity of a Griffith crack in a strip of finite width

Abstract The problem of determining the stress field in an elastic strip of finite width when pressure is applied to the faces of a Griffith crack situated symmetrically within it is considered. It is assumed that the faces of the strip are themselves free from stress. The problem in which the applied pressure is constant is considered in some detail and the results compared with those obtained for the simpler model in which the normal component of displacement of the surface of the strip is assumed to vanish.

The distribution of stress in the vicinity of an external crack in an infinite elastic solid

Abstract A solution is derived of the equations of equilibrium appropriate to the application of pressure to the faces of a plane crack covering the outside of a circle in an infinite elastic body. The solution corresponding to axisymmetric loading of the crack is deduced and the results illustrated by the consideration of some particular cases.

A note on the distribution of stress in a cylinder containing a penny-shaped crack

Abstract This note contains an analysis of the distribution of stress in a long circular cylinder of elastic material when it is deformed by the application of pressure to the inner surfaces of a penny-shaped crack situated symmetrically at the centre of the cylinder. It is assumed that the cylindrical surface is free from Stress. The equations of the classical theory of elasticity are solved in terms of an unknown function which is then shown to be the solution of a Fredholm integral equation of the second kind previously derived by W.D. Collins. The solutions of this equation for constant pressure and for various values of the radius of the crack to that of the cylinder, derived using a high-speed computer, are discussed and quantities of physical interest calculated. The calculations were repeated for the case of a variable pressure following a parabolic law and they are also reported.

The stress intensity factor at the tip of an edge crack in an elastic half-plane☆

Abstract The problem of determining the stress and displacement fields in an elastic half-plane containing an edge crack normal to the free surface when the crack faces are subjected to normal pressure is reduced to a mixed boundary value problem for the quarter plane. The theory of dual integral equations is used to reduce the boundary value problem to that of solving a pair of simultaneous integral equations of Fredholm type; these in turn are reduced to a single integral equation of Fredholm type with non-singular kernel. The expressions for the stress intensity factor and the crack energy are transformed to formulae involving only the solution of this integral equation. The numerical solution of the governing integral equation is discussed and the numerical results for the case of a pressure which is distributed uniformly presented.

The Reissner-Sagoci problem

The statical Reissner-Sagoci problem [1, 2, 3] is that of determining the components of stress and displacement in the interior of the semi-infinite homogeneous isotropic elastic solid z ≧ 0 when a circular area (0 ≦ p ≦ a, z = 0) of the boundary surface is forced to rotate through an angle a about an axis which is normal to the undeformed plane surface of the medium. It is easily shown that, if we use cylindrical coordinates ( p , φ, z ), the displacement vector has only one non-vanishing component u φ ( p, z ), and the stress tensor has only two non-vanishing components, σ ρπ ( p, z ) and σ πz ( p, z ). The stress-strain relations reduce to the two simple equations where μ is the shear modulus of the material. From these equations, it follows immediately that the equilibrium equation is satisfied provided that the function u π ( ρ, z ) is a solution of the partial differential equation The boundary conditions can be written in the form where, in the case in which we are most interested, f(p) = αρ. We also assume that, as r → ∞, u π , σ ρπ and σ πz all tend to zero.