M. A. Jaswon

Integral equation methods in potential theory. II

This paper makes a short study of Fredholm integral equations related to potential theory and elasticity, with a view to preparing the ground for their exploitation in the numerical solution of difficult boundary-value problems. Attention is drawn to the advantages of Fredholm ’s first equation and of Green’s boundary formula. The latter plays a fundamental and hitherto unrecognized role in the integral equation formula of biharm onic problems.

Distribution of Solute Atoms Round a Slow Dislocation

The factors determining the average velocities of solute atoms attracted to a dislocation are discussed, and an equation is set up for the concentration of solute round the dislocation. This leads to a symmetrical distribution of a Maxwell-Boltzmann type round a stationary dislocation. In the case of a slowly moving dislocation the distribution is given as a series solution in Mathieu functions, and the non-symmetry of this distribution causes a force on the dislocation, opposing its motion. This force is calculated and shown to increase linearly with the speed of the dislocation, at low speeds, and a critical range exists above which the motion is unstable and the dislocation accelerates. The characteristics of the plastic flow below this critical range are compared with those of micro-creep in tin crystals and are shown to be similar. Quantitative agreement can be obtained by assuming plausible values for the density of dislocations and the rate of diffusion of solute atoms in tin, but the need for further experiments on micro-creep is emphasized.

Factors Controlling Dislocation Widths

The work of Peierls and Nabarro is extended to a family of edge dislocations of greater widths. The weaker the shearing forces between adjacent atoms in the slip plane, the wider is the dislocation. The external shear stress required to move the dislocations is extremely sensitive to the width, becoming vanishingly small at widths of the order of three atomic spacings. The theory is applied to bubble raft dislocations, and satisfactory agreement is found with experiment.

An Integral Equation Solution of the Torsion Problem

The classical torsion problem of St Venant is formulated mathematically as a Neumann boundary-value problem for the warping function. This can be found numerically on the boundary by means of an integral equation method applicable to cross-sections of any shape or form. A single digital computer program assembles and solves the relevant equations, yields the torsional rigidity and boundary shear stress, and evaluates the warping function and stress components at any selected array of points throughout the cross-section. An accuracy of 1% in the torsional rigidity and maximum shear stress can be attained without undue effort.

Two-dimensional elastic inclusion problems

An account is given of Eshelbys point-force method for solving elastic inclusion problems, and of his equations relating an in homogeneity to its equivalent inclusion. The introduction of complex variable formalism enables explicit solutions to be found in various two-dimensional cases. Strain energies are calculated. The equilibrium shape of an elliptic inclusion exhibits an interesting feature not previously expected. A fresh analysis of stress magnification effects is developed.

Elastic Inclusion Problems

These problems arise naturally in some solid-state phase changes of metallurgical importance, e.g. the austenite-martensite transformation which lies at the heart of steel manufacture. Austenite is a high-temperature FeC alloy which generates discrete plate-like structures embedded within the parent austenite matrix, consequent upon rapid quenching, so transforming the austenite into martensitic steel. Each martensite plate can be regarded as an elastic inclusion within the matrix, which generates misfit stresses analogous to the generation of classical thermoelastic stresses by temperature gradients, and with equally significant physical effects. Eshelby [1] formulated the stress field generated by an ellipsoidal elastic inclusion within an infinite elastic continuum, using imagined operations of cutting, space filling and welding. In effect, however, these operations amount to an intuitive application of Somiglianas formula as implied by Jaswon & Symm [2]. If so, BEM would be well suited to achieve efficient numerical solutions for a wide variety of problems involving a thin interface between a finite and infinite medium.

What is a dislocation

A Volterra dislocation is the elastostatic analogue of a uniform magnetic shell or vortex-equivalent sheet. Just as these may be regarded mathematically as uniform dipole sheets, so dislocations may be regarded as specialised traction sheets. This model is briefly explained and connected up with the theory of Taylor dislocations in a crystal.

Scalar and Vector Potential Theory

This provides a specialised representation for harmonic functions, which proves to be particularly convenient for solving certain boundary-value problems. Physically speaking, it models the properties of continuous electrostatic charge distributions over closed conductors, so providing an easy entry into the theory. Thus, if charges are introduced on a smooth, closed, conducting surface ∂B, we posit a continuous charge density σ(q) at every q ⊂ ∂B. It is convenient to write dq for the area element at q, in which case σ(q)dq defines the charge strength associated with dq. This generates an electrostatic potential g(p,q)σ(q)dq at any point p of space, where

The crystallography of deformation twinning

[{\text{g(}}\underline {\text{p}} {\text{,}}\underline {\text{q}} {\text{) = g(}}\underline {\text{q}} {\text{,}}\underline {\text{p}} {\text{) = |}}\underline {\text{p}} {\text{ - }}\underline {\text{q}} {{\text{|}}^{{\text{ - 1}}}}