Alan Cottrell

The spread of plastic yield from a notch

A calculation is made of the length of plastic zone needed to accommodate a given plastic displacement at the root of a notch in a uniformly stressed solid. In an optimum range of stress this zone is about 1000 times larger than the plastic displacement and about five times longer than the notch. The distribution of plastic-elastic strain in the yielded region can be represented by an inverted pile-up of dislocations. The results are related to the problem of notch-brittleness in steel and it is concluded that a condition of ‘ far-reaching ’ yield should replace the condition of general yield for starting a fracture. Various factors important to notch-brittleness are briefly discussed.

Ductile and brittle crystals

Abstract An attempt is made to assess whether a crystal can break in a fully brittle manner or whether some plastic flow must accompany fracture. The criterion proposed is that, if the ratio of the largest tensile stress to the largest shear stress close to the tip of an equilibrium crack in the crystal is greater than the ratio of the ideal cleavage stress to the ideal shear stress, then a fully brittle fracture is possible. If the converse is the case, the crystal must always break with some plastic flow. Calculations are presented for sodium chloride, a Lennard-Jbnes solid, diamond, copper, silver, gold, nickel, iron and tungsten. The criterion appears to give results in broad qualitative agreement with experiment.

Plastic Yielding from Sharp Notches

In a previous paper a model of a relaxed crack has been treated in which the plastic relaxation round an isolated crack is represented by an array of dislocations collinear with the crack itself. A similar model is used here to consider the behaviour of an infinite row of relaxed cracks subject to a uniform stress at infinity. The model is used to represent a plastic notch in a plate of finite thickness and the results compared with numerical calculations for the same problem using the macroscopic theory of plasticity. Compared with the dislocation model of the isolated crack, the relaxation spreads further in the presence of other cracks and the length of the relaxed zone required to accommodate a given plastic displacement at a tip is increased by about 10 %. Some applications of the results to the theory of notch brittleness and to high-strain fatigue are briefly discussed.

Stored Energy in the Graphite of Power-Producing Reactors

The effects of the atomic displacements produced in graphite by the collisions of fast neutrons are of great importance in the technology of graphite-moderated nuclear power reactors. This paper describes experimental and theoretical work on the stored energy associated with such displacements, the rate of release of this stored energy as a function of temperature, and the thermal conductivity of graphite, over a range of irradiation and annealing conditions. Some large effects are observed, particularly in the rate of accumulation of radiation damage at various temperatures of irradiation, as well as many other effects of fundamental interest. After high doses at low irradiation temperatures the rate of release of stored energy with increasing temperature can approach the specific heat of unirradiated graphite, but a relatively small increase in the irradiation temperature brings about a drastic reduction in the amount of energy stored. The conditions under which this radiation damage is produced are very complicated, for they involve not only sequences of spontaneous rearrangements amongst the displaced atoms during and after the irradiation treatment but also the total intensity and the energy spectrum of the neutrons, and it is impossible to separate all the variables in a strict sense. Various experiments have however been made, and theoretical and semi-empirical procedures developed, which enable results to be correlated satisfactorily over a range of conditions including those in both graphite-moderated and water-moderated reactors. The paper shows how the rate of stored energy release may be estimated for practical values of temperature and irradiation dose.

The Bakerian Lecture, 1963. Fracture

Under a large tensile stress the atoms in a solid may slide past one another or they may pull apart. The great contrast between ductile solids such as gold and brittle ones such as diamond has led to a widely held belief that yielding and fracture are independent processes, each occurring at its own characteristic stress, and that a solid is ductile when its yield stress lies below its fracture stress and is brittle conversely. Recent work has shown, however, that they are usually closely interlinked processes, even in the most unlikely solids. One of my aims in this lecture is to show how various types of fracture all depend in some way or other on plastic yielding.

Constitutional vacancies in NiAl

Abstract An explanation is offered, based on a simplified form of many-atom bonding theory, of why NiAl forms constitutional vacancies on its Ni sublattice when excess Al is present. The resultant increase in the number of Al atoms thereby able to bond with Ni atoms is only partially offset by the weakening of individual AlNi bonds, so that there is a net gain. Also, the NiNi bonds, being weak in NiAl, claim only a minor penalty when an Ni site is vacated. Other defects, i.e. antisite Ni atoms and triple defects, as well as ordering of vacancies, are briefly discussed.

Mechanics of Fracture in Large Structures

A dislocation theory of cracks, which takes account of elastic-plastic and other non-linear modes of deformation, is used to discuss the geometrical and physical conditions for unstable fracture. The importance of discontinuous cracking in providing an unstable mode of plane strain, semi-brittle, tensile fracture is emphasized. The general formulae reduce to those of Griffith, Orowan, and Irwin, at low stresses. Size effects and their relation to microscopic processes of deformation and fracture, especially twinning, are discussed and the concept of the crack arrest temperature is considered.

The natural philosophy of engines

Abstract Summary An elementary account is given of the circumstances under which mechanical and thermodynamical systems of various kinds—physical, chemical, biological; natural or man-made-are able to extract, concentrate and store free energy by acting on an energy flow. The essential condition, which has recently been analysed in rather general forms by Prigogine, is that a system should become unstable in one of its modes of resistance to the forces driving the energy flow. This may take many forms: buckling collapse in compressed elastic struts, convection cells in heated fluids, auto-catalytic accelerations of chemical reactions, biological growth and multiplication. All such systems, whether natural or man-made, are engines. Under certain circumstances the free energy extracted by an engine may be used to produce more such engines. Multiplication, competitive growth, and evolution of improved forms may then occur. The long history of this, in biology, has led to the emergence of organization and pur...

Criticality in Andrade creep

The transition from instantaneous plastic yielding to rapid creep, under constant applied stress, is a form of critical state behaviour, analogous to the avalanching of sandhills of critical slope....

Vacancies in FeAl and NiAl

Abstract The B2 compound NiAl extends substantially into Al-rich compositions where it forms constitutional vacancies abundantly, but the corresponding FeAl does not. However, FeAl forms thermal vacancies abundantly. This difference is traced to the smaller heat of formation of FeAl, so that the enthalpy of formation of constitutional vacancies, from the stoichiometric compound in the presence of excess Al, is favourable for NiAl but slightly unfavourable for FeAl. At high temperatures, the effect of mixing entropy enables FeAl to overcome this small enthalpy disadvantage, despite the competition of the next phase (FeAl 2 ) for the excess Al. A corresponding calculation of the NiAl and Ni 2 Al 3 equilibrium enables the Al-rich phase limit of NiAl to be estimated. The energy of formation of triple defects is calculated to be appreciably lower in FeAl than in NiAl, so that these become abundant at high temperatures in stoichiometric FeAl.