Bruce Alexander Bilby

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.

Continuous Distributions of Dislocations: A New Application of the Methods of Non-Riemannian Geometry

When describing a crystal containing an arbitrary distribution of dislocation lines it is often convenient to treat the distribution as continuous, and to specify the state of dislocation as a function of position. Formally, however, there is then no ‘good crystal’ anywhere, and difficulties arise in defining Burgers circuits and the dislocation tensor. The dislocated state may be defined precisely by relating the local basis at each point to that of a reference lattice. The dislocation density may then be defined; it is important to distinguish this from the local dislocation density. The geometry of the continuously dislocated crystal is most conveniently analyzed by treating the manifold of lattice points in the final state as a non-Riemannian one with a single asymmetric connexion. The coefficients of connexion may be expressed in terms of the generating deformations relating the dislocated crystal to the reference lattice. The tensor defining the local dislocation density is then the torsion tensor associated with the asymmetric connexion. Some properties of the connexion are briefly discussed and it is shown that it possesses that of distant parallelism, in conformity with the requirement that the dislocated lattice be everywhere unique.

The theory of the crystallography of deformation twinning

The crystallographic characteristics of deformation twinning are derived by considering the atomic movements which occur at the moving interface as a twin propagates. This is facilitated by making use of the notation of the tensor calculus, and general expressions, valid for all crystal structures, are obtained giving the magnitude of the twinning shear and relating the twinning elements for both type I and type II twinning. The atomic shuffles, which in general must accompany the twinning shear in both single and multiple lattice structures, are examined in detail and expressions are derived for their magnitudes and directions for the cases of the four classical orientation relationships associated with deformation twinning. The use of these expressions in predicting operative twinning modes is described and the relations between this theory and other recent theories of the crystallography of deformation twinning are discussed.

Continuous Distributions of Dislocations. III

The theory of the continuously dislocated crystal is extended. It is shown that the general equations of the theory of Nye are applicable in full only when the lattice curvature (and so also the dislocation density) is small. The correct equations for large curvatures are formulated. The relation between the fundamental node theorem of dislocation theory and certain theorems of the theory of generalized space is considered in some detail. It is also shown that when the lattice lines of the dislocated crystal are treated as a system of independent congruences of curves, the tensor describing the local dislocation density may be expressed in terms of certain invariants connected with the system of congruences. For the rotation case, a relation between the dislocation density and Ricci’s coefficients of rotation is derived.

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.

The finite deformation of an inhomogeneity in two-dimensional slow viscous incompressible flow

The theory of the elastic fields round ellipsoidal inclusions and inhomogeneities together with the well-known analogy between linear elasticity and slow incompressible viscous flow are used to develop the governing equations for the finite deformation of a viscous ellipsoidal inhomogeneity in a viscous matrix undergoing a general linear time-dependent flow at infinity. The governing equations are then solved for an inhomogeneity in the form of an elliptic cylinder in a linear two-dimensional flow whose stream lines at infinity are steady. The behaviour of the inhomogeneity under pure shear and simple shear is considered in detail and it is shown that the boundaries of certain deforming inhomogeneities remain unchanged during simple shear. These steady inhomogeneities can appear also in general linear two-dimensional applied flows. In such flows the behaviour is influenced both by the initial shape and orientation of the inhomogeneity and by its viscosity. Inhomogeneities which are rather viscous or subject to an applied flow with high vorticity deform periodically, while most others elongate indefinitely. The patterns of behaviour may be described in terms of a number of regimes which can be classified by considering the singularities of the differential equations governing the variations of shape and orientation of the inhomogeneity, or, equivalently, by studying the invariants of the corresponding one-parameter Lie groups. Finally, some obvious extensions of the treatment are indicated. These make it possible to consider inhomogeneities (such as holes) whose volume does not remain constant, and which have constitutive relations more general than those of a linear viscous material.

Representation of Plasticity at Notches by Linear Dislocation Arrays

Further work is reported on the use of linear dislocation arrays to represent plastic relaxation round notches. In previous papers the relaxation in an infinite medium by arrays collinear with the notch was treated, both for a single notch and for an infinite sequence of notches. In antiplane strain these models provide a representation of relaxation round a surface notch in a semi-infinite medium and in a finite plate respectively. In the application of this work to the fracture of large structures the fracture criterion used involves the achievement of a critical displacement at the notch root. Further numerical calculations have now been completed which enable the critical displacement to be related explicitly to the applied stress, the yield stress, the notch size, the size of the structure and the extent of the relaxation, so that predictions about the dangerous stresses for a given notched structure, and the dangerous notch sizes for a structure subjected to a given stress can be obtained. A model has also been treated which includes a representation of work hardening during the relaxation process. Finally, preliminary results are reported on another extension of the model in which the relaxation takes place on two slip planes inclined symmetrically to the notch.

Crack growth in notch fatigue

A dislocation model which is appropriate for discussing the propagation of fatigue cracks from notches of elliptical or V-shaped profiles is presented. The principal conclusion is that the rate of crack growth is proportional to the fourth power of the stress intensity factor, deduced from the notch geometry. Detailed predictions are made about the variation of the rate of crack growth with the parameters governing the notch shape.

Continuous Distributions of Dislocations. IV. Single Glide and Plane Strain

The theory of the continuously dislocated crystal is applied to discuss the behaviour of a crystal which has deformed by single glide. When the single glide is also in one crystallographic direction only the deformation is two-dimensional, and this situation is treated in detail. The analysis assumes that the dislocations arrange themselves so that they cause no far-reaching stress. In single glide, the lattice lines originally normal to the slip plane are the orthogonal trajectories of a family of surfaces, the glide surfaces. It is shown that these surfaces are developable, and that this imposes additional restrictions on the dislocation density, which are not considered by Nye (1953) and which provide a basis for the classification of the possible states of dislocation. A convenient analytical solution is given of the general equations for single glide when the lattice rotations are everywhere small. The discussion of plane strain begins by treating the lattice deformation. The general equation governing the lattice rotation is given, and the necessary boundary conditions and methods of solution are discussed. The analysis is compared with that of Nye and illustrated by some examples of practical interest. More realistic problems are posed when the boundary conditions specify changes of shape of the specimens. To discuss these the relationship between the lattice, shape and dislocation deformations is considered, and it is shown how all these deformations can be found from the necessary boundary conditions on the shape deformation. The general solution, which is valid for large strains, is applied to a general class of bending problem, and equations relating the geometrical features of the lattice, and of lines scribed on the crystal are derived. The general analysis is illustrated by examples emphasizing certain confusing degeneracies arising in the very symmetrical situation of uniform plane bending (Nye 1953), and revealing the meaning of many valued solutions for the lattice rotation.

Continuous distributions of dislocations VI. Non-metric connexions

The quasi-static plastic-elastic deformation of a crystalline solid containing a continuous distribution of both dislocations and extra-matter is reexamined, retaining the assumption that the lattice structure is everywhere uniquely defined. The various distortions are considered to occur in a formal sequence and the solid is assumed to pass as a continuous whole through a series of generalized spaces as the distortions occur in succession. The principles used in associating the appropriate differentiable manifold with each state of the solid are indicated, and it is emphasized that the theory of non-metric connexions is required if the dislocation density is always to be measured by the torsion of the manifold. The importance of this association is emphasized, since real crystalline matter does in fact deform plastically by the motion of dislocations. It is also shown that the theory of non-metric linear connexions is especially suitable for describing the lattice geometry, and particularly the spatial increments of pure strain and rotation. The governing equations are formulated in such a way that any of the manifolds may be regarded as primary, and the relation of the analysis to treatments of other workers using different coordinate systems is discussed.