VECTOR FIELDS IN R2 WITH MAXIMAL INDEX

We consider the method of Poincaré to investigate thelocal index of vector fields in the plane. If m is the degreeof the first non-zero jet, X_m, of the vector field X at an isolatedzero, we explore the geometry of the pencil generated by thecoordinate functions of X_m when the absolute value of the indexof X, |ind (X)|, is m. We also find necessary and sufficientconditions for |ind (X)| to be m.     

The plasticity of pure single crystals

The theory of the work-hardening curve of a pure single crystal is discussed. An outline is given of the experimental methods available for the study of the mechanism of plastic deformation and work hardening. An attempt is made to correlate existing theories of work hardening. The theory is illustrated by discussions of copper, of other face-centred cubic metals, of the diamond and sphalerite structures, of hexagonal close-packed metals, of ionic face-centred cubic crystals and of body-centred cubic metals.     

Grain size, stress, and creep in polycrystalline solids

If a stress σ is applied to a polycrystal of grain size L, the mode of creep deformation depends on the answers to the following questions: (I) Does σ exceed the Peierls stress σ_p; (II) Does L exceed the dislocation spacing in a Taylor lattice stabilized by σ_p; (III) Does Lσ exceed the value required for a Frank-Read or Bardeen-Herring source to operate within the grain? (IV) Does L ^1/2σ exceed the Hall-Petch value required for slip to propagate across a grain boundary? The (L, σ) plane is thus partitioned into regions in which different creep modes predominate.     

One-dimensional models of thermal activation under shear stress

The analysis of thermal activation under shear stress in three- and even two-dimensional models presents unresolved problems. The analysis of one-dimensional models presented here may illuminate the study of more realistic models. For the model in which as many dislocations are poised for backward jumps as for forward jumps, the experimental activation volume V ^?e(σ_a) under applied stresses close to σ_a is different from the true activation volume V(σ) evaluated at σ = σ_a. The relations between the two are developed. A model is then discussed in which fewer dislocations are available for backward than for forward jumps. Finally, the appropriateness of the hyperbolic sine approximation for moderately low stresses is defined and shown to be very limited.