Discontinuous yielding and fracture initiation near a notch in mild steel

This thesis presents the results of an experimental investigation of the initiation of brittle fracture and the nature of discontinuous yielding in small plastic enclaves in an annealed mild steel. Upper and lower yield stress data have been obtained from unnotched specimens and nominal fracture stress data have been obtained from specimens of two scale factors and two grain sizes over a range of nominal stress rates from 10^2 to 10^7 lb/in.^2 sec at -111°F and -200°F. The size and shape of plastic enclaves near the notches were revealed by an etch technique.

A stress analysis utilizing slip-line field theory in the plastic region has been developed for the notched specimen geometry employed in this investigation. The yield stress of the material in the plastic enclaves near the notch root has been correlated with the lower yield stress measured on unnotched specimens through a consideration of the plastic boundary velocity under dynamic loading. A maximum tensile stress of about 122,000 lb/in.^2 at the instant of fracture initiation was calculated with the aid of the stress analysis for the large scale specimens of ASTM grain size 8 1/4.

The plastic strain state adjacent to a plastic-elastic interface has been shown to cause the maximum shear stress to have a larger value on the elastic than the plastic side of the interface. This characteristic of dis continuous yielding is instrumental in causing the plastic boundaries to be nearly parallel to the slip-line field where the plastic strain is of the order of the Lüder's strain.

On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics

This investigation deals with certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamical theory of not necessarily homogeneous nor isotropic elastic solids. First, the regularity assumptions underlying the foregoing theorem are relaxed by admitting stress fields with suitably restricted finite jump discontinuities. Such singularities are familiar from known solutions to dynamical elasticity problems involving discontinuous surface tractions or non-matching boundary and initial conditions. The proof of the appropriate uniqueness theorem given here rests on a generalization of the usual energy identity to the class of singular elastodynamic fields under consideration.

Following this extension of the conventional uniqueness theorem, we turn to a further relaxation of the customary smoothness hypotheses and allow the displacement field to be differentiable merely in a generalized sense, thereby admitting stress fields with square-integrable unbounded local singularities, such as those encountered in the presence of focusing of elastic waves. A statement of the traction problem applicable in these pathological circumstances necessitates the introduction of "weak solutions'' to the field equations that are accompanied by correspondingly weakened boundary and initial conditions. A uniqueness theorem pertaining to this weak formulation is then proved through an adaptation of an argument used by O. Ladyzhenskaya in connection with the first boundary-initial value problem for a second-order hyperbolic equation in a single dependent variable. Moreover, the second uniqueness theorem thus obtained contains, as a special case, a slight modification of the previously established uniqueness theorem covering solutions that exhibit only finite stress-discontinuities.