William E. Warren

Foam mechanics: the linear elastic response of two-dimensional spatially periodic cellular materials☆

Abstract A theoretical model for the linear elastic response of two-dimensional, spatially periodic cellular materials in simple shear and pure strain is developed. Hexogonally symmetric network morphology, with three identical films intersecting at each junction, is assumed. Film thickness is permitted to vary with distance from the joint. Such hexagonal networks are transversely isotropic; the effective stress does not depend upon the principal directions of strain. The effective in-plane elastic constants are explicitly related to the film compliances for bending and stretching. Film bending is the primary deformation mechanism for low-density, two-dimensional foams, as observed by other investigators. The hexagonal symmetry that is responsible for mechanical isotropy is favored by a foam synthesis route that includes a liquid state. It is shown that a ‘square’ cellular network is neither elastically isotropic nor energetically preferred during formation.

Spall fracture in aluminum monocrystals: a dislocation‐dynamics approach

High‐purity monocrystalline aluminum disks of three crystallographic orientations were subjected to carefully controlled planar impact producing low levels of spall damage. This damage was observed by optical and scanning electron microscopy of sections through the recovered disks, and was found to consist of voids of essentially octahedral form having {111} planes as faces. To describe the growth of these voids we propose a kinematical model based on the motion of edge dislocations. Dynamical equations describing the rate of growth of an individual void are obtained by applying established concepts of dislocation mechanics to the kinematical model. Finally, the dynamical void growth model is combined with an empirically established nucleation model to yield equations for calculating the total volume growth rate in a spalling sample. Extension of these results to other ductile fracture phenomena is suggested.

The edge dislocation inside an elliptical inclusion

Abstract The plane elasticity problem of an edge dislocation located within an elliptical inclusion in an unbounded matrix is considered. A general solution to this problem is obtained in terms of complex potential functions and all coefficients in the series representation of these potential functions are explicitly obtained. Convergence of these series is thus assured. Two specific cases of this general solution are considered in detail. The first case considers a very long, thin inclusion corresponding physically to the geometry of typical crazes in glassy polymers. the second case considers the almost circular inclusion representative of a number of crystal defects and imperfections. For the second case, expressions for the resultant force on the dislocation agree with results previously obtained for the limiting circular inclusions.

Electrostatic forces between conducting spheres at constant potentials

A derivation in the bispherical coordinate system is presented for the electrostatic force between conducting spheres at a constant potential difference. The theoretical results are compared with experimentally measured forces.

Interaction of dislocations with surface notches and protrusions

Abstract Plane strain linear elasticity is utilized to develop expressions for the force on edge and screw dislocations in a half-plane whose stress-free surface is deformed by a protrusion or notch. Representative force fields are portrayed graphically, and the limiting case of a surface crack is investigated in detail. Results indicate that material directly ahead of a notch or crack tends to be depleted of edge dislocations whose Burgers vector is normal to the crack and to accumulate those with Burgers vector parallel to the crack. No stable equilibrium positions in the material where dislocation pile-up may occur are found.

The stress and displacement fields at the tip of crazes in glassy polymers

Abstract This analysis models a craze region in glassy polymers as an elastic transversely isotropic homogeneous inclusion of thin elliptical shape with different elastic properties from the bulk polymer. The plane elasticity problem for an applied uniform stress field is solved and the results dimensionalized with respect to the craze tip radius. Stress and strain enhancements of several times far field values are found to occur at the craze tip and are independent of craze tip radius. These results are consistent with experimentally observed characteristics of craze growth and should be important in assessing the relative merits of different criteria that have been proposed for craze growth in glassy polymers.

Theoretical Approach to Enhanced Pressure Apparatus Design

The possibility of obtaining an enhanced stress level within a solid is investigated for a spherical configuration consisting of a solid homogeneous isotropic core encapsulated by a nonhomogeneous isotropic shell. Utilizing a linear elastic analysis, the enhancement is maximized (1) for the case when the nonhomogeneous shell is composed of a number of concentric homogeneous shells; and (2) for the case where the elastic shell modulus varies as a power of the radius. The dependence of the enhancement on the various parameters is depicted graphically and the results are discussed in terms of presently available materials. It is felt that the attainable enhancement may be significant only in the region up to 10 kbar and very limited above 30 kbar.

Thermoelastic wave propagation from cylindrical and spherical cavities in the two‐temperature theory

A linear two‐temperature theory of thermoeleasticity is used to investigate cylindrical and spherical wave propagation emanating from a cavity. Transform methods of solution to specific boundary‐value problems are employed and interest is directed to wave speeds and traveling discontinuities. A comparison with classical one‐temperature coupled thermoelastic effects is presented.

Void growth during spall fracture of aluminum monocrystals

High-purity monocrystalline aluminum disks of three crystallographic orientations were subjected to carefully controlled planar impact producing low levels of spall damage. This damage was observed to consist of voids of essentially octahedral form having {111} planes as faces. To describe the growth of these voids we propose a kinematical model based on the motion of edge dislocations. Dynamical equations describing the rate of growth of an individual void are obtained by applying established concepts of dislocation mechanics to the kinematical model.

Electrostrictive stress singularities in angular corners of plates

ZusammenfassungDie ungekoppelte Theorie der zweidimensionalen Elektrostriktion wird verwendet, um die Hauptspannungssingularität zu untersuchen, die am Scheitel eines elastischen dielektrischen Sektors in einem ziemlich allgemeinen clektrischen Feld entsteht. Die (radialen) Ränder des Sektors sind entweder spannungslos oder verschiebungslos. Es wurde festgestellt, dass für Scheitelwinkel >180° diese elektrostriktive Singularität von höherer Ordnung ist als die Hauptsingularität, welche durch mechanische und thermische Wirkungen gekennzeichnet ist. Kein singuläres Verhalten ist möglich bei Scheitelwinkeln <180°.