Xanthippi Markenscoff

Effect of surface stress on the natural frequency of thin crystals

Within the framework of classical beam theory it is shown that a strain‐independent surface stress has no effect on the natural frequency of a thin cantilever beam. Therefore, the experimental results of Lagowski, Gatos, and Sproles must have a different explanation.

High−frequency vibrations of crystal plates under initial stresses

A system of six two−dimensional plate equations is derived for motions of small−amplitude waves or vibrations superimposed on finite, elastic deformations due to static, initial stresses. In the stress−strain relations, the nonliner terms associated to the third−order elastic stiffness coefficients are included. These equations accomodate the coupling of the six lowest modes of vibration, i.e., the flexure, extension, face−shear, thickness−shear, thickness−twist, and thickness−stretch modes, and all their anharmonic overtones. The new equations are applied to the rotated Y cuts of quartz in studying the thickness−shear and flexural vibrations. The changes in the resonance frequencies of the fundamental thickness−shear vibrations are computed as functions of the direction of initially applied force and of the angle of rotated Y cuts. The predicted results are compared with experimental data and with existing computed results. An explicit formula is obtained for the change of fundamental thickness−shear fre...

Conservation integrals in couple stress elasticity

Abstract Noether’s theorem on invariant variational principles is applied in the case of infinitesimal couple stress elasticity, thereby extending the analysis of Knowles and Sternberg (1972. On a class of conservation laws in linearized and finite elastostatics. Arch. Ration. Mech. Anal. 44, 187–211) beyond the range of classical elasticity. Two conserved integral quantities are deduced which generalize the J -integral and L -integral in the notation of Budiansky and Rice (1973: Budiansky, B. and Rice, J. R. (1973) Conservation laws and energy-release rates. J. Appl. Mech. 40, 201–203 ). An expression for an M -integral is also obtained, but it is demonstrated that there is no corresponding conservation law for this integral. Relationships of the derived path integrals to other similar quantities for couple stress elasticity which have appeared in the literature are discussed.

A Green’s Function Formulation of Anticracks and Their Interaction With Load-Induced Singularities

This paper provides a Greens function formulation of anticracks (rigid lamellar inclusions of negligible thickness that are bonded to the surrounding elastic material). Apart from systematizing several previously known solutions, the article gives the pertinent fields for concentrated forces, dislocations, and a concentrated couple applied on the line of the anticrack

On the shape of the Eshelby inclusions

It is shown, based on properties of analytic functions, that for inclusions of constant eigenstrain and eigenstress that the shape of the inclusion is restricted and any part of a plane (i.e. polyhedral inclusion) is prohibited.

Inclusions with constant eigenstress

The Eshelby conjecture, stating that the only inclusions of constant eigenstrain that may sustain constant eigenstress are ellipsoidal shaped, is being considered by a geometric approach. It is established that the class of shapes that may sustain constant eigenstresses form a 9-dimensional manifold embedded in the space of all possible shapes. In particular, it is shown that the only infinitesimal perturbations of an ellipsoidal inclusion that preserve the constancy of eigenstresses are those that perturb the ellipsoid into another ellipsoid.

On the absence of eshelby property for non-ellipsoidal inclusions

It is shown that the Eshelby property does not hold for any inclusion bounded by a polynomial surface of higher than the second-degree, or any inclusion bounded by a non-convex surface. Inclusions bounded by segments of two or more different surfaces are also precluded. The absence of the Eshelby property for non-ellipsoidal inclusions is then discussed.

Elastic precursor decay and radiation from nonuniformly moving dislocations

Abstract P recursor decay in plate impact experiments on single crystals is re-examined from the viewpoint of the elastodynamics of moving dislocations. Superposition of solutions for many dislocations set in motion by an incident plane wave is used to relate the decay of the wave amplitude at the front of the plane wave to the density and velocity of dislocations at the wavefront. The resulting precursor decay relation is the same as the one derived from an elastic/visco-plastic model of the material, except for a small correction due to differences between the effects of forward and backward propagating dislocations. Motivated by this added support for the validity of the precursor decay equation, the values used for the quantities in this equation are re-examined. Recent experimental results and the elastodynamics analysis are interpreted as indicating that the commonly-used values of dislocation velocity are probably satisfactory, but that the values used for dislocation density are several orders of magnitude too small near the lapped surfaces of the crystal. These large dislocation densities are identified as the probable dominant cause of the lower-than-predicted precursor amplitudes that are recorded in experiments. More accurate experimental data and inclusion of the non-linear elasticity effects are essential in determining whether or not the observed precursor decay in the bulk of the specimen can be explained by the motion of dislocations present initially. Calculations of energy radiated from screw and edge dislocations that start from rest and move thereafter at constant velocity confirm that dislocation drag forces due to continuum elasticity effects are small for dislocation velocities less than, say, 80% of the elastic shear wave speed. At supersonic speeds the continuum drag effects become so large that sustained supersonic motion of dislocations appears unlikely.

The nonuniformly moving edge dislocation

Abstract The motion of an edge dislocation starting from rest and moving thereafter nonuniformly on its slip-plane is analyzed by means of Laplace transforms in space and time, with detailed treatment of the singularities involved. The stress and particle velocities are derived in the form of an integral over the history of the motion in the case of a general motion and in closed form in the special case of a dislocation starting from rest and moving with constant velocity thereafter.

Invariance of stresses under a change in elastic compliances

A recent discovery by Cherkaev, Lurie and Milton shows that, in plane elasticity involving a material with varying compliances, it is possible to keep the stress state the same by compensating for a constant shift in the bulk compliance with the opposite shift in the shear compliance. The present article demonstrates that, depending on the connectivity of the domain and some side conditions, the stress state may also be invariant upon a linear shift in the elastic compliances. Discontinuous compliances, such as encountered in composite materials, and various interface conditions, as well as global compatibility are discussed in detail.