Stress distributions in elastic and viscoelastic plates subjected to symmetrical rigid indentations

Abstract

In a previous paper [1], the static and dynamic tensile strengths of glass blocks were measured by the methods of spherical indentation and impact. That paper was concerned with the apparent dependence of the values of the fracture strength on the radius of the spherical indenter. The theory developed there critically depends upon the validity of the Hertz treatment for an elastic half space [2], [3], In later work [4] the nature of fractures produced in glass blocks by impact was studied by measuring the fracture stress waves as a function of impact velocities. From the stress waves experimentally observed and a knowledge of wave propagation, it could be estimated that at the highest velocities of impact, fractures occurred only a few microseconds after the glass blocks were first compressed. At the instant of occurrence of fracture a thin layer of glass was stressed while the rest of the glass block was still free from traction. The physical situation would appear to be that of a thin layer of stressed material resting on a rigid foundation. It was also observed [4] that at high impact velocity large radial cracks were produced inside the material. This pattern of fractures, however, was not observed at lower impact velocities, where a relatively thicker layer of glass was stressed at the instant of fracture. This velocity effect on the patterns of fractures would lead one to consider the possible effect that the thickness of the stressed material has on the local stress distribution. It is with these considerations and with the practical applicability of the Hertz solution, for plates of finite thickness, that the present investigation is concerned.

Symmetrical contact problem of a thick transversely isotropic plate

“…(21) are applied over Eq. (20) [7]. The results can be written in an integral equation for determining p(r) as follows:…”

“…The results obtained in Ref. 7 were later employed in an experimental investigation [8]. In the present work, the techniques of the Hankel transform are used to solve the symmetrical problem of the contact between a spherical indenter and a transversely isotropic plate of finite thickness.…”

“…The stress distributions in elastic and viscoelastic plates subjected to symmetrical rigid indentations were studied in terms of the plate thicknesses [7]. An elastic plate of thickness H, which is subjected to an axisymmetrical indentation on its upper surface and which rests on a rigid smooth foundation, is mathematically equivalent to a plate of thickness 2H subjected to equal indentations on its two surfaces.…”

Viscoelastic effects in the propagation of a central crack

“…It appears interesting to extend the earlier work [1] to viscoelastic materials. The stress-strain relationships for viscoelastic materials are time-dependent and can be represented to a certain extent by the behaviors of analog models [2][3][4][5]. The propagation of a crack in viscoelastic materials involves the combined effect of both time-dependent material properties and inertias of material particles.…”

“…The equations of motion to be satisfied and the strain-displacement relationships are the same as those for the corresponding elastic solutions [1]. The main difference is in the timedependent stress-strain relationships as follows [2,3] f'[ ';' 1 /i eij (7)+~iik(t T)c~rekk (7) dr…”