Another Special Case of Vibrations of a Rolling Tire

We investigate a special case of vibrations of a loaded tire rolling at constant speed. A previously proposed analytical model of a radial tire is considered. The surface of the tire is a flexible tread combined with elastic sidewalls. In the undeformed state, the tread is a circular cylinder. The tread is reinforced with inextensible cords. The tread is the part of the tire that makes actual contact with the ground plane. In the undeformed state, the sidewalls are represented by parts of two tori and consist of incompressible rubber described by the Mooney –Rivlin model. The previously obtained partial differential equation which describes the tire radial in-plane vibrations about steady-state regime of rolling is investigated. Analyzing the discriminant of the quartic polynomial, which is the function of the frequency of the tenth degree and the function of the angular velocity of the sixth degree, the rare case of a root of multiplicity three is discovered. The angular velocity of rotation, the tire speed and the natural frequency, corresponding to this case, are determined analytically. The mode shape of vibration in the neighborhood of the singular point is determined analytically.

Evaluation of Grain Deformation in Polycrystals

Method of local strain estimation based on evaluation of relative surface area of grain boundaries in deformed and undeformed state is used very often [. Unfortunately, this method requires information about the parameter of structure in case of zero value of initial deformation. Mentioned parameter is unknown in most cases. In addition, value of parameter of structure depends on grain size and it can change in volume of material.

Kinematics and Stresses of Deformable Bodies

The form of kinematics to be used for the description of a deformation process is largely determined by the kind of mechanical response that is being described. To describe the mechanical response of purely viscous fluids it is convenient to use coordinates, which are fixed in space, since purely viscous fluids have no past memory and therefore remain in the deformed state when loads are removed. In other words, the mechanical response of purely viscous fluids is determined solely by the instantaneous values of the time rate of deformation. However, in order to describe the deformation of a viscoelastic fluid it is necessary to follow a given material element with time as it moves to define a suitable measure of deformation that always refers to the same material element as time varies. The reason is that when a material element undergoes a finite deformation the coordinate positions of the given material element (with respect to a fixed origin) will vary. Hence, any measure of deformation defined in terms of infinitesimal deformation of fixed coordinate positions loses its physical significance since it will not always be associated with the same material element. In this chapter, we introduce some basic concepts of the kinematics and stresses of a deformable body from the point of view of continuum mechanics, and discuss various representations of a deformation process in terms of the deformation (or strain) tensor and the rate-of-deformation (or rate-of-strain) tensor. In order to help the readers follow the material in the text, the elementary properties of second-order tensors are presented in Appendix 2A. In this section, we briefly describe the motion of a body, which consists of a set of particles (or “elements”), sometimes called “material points” (or “material elements”) (Jaunzemis 1967). Let X(Xi ; i = 1, 2, 3) be the particles P of the body B in some reference configuration κ at time t = 0 (i.e., undeformed state) and then we have. . . X = κ(P). . .in which κ describes the shape of the body B in the undeformed state, which in general is known to an observer.

A Theoretical Analysis of the Effect of External Polymers on the Wrinkle Recovery of Fabrics

The improvement observed in the wrinkle recovery of wool fabrics treated with certain external elastomeric polymers has previously been attributed to the presence of interfiber bonds, which considerably reduce the frictional component of wrinkling and provide an additional elastic restoring moment, aiding the return of the fabric to its undeformed state. These mechanisms are shown to be quantitatively consistent with the additional experimental evidence now available, including the effects of polymer treatment on a series of wool/polyester blend fabrics and the observation that washing of a polymer-treated fabric can sometimes improve its wrinkle recovery. The theory is used to separate the effects due to energy stored in the elastomer and those due to increased fiber strains.

Results of Preliminary Experiments on the Plasticity of Ice Crystals

The growth of single crystals of ice for the study of plasticity has to be done with care, since a disturbing lineage structure is easily generated. The orientation of specimens can be found from growth figures (dendrites) and Tyndall’s “flowers”; these give the a-axis, as the branches in both cases point in the [1120] direction. Slip occurs on the basal plane without following a definite glide direction. In pure shear two stages of creep exist, the first for a glide of 10–20 per cent in an undeformed crystal, the second for greater shears having a higher creep velocity. Both stages obey the law , n being 2.3–4 in the first case and 1.3–1.8 in the second. The deformed state above the transition remains stable and no recrystallization takes place in pure shear. Restoration of the undeformed state by recrystallization occurs only in inhomogeneously deformed parts. This observation may have an influence in interpreting measurements of glacier movements and tests on polycrystalline specimens in the laboratory.

Large elastic deformations of isotropic materials VIII. Strain distribution around a hole in a sheet

The deformation produced by radial forces in a thin circular sheet of incompressible highly elastic material, isotropic in its undeformed state, containing a central circular hole, is studied theoretically, and results calculated on the basis of the theory are compared with those obtained experimentally employing a vulcanized natural rubber compound as the highly elastic material.