Thermal Reaction During Thermal Shock of a Massive Body with an Internal Cylindrical Cavity

The paper examines mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to specific cases during intense heating of a solid boundary. We introduce a stress compatibility equation for dynamical problems, generalizing the well-known Beltrami --- Mitchell relation for quasi-static cases. It is convenient to use this relation when considering numerous special cases in the theory of heat shock in Cartesian coordinates both for bounded bodies of a canonical form, i.e., an infinite plate, and for partially bounded bodies, i.e., space bounded from the inside by a flat surface. In the latter case, the obtained analytical solutions of dynamic problems of thermoelasticity lead to visual and convenient for physical analysis functional structures describing the kinetics of thermal stresses. For cylindrical and spherical coordinate systems, we propose a compatibility equation in displacements, which is convenient for studying the problem of thermal shock in bodies with a radial heat flux and under conditions of central symmetry. In the study, we singled out a class of problems in which the consideration of the geometric dimensions of a structure investigated for a thermomechanical reaction under conditions of intense heating concerns mainly the near-surface layers. According to the experimental results, it is these layers that absorb the main amount of heat during a time close to the beginning of heating, which corresponds to the time of the microsecond duration of the inertial effects. We investigated the thermal reaction of a massive body with an inner cylindrical cavity within the framework of dynamic thermoelasticity under various modes of intense heating of the cavity surface. Finally, we carried out numerical experiments and described the wave character of thermal stresses with the corresponding quasi-static values, and established the role of inertial effects in mathematical models of the theory of thermal shock

Model representations of heat shock in terms of dynamic thermal elasticity

This article is devoted to mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to the specific conditions of intensive heating and cooling of solids. A scheme is proposed for deriving the compatibility equation in voltages for dynamic problems, which generalizes the well-known Beltrami-Mitchell relation for quasistatic cases. The proposed relation can be used to consider numerous special cases in the theory of thermal shock in Cartesian coordinates for both bounded canonical bodies and partially bounded ones. As a detailed study, the latter case was considered under conditions of abrupt temperature heating and cooling, thermal heating and cooling, and medium heating and cooling. Numerical experiments were carried out, and the wave nature of the propagation of thermoelastic waves was described. The effect of relaxation of the solid boundary on sudden heating and sudden cooling, which has been little studied in thermomechanics, is described. It is established that this effect influences maximum of internal temperature stresses, which depend on the parameters characterizing the elastic and thermal properties of materials, as well as the heating time and cooling time. A “compatibility equation” in displacements was proposed to study the problem of thermal shock in cylindrical and spherical coordinate systems in bodies with a radial heat flow and central symmetry. The formulation of a generalized problem in the theory of thermal shock is formulated, which is of practical and theoretical interests for many areas of science and technology.

A Theory of Autofrettage for Open-Ended, Polar Orthotropic Cylinders

Autofrettage is a widely used process to enhance the fatigue life of holes. In the theoretical investigation presented in this article, a semi-analytic solution is derived for a polar, orthotropic, open-ended cylinder subjected to internal pressure, followed by unloading. Numerical techniques are only necessary to solve a linear differential equation and evaluate ordinary integrals. The generalized Hooke’s law connects the elastic portion of strain and stress. The flow theory of plasticity is employed. Plastic yielding is controlled by the Tsai–Hill yield criterion and its associated flow rule. It is shown that using the strain rate compatibility equation facilitates the solution. The general solution takes into account that elastic and plastic properties can be anisotropic. An illustrative example demonstrates the effect of plastic anisotropy on the distribution of stresses and strains, including residual stresses and strain, for elastically isotropic materials.

Effect of Geometric Error of Inner Raceway on Motion of Cylindrical Roller Bearings Under Radial Load

The motion error of bearing depends highly on the geometric profile of bearing components. Therefore, it is crucial to establish a correlation between the geometric error of bearing components and the motion error of an assembled bearing, which is required for designing and manufacturing bearings with high accuracy of motion. In this paper, authors derived a geometric compatibility equation for cylindrical roller bearing considering the geometric error of bearing inner raceway. Based on the load balance and the geometric compatibility derived, a mathematical model of motion accuracy is established, and the model is also validated. The effect of geometric error such as the amplitude of roundness error and dimension error of bearing inner raceway, and radial clearance on the bearing motion error is investigated. Results show that the motion error of the bearing increases with the amplitude of the roundness error of inner raceway, and reduces with the increase of radial load. The results indicated that the motion accuracy can be improved by controlling the distribution of machining tolerance of bearing components.

Free Vibration of Thin Shallow Elliptical Shells

This research presents a study of the free vibration of thin, shallow elliptical shells. The equations of motion for the elliptical shell, which are developed from Love's equations, are coupled and nonlinear. In this research, a new approach is introduced to uncouple the transverse motion of the shallow elliptical shell from the surface coordinates. Through the substitution of the strain-compatibility equation into the differential equations of motion in terms of strain, an explicit relationship between the curvilinear surface strains and transverse strain is determined. This latter relationship is then utilized to uncouple the spatial differential equation for transverse motion from that of the surface coordinates. The approach introduced provides a more explicit relationship between the surface and transverse coordinates than could be obtained through use of the Airy stress function. Angular and radial Mathieu equations are used to obtain solutions to the spatial differential equation of motion. Since the recursive relationships that are derived from the Mathieu equations lead to an infinite number of roots, not all of which are physically meaningful, the solution to the eigenvalue problem is used to determine the mode shapes and eigenfrequencies of the shallow elliptical shell. The results of examples demonstrate that the eigenfrequencies of the thin shallow elliptical shell are directly proportional to the curvature of the shell and inversely proportional to the shell's eccentricity.

The power of weak-field GR gravity

While general relativity (GR) is our premier theory of gravity, galactic dynamics from the outset has been studied with Newtonian gravity (NG), guided by the long-held belief in the idea of the “Newtonian-limit” of GR. This maintains that when the gravitational field is weak and the velocities are nonrelativistic, NG is the appropriate theory, apart from small corrections at best (such as in GPS tracking). However, there are simple examples of phenomena where there is no NG counterpart. We present a particularly simple new example of the stark difference that NG and weak-field GR exhibit for a modified van Stockum source, which speaks to the flat galactic rotation curve problem. We note that the linear GR compatibility equation in the literature is incomplete. Its completion is vital for our case, leading to a stark contrast between GR and NG for totally flat van Stockum rotation curves.

On the Exact Solutions of Couple Stress Fluids

Exact solutions of the momentum equations of couple stress fluid are investigated. Making use of stream function, the two-dimensional flow equations are transformed into non-linear compatibility equation, and then it is linearized by vorticity function. Stream functions and velocity distributions are discussed for various flow situations.

Rod Dynamics With Large Stretch

In the last few decades, the use of synthetic fiber line in mooring systems has become increasingly popular, for instance in composite moors consisting of wire rope, polyester line and chain. Synthetic fiber lines are noted for their large stretch which can under high load be as much as 10 to 20 percent of their unloaded length or more. In developing a consistent model for the motions of a moored offshore platform using composite moors, it is necessary to model the dynamics of the moor recognizing that some elements may exhibit large stretch. The model for the dynamics of a rod without stretch was developed by Garrett (1982). This model has been frequently extended to the case with small stretch by linearizing the stretch term in the compatibility equation, for instance, Paulling & Webster (1986). The research presented here is an extension of Garrett’s theory to include the possibility of large stretch. With the adoption of a simple assumption concerning the character of the stretch, and with the incorporation of visco-elastic behavior of the large-stretch elements given by Kim, Kyoung & Sablok (2010), large stretch can be introduced consistently with few changes in the traditional finite-element scheme. Finally, the effects of large stretch on the physical properties and dynamics of the rod are also discussed.