The impact of a rigid sphere with an elastic layer of finite thickness

Summary The problem of the impact of a rigid sphere with an elastic isotropic layer is considered in the initial stage of dynamic interaction. The initial stage is characterized by the fact that the velocity of the displacement of the intersection points of the sphere with the upper boundary of the layer is larger than the velocity of longitudinal waves, hence the free surface normal to the contact domain with the body is undisturbed. The method of successive approximations as well as the ray method, according to which the solution behind the fronts of incident and reflected waves is constructed in terms of power series (ray expansions), are used as methods of solution. In the problem under consideration, we used one-term ray expansions whereby the main characteristics of the shock interaction have been obtained, and the possibility of localized damage of the material of the layer at the points lying along the central ray has been examined.     

THE RAY METHOD FOR SOLVING BOUNDARY-VALUE PROBLEMS CONNECTED WITH THE PROPAGATION OFTHERMOELASTIC SHOCK WAVES OF FINITE AMPLITUDE

The ray method for solving dynamic boundary value problems for nonlinear thermo-elastic media, wherein heat propagates with a finite speed, is developed. By the action of initial and boundary conditions, two types of finite amplitude shock wave propagate in such media: quasi-thermal wave (fast wave) and quasi-longitudinal wave (slow wave). Behind the wave fronts the solution for the desired functions is constructed along the rays in terms of power series (ray series) whose coefficients are the discontinuities in various orders of partial derivatives of the functions to be found with respect to time, but a variable value is the time needed for a disturbance to propagate along the ray from the point under consideration up to the wave front; in so doing the power of the variable value corresponds to the order of the partial time-derivative of the desired function.     

THE IMPACT OF A THERMOELASTIC ROD AGAINST A HEATED BARRIER WITH ACCOUNT OF FINITE SPEED OF HEAT PROPAGATION

This paper examines the impact of a thin thermoelastic bar of finite length with a heat-insulated lateral surface against a heated massive barrier. The hyperbolic equation of heat conduction is used. During impact, thermoelastic waves propagate along the bar, and behind the wave fronts a solution is constructed in terms of ray series. The stresses, temperature, displacement velocity, and heat flux profiles are obtained. Also, the time-dependent contact stress and temperature and the contact time as a function of the temperature of wall heating, heat flow relaxation time, impact velocity, and coefficient of convective heat exchange from the wall to the rod are investigated.     

D'Alembert's Solution in Thermoelasticity – Impact of a Rod Against a Heated Barrier: Part II. A Case of Coupled Strain and Temperature Fields

The problem of impact of a thermoelastic rod against a heated rigid barrier is considered, in so doing lateral surfaces and free end of the rod are heat insulated, and free heat exchange between the rod and the rigid obstacle or ideal thermal contact occurs within contacting end. The rod's thermoelastic behavior is described by the Green–Naghdi theory of thermoelasticity. D'Alembert's method, which is based on the analytical solution of equations of the hyperbolic type describing the dynamic behavior of the thermoelastic rod, is used as the method of solution. This solution involves four arbitrary functions which are determined from the initial and boundary conditions and are piecewise constant functions. The procedure developed enables one to analyze the influence of thermoelastic parameters on the values to be found and to investigate numerically the longitudinal coordinate dependence of the desired functions at each fixed instant of the time beginning from the moment of the rod's collision with the barrier up to the moment of its rebound both without account for the stress and temperature fields coupling (in the companion paper, Part I) and in the case of coupling thermoelasticity (in this paper). As a numerical example, the impact of a thermoelastic rod against a heated barrier is considered with a small parameter of coupling between the strain and temperature fields. It has been shown that the presence of small coupling results in the generation of a new shock wave of small amplitude, namely: the reflected thermal wave from the incident elastic wave at the free rod's end. The rod's rebound may occur either at the moment of simultaneous arrival at the contact place of two reflected waves: elastic wave from the incident thermal wave and thermal wave from the incident elastic wave—or at the time when the reflected elastic wave from the incident elastic wave reaches the contact point.     

Methods for solving one-dimensional boundary-value problems in a nonlinear elastic medium

Summary One-dimensional problems connected with a mechanical exposure to the boundary of a nonlinear elastic half-space, which leads to a constantly accelerated motion of this boundary, are considered. The value , equal to the square root of the ratio between the velocity of the boundary motion and the velocity of longitudinal wave propagation in a linear elastic medium, is used as the value characterizing the intensity of this exposure. It is shown that as a result of such an exposure shock waves of small or finite amplitude may propagate in the half-space. The asymptotic matching principle and the ray method are used as methods of solution. The merits and demerits of each method are analyzed. It has been inferred that the matched asymptotic method can be applied to waves of small amplitude, and the ray method is usable when investigating the propagation of shock waves not only of small amplitude, but of finite amplitude as well if the time of a consideration of the shock process is not long. The results obtained by the two methods for shock waves of small amplitude are in close agreement. It has been demonstrated that the ray method is adaptable for solving more intricate boundary-value problems resulting in the propagation of several shock waves of finite amplitude at a time. The problem connected with the constantly accelerated motion of one of the boundaries of an initially deformed elastic layer provides an example.     

The method of ray expansions for investigating transient wave processes in thin elastic plates and shells

Summary In the present paper, starting from equations of three-dimensional theory of elasticity, the system of the recurrent equations of the ray method is obtained allowing one to describe the dynamic behavior of thin solids. It is shown that the implementation of the recurrent equations of the ray method and the ray series derived from these equations is preferred over the hyperbolic equations of the Timoshenko type for finding the solution for the problems resulting in the generation and propagation of surfaces of discontinuity in thin bodies. This is due to the fact that in the theories of the Timoshenko type there occur two shear waves, on which shear disturbances along and transverse to the plate's middle surface propagate with different velocities, but within the framework of the proposed theory only one shear wave propagates, on which shear takes place both along the middle surface and transversely to the middle surface. The inconsistency of shear wave splitting into two waves is most pronounced in the problems wherein disturbances propagate along the rays possessing not only curvature but torsion as well, for example, along the helical lines located on the surface of a cylindrical shell.     

Hydrostatic pressing for the manufacture of magnesite crucibles for melting nickel alloys

Conclusions The main technological parameters of the hydrostatic pressing of crucibles of technical periclase have been determined: bonding, 3–4% K-9 silicoorganic resin; duration of vibrocompaction, 5 min at 130 vib/min and an amplitude of 0.9 mm; and pressing pressure, 75 MPa.It has been established that the crucibles made by hydrostatic pressing have the same density throughout before and after firing. Depending on the melting conditions of the nickel alloys, the resistance of the crucibles is 30–100 melts.Translated from Ogneupory, No. 2, pp. 58–60, February, 1980.