Fracture kinetics for fuel element cladding tubes made from zirconium alloy in an iodine medium

We present a technique for measuring the growth rate of stress corrosion cracks in cladding tubes made of Zr + 1%Nb alloy and the results of the corresponding investigations at T=380C in an iodine medium for a concentration of 0.1 mg/cm². We show that in analyzing the results we need to carefully take into account the experimental conditions and the geometric parameters of the cracks. This implies that it is impossible to simultaneously consider brittle cleavage cracks and creep cracks. We also need to take into account the relations between the depth and length of the cracks. Rough calculations have confirmed that in fuel element cladding tubes, predominantly short cracks should arise for which it is of foremost importance to take into account the relations mentioned above in calculating the stress intensity factor. We can plot a common kinetic curve only for very long cracks. In order to conservatively estimate the lifetime of the cladding tubes, we should consider K₁ for the most probable short cracks.Translated from Problemy Prochnosti, No. 9, pp. 30–39, September, 1994.     

Investigation of surface instability in an elastic anisotropic cone by the use of surface waves of weak discontinuity

Summary The problem of surface instability of a right circular cone with an arbitrary opening made of a hexagonal single crystal (the cone axis coincides with the crystal's axis of isotropy) is investigated. The surface of the cone is free from normal and tangential stresses, but in the layer near the surface initial constant tensile or compressive stresses act in the hoop direction and in the direction of the cone's generators. Surface instability is analyzed by the use of weak nonstationary disturbances which propagate along the surface of the cone in the form of the two types of surface waves: the nonstationary Rayleigh waves polarized in the sagittal plane, and the nonstationary wave of the whispering gallery type polarized perpendicular to the sagittal plane. The weak nonstationary surface waves are interpreted as the lines of discontinuity (diverging circles) on which partial derivatives of the stress and strain tensor components with respect to coordinates and time have a discontinuity, but the components of these tensors are continuous. Each of the lines of discontinuity propagates with a constant normal velocity along the cone's surface in the direction of its generators and is obtained as a result of the exit onto the cone surface either of two conic complex wave surfaces of weak discontinuity intersecting along this line (Rayleigh wave) or of one real conic wave surface of weak discontinuity (wave of the whispering gallery type). The analysis is carried out within the framework of the theory of discontinuities based on the kinematic, geometric and dynamic conditions of compatibility; using them the velocities of the surface wave propagation and their intensities have been found. It has been shown that the surface wave velocities are dependent only on the initial stress acting in the direction of the propagation of a surface disturbance whereas the damping coefficients for the intensities of the surface waves are dependent not only on this stress but also on the initial stress acting in the hoop direction as well. The relationships for two critical magnitudes of the force compressive in the hoop direction have been obtained, and it has been shown that under the hoop compressive forces in excess of one of these magnitudes the intensity of the Rayleigh wave or the surface wave of the whispering gallery type begins to increase without bounds during its propagation, i.e., the surface of the cone loses stability with respect to either of two types of weak nonstationary disturbances.     

Nonstationary Rayleigh waves on the thermally-insulated surfaces of some thermoelastic bodies of revolution

Summary The influence of heat conduction and thermal relaxation on the propagation of the surface waves polarized in the sagittal plane along the heat-insulated surfaces of the following thermoelastic bodies of revolution: a cylinder, a sphere, a torus, and a cone is investigated. The modified Maxwell law is used as the law of heat conduction, which allows one to take a finite speed of heat propagation into account. The nonstationary surface waves are interpreted as lines (a straight line or a diverging or converging circumference) on which the temperature and the components of the stress and strain tensors experience a discontinuity. Each of the discontinuity lines propagates with a constant normal velocity across the free from stresses and thermally-insulated surface of the body of revolution along the corresponding lines of curvature and is obtained by coming onto the body's surface of the three strong discontinuity complex wave surfaces which intersect along this line: quasi-thermal, quasi-longitudinal and quasi-transverse volume waves. By applying the theory of discontinuities, the velocities and the intensities of the surface waves have been found. It has been shown that the attenuation of the surface wave intensity is determined by the two factors: the coupling between the related strain and temperature fields and the change in curvature of the surface wave with time if the wave is a curvilinear one.     

Analysis of a Hyperbolic System Modelling the Thermoelastic Impact of Two Rods

The problem on collinear collision of two thermoelastic rods possessing the same rheological parameters but of different length and temperature is considered, in so doing before the impact the rods move unidirectionally along a common longitudinal axis with distinct constant velocities. Lateral surfaces and free ends of both rods are heat insulated, and free heat exchange between the rods occurs within contacting ends. The rods' thermoelastic behavior is described by the Green–Naghdy theory. Two methods are used as the methods of solution, namely: D'Alembert's method and the ray method. D'Alembert's method is based on the analytical solution of equations of the hyperbolic type describing the dynamic behavior of the thermoelastic rods. This solution involves four arbitrary functions which are determined from the initial and boundary conditions and are piecewise constant functions. The ray method is based on the theory of discontinuities and also allows one to obtain the solution involving four arbitrary constants. Both solutions complement each other, since D'Alembert's solution enables one to analyze the influence of thermoelastic parameters on the values to be found, while the solution via the ray method is best suited for numerical investigation of the longitudinal coordinate dependence of the desired values at each fixed instant of the time beginning from the moment of rods' collision up to the moment of their rebound. To illustrate the both approaches, the problem of the rods' collision without account for the stress and temperature fields coupling is also considered.     

The method of ray expansions for solving boundary-value dynamic problems for spatially curved rods of arbitrary cross-section

The method of ray expansions is developed for solving boundary-value problems connected with the propagation of planes of strong and weak discontinuity in spatially curved linearly elastic rods of arbitrary cross section. The equations of the three-dimensional theory of elasticity are utilized, which are written on the wave surface using the theory of discontinuities and then are integrated over the cross-sectional area. It is assumed that the plane of discontinuity remains perpendicular to the centroidal axis of the rod all the time during its propagation; the discontinuities in the normal stresses in the sections with the normals perpendicular to the centroidal axis can be ignored as compared to the discontinuities in stresses in the sections with the normals parallel to the centroidal axis. The cross-sections of the rod remain plane during the process of the rod deformation. These assumptions lead to the generation of two wave surfaces propagating in the spatially curved rod with the velocities of longitudinal-flexural and transverse-torsional waves of elastic rods. As this takes place, on the longitudinal-flexural wave, the bulk deformations experience a discontinuity not only at the sacrifice of shortening-elongation of the medium’s element locating along the centroidal axis, but also at the expense of thickening–thinning of this element in the directions of the principle axes of the rod cross-section. For the transverse-torsional wave, there exist discontinuities in the components of the velocities directed along the principle axes of the cross-section, in the angular velocity of the cross-section rotation as a rigid whole with respect to the centroidal axis, as well as in transverse deformations occurring due to the inhomogeneity of the transverse displacements. During the solution of boundary-value problems, the values to be found are represented in terms of the power series, the coefficients of which are the discontinuities in arbitrary order partial time-derivatives of the desired functions, while the time of arrival of the wave front is the independent variable; in so doing the order of the partial time-derivative coincides with the power exponent of the independent variable. The ray series coefficients are determined from the recurrent equations of the ray method within the accuracy of arbitrary constants, while the arbitrary functions themselves are found from the boundary conditions. Examples illustrating the efficiency of the ray method for solving the problems of dynamic contact interaction, resulting in the propagation of transient waves of strong discontinuity in spatially curved rods, are presented.     

Modelling of the collision of two viscoelastic spherical shells

In the present paper, the collision of two viscoelastic spherical shells is investigated using the wave theory of impact. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. Since the local bearing of the materials of the colliding viscoelastic shells is taken into account, the solution in the contact domain is found via the modified Hertz contact theory involving the operator representation of viscoelastic analogs of Young’s modulus and Poisson’s ratio. The collision of two elastic spherical shells is considered first, and then using Volterra correspondence principle, according to which the elastic constants in the governing equations should be replaced by the corresponding viscoelastic operators, the solution obtained for elastic shells is extended over the case of viscoelastic shells.     

Analysis of Rheological Equations Involving More Than One Fractional Parameters by the Use of the Simplest Mechanical Systems Based on These Equations

The behaviour of rheological models containing more than onefractional derivative or fractional operator of fractional orders areinvestigated. All rheological models discussed can be separated intothree groups depending on magnitudes of the value_*/_* (where_* and _* are the orders ofsenior fractional derivatives of stress and strain, respectively): themodels are thermodynamically admissible only when_*/_* = 1 (the first group),thermodynamically compatible only for_*/_* 1 (the secondgroup) and, finally, thermodynamically well-conditioned both at_*/_* 1 and_*/_* > 1 (the third group).It is shown that, under nonstationary excitations, thebehaviour of the simplest mechanical systems (mechanical oscillators,finite and semi-infinite viscoelastic rods), based on the consideredrheological models, may be different (from the point of view ofthermodynamics) from that of the underlying rheological models. Thus,under impulse excitations, the mechanical models based on rheologicalmodels of the first and second groups become thermodynamicallyadmissible not only at_*/_* = 1 but alsowhen _*/_* < 1(mechanical models of group I), but mechanical models based onrheological models of the third group remain thermodynamicallywell-conditioned at the same magnitudes of rheological parameters as thecorresponding rheological models do (mechanical models of group II). Asthis takes place, group I mechanical models possess diffusion-wavefeatures, that is at_*/_*=1 the stress waves ina semi-infinite rod propagate at a finite speed, and the roots ofcharacteristic equations (for nonstationary vibrations of a mechanicaloscillator or a rod of finite length) as functions of the relaxation orretardation times, behave in a way similar to the characteristicequation roots of rheological models possessing instantaneous elasticity(models of the Maxwell type). When_*/_*<1, the stress wavesin a semi-infinite rod propagate instantaneously at infinitely largespeeds, and the roots of characteristic equations (under nonstationaryvibrations of a mechanical oscillator or a rod of finite length) asfunctions of relaxation times behave in a way similar to thecharacteristic equation roots of rheological models lackinginstantaneous elasticity (models of the Kelvin–Voigt type).Mechanical models from group II possess pure wave or pure diffusionfeatures at all magnitudes of_*/_*.     

The impact of a thermoelastic rod against a rigid heated barrier

The impact of a thermoelastic rod with a heat-insulated lateral surface against a rigid heated barrier is considered. The heat exchange between the rod and the wall occurs at one of its ends contacting with the wall, while the other end is heat-insulated and free from external forces. The behaviour of the rod during the impact process is described by the Green-Naghdy theory which allows one to take finite speed of heat propagation into account, neglecting therewith thermal relaxation. The Laplace integral transform with the subsequent expansion of the found images in terms of the natural functions of the problem is used as a method of solution, which is found in explicit exact closed form. The analytical time-dependence of displacements, stresses, and temperature at each rod particle is obtained. The emphasis is on the analysis of the contact stress, the temperature of the colliding bodies during their contact interaction, and on the detection of the duration of contact of the rod with the rigid wall. It is shown that the contact time essentially depends on the relationship between the mechanical and thermal values.     

Fractional-derivative viscoelastic model of the shock interaction of a rigid body with a plate

The impact of a rigid body upon an elastic isotropic plate is investigated for the case when the equations of motion take rotary inertia and shear deformation into account. The impactor is considered as a mass point, and the contact between it and the plate is established through a buffer involving a linear-spring–fractional-derivative dashpot combination, i.e., the viscoelastic features of the buffer are described by the fractional-derivative Maxwell model. It is assumed that a transient wave of transverse shear is generated in the plate, and that the reflected wave has insufficient time to return to the location of the spring’s contact with the plate before the impact process is completed. To determine the desired values behind the transverse-shear wave front, one-term ray expansions are used, as well as the equations of motion of the impactor and the contact region. As a result, we are led to a set of two linear differential equations for the displacements of the spring’s upper and lower points. The solution of these equations is found analytically by the Laplace-transform method, and the time-dependence of the contact force is obtained. Numerical analysis shows that the maximum of the contact force increases, tending to the maximal contact force when the fractional parameter is equal to unity.