Planar contact problem for a body of rectangular cross section in a prestressed state

Scientific Engineering Institute of Mechanics and Applied Mathematics, Rostov State University, Rostov-on-Don. Translated from Prikladnaya Mekhanika, Vol. 26, No. 12, pp. 81–89, December, 1990.     

Sub-doppler laser cooling of thulium atoms in a magneto-optical trap

Sub-Doppler laser cooling in a magneto-optical trap for thulium atoms at a wavelength of 410.6 nm has been experimentally studied. Without any dedicated molasses period of sub-Doppler cooling, the cloud of 3 × 10⁶ atoms at a temperature of 25(5) μK was observed. The measured temperature is significantly lower than the Doppler limit of 240 μK for the cooling transition at 410.6 nm. The high efficiency of the sub-Doppler cooling process is due to a near-degeneracy of the Landé g-factors of the lower 4f ¹³6s ² (J = 7/2) and the upper 4f ¹²5d _3/26s ² (J = 9/2) cooling levels.     

On a method of solving dual integral equations : PMM vol. 37, n6, 1973, pp. 1087–1097

We expound a method of reducing a class of dual integral equations which find important practical application to infinite algebraic systems of the first kind. The latter system can be reduced to the systems of the second kind by exact inversion of the principal singular part, and the second kind systems can be solved using the method of consecutive approximations [1–6], The dual integral equations generated by the Kontorovich-Lebedev and Mehler-Fock integral transforms are considered as examples as well as the problems of torsion of a truncated elastic sphere by a punch and that of a circular crack in an elastic space.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Laser cooling of thulium atoms

We demonstrated laser cooling and trapping of thulium atoms at sub-Doppler temperatures in a magneto-optical trap (MOT). Up to 3 × 10⁶ thulium atoms were trapped in the MOT at temperatures down to 25(5) μK which is approximately 10 times lower than the Doppler limit. The lifetime of atoms in the MOT varied between 0.3–1.5 s and was restricted mostly by optical leaks from the upper cooling level. The lower limit for the leaking rate was estimated to be 22(6) s⁻¹. Due to a big magnetic moment of Tm atoms, a part of them were trapped in a magnetic trap from the quadrupole field of the MOT. We observed about 3 × 10⁴ purely magnetically trapped atoms at temperature of 25 μK with a lifetime in the trap of 0.5 s. Also we set up a “dark” MOT consisting of six crossed hollow beams which increased the number of trapped atoms by a factor of 5 leading to 1.5 × 10⁷ atoms at the expense of higher temperature.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

Contact strength of a two-layer covering under friction forces in the contact region

Friction and antifriction composite materials of multilayer structure [1] have recently become very popular in the engineering industry. Antifriction materials are widely used in sliding bearings, and friction materials are widely used in brakes. In the first case, the friction forces between the contacting surfaces are negligible, but in the second case, they are rather large. We use two examples of two plane problems from the theory of elasticity concerning the interaction between a die and a base formed by two elastic layers with different mechanical properties, which are rigidly connected with each other and with an undeformable support, to study how the geometric and mechanical parameters of the problem affect the stress-strain state of such a base, both on its surface and at its internal points, and to find the optimal parameters ensuring the required operation resources of the friction units thus modeled. We assume that the die foot is parabola-shaped or plane, the normal and tangential stresses in the contact region are related to each other by the Coulomb law, and the die is subjected to normal and tangential forces. In this case, the die-two-layer base is in the limit equilibtrium, and the die does not rotate in the process of deformation of the layer. In this setting, the problems were studied in [2] by solving the integral equations (IE) by the asymptotic method of large λ (see [3–7], etc.), which permits finding the effective solution only for relatively large thicknesses of layers compared with the dimensions of the contact region. But in real friction units mentioned above, the layers can have rather small relative thicknesses, and the large λ method cannot be used. We note that the other asymptotic methods (e.g., see [3]) efficient in the case of relatively small thicknesses of layers cannot yet be adapted to the case of friction forces in the contact region. In the present paper, we propose to use the collocation method following the scheme given in [8] to solve the corresponding integral equation of the first kind with logarithmic kernel. This method allows one to obtain sufficiently exact solutions practically for all values of the parameters of the problem with relatively small expenditure of the computer time for modern computers. The contact problem for a two-layer base was used in [9] for a close statement of the problem without friction forces in the contact region.