**Journal of Theoretical and Applied Mechanics**

Volume 31, Number 2, 2001

Contents

**J. N. Boyd, R. G. Hudepohl, P. N. Raychowdhury**

Department of Mathematical Sciences, Virginia Commonwealth University,

Richmond, VA 23284-2014, U.S.A.,

*e-mail: *praychow@saturn.vcu.edu

**Frequencies of Vibration for a Circular Array of Masses Coupled Through
Their Displacements and Velocities**

Abstract: In this paper, we obtain in closed form the natural
frequencies of vibration for a symmetric arrangement of pairs of masses, *m* and *M,
*on a fixed circle. Each mass is attached to its two nearest neighbours by identical,
ideal springs which possess extended mass m . The springs
couple the motions of the vibrating masses through their displacements on the circle. The
effect of attributing mass to the springs is to couple the motions of the masses through
their velocities as well. We write the Lagrangian for the array and then transform the
Lagrangian with a unitary transformation derived from the rotations of a regular polygon.
The purpose of the transformation is to decouple the equations of motion to the maximum
extent possible from the symmetry of the array. We consider in some detail the case for
four pairs of interconnected masses and then present results for the general case with *N*
pairs of masses. We have employed *Mathematica* to perform many of the
straightforward but lengthy calculations.

Key words: harmonic oscillator, natural frequency, Lagrangian function, symmetry group, group representation, unitary transformation, Mathematica.

**P. V. Atanasova**

*Institute of Mechanics, Bulgarian Academy of Sciences,Acad. G. Bonchev Str. Bl. 4, 1113
Sofia, Bulgaria*

**Rotating of the satellite-gyrostat around an axis fixed in the orbital
frame**

Abstract. This paper describes the state of a satellite-gyrostat rotating around an axis fixed in the orbital frame. It is assumed that the gyrostat is on a circular equatorial orbit and it is under the influence of gravitational and magnetic torques as well as time-constant internal and external ones.

Key words: attitude control, gyrostat.

**N. D. Nikolov, J. D. Nikolova **

University “Prof. Dr. Assen Zlatarov” – Bourgas,

*1, Prof. Yakimov Str., 8010 Bourgas, Bulgaria,*

e-mail: nnikolov1@hotmail.com

**L. P. Bojinov**

Technical University – Varna,

1, Studentska Str., 9010 Varna, Bulgaria,

*e-mail*: lbojinov@hotmail.com

Ultrafiltration of oil emulsions by pulsating pressure

**in a round channel**

Abstract. A process of ultrafiltration with a continuous withdrawal of superconcentrate from the existing concentration zone at a round cross section of the channel at pulsating flow in laminar region is carried out. Acetatecellulose tubular membranes with inside diameter of 12,5 mm are used. An increased relative productivity is established versus the case of utilization without superconcentrate receptacle.

Key words*:* ultrafiltration, concentration polarization, laminar
flow, tubular membranes, superconcentrate.

- Baltov, N. Bontcheva
**Kouyumdjiev, I. Ivanov**

Institute of Mechanics, Bulgarian academy of sciences,

Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria,

*e-mail: *boncheva@bgcict.acad.bg*,
*Baltov@eagle.bas.bg*
*

V. Kafka

Institute of Theoretical and Applied Mechanics,

Academy of Sciences of The Czech Republic,

76, Prosecka Str., 19000 Praha – 9, Czech Republic,

*e-mail: *kafka@itam.cas.cz*
*

Localization of plastic deformation and of continuum damage: different points of view

Abstract. A parallel description of two approaches to determining localization of plastic deformation is given. Approaches on macrolevel and on mesolevel are considered. The macroscopic approach is based on the rigid plastic model and on bifurcation of the process parameters. The method is applied to metals, taking into account influence of temperature, damage and porosity. The mesoscopic approach is based on a two-component elastic-plastic model. The material properties on mesolevel are derived from the microscopic observations and analyses. The criterion for localization bands initiation is given on mesolevel.

Key words: plastic localization, bifurcation, composite microstructure.

M. Datcheva

Institute of Mechanics, Bulgarian Academy of Sciences

Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria

*e-mail: *datcheva@imbm.bas.bg

Constitutive Modelling of the Creep Behaviour of Chalk

Abstract. Constitutive equations for chalk time dependent behaviour are determined from available experimental data by finding the failure surface and the evolution of irreversible stress work and using the procedure proposed by Cristescu [5] In the present paper the transient creep is only studied, since in the time intervals considered and experimentally investigated it was the only one exhibited by the chalk. The different failure properties under compression and extension are taken into account. The loading function determination general procedure is adopted for conventional tests data.

Key words: creep, viscoplasticity, porous rocks, chalk, modelling.

Al. Zolochevsky, Y. Obataya, T. Itoh

Fukui University, Department of Mechanical Engineering, Faculty of Engineering,

9-1, Bunkyo 3-chome, Fukui, 910-8507, Japan,

e-mail: alex@mech.fukui-u.ac.jp

A COMBINED CRITICAL PLANE AND STRAIN-BASED APPROACH WITH TWO FAMILIES OF MICROCRACKS TO MULTIAXIAL LOW CYCLE FATIGUE DAMAGE

Abstract. This paper proposes a combined equivalent strain and critical plane concept for multiaxial low cycle fatigue damage that considers different physical mechanisms for a shear and tensile types of failure. In general, each potential failure plane (critical plane) contains two families of parallel microcracks. The expression for the equivalent strain in the damage growth equation includes three material parameters as well as the strain intensity for the amplitudes, and joint invariants for the strain amplitudes and for the two unit vectors associated with the directions of microcracks. The determination of material parameters required in the proposed approach, from a series of basic experiments, is shown. Theoretical predictions of multiaxial low cycle fatigue lives are compared with those obtained from fatigue tests using cruciform specimens.

Key words*:* damage accumulation, low cycle fatigue, multiaxial
fatigue, fatigue modelling, critical plane approach.

“An. Kanchev” University of Rousse,

8, Studentska Str., 7017 Rousse, Bulgaria,

*e-mail: *hristo@mech.ru.acad.b*,
*ivelin@mech.ru.acad.bg*
*

**A. Baltov**

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria,

*e-mail: *baltov@eagle.cu.acad.bg

**Mathematical Models of Thin Interlayer between Contacting Solids**

Abstract. A number of works are dedicated to determination of the stress-strain state of solids interacting by sliding friction in the presence of thin layers formed on them. During modelling a linear distribution of displacements along the layer thickness is usually assumed.

A method of assessing applicability of this linear model as well as of a quadratic model is proposed in the present work. The results show the influence on the models errors of two parameters: the relative thickness of the layer and the ratio of mechanical characteristics of layer and solid. They allow a better-grounded choice of one of the models when developing new specialized finite elements for solving the above problems.

Key words: stress analysis, sliding friction, thin layer coatings, interaction of solids with interlayers.

**Ji-Huan He**

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University,

Shanghai 200072, P.R. China,

*e-mail: jhhe@mail.shu.edu.cn*

**Minimum Energy Principles and Generalized Variational Principles of
Nonlinear Piezoelectricity**

Abstract. Variational theory is the base of the finite element methods. By the semi-inverse method proposed by He, Hu-Washizu-like principle and Hellinger-Reissner-like principle of nonlinear piezoelectricity are derived from the field equations and boundary conditions without using the Lagrange multiplier method. By constraining the obtained generalized functionals, two minimal potential functionals are obtained, which reduce to the minimum potential energy principle and the minimum complementary energy principle of elasticity when the effect of electricity is ignored.

Key words: piezoelectricity, semi-inverse method, variational principle, finite element method.