Vassil M. Vassilev

Abstract: The parametric equations of the plane curves determining the equilibrium shapes that a uniform inextensible elastic ring or tube could take subject to a uniform hydrostatic pressure are presented in an explicit analytic form. The determination of the equilibrium shape of such a structure corresponding to a given pressure is reduced to the solution of two transcendental equations. The shapes with points of contact and the corresponding (contact) pressures are determined by the solutions of three transcendental equations. The analytic results presented here confirm many of the previous numerical results on this subject but the results concerning the shapes with lines of contact reported up to now are revised. (C) 2011 Elsevier Ltd. All rights reserved.

Abstract: The Sturm spirals which can be introduced as those plane curves whose curvature radius is equal to the distance from the origin are embedded into one-parameter family of curves. Explicit parametrization of the ordinary Sturmian spirals along with that of a wider family of curves are found and depicted graphically.

Abstract: The junctions of carbon nanotubes and flat graphene sheets or between nanotubes are structures with various potential applications in building nano- and micro-electromechanical systems. Here we report the results of a continuum approach to the determination of the possible shapes of such junctions based on the continuum limit of the Lenosky potential. For various geometric parameters of the tubes and sheets families of surfaces that are local extrema of the bending energy functional are obtained numerically. Each such surface may be thought of as a possible junction between a carbon nanotube and a flat graphene sheet or between co-axial carbon nanotubes.

Abstract: The dynamic stability of a cantilevered Timoshenko beam lying on ail elastic foundation of Winkler type and subjected to a tangential follower force is studied. Two models describing this phenomenon are examined and their predictions are compared in several special cases. For the values of the beam parameters considered here, the critical compressive forces obtained using these models differ substantially only for short beams as has already been established in other cases. Both models are found to predict dynamic instability of cantilevers tinder tension unlike the Bernoulli-Euler beam theory. For a beam of intermediate slenderness the Winkler foundation is found to reduce the critical tensile force. (c) 2007 Elsevier Ltd. All rights reserved.

Abstract: Within the framework of the well-known curvature models, a fluid lipid bilayer membrane is regarded as a surface embedded in the three-dimensional Euclidean space whose equilibrium shapes are described in terms of its mean and Gaussian curvatures by the so-called membrane shape equation. In the present paper, all solutions to this equation determining cylindrical membrane shapes are found and presented, together with the expressions for the corresponding position vectors, in explicit analytic form. The necessary and sufficient conditions for such a surface to be closed are derived and several sufficient conditions for its directrix to be simple or self-intersecting are given.

Abstract: The dynamic stability of straight cantilevered viscoelastic pipes conveying inviscid fluid and lying on an elastic foundation of variable modulus is studied. The corresponding eigenvalue problem is solved using both Galerkin and shooting methods. It is found that certain combinations of the pipe parameters (the elastic foundation modulus, mass ratio and internal damping coefficient) can destabilize the pipe. (c) 2006 Elsevier Ltd. All rights reserved.

Abstract: In the present paper, a class of partial differential equations governing various rod and plate theories of Bernoulli-Euler and Poisson-Kirchhoff type is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived and the general statement of the associated group-classification problem is given. A simple relation is deduced allowing to recognize easily the variational symmetries among the �ordinary� symmetries of a self-adjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Bessel-Hagen�s extension of Noether�s theorem) conservation laws are suggested. Solutions of group-classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods, fluid conveying pipes and plates resting on variable elastic foundations. The obtained group-classification results are used to derive conservation laws and group-invariant solutions readily applicable in rod dynamics and plate statics and dynamics. New generalized symmetries and conservation laws for the theories of Timoshenko beams, Reissner-Mindlin plates and three-dimensional elastostatics are presented. (C) 2002 Elsevier Science Ltd. All rights reserved.

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Abstract: The present paper is concerned with the classical von Karman equations for large deflection of thin elastic plates both in 2 (space) and in 2+1 (space-time) dimensions. All infinitesimal divergence symmetries of two action functionals whose Euler-Lagrange equations coincide with the von Karman equations are established and the corresponding conservation laws obtainable through Noether�s theorem are determined. Wave phenomena in infinite thin elastic plates are studied using group-invariant solutions to the time-dependent von Karman equations, the derived conservation laws being taken into consideration.

Abstract: The parametric equations of the plane curves determining the equilibrium shapes that a uniform inextensible elastic ring could take subject to a uniform hydrostatic pressure are presented in an explicit analytic form. The determination of the equilibrium shape of such a structure corresponding to a given pressure is reduced to the solution of two transcendental equations. The shapes with points of contact and the corresponding (contact) pressures are determined by the solutions of three transcendental equations. The analytical results presented here confirm many of the previous numerical results on this subject but the results concerning the shapes with lines of contact reported up to now are revised.

Abstract: The work is concerned with the determination of explicit parametric equations of several plane curves whose curvature depends solely on the distance from the origin. Here we suggest and exemplify a simple scheme for reconstruction of a plane curve if its curvature belongs to the above-mentioned class. Explicit parameterizations of generalized Cassinian ovals including also the trajectories of a charged particle in the field of a magnetic dipole are derived in terms of Jacobian elliptic functions and elliptic integrals.

Abstract: The work is concerned with the determination of the mechanical behaviour of cell membranes under uniform hydrostatic pressure subject to micro-injections. For that purpose, assuming that the shape of the deformed cell membrane is axisymmetric a variational statement of the problem is developed on the ground of the so-called spontaneous curvature model. In this setting, the cell membrane is regarded as an axisymmetric surface in the three-dimensional Euclidean space providing a stationary value of the shape energy functional under the constraint of fixed total area and fixed enclosed volume. The corresponding Euler-Lagrange equations and natural boundary conditions are derived, analyzed and used to express the forces and moments in the membrane. Several examples of such surfaces representing possible shapes of cell membranes under pressure subjected to micro injection are determined numerically.

Abstract: The consideration of the variational problem for minimization of the bending energy of membrane surfaces under some natural geometric constraints leads to the so called general shape equation. The available analytical solutions of this equation including a new one are reviewed in some detail.

Abstract: Here we suggest and have exemplified a simple scheme for reconstruction of a plane curve if its curvature belongs to the class specified in the title by deriving explicit parametrization of Bernoulli�s lemniscate and newly introduced co-lemniscate curve in terms of the Jacobian elliptic functions. The relation between them and with the Bernoulli elastica are clarified.

Abstract: Starting from first principles, the curvature of the plane elastic line is expressed in terms of the slope of its tangent. This is used further for finding its intrinsic equation and for deriving a new explicit parametrization of the Euler elastica.

Abstract: A class of differential equations governing the dynamic stability of fluid conveying pipes is studied by Lie group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived and the general statement of the associated group-classification problem is given. A simple relation is deduced allowing to recognize easily the Variational symmetries, and explicit formulae for the conserved currents of the corresponding (via Noether�s theorem) conservation laws are suggested. Solutions of group-classification problems are presented for differential equations of the foregoing type governing also vibrations of rods resting on a variable elastic foundation and rods compressed by axial forces. The obtained group-classification results are used to derive collections of conservation laws readily applicable in the dynamics of pipes and rods.