Conservation and Balance Laws in Linear Elasticity of Grade Three

In the present paper we investigate conservation and balance laws in the framework of linear elastodynamics considering the strain energy density depending on the gradients of the displacement up to the third order, as originally proposed by Mindlin (Int. J. Solids Struct. 1, 417–438, 1965). The conservation and balance laws that correspond to the symmetries of translation, rotation, scaling and addition of solutions are derived using Noether’s theorem. Also, the formulas of the dynamical J,L and M-integrals are presented for the problem under study. Moreover, the balance law of addition of solutions gives rise to explore the dynamical reciprocal theorem as well as the restrictions under which it is valid.      

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

Complete Integrability of Shock Clustering and Burgers Turbulence

We consider scalar conservation laws with convex flux and random initial data. The Hopf–Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group; (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for N-dimensional approximations of the Lax equations, and yield formulas for the limit N → ∞. The main observation is that the Lax equations and zero-curvature equations are a Markovian analog of known integrable systems (geodesic flow on Lie groups and the N-wave model respectively). This allows us to introduce a variety of methods from the theory of integrable systems.     

Complete systems of conservation laws for two-layer shallow water models

A well-posedness criterion for a complete system of conservation laws is proposed that assumes maximum compatibility of the convexity domain of the closing conservation law with the domain of hyperbolicity of the model used. This criterion is used to obtain well-posed complete systems of conservation laws for the models of two-layer shallow water with a free-surface (model I) and with a rigid lid (model II). Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 23–32, September–October, 1999.     

Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives

In this paper, we present a new kind of fractional dynamical equations, i.e., the fractional generalized Hamiltonian equations in terms of combined Riesz derivatives, and it is proved that the fractional generalized Hamiltonian system possesses consistent algebraic structure and Lie algebraic structure, and the Poisson conservation law of the fractional generalized Hamiltonian system is investigated. Then the conditions, which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given. Further, the conserved quantities of a fractional dynamical system are given to illustrate the method and results of the application. At last, a new fractional Volterra model of the three species groups is presented and its conserved quantities are obtained, by using the method of this paper.     

Multibody Formulation with Large Bending Finite Elements (LBFE)

The goal of this paper is to present a flexible multibody formulation for Euler-Bernoulli beams involving large displacements. This method is based on a discretisation of internal and kinetic energies. The beam is represented by its line of centroids and each section is oriented by a frame defined by three Euler angles. We apply a finite element formulation to describe the evolution of these angles along the neutral fibre and describe the internal energy. The kinetic energy is approximated as the one of two rigid bars tangent to the neutral fibre at the ends of the beam. We derive the equations of motion from a Lagrange formulation. These equations are solved using the Newmark method or/and the Newton-Raphson technique. We solve some very classic problems taken from the literature as the curved beam presented by Simo [Simo, J. C., ‘A three-dimensional finite-strain rod model. the three-dimensional dynamic problem. Part I’, Comput. Meths. Appl. Mech. Engrg. 49, 1985, 55–70; Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model, Part II: Computationals aspects’, Comput. Meths. Appl. Mech. Engrg. 58, 1988, 79–116] and Lee [Lee, Kisu, ‘Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors’, Commun. Numer. Meth. Engrg. 13, 1997, 987–997] or the rotational rod presented by Avello [Avello, A., Garcia de Jalon, J., and Bayo, E., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’, Int. J. Num. Methods in Engineering 32, 1991, 1543–1563] and Simo [Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part I’ Jour. of Appl. Mech. 53, 1986, 849–854; Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part II’, Jour. of Appl. Mech. 53, 1986, 855–863].     

Theorem on perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications

We study the problem of perturbations of quasiperiodic motions on coisotropic invariant tori in a class of locally Hamiltonian systems. We prove a general KAM-theorem on the perturbation of coisotropic invariant tori for locally Hamiltonian systems. As applications of this theorem, we consider the motion of an electron on a two-dimensional torus under the action of an electromagnetic field and extend results concerning the bifurcation of a Cantor set of coisotropic invariant tori to the case of locally Hamiltonian systems. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 490–515, October–December, 2005.     

Elastodynamics and Elastostatics by a Unified Method of Potentials for x 3-Convex Domains

A new general solution in terms of two scalar potential functions for classical elastodynamics of x ₃-convex domains is presented. Through the establishment and usage of a set of basic mathematical lemmas, a demonstration of its connection to Kovalevshi–Iacovache–Somigliana elastodynamic solution, and thus its completeness, is realized with the aid of the theory of repeated wave equations and Boggio’s theorem. With the time dependence of the potentials suppressed, the new decomposition can, unlike Lamé’s, degenerate to a complete solution for elastostatic problems.      

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.      

Fractal First-Order Partial Differential Equations

The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton–Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton–Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton–Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.     

On Time Reversible Description of the Process of Coagulation and Fragmentation

The process of coagulation is associated with scalar conservation laws, where the adhesion particle dynamics results from shock waves. Conversely, the fragmentation of a massive particle into a number of smaller ones, or into a continuous (dust) distribution, is associated with rarefaction waves. It is generally agreed that a reversible solution of a conservation law can include neither shock waves nor the spontaneous emergence of rarefaction waves. The present paper is an attempt to demonstrate that both coagulation and fragmentation may coexist for a reversible solution, under a natural generalization of the system of conservation law. This is done by introducing an action principle which includes, in addition to the inertial (kinetic energy) term, also an appropriately defined internal energy. The above generalization of the system of conservation law appears as the Euler–Lagrange equations for this action.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Formulation of Multidimensional Scalar Conservation Laws

We show that Kruzhkov’s theory of entropy solutions to multidimensional scalar conservation laws (Kruzhkov in Mat Sb (N.S.), 81(123), 228–255, 1970) can be entirely recast in L ² and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Brenier (in C R Acad Sci Paris Ser I Math, 292, 563–566, 1981; in J Diff Equ, 50, 375–390, 1983; in SIAM J Numer Anal, 21, 1013–1037; in Methods Appl Anal, 11, 515–532, 2004), Giga and Miyakawa (in Duke Math J, 50, 505–515, 1983), and Tsai et al. (in Math Comp, 72, 159–181, 2003).     

A Material Momentum Balance Law for Rods

A material momentum balance law is presented in this paper where it is also specialized for a variety of rod and string theories. The local form of the law is assumed to be identically satisfied, while the jump condition provides an extra equation which is often needed to solve problems involving the application of rod and string theories. The balance law is also related to several existing conservation laws for strings and rods, including Kelvin’s circulation theorem. A novel identity for the singular sources at a discontinuity is also established. Dedicated to James N. Flavin, my friend and mentor, on the occasion of his 70th Birthday.     

On the instability of steady motion

This paper deals with the instability of steady motions of conservative mechanical systems with cyclic coordinates. The following are applied: Kozlov’s generalization of the first Lyapunov’s method, as well as Rout’s method of ignoration of cyclic coordinates. Having obtained through analysis the Maclaurin’s series for the coefficients of the metric tensor, a theorem on instability is formulated which, together with the theorem formulated in Furta (J. Appl. Math. Mech. 50(6):938–944, 1986), contributes to solving the problem of inversion of the Lagrange-Dirichlet theorem for steady motions. The cases in which truncated equations involve the gyroscopic forces are solved, too. The algebraic equations resulting from Kozlov’s generalizations of the first Lyapunov’s method are formulated in a form including one variable less than was the case in existing literature.     

Frictional passage of fluid through inhomogeneous porous system: a variational principle

A Fermat-like principle of minimum time is formulated for nonlinear steady paths of fluid flow in inhomogeneous isotropic porous media where fluid streamlines are curved by a location dependent hydraulic conductivity. The principle describes an optimal nature of nonlinear paths in steady Darcy’s flows of fluids. An expression for the total path resistance leads to a basic analytical formula for an optimal shape of a steady trajectory. In the physical space an optimal curved path ensures the maximum flux or shortest transition time of the fluid through the porous medium. A sort of “law of bending” holds for the frictional fluid flux in Lagrange coordinates. This law shows that—by minimizing the total resistance—a ray spanned between two given points takes the shape assuring that a relatively large part of it resides in the region of lower flow resistance (a ‘rarer’ region of the medium).     

Anisotropy of elastic properties of materials

Papers dealing with the generalized Hooke’s law for linearly elastic anisotropic media are reviewed. The papers considered are based on Kelvin’s approach disclosing the structure of the generalized Hooke’s law, which is determined by six eigenmoduli of elasticity and six orthogonal eigenstates. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 131–151, November–December, 2008.     

Nonlinear gas oscillations in a closed tube driven by the aperiodic motion of a piston

Nonlinear gas oscillations in a closed tube driven by the aperiodic motions of a piston as a result of the action of the external and internal pressure drop are studied. The external pressure takes two values alternating at the moment of change of direction of motion of the piston. Two models of the motion of the gas are considered. Model 1 is formed by a system of equations representing the mass, momentum, and entropy conservation laws. As distinct from model 1, model 2 includes the total energy conservation law in place of the entropy conservation laws. Kazan’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 134–142, March–April, 1998. The work was carried out with partial support from the Russian Foundation for Fundamental Research (project No. 96-01-00484).     

On positive periodic solutions of nonlinear impulsive functional differential equations

Using the Krasnosel’skii theorem on a fixed point of a mapping in a cone, we obtain conditions for the existence of positive, piecewise-smooth, periodic solutions of impulsive functional differential equations. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 501–511, October–December, 2008.     

Variational principles and conservation laws to the Burridge–Knopoff equation

We generate conservation laws for the Burridge–Knopoff equation which model nonlinear dynamics of earthquake faults by a new conservation theorem proposed recently by Ibragimov. One can employ this new general theorem for every differential equation (or systems) and derive new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to the Burridge–Knopoff equation.