On conservation laws in geometrically nonlinear elasto-dynamic field of non-homogenous materials

By applying Noether’s theorem to the Lagrangian density of non-homogenous elastic materials in the so-called Lagrangian framework, conservation laws in geometrically nonlinear elasto-dynamic field have been studied, and a clear picture of relations between the conservation laws in material space and the material balance laws is given. It is found that the mass density and Lamé’s moduli have to satisfy a set of first-order linear partial differential equations, which contain all the symmetry-transformations of space–time based on Newtonian viewpoint of mechanics. The existence and existent forms of conservation laws in material space are governed by these equations. Especially, translation and rotation of coordinates are symmetry-transformations of the Lagrangian density for obtaining both the conservation laws of homogenous material and the material balance laws of non-homogenous material, but change of coordinate scale is not. However, if the mass density and Lamé’s moduli satisfy special equations simplified from those partial differential equations, change of coordinate scale becomes a symmetry-transformation of the Lagrangian density from which a conservation law follows, whereas the associated material balance law does not exist still. An insight into the usability of those equations for constructing conservation laws is presented, and all the non-trivial conservation laws of the functionally graded material (FGM) layer bonded to a substrate are given for mechanical analysis.     

On conservation laws in generalized dynamic thermoelasticity

The differential equations of generalized dynamic thermoelasticity constitute a coupled non-self-adjoint system of PDEs, which in its variational form does not admit a Lagrangian. However following exterior calculus methodology, we augment the space of dependent variables, accompanying the initial differential equations with suitable adjoint equations and additional auxiliary “fields”. So the construction of a Lagrangian is now possible along with the realization of Noether's theory for finding conservation laws. Finally, in our case, it is possible to characterize the additional dependent variables in terms of the physical fields themselves and obtain then conservation laws expressed via the original known thermoelastic fields.     

Potential systems for PDEs having several conservation laws

A generalization of the usual procedure for constructing potential systems for systems of partial differential equations with multidimensional spaces of conservation laws is considered. More precisely, for the construction of potential systems with a multi-dimensional space of local conservation laws, instead of using only basis conservation laws, their arbitrary linear combinations are used that are inequivalent with respect to the equivalence group of the class of systems or symmetry group of the fixed system. It appears that the basis conservation laws can be equivalent with respect to groups of symmetry or equivalence transformations, or vice versa; in this sense the number of independent linear combinations of conservation laws can be grater than the dimension of the space of conservation laws. The first possibility leads to an unnecessary, often cumbersome, investigation of equivalent systems, the second one makes possible that a great number of inequivalent potential systems are missed. Examples of all these possibilities are given.     

Canonical equations of hydrodynamics of quantum liquids

A method of constructing canonical equations for quantum liquids is proposed. For superfluid ⁴He the forms of the Hamiltonian equations of hydrodynamics and conservation laws are obtained. The kinetic terms in the equations of hydrodynamics are written out. The set of equations obtained is valid up to the point. For anisotropic quantum liquid ³He-A canonical equations both for spin and orbital hydrodynamics are found with the spin and orbital moments taken into account. The forms of the laws of the conservation of momentum, angular momentum, mass, and energy are obtained. The kinetic terms in the equations of hydrodynamics are considered, and both the dissipative and reactive coefficients are classified. A transfer equation for the order parameter near the A transition point is proposed with the dissipative term taken into account.     

Reductions and new exact solutions of the density-dependent Nagumo and Fisher equations

We show that conservation laws and their corresponding multipliers can be used to reformulate systems of partial differential equations and obtain some new classes of solutions. By applying the generalized double-reduction theorem to these systems via associated symmetries, one can construct exact solutions. This procedure is applied to specific cases of the density-dependent Nagumo and Fisher equations for which we obtain new solutions that differ in form from the well known analytic and numerical solutions.     

Conservation laws for the Navier-Stokes equations

A variety of conservation laws for the Navier-Stokes equations are derived. The conserved currents are described in terms of Lie symmetries and of adjoint variables, which give rise to a complementary variational principle for the Navier-Stokes equations.     

ISovector fields and self-similar solutions for power law creep

Isovector methods are a recently developed technique for systematically investigating properties of the solutions of systems of differential equations. These methods are applied to the nonlinear equations of power law creep with elastic strains under conditions of plane and antiplane strain. Among the results are a family of self-similar solutions which are shown to be the only ones extant for the cases investigated. Some light is also shed upon the existence of conservation laws for these equations, and upon the existence of mappings which transform the governing equations into a set of equivalent linear equations.     

Numerical models of steady rolling for non-linear viscoelastic structures in finite deformations

A Lagrangian formulation of constitutive laws for a viscoelastic material based on irreversible thermodynamics is first presented. These laws are expressed by a non-linear differential equation governing the evolution of an internal variable. Then equations describing the steady rolling of an axisymmetric viscoelastic structure are obtained from the conservation laws of continuum mechanics. A finite element approximation and a solution technique of the algebraic system is proposed. The eiimination of the internal variable allows one to keep an elastic-like algorithm with an independent solution of the viscoelastic equation. Numerical tests show the feasibility and the efficiency of the proposed methods in large three-dimensional situations.     

Thermodynamics of irreversible processes and derivation of a system of differential equations for molecular transfer

On the basis of the linear laws of the thermodynamics of irreversible processes and the law of conservation of matter, a system of differential equations is derived for molecular transfer in the presence of n interrelated flows of generalized charges.     

Analysis of a class of nonconservative systems reducible to pseudoconservative ones and their energy relations

This paper analyzes a class of nonconservative systems, whose Lagrangian equations can be reduced to Euler–Lagrangian equations by introducing a new Lagrangian, which is equal to a product of some function of time f(t) and the primary Lagrangian. These equations formally have the same form as for the systems with potential forces, while the influence of nonconservative forces is contained in the factor f(t), and such systems are called pseudoconservative. It is further shown that the requirement for a nonconservative system to be considered as a pseudoconservative is the existence of at least one particular solution of a system of differential equations with unknown function f(t), or their linear combination with suitably chosen multipliers. Further on, the energy relations and corresponding conservation laws of those systems are analyzed from two aspects: directly, on the basis of the corresponding Lagrangian equations and via modified Emmy Noether’s theorem. So, it has been shown, even in two different ways, that there are two types of the integrals of motion, in the form of the product of an exponential factor and the sum of the generalized energy (energy function) and an additional term. For the existence of these integrals of motion, it is necessary and sufficient that there exists at least one particular solution of a partial differential equation, which is in accordance with the Lagrangian equations for the observed problem. The obtained results are equivalent to so-called energy-like conservation laws, obtained via Vujanovi?-Djuki?’s generalized Noether’s theorem for nonconservative systems (Vujanovi? and Jones in: Variational Methods in Nonconservative Phenomena (monograph). Acad. Press, Boston, 1989).     

Implicit dissipative schemes for solving systems of conservation laws

Summary New implicit schemes for solving a system of conservation laws in one space dimension are obtained by using the cubic-spline technique. By making use of certain perturbation terms, these implicit schemes have been transformed to dissipative schemes. The nonlinear instabilities appearing in the solution in the narrow shock region have been damped by applying the automatic switched Shuman-filter method. Four test examples with continuous and discontinuous initial conditions have been solved to illustrate the theory. The proposed method has been extended to solve a system of conservation laws in two space dimensions.     

Derivatives of differential sequences

One way to construct a differential sequence is to take the recursion operator of the corresponding evolution partial differential equation and use it on some suitable seed equation such as the right side of the seed evolution partial differential equation. The differential sequence inherits the appropriate properties of the hierarchy such as integrating factors and an infinite number of conservation laws (rendered somewhat meaningless by the ability to remove derivatives higher than those occurring in the differential equation itself). We consider some variations which lead to sequences of ordinary differential equations with (at least sometimes) interesting properties.     

Application of the Hamilton-Jacobi method to the study of rheo-linear oscillators

Summary In this paper we demonstrate that the well-known Hamilton-Jacobi method can be used in study of the rheo-linear (i.e. time dependent) harmonic oscillator with a single degree of freedom. It will be shown that the quadratic conservation laws (exact invariants) together with corresponding auxiliary equations follow immediately from the complete integral of Hamilton-Jacobi partial differential equation by application of the Jacobi theorem. Attention is also paid to the study of linear conservation laws and to the motion of rheo-linear dynamical systems.     

Amplitude and phase representation of monochromatic fields in physical optics

The conservation equation for a monochromatic field with arbitrary polarization propagating in an inhomogeneous transparent medium is expressed in terms of amplitude and phase variables. The expressions obtained for linearly polarized fields are compared with the results obtained in the eikonal approximation. The electric field wave equation is written in terms of intensity and phase variables. The transport equations for the irradiance and the phase are shown to be particular cases of these derivations. The conservation equation arising from the second-order differential wave equation is shown to be equivalent to that obtained from Poynting's theorem.     

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi‐implicit and explicit 2D and 3D non‐linear transient dynamic analysis of an impact problem and a bulk forming process are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Plane-parallel nonisothermal filtration of a gas: the role of heat transfer

The influence of the parameters of a mathematical model and of the type of boundary conditions on the dynamics of pressure and temperature fields in nonisothermal gas filtration has been investigated in a computational experiment. To describe the process, the nonlinear system of partial differential equations obtained from the mass and energy conservation laws and Darcy law were used, with the physical and caloric equations of state employed as closing relations. The boundary conditions correspond to gas injection at a given mass flow rate of different intensities.     

Energy integrals for the systems with nonholonomic constraints of arbitrary form and origin

This work analyzes energy relations between nonholonomic systems, whose motion is restricted by nonholonomic constraints of arbitrary form and origin. Such constraints can be natural, originating from spontaneous formulation of the problem, or artificial, expressing some program motion in control theory. On the basis of corresponding Lagrange’s equations, a general law of the change in energy d?/dt was formulated for such systems by the help of which it has been shown that here there exist two types of laws of conservation of energy, depending on the structure of work of these reaction forces. Also, the condition for existence of this second type of the law of conservation of energy has been formulated in the form of the system of differential equations. The results obtained are illustrated by a number of examples, with natural nonlinear constraints, as well as with artificial ones that express some program motion.     

Solution of the laminar transport equations in complex geometries by means of curvilinear coordinates

The standard coordinate systems (i.e. cartesian, cylindrical-polar, spherical-polar or combinations theoreof) which are usually employed in the numerical solution of the transport differential equations, are suitable for simple geometries only. In the present study the momentum, mass and energy conservation equations are expressed in terms of general curvilinear-orthogonal coordinates, which enable the irregularly shaped solution domains, encountered in practice, to be completely mapped i.e. all boundaries to coincide with, coordinate surfaces. A mixed coordinate system is also introduced, which, is curvilinear-orthogonal in, the two of the three directions and rectilinear in the third direction. The latter system gives simpler equations and is suitable for straight flow passages of arbitrary cross sectional shape. The momentum, mass and energy conservation, differential, equations are transformed to finite-difference ones by integration over six-sided control volumes formed by coordinate surfaces and are then solved by an iterative procedure. The method is tested successfully in various flow and heat transfer cases.     

Conservation laws for 3-dimensional compressible Euler equations

The generators of infinitesimal symmetry transformations for the Euler equations, and the corresponding set of adjoint variables are derived. The associated conservation laws are then discussed. A detailed analysis of 1-dimensional flows brings into evidence the connections with current alternative approaches to conservation laws.     

On the duality principle of conservation laws in finite elastodynamics

The duality principle of conservation laws which holds in finite elastodynamics is studied using the two-point tensor method. Based on the general Noether's theorem, two basic equations of variational invariance are first derived, which correspond to the action integrals given, respectively, in Lagrangian and Eulerian representations for a finite motion of an elastic body. The dual relations between the conservation laws in both representations are given. The procedure for constructing these dual relations is to apply simultaneously the same infinitesimal transformation of either time or position coordinates as well as field variables to the dual equations of variational invariance, where the position coordinates could be taken either from the reference configuration or from the deformed configuration of the material body. Based on these dual relations it is shown that the conservation equations of material momentum and moment of material momentum possess the same structure as those of physical momentum and physical moment of momentum. Furthermore, three pairs of dual relations between stress tensors and material momentum tensors of various kinds are derived based on the duality principle by using the two-point tensor method. Finally, using the dual integral forms of conservation laws the concepts of dynamic material force and moment acting on defects are introduced and analyzed. The force and moment can be decomposed into a pure kinetic part and a pure deformation part, the latter corresponding to the path-independent integral as suggested in elastostatics.