Conservation laws in elasticity

In this paper the basic results involved in the application of Noether's theorem relating symmetry groups and conservation laws to the variational problems of homogeneous elastostatics are outlined. General methods and conditions for the existence of variational and generalized symmetries are presented. Applications will be considered in subsequent papers in this series.The research reported here was supported in part by the U. S. National Science Foundation, Grant NSF MCS 81-00786.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

The purpose is to reestablish the coupled conservation laws, the local conservation equations and the jump conditions of mass and inertia for polar continuum theories. In this connection the new material derivatives of the deformation gradient, the line element, the surface element and the volume element were derived and the generalized Reynolds transport theorem was presented. Combining, these conservation laws of mass and inertia with the balance laws of momentum, angular momentum and energy derived in our previous papers of this series, a rather complete system of coupled basic laws and principles for polar continuum theories is constituted on the whole. From this system the coupled nonlocal balance equations of mass, inertia, momentum, angular momentum and energy may be obtained by the usual localization. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

Conservation laws in linear viscoelastodynamics

Noether's theorem on variational principles invariant under a group of infinitesimal transformations is used to obtain two conservation laws associated with linear viscoelastodynamics. These laws represent viscoelastic generalizations of two conservation laws in elasticity.     

Conservation laws in linear elastodynamics

Noether's theorem on variational principles invariant under a group of infinitesimal transformations is used to obtain a class of conservation laws associated with linear elastodynamics. These laws represent dynamical generalizations of certain path-independent integrals in elastostatics which have been of considerable recent interest. It is shown that the conservation laws obtained here are the only ones obtainable by Noether's theorem from invariance under a reasonably general group of infinitesimal transformations.     

Conservation laws in non-linear dissipative systems

In Hamiltonian theory, Noether's theorem commonly is used to show the conservation of linear momentum and energy as a consequence of symmetry properties. The possibility of enclosing Hamiltonian theory in a wider context by use of Gibbs-Falkian thermodynamical methods, offering the opportunity to cover mechanical and thermodynamical systems with the same mathematical tools, is considered. Consequently it is shown how Noether's identity can be extended for dissipative systems which are appropriate to describe real life phenomena. By use of the principle of least action an extended version of Noether's theorem is calculated, from which the conservation of linear momentum and total energy can be derived. Additionally, the condition of absolute invariance is shown to be too restrictive for physical applications.     

On consistent plate theories

Summary  Applying the uniform-approximation technique, consistent plate theories of different orders are derived from the basic equations of the three-dimensional linear theory of elasticity. The zeroth-order approximation allows only for rigid-body motions of the plate. The first-order approximation is identical to the classical Poisson-Kirchhoff plate theory, whereas the second-order approximation leads to a Reissner-type theory. The proposed analysis does not require any a priori assumptions regarding the distribution of either displacements or stresses in thickness direction. Received 10 January 2002; accepted for publication 16 April 2002     

Conservation laws in the dynamics of rods

Conservation laws and associated integrals of motion for the dynamics of rods are derived. The classic conservation laws are those of total linear and angular momentum, and, for hyperelastic rods, conservation of energy. It will here be shown that an additional conservation law arises in each of two cases. The first case is that of uniform, hyperelastic rods, the second is that of a class of transversely isotropic rods. AMS(MOS) 73C50, 73K05.The research reported in this paper was partially supported by grants from the US Air Force Office of Scientific Research.     

Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.     

Conservation laws of multidimensional diffusion--convection equations

All possible linearly independent local conservation laws for n-dimensional diffusion–convection equations u _t=(A(u)) _ii +(B ⁱ(u)) _i were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O.~Popovych, N.M. Ivanova, J. Math. Phys. 46, 2005, 043502 (math-ph/0407008)] allows us to formulate the result in explicit closed form. Action of the symmetry groups on the conservation laws of diffusion equations is investigated and generating sets of conservation laws are constructed.     

Conservation laws of an electro-active polymer

Ionic electro-active polymer is an active material consisting in a polyelectrolyte (for example Nafion). Such material is usually used as thin film sandwiched between two platinum electrodes. The polymer undergoes large bending motions when an electric field is applied across the thickness. Conversely, a voltage can be detected between both electrodes when the polymer is suddenly bent. The solvent-saturated polymer is fully dissociated, releasing cations of small size. We used a continuous medium approach. The material is modelled by the coexistence of two phases; it can be considered as a porous medium where the deformable solid phase is the polymer backbone with fixed anions; the electrolyte phase is made of a solvent (usually water) with free cations. The microscale conservation laws of mass, linear momentum and energy and the Maxwell’s equations are first written for each phase. The physical quantities linked to the interfaces are deduced. The use of an average technique applied to the two-phase medium finally leads to an Eulerian formulation of the conservation laws of the complete material. Macroscale equations relative to each phase provide exchanges through the interfaces. An analysis of the balance equations of kinetic, potential and internal energy highlights the phenomena responsible of the conversion of one kind of energy into another, especially the dissipative ones : viscous frictions and Joule effect.     

Conservation laws of two coupled non-linear oscillators

This paper presents a novel approach to obtaining a complete set of time-dependent expressions for approximate conservation laws of two weakly non-linear coupled oscillators. The procedure developed for a non-resonant case is based on the field method concept of deriving a conservation law from an incomplete solution of a partial differential equation. Due to the non-linearity of the system being considered, this concept is combined with the multiple variable expansion procedure.     

Conservation laws of nonholonomic nonconservative dynamical systems

In this paper conservation laws of nonholonomic nonconservative dynamical systems are studied by using the differential variational principles of Jourdain and the generalized Noether's identities of nonconservative systems subject to first order nonlinear nonholonomic constraints are provided. The project is supported by the National Natural Science Foundation of China.     

Conservation laws in linear elasticity based upon divergence transformations

The existence of conservation laws in linear elasticity based upon divergence transformations of the Lagrangian density function is investigated. It is found that there exist a set of conservation laws which correspond to infinitesimal homogeneous perturbations of the strain and velocity fields. These conservation laws have a unique feature not shared by other conservation laws in linear elasticity in that they contain an arbitrary free parameter.     

Conservation of charge and second-order gauge-tensor field theories

In a space of four-dimensions I determine all possible second-order gauge-tensor field equations which are derivable from a variational principle, compatible with the notion of conservation of charge, and in agreement with the Yang-Mills field equations in flat space. The Lagrangian which yields these fields equations differs from the Lagrangian of the Einstein-Yang-Mills field theory (with cosmological term included) by one term.     

ANALYTICAL RELATIONS BETWEEN EIGENVALUES OF CIRCULAR PLATE BASED ON VARIOUS PLATE THEORIES

Based on the mathematical similarity of the axisymmetric eigenvalue problems of a circular plate between the classical plate theory(CPT), the first-order shear deformation plate theory(FPT) and the Reddy's third-order shear deformation plate theory (RPT), analytical relations between the eigenvalues of circular plate based on various plate theories are investigated. In the present paper, the eigenvalue problem is transformed to solve an algebra equation. Analytical relationships that are expressed explicitly between various theories are presented. Therefore, from these relationships one can easily obtain the exact RPT and FPT solutions of critical buckling load and natural frequency for a circular plate with CPT solutions. The relationships are useful for engineering application, and can be used to check the validity, convergence and accuracy of numerical results for the eigenvalue problem of plates.     

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids

Conservation laws and constitutive relations for a density-gradient-dependent viscous fluid as a multipolar continuum obeying an entropy inequality with generalized entropy flux and supply density are considered in this paper. A decomposition of the rate of work of dipolar stress, which reveals the contribution of various parts of this stress to the energy equation, is used to discuss the inconsistencies between the results obtained here and those obtained by Bluestein and Green [1] on the basis of the pioneering work of Green and Rivlin [8]. Furthermore, we discuss the connection between the model presented here and the materials of Korteweg type considered by Dunn and Serrin [6]. In particular, we relate the rate of work of dipolar stress and the interstitial working introduced by Dunn and Serrin [6].Received: 3 April 2004, Accepted: 6 December 2004, Published online: 4 March 2005 Correspondence to: M.M. Mehrabadi