Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media

In the present paper we investigate the mechanics of systems of affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry. Certain physical applications are possible in modelling of molecular crystals, granular media, and other physical objects. Particularly interesting are dynamical models invariant under the group underlying geometry of degrees of freedom. In contrary to the single body case there exist nontrivial potentials invariant under this group (left and right acting). The concept of relative (mutual) deformation tensors of pairs of affine bodies is discussed. Scalar invariants built of such tensors are constructed. There is an essential novelty in comparison to deformation scalars of single affine bodies, i.e., there exist affinely-invariant scalars of mutual deformations. Hence, the hierarchy of interaction models according to their invariance group, from Euclidean to affine ones, can be considered.Comment: 50 pages, 4 figure

A symmetry classification for a class of (2+1)-nonlinear wave equation

In this paper, a symmetry classification of a $(2+1)$-nonlinear wave equation $u_{tt}-f(u)(u_{xx}+u_{yy})=0$ where $f(u)$ is a smooth function on $u$, using Lie group method, is given. The basic infinitesimal method for calculating symmetry groups is presented, and used to determine the general symmetry group of this $(2+1)$-nonlinear wave equation

Thermoelastic Damping in Micro- and Nano-Mechanical Systems

The importance of thermoelastic damping as a fundamental dissipation mechanism for small-scale mechanical resonators is evaluated in light of recent efforts to design high-Q micrometer- and nanometer-scale electro-mechanical systems (MEMS and NEMS). The equations of linear thermoelasticity are used to give a simple derivation for thermoelastic damping of small flexural vibrations in thin beams. It is shown that Zener's well-known approximation by a Lorentzian with a single thermal relaxation time slightly deviates from the exact expression.Comment: 10 pages. Submitted to Phys. Rev.

On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models

Higher order gradient continuum theories have often been proposed as models for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent comparison of the solution to a boundary value problem using both the exact microscopic model and the corresponding approximate higher order gradient macroscopic model.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42682/1/10659_2004_Article_BF00043251.pd

19.—Propagation of Weak Discontinuities in a Layered Hyperelastic Half-space

SYNOPSISThis paper investigates the one-dimensional propagation of weak discontinuities, that is acceleration waves, in a homogeneous and isotropic half-space composed of an arbitrary number of non-linearly hyperelastic layers. The transmission and reflection coefficients are evaluated in terms of the initial condition at the boundary, and the steepening of the waves to form a shock is discussed. The results are specialised to the case of periodic layering.</jats:p

Possible configurations for Weiss domains in uniaxial ferroelectric crystals

The variational principle proposed in another paper of ours [Internat. J. Engrg. Sci. 30 (1992), no. 12, 1715--1729; MR1185410 (93h:82078)] is here applied to derive the configuration of Weiss domains in uniaxial crystals. It is proved that the configuration which was proposed by L. Landau and E. Lifshits [Phys. J. Sowjet. 8 (1935), no. 2, 153--169] is not possible even in the presence of an electric field. Moreover, another possible configuration is proposed in the absence of an electric field