Physical relations for an orthotropic material in the local strains theory

Compressible and incompressible orthotropic materials are investigated. A method of constructing variants of the local strains theory describing the deformation properties of an orthotropic material is developed on the basis of the concept of an elastic potential.Institute of Polymer Mechanics, AS LatSSR, Riga. Translated from Mekhanika Polimerov, Vol. 4. No. 6, pp. 970–975, November–December, 1968.     

The duality theory of a finite dimensional discrete quantum group

Suppose that is a finite dimensional discrete quantum group and is a Hilbert space. This paper shows that if there exists an action of on so that is a modular algebra and the inner product on is -invariant, then there is a unique C*-representation of on supplemented by the The commutant of in is exactly the -invariant subalgebra of . As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

     

Symmetry analysis for uniaxial compression of a hypoplastic granular material

A variety of modelling approaches currently exist to describe and predict the diverse behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or “optimal system” of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically. Received: February 3, 2004; revised: June 2, 2004     

Symmetry analysis for uniaxial compression of a hypoplastic granular material

A variety of modelling approaches currently exist to describe and predict the diverse behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or “optimal system” of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically.     

Oscillation theory at a finite singularity

It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).     

On consistent tangent operator derivation and comparative study of rubber-like material models at finite strains

Rubber-like materials can deform largely and non-linearly upon loading and return to the initial configuration when the load is removed. Such rubber elasticity is achieved due to very flexible long chain molecules and a three-dimensional network structure that is formed via cross-linking or entanglements between molecules. Over the years, to model the mechanical behaviour of such randomly-oriented micro-structure, several phenomenological and micro-mechanically motivated network models for nearly incompressible hyperelastic polymeric materials have been proposed in the literature. For comparison of all selected models in reproducing the well-known Treloar data, the analytical expressions for the three homogeneous defomation modes, i.e. uniaxial tension, equibiaxial tension and pure shear have been derived and the performances of the models are analysed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element method for granular flow through a frictional boundary

A finite element method for the flow of dry granular solids through a domain involving a frictional contact boundary is formulated. The granular material is assumed as a compressible viscous-elastic–plastic continuum. Based on the principles of continuum mechanics, a complete set of equations is developed. The resulting boundary value problem is solved by the finite element method in space and by the finite difference method in time. The derivation of the finite element equations and the mathematical framework of the numerical technique are presented, together with two illustrative examples to demonstrate the validity of the technique.     

Riemann-Roch and Abel-Jacobi theory on a finite graph

It is well known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.     

The construction of a model of the growth of a defect for finite strains: Non-local criteria

A version of a model of the growth of a defect (an inclusion or cracks of non-zero opening) is presented. When constructing the model, the axiomatic of the mechanics of a deformable solid is used, as well as the idea of a zone of pre-existing imperfection as part of the body, in which, due to the action of external loads, applied to the body, a change in the properties of the material occurs. The apparatus of the theory of multiple superposition of large deformations and certain strength criteria are used.     

The foundations of enumeration theory for finite nilpotent groups

This paper is the first in a series of papers that lay the foundations of enumeration theory for finite groups including the classical inversion calculus on segments of the natural series and on lattices of subsets of finite sets. Since it became possible to calculate the Möbius function on all subgroups of finite nilpotent groups, the Möbius inversion on these groups began to play a decisive role. The efficiency of the inversion method as a regular technique suitable for solution of enumeration problems of group theory is illustrated with a number of concrete and very important enumerations. Bibliography: 13 titles.     

Application of a Tauberian theorem to finite model theory

An extension of a Tauberian theorem of Hardy and Littlewood is proved. It is used to show that, for classes of finite models satisfying certain combinatorial and growth properties, Cesàro probabilities (limits of average probabilities over second order sentences) exist. Examples of such classes include the class of unary functions and the class of partial unary functions. It is conjectured that the result holds for the usual notion of asymptotic probability as well as Cesàro probability. Evidence in support of the conjecture is presented.     

Adaptive finite element procedures for elastoplastic problems at finite strains

A major difficulty in the context of adaptive analysis of geometrically nonlinear problems is to provide a robust remeshing procedure that accounts both for the error caused by the spatial discretization and for the error due to the time discretization. For stability problems, such as strain localization and necking, it is essential to provide a step–size control in order to get a robust algorithm for the solution of the boundary value problem. For this purpose we developed an easy to implement step–size control algorithm. In addition we will consider possible a posteriori error indicators for the spatial error distribution of elastoplastic problems at finite strains. This indicator is adopted for a density–function–based adaptive remeshing procedure. Both error indicators are combined for the adaptive analysis in time and space. The performance of the proposed method is documented by means of representative numerical examples.