Constraints as models of bodies possessing non-smooth constitutive characteristics (original) (raw)

On Mathematical Aspects of Dual Variables in Continuum Mechanics. Part 1: Mathematical Principles

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1996

STEIN on the occasion of his 65th birthday In diesem: aus zwei Teilen bestehendem Aufsatt werden mathematische Gesichtspunkte won dualen Variablen behandelt? die in der Kontinuumsmechanik auftreten. Dabei wird der Tensorkalkiil auf Mannigfaltigkeiten venuendet. wie er iion l l. 4~s n~s und HUGHES [l] in die Kontinuumsmechanik eingefuhr? wurde. Dieser mathematische Formalismus fihrt Z I L zusatzlicher Stmktur in kontinuumsmechanischen Theorien. Insbesondere ergibt die Invarianz bestimmter Bilinearformen eindeutige Transformationsreyeln f i r Tensoren zwischen der Referenz-und der Momentankonfiguration. Diese Transformationsregeln werden durch die push-forward-bzw. pull-back-Operatoren festgelegt.-In Teil 1 stellen wir die mathemataschen Grundlagen unseres Vorgehens vor. Ein wesentlicher Aspekt besteht darin, sor,qfiltig zwischen inneren und skalaren Produkten zii unterscheiden. Diese Unterscheidung wird physikalisch motiviert und mathematzsch formuliert. Innere Produkte kiinnen nur f'iir solche Objekte gebildet werden, die in ein und demselben Vektorraum leben. Dageyen werden Skalarprodukte aus Objekten gebildet, die in verschiedenen Vektorriiumen leben. Die Un.tersch,eidung von inneren und skalaren Produkten fihrt zu einer Unterscheidung zwischen transponierten und dualen Tensoren. Entsprechend wird zwischen Symmetrie und Selbstdualitiit unterschieden. Ein wichtiges Ergebnis der Untersuchungen sand neue Beziahungen fur die Berechnung des push-forwards bzw. pull-backs won Tensoren zweiter Stufe. Sie werden, aiL,s Inrinriantforderungen fur bestimmte innere bzw. skalare Produkte hergeleitet. I m Gegensatz t u den aus der Literatur bekannv ten Beziehungen bleibt bei diesen Formeln die Symmetrie gemischter Tensoren erhalten. In this paper consisting of two parts we consider mathematical aspects of dual variables appearing in continuum mechanics. Tensor calculus on manifolds as introduced into continuum mechanics by MARSDEN and HWHES [l] is wed as u point of departure. This mathematical formalism leads to additional structure of continuum mechanical theories. Specifically iniiariance of certain bilinear forms renders unambiguous transformation rules for tensors between the reference and the current configuration. These transformation rules are determined by push-forwards and pull-backs, respectzaely.-In Part 1 we consider the basic mathematical features of our theory. The key aspect of our approach is that, contrary to the usual considerations in this field, we distinguish carefully between inner products and scalar products. This discrrmination is motivated by physical considerations and is subsequently given a firm mathematical basis. Inner products can only be formed with objects living in one and the same vector space. Scalar products, on the other hand, are formed between objects living in different spaces. The distinction, between inner and scalar products leads to a distinction between transposes and duals of tensors. Therefore, we distinguish between symmetry and se2f-duality. An, im'portant result of this approach are new formulae for the computation of push-forwa.rds and pull-backs, respectively, of second-order tensors, which are derived from invariance requirements of inner and scalar products, respectively. In contrast to prior approaches these new formulae preserve symmetry of symmetric mixed tensors.

On the Representation of Energy and Momentum in Elasticity

Mathematical Models and Methods in Applied Sciences, 2000

In order to clarify common assumptions on the form of energy and momentum in elasticity, a generalized conservation format is proposed for finite elasticity, in which total energy and momentum are not specified a priori. Velocity, stress, and total energy are assumed to depend constitutively on deformation gradient and momentum in a manner restricted by a dissipation principle and certain mild invariance requirements. Under these assumptions, representations are obtained for energy and momentum, demonstrating that (i) the total energy splits into separate internal and kinetic contributions, and (ii) the momentum is linear in the velocity. It is further shown that, if the stress response is strongly elliptic, the classical specifications for kinetic energy and momentum are sufficient to give elasticity the standard format of a quasilinear hyperbolic system.

The Scalar, Vector and Tensor Fields in Theory of Elasticity and Plasticity

Transactions of the VŠB - Technical University of Ostrava, Mechanical Series

This article is devoted to an analysis of scalar, vector and tensor fields, which occur in the loaded and deformed bodies. The aim of this article is to clarify and simplify the creation of an understandable idea of some elementary concepts and quantities in field theories, such as, for example equiscalar levels, scalar field gradient, Hamilton operator, divergence, rotation and gradient of vector or tensor and others. Applications of those mathematical terms are shown in simple elasticity and plasticity tasks. We hope that content of our article might help technicians to make their studies of necessary mathematical chapters of vector and tensor analysis and field theories easier.

A Primer in Elasticity

Journal of Elasticity - J ELAST, 2000

Let E be a 3-dimensional Euclidean space, and let V be the vector space associated with E. We distinguish a point p ∈ E both from its position vector p(p) := (p − o) ∈ V with respect to a chosen origin o ∈ E and from any triplet (ξ 1 , ξ 2 , ξ 3 ) ∈ IR 3 of coordinates that we may use to label p. Moreover, we endow V with the usual inner product structure, and orient it in one of the two possible manners. It then makes sense to consider the inner product a · b and the cross product a × b of two elements a, b ∈ V; in particular, we define the length of a vector a to be |a| = (a · a) 1/2 , and denote by U := {v ∈ V | |v| = 1} the sphere of all vectors having unit length. When needed or simply convenient, we think of E as equipped with a Cartesian frame {o; c 1 , c 2 , c 3 } with orthogonal basis vectors c i ∈ U (i = 1, 2, 3); the Cartesian components of a vector v ∈ V are then v i := v · c i and, in particular, the triplet (p 1 , p 2 , p 3 ) ∈ IR 3 , p i := p(p) · c i , of components of the position vector are the Cartesian coordinates of a point p ∈ E.

Mechanics of deformations in terms of scalar variables

Continuum Mechanics and Thermodynamics, 2017

Theory of particle and continuous mechanics is developed which allows a treatment of pure deformation in terms of the set of variables "coordinate-momentum-force" instead of the standard treatment in terms of tensor-valued variables "strain-stress." This approach is quite natural for a microscopic description of atomic system, according to which only pointwise forces caused by the stress act to atoms making a body deform. The new concept starts from affine transformation of spatial to material coordinates in terms of the stretch tensor or its analogs. Thus, three principal stretches and three angles related to their orientation form a set of six scalar variables to describe deformation. Instead of volume-dependent potential used in the standard theory, which requires conditions of equilibrium for surface and body forces acting to a volume element, a potential dependent on scalar variables is introduced. A consistent introduction of generalized force associated with this potential becomes possible if a deformed body is considered to be confined on the surface of torus having six genuine dimensions. Strain, constitutive equations and other fundamental laws of the continuum and particle mechanics may be neatly rewritten in terms of scalar variables. Giving a new presentation for finite deformation new approach provides a full treatment of hyperelasticity including anisotropic case. Derived equations of motion generate a new kind of thermodynamical ensemble in terms of constant tension forces. In this ensemble, six internal deformation forces proportional to the components of Irving-Kirkwood stress are controlled by applied external forces. In thermodynamical limit, instead of the pressure and volume as state variables, this ensemble employs deformation force measured in kelvin unit and stretch ratio. Keywords Mechanics on torus • Finite deformation as scalar • Constitutive equations • Hyperelastisity • Isotension ensemble Communicated by Andreas Öchsner.