Lodge Body Tensor Fields in Continuum Mechanics

With Applications to Polymer Rheology
1. Auflage 2014
ISBN: 978-1-4832-6299-4
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 336 Seiten

Body Tensor Fields in Continuum Mechanics: With Applications to Polymer Rheology aims to define body tensor fields and to show how they can be used to advantage in continuum mechanics, which has hitherto been treated with space tensor fields. General tensor analysis is developed from first principles, using a novel approach that also lays the foundations for other applications, e.g., to differential geometry and relativity theory. The applications given lie in the field of polymer rheology, treated on the macroscopic level, in which relations between stress and finite-strain histories are of central interest. The book begins with a review of mathematical prerequisites, namely primitive concepts, linear spaces, matrices and determinants, and functionals. This is followed by separate chapters on body tensor and general space tensor fields; the kinematics of shear flow and shear-free flow; Cartesian vector and tensor fields; and relative tensors, field transfer, and the body stress tensor field. Subsequent chapters deal with constitutive equations for viscoelastic materials; reduced constitutive equations for shear flow and shear-free flow; covariant differentiation and the stress equations of motion; and stress measurements in unidirectional shear flow.

Autoren/Hrsg.

Lodge, Arthur S.

Leseproben

1

SOME MATHEMATICAL PREREQUISITES

Publisher Summary

This chapter discusses a mathematical formalism capable of describing the behavior and properties of a continuous geometric manifold. Formalisms are developed in mathematics by choosing certain primitive concepts, or undefined elements, that are to be accepted without further logical analysis. Properties are then assigned to these primitive concepts by means of axioms, or unproved propositions, from which various theorems are deduced. The primitive concepts might invite deeper logical analysis; however, such an analysis would be a part of a different activity. The primitive concepts need not necessarily be the smallest number that could be used; in fact, they are chosen to enable the development of the required formalism quickly. Primitive concepts are set of distinguishable elements such as; point, particle, place; order, relation, correspondence; and rigid body, mass, force, time. The chapter discusses the properties of matrices and determinants.

1.1 Primitive concepts

Our aim is to build a mathematical formalism capable of describing the behavior and properties of a continuous geometric manifold. One develops formalisms in mathematics by choosing certain (or ) which are to be accepted without further logical analysis. One then assigns properties to these primitive concepts by means of (or unproved propositions), and then deduces various theorems. The primitive concepts may invite deeper logical analysis, but such analysis would be part of a different activity. The primitive concepts need not necessarily be the smallest number which could be used; in fact, we shall choose them to enable us to get on with the development of the required formalism as quickly as possible. It is necessary, however, to list the primitive concepts to be used (Robison, 1969, p. 2). Our list is as follows.

(1) Primitive concepts

Set of distinguishable elements; point, particle, place; order, relation, correspondence; rigid body, mass, force, time.

We shall also use the standard formalisms of analysis, calculus, and geometry. “Set” is synonymous with “collection, aggregate, class.” “Correspondence” is synonymous with “mapping, function.”

We use the standard notation {} for a set of elements of which is a member. When it is necessary, we include the condition for membership after a vertical line, and we use the symbol e to mean “is a member of the set …”; thus, for example, if denotes the set of integers, then {2|?} denotes the set of even numbers.

Let = {} and = {} denote sets. A from into , written ? , is a rule which assigns to element ? a unique element ? , written = or = () or : ? . For example, if = = (the set of all real numbers), we could have = sin , in which case = sine. Other words used for function are mapping, map, transformation, correspondence, and operator.

The most familiar use of the term function is that in which = = . It is essential for our purposes to realize that the idea “function” is in no way restricted to such applications and can be applied to any sets of distinguishable elements. Thus, for example, we shall define a coordinate system : P ? ? as a correspondence or function which assigns to each particle P of a body = {P} a unique ordered set ? of three real numbers ?1, ?2, ?3. Thus we may write : ? 3, where 3 denotes the arithmetic three-space, the set of all ordered sets of three real numbers. The reader will notice that we have defined a coordinate system in terms of the primitive concepts: particle, correspondence, order. One might feel uneasy about defining things in terms of undefined elements (or primitive concepts), but such a procedure is in fact valid and unavoidable at the outset; one simply gets used to it.

For any function : ? , with = , the set is called the of (i.e., it is the set of all values of for which is defined), and the set {| ? } is called the of (i.e., it is the set of all values of = ). If = , i.e., if is composed entirely of elements , we say that maps ; otherwise, maps If for every ? there is only one such that = , then the function or correspondence is said to be , and the inverse -1 exists which maps onto In the example given above, a coordinate system : ? 3 has a subset of 3 as range and the body as domain, and the correspondence is one-to-one.

An integral =?abf(t') dt' is an example of an important type of mapping called a : Such an integral assigns a unique real number to each integrable real function with domain (). More generally, a functional : ? , where denotes the set of all functions ? [usually subject to some restriction, such as integrability, and having some specified domain, (), say] is a rule which assigns to each a unique number = . When it is necessary to make explicit the domain of , we write =F{f(t')bb}.. We shall need more elaborate functionals in which () is replaced by a 3 × 3 matrix function of and the value is a 3 × 3 matrix.

A set = {} is called a if there exists a rule which assigns to each ordered pair of elements in an element, written ’ in with the following properties: if ? , then () = (); there exists a unit element ? such that = = for every ? ; there exists an inverse element -1 ? such that -1 = -1 = for every ? . The notation () means: form first, and then take the element () which corresponds to element and , in that order.

1.2 Linear spaces

(1) Definition

A set = {} of elements , … is called a if operations of addition and multiplication by numbers , … are defined with the following properties.

(i) Addition

To any ? and any ? there corresponds a unique element, written + , in , such that

+y=y+x (addition is commutative),(x+y)+z=x+(y+z)      (addition is associative),x+0=x    (a unique zero element 0 ? x exists),x+(-x)=0   (each x has a unique inverse, -x).

(ii) Multiplication by a number

To any element and any number there corresponds a unique element, written or , which is also in the set {} and has the following properties :

(x+y)=ax+ay;

(bx)=(ab)x;

x=x;

-1)x=-x;

a+b)x=ax+bx.

The linear space {} is said to be a if , … range through...