3DElasticity_01_3D_Index

3DElasticity_01_3D_Index - Section 7.1 Solid Mechanics Part...

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Unformatted text preview: Section 7.1 Solid Mechanics Part II Kelly 104 7.1 Vectors, Tensors and the Index Notation The equations governing three dimensional mechanics problems can be quite lengthy. For this reason, it is essential to use a short-hand notation called the index notation 1 . Consider first the notation used for vectors. 7.1.1 Vectors Vectors are used to describe physical quantities which have both a magnitude and a direction associated with them. Geometrically, a vector is represented by an arrow; the arrow defines the direction of the vector and the magnitude of the vector is represented by the length of the arrow. Analytically, in what follows, vectors will be represented by lowercase bold-face Latin letters, e.g. a , b . The dot product of two vectors a and b is denoted by b a and is a scalar defined by cos b a b a = . (7.1.1) here is the angle between the vectors when their initial points coincide and is restricted to the range . Cartesian Coordinate System So far the short discussion has been in symbolic notation 2 , that is, no reference to axes or components or coordinates is made, implied or required. Vectors exist independently of any coordinate system. The symbolic notation is very useful, but there are many circumstances in which use of the component forms of vectors is more helpful or essential. To this end, introduce the vectors 3 2 1 , , e e e having the properties 1 3 3 2 2 1 = = = e e e e e e , (7.1.2) so that they are mutually perpendicular, and 1 3 3 2 2 1 1 = = = e e e e e e , (7.1.3) so that they are unit vectors. Such a set of orthogonal unit vectors is called an orthonormal set, Fig. 7.1.1. This set of vectors forms a basis, by which is meant that any other vector can be written as a linear combination of these vectors, i.e. in the form 3 3 2 2 1 1 e e e a a a a + + = (7.1.4) where 2 1 , a a and 3 a are scalars, called the Cartesian components or coordinates of a along the given three directions. The unit vectors are called base vectors when used for 1 or indicial or subscript or suffix notation 2 or absolute or invariant or direct or vector notation Section 7.1 Solid Mechanics Part II Kelly 105 this purpose. The components 2 1 , a a and 3 a are measured along lines called the 2 1 , x x and 3 x axes , drawn through the base vectors. Figure 7.1.1: an orthonormal set of base vectors and Cartesian coordinates Note further that this orthonormal system { } 3 2 1 , , e e e is right-handed , by which is meant 3 2 1 e e e = (or 1 3 2 e e e = or 2 1 3 e e e = ). In the index notation, the expression for the vector a in terms of the components 3 2 1 , , a a a and the corresponding basis vectors 3 2 1 , , e e e is written as = = + + = 3 1 3 3 2 2 1 1 i i i a a a a e e e e a (7.1.5) This can be simplified further by using Einsteins summation convention , whereby the summation sign is dropped and it is understood that for a repeated index ( i in this case) a...
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 2 taught by Professor Staff during the Fall '11 term at Auckland.

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3DElasticity_01_3D_Index - Section 7.1 Solid Mechanics Part...

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