Vectors_Tensors_08_Tensors

Vectors_Tensors_08_Tensors - Section 1.8 1.8 Tensors Here...

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Section 1.8 Solid Mechanics Part III Kelly 67 1.8 Tensors Here the concept of the tensor is introduced. Tensors can be of different orders – zeroth- order tensors, first-order tensors, second-order tensors, and so on. Apart from the zeroth and first order tensors (see below), the second-order tensors are the most important tensors from a practical point of view, being important quantities in, amongst other topics, continuum mechanics, relativity, electromagnetism and quantum theory. 1.8.1 Zeroth and First Order Tensors A tensor of order zero is simply another name for a scalar α . A first-order tensor is simply another name for a vector u . 1.8.2 Second Order Tensors Notation Vectors: lowercase bold-face Latin letters, e.g. a , r , q 2 nd order Tensors: uppercase bold-face Latin letters, e.g. F , T , S Tensors as Linear Operators A second -order tensor T may be defined as an operator that acts on a vector u generating another vector v , so that v u T = ) (, o r 1 v Tu v u T = = or Second-order Tensor (1.8.1) The second-order tensor T is a linear operator (or linear transformation ) 2 , which means that () Tb Ta b a T + = + distributive ( ) Ta a T = associative This linearity can be viewed geometrically as in Fig. 1.8.1. Note: the vector may also be defined in this way, as a mapping u that acts on a vector v , this time generating a scalar α , = v u . This transformation (the dot product) is linear (see properties (2,3) in §1.1.4). Thus a first-order tensor (vector) maps a first-order tensor into a zeroth-order tensor (scalar), whereas a second-order tensor maps a first-order tensor into a first-order tensor. It will be seen that a third-order tensor maps a first-order tensor into a second-order tensor, and so on 1 both these notations for the tensor operation are used; here, the convention of omitting the “dot” will be used 2 An operator or transformation is a special function which maps elements of one type into elements of a similar type; here, vectors into vectors
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Section 1.8 Solid Mechanics Part III Kelly 68 Figure 1.8.1: Linearity of the second order tensor Further, two tensors T and S are said to be equal if and only if Tv Sv = for all vectors v . Example (of a Tensor) Suppose that F is an operator which transforms every vector into its mirror-image with respect to a given plane, Fig. 1.8.2. F transforms a vector into another vector and the transformation is linear, as can be seen geometrically from the figure. Thus F is a second-order tensor.
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Vectors_Tensors_08_Tensors - Section 1.8 1.8 Tensors Here...

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