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9780203009512.ch1

# 9780203009512.ch1 - 0749_Frame_C01 Page 1 Wednesday 4:55 PM...

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1 Mathematical Foundations: Vectors and Matrices 1.1 INTRODUCTION This chapter provides an overview of mathematical relations, which will prove useful in the subsequent chapters. Chandrashekharaiah and Debnath (1994) provide a more complete discussion of the concepts introduced here. 1.1.1 R ANGE AND S UMMATION C ONVENTION Unless otherwise noted, repeated Latin indices imply summation over the range 1 to 3. For example: (1.1) (1.2) The repeated index is “summed out” and, therefore, dummy. The quantity a ij b jk in Equation (1.2) has two free indices, i and k (and later will be shown to be the ik th entry of a second-order tensor). Note that Greek indices do not imply summation. Thus, a α b α = a 1 b 1 if α = 1. 1.1.2 S UBSTITUTION O PERATOR The quantity, δ ij , later to be called the Kronecker tensor, has the property that (1.3) For example, δ ij v j = 1 × v i , thus illustrating the substitution property. 1 a b a b a b a b a b i i i i i = = + + = 1 3 1 1 2 2 3 3 a b a b a b a b ij jk i k i k i k = + + 1 1 2 2 3 3 δ ij i j i j = = 1 0 © 2003 by CRC CRC Press LLC

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2 Finite Element Analysis: Thermomechanics of Solids 1.2 VECTORS 1.2.1 N OTATION Throughout this and the following chapters, orthogonal coordinate systems will be used. Figure 1.1 shows such a system, with base vectors e 1 , e 2 , and e 3 . The scalar product of vector analysis satisfies (1.4) The vector product satisfies (1.5) It is an obvious step to introduce the alternating operator, ε ijk , also known as the ijk th entry of the permutation tensor: (1.6) FIGURE 1.1 Rectilinear coordinate system. 3 2 1 v 1 v 2 v e 1 e 2 e 3 v 3 e e i j ij = δ e e e e 0 i j k k i j ijk i j ijk i j × = = and in right-handed order and not in right-handed order ε ijk i j k ijk ijk ijk = × = [ ] e e e 1 1 0 distinct and in right-handed order distinct but not in right-handed order not distinct © 2003 by CRC CRC Press LLC
Mathematical Foundations: Vectors and Matrices 3 Consider two vectors, v and w . It is convenient to use two different types of notation. In tensor indicial notation , denoted by (*T), v and w are represented as *T) (1.7) Occasionally, base vectors are not displayed, so that v is denoted by v i . By displaying base vectors, tensor indicial notation is explicit and minimizes confusion and ambiguity. However, it is also cumbersome. In this text, the “default” is matrix-vector (*M) notation, illustrated by *M) (1.8) It is compact, but also risks confusion by not displaying the underlying base vectors. In *M notation, the transposes v T and w T are also introduced; they are displayed as “row vectors”: *M) (1.9) The scalar product of v and w is written as *T) (1.10) The magnitude of v is defined by *T) (1.11) The scalar product of v and w satisfies *T) (1.12) in which θ vw is the angle between the vectors v and w . The scalar, or dot, product is *M) (1.13) v e w e = = v w i i i i v w = = v v v w w w 1 2 3 1 2 3 v w T T = = { } { } v v v w w w 1 2 3 1 2 3 v w e e e e = = = = ( ) ( ) v w v w v w v w i i j j i j i j i

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