tailieunhanh - Finite Element Analysis: Thermomechanics of Solids

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. | Mathematical Foundations Vectors and Matrices 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. Range and Summation Convention Unless otherwise noted repeated Latin indices imply summation over the range 1 to 3. For example 3 ab. -Y i-1 ữịbị - a1b1 a2b2 a3b3 aijbjk - ai1b1k a2b2k ai3b3k The repeated index is summed out and therefore dummy. The quantity aii jk in Equation has two free indices i and k and later will be shown to be the ikth entry of a second-order tensor . Note that Greek indices do not imply summation. Thus aaba - a1b1 if a - 1. Substitution Operator The quantity 8j later to be called the Kronecker tensor has the property that j p 10 i- j i j For example SjjVj - 1 X vi thus illustrating the substitution property. 1 2003 by CRC CRC Press LLC 2 Finite Element Analysis Thermomechanics of Solids FIGURE Rectilinear coordinate system. VECTORS Notation Throughout this and the following chapters orthogonal coordinate systems will be used. Figure shows such a system with base vectors e1 e2 and e3. The scalar product of vector analysis satisfies ei e j 8ij The vector product satisfies ek i j and ijk in right-handed order ei X ej -ek i j and ijk not in right-handed order 0 i j It is an obvious step to introduce the alternating operator eijk also known as the ijkth entry of the permutation tensor E. e. X e l e ijk - i jJ k 1 ijk distinct and in right-handed order -1 ijk distinct but not in right-handed order . 0 ijk 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 v vt e w we Occasionally base vectors are not displayed so that v .