Finite Element Analysis: Thermomechanics of Solids

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 RANGE AND SUMMATION CONVENTION Unless otherwise noted, repeated Latin indices imply summation over the range 1 to 3. For example: 3 aib = aib = a1b + a2b2 + a3b3 (1.1) i=1 aijbjk = ai1b k + a2b2k + a3b k (1.2) The repeated index is “summed out” and, therefore, dummy. The quantity aijbjk in Equation (1.2) has two free indices, i and k (and later will be shown to be the ik entry of a second-order tensor). Note that Greek indices do not imply summation. Thus, aaba = a1b1 if a = 1. 1.1.2 SUBSTITUTION OPERATOR The quantity, dij, later to be called the Kronecker tensor, has the property that dij = 1 i = j i ¹ j (1.3) For example, dijvj = 1 ´ vi, thus illustrating the substitution property. 1 © 2003 by CRC CRC Press LLC 2 Finite Element Analysis: Thermomechanics of Solids 3 v3 v e3 e2 v2 2 e1 1v1 FIGURE 1.1 Rectilinear coordinate system. 1.2 VECTORS 1.2.1 NOTATION Throughout this and the following chapters, orthogonal coordinate systems will be used. Figure 1.1 shows such a system, with base vectors e1, e2, and e3. The scalar product of vector analysis satisfies ei ×ej = dij (1.4) The vector product satisfies  ek ei ´ ej = −ek  0 i ¹ j and ijk in right-handed order i ¹ j and ijk not in right-handed order (1.5) i = j It is an obvious step to introduce the alternating operator, eijk, also known as the ijk entry of the permutation tensor: eijk =[ei ´ ej ]×ek  1 ijk distinct and in right-handed order = −1 ijk distinct but not in right-handed order  0 ijk not distinct (1.6) © 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 = viei w = wiei (1.7) Occasionally, base vectors are not displayed, so that v is denoted by vi. 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 v  w  *M) v = v2 w = w2 (1.8) v3 w3 It is compact, but also risks confusion by not displaying the underlying base vectors. In *M notation, the transposes vT and wT are also introduced; they are displayed as “row vectors”: *M) vT ={v1 v2 v3} wT ={w1 w2 w3} (1.9) The scalar product of v and w is written as v×w = (viei )×(wjej ) = viwjei ×ej *T) = viwjdij = viwi (1.10) The magnitude of v is defined by *T) v = v×v (1.11) The scalar product of v and w satisfies *T) v×w = v w cosqvw (1.12) in which qvw is the angle between the vectors v and w. The scalar, or dot, product is *M) v×w ® vTw (1.13) © 2003 by CRC CRC Press LLC 4 Finite Element Analysis: Thermomechanics of Solids The vector, or cross, product is written as v ´ w = viwjei ´ ej *T) = eijkviwjek (1.14) Additional results on vector notation are presented in the next section, which introduces matrix notation. Finally, the vector product satisfies *T) v ´ w = v w sinqvw (1.15) and vxw is colinear with n the unit normal vector perpendicular to the plane containing v and w. The area of the triangle defined by the vectors v and w is given by 2 v ´ w. 1.2.2 GRADIENT, DIVERGENCE, AND CURL The derivative, dj/dx, of a scalar jwith respect to a vectorx is defined implicitly by *M) dj = dj dx (1.16) and it is a row vector whose ith entry is dj/dxi. In three-dimensional rectangular coordinates, the gradient and divergence operators are defined by  ¶( )  ¶x  *M) Ñ( ) =  ¶( ) (1.17)  ¶( ) ¶z and clearly, *M)  dxT( ) = Ñ( ) (1.18) The gradient of a scalar function j satisfies the following integral relation: òÑjdV = ònjdS (1.19) The expression ÑvT will be seen to be a tensor (see Chapter 2). Clearly, ÑvT =[Ñv1 Ñv2 Ñv3] (1.20) © 2003 by CRC CRC Press LLC Mathematical Foundations: Vectors and Matrices 5 from which we obtain the integral relation òÑvTdV = ònvTdS (1.21) Another important relation is the divergence theorem. Let V denote the volume of a closed domain, with surface S. Let n denote the exterior surface normal to S, and let v denote a vector-valued function of x, the position of a given point within the body. The divergence of v satisfies *M) òdv dV = ònTvdS (1.22) The curl of vector v, Ñ ´ v, is expressed by (Ñ´ v)i = eijk ¶¶j vk (1.23) which is the conventional cross-product, except that the divergence operator replaces the first vector. The curl satisfies the curl theorem, analogous to the divergence theorem (Schey, 1973): òÑ´ vdV = òn´ vdS (1.24) Finally, the reader may verify, with some effort that, for a vector v(X) and a path X(S) in which S is the length along the path, òv×dX(S) = òn×Ñ´ vdS . (1.25) The integral between fixed endpoints is single-valued if it is path-independent, in which case n × Ñ ´ v must vanish. However, n is arbitrary since the path is arbitrary, thus giving the condition for v to have a path-independent integral as Ñ´ v = 0. (1.26) 1.3 MATRICES An n ´ n matrix is simply an array of numbers arranged in rows and columns, also known as a second-order array. For the matrix A, the entry a occupies the intersection of the ith row and the jth column. We may also introduce the n ´ 1 first-order array a, in which ai denotes the ith entry. We likewise refer to the 1 ´ n array, a , as first-order. © 2003 by CRC CRC Press LLC ... - tailieumienphi.vn