APPENDIX A TENSORS, TENSOR PRODUCTS AND TENSOR OPERATIONS IN THREE-DIMENSIONS A.I Vectors and vector operations The purpose of this section is to summarize some basic properties of vectors and vector operations. Complete descriptions of linear vector spaces can be found in standard texts on linear algebra (Noble, 1969) and a more complete summary of the use of vectors in continuum mechanics can be found in (Sokolnikoff, 1964). For the present purpose it is sufficient to recall that a vector in three-dimensional space is usually identified with an arrow connecting two points in space. This arrow has a magnitude and a specific direction. R p.:: Fig. A.I.l Parallelogram rule of vector addition. Some basic vector operations can be summarized as follows. If a,b are two vectors, then the quantity c=a+b, (A. 1.1) is also a vector which is defined by the parallelogram law of addition (see Fig. A.I.l). Furthermore, the operations a+b=b+a (commutative law) , (a+b)+c=a+(b+c) (associative law) , aa=aa (multiplication by a real number) , a-b=b-a (commutative law) , a-(b+c)=a-b+a-c (distributive law) , a(a-b)=(aa)-b (associative law) , axb=-bxa (lack of commutativity) , ax(b+c)=axb+axc (distributive law) , a(axb)=(aa)xb (associative law) , (A.I.2) are satisfied for all vectors a,b,c and all real numbers a, where a - b denotes the dot product (or scalar product), and a x b denotes the cross product (or vector product) between the vectors a and b. 429 430 APPENDIX A A.2 Tensors as linear operators Scalars (or real numbers) are referred to as tensors of order zero, and vectors are referred to as tensors of order one. Here, higher order tensors are defined inductively starting with the notion of a vector. Tensor of Order M: The quantity T is called a tensor of order two (or a second order tensor) if it is a linear operator whose domain is the space of all vectors v and whose range Tv or vT is a vector. Similarly, the quantity T is called a tensor of order three if it is a linear operator whose domain is the space of all vectors v and whose range Tv or vT is a tensor of order two. Consequently, by induction, the quantity T is called a tensor of order M (M~I)(M~l) if it is a linear operator whose domain is the space of all vectors v and whose range Tv or vT is a tensor of order (M-I). Since T is a linear operator, it satisfies the following rules T(v + w) = Tv + Tw , a(Tv) = (aT)v = T(av) , (v + w)T = vT + wT , a(vT) = (av)T = (vT)a , (A.2.I)(A.2.1) where v,w are arbitrary vectors and a is an arbitrary scalar. Notice that the tensor T can be operated upon on its right [e.g. (A.2.1)J,2](A.2.1)l,2l or on its left [e.g. (A.2.I)3,4](A.2.1»),4l and that in general, operation on the right and on the left is not commutative Tv '#;f. vT (Lack of commutativity in general) . (A.2.2) Zero Tensor of Order M: The zero tensor of order M is denoted by OeM) and is a linear operator whose domain is the space of all vectors v and whose range O(M-I)O(M-J) is the zero tensor of order M-I. OeM) v = v OeM) = O(M-I) . (A.2.3) Notice that these tensors are defined inductively starting with the known properties of the real number 0, which is the zero tensor 0(0) of order O. Often, for simplicity in writing a tensor equation, the zero tensor of any order is denoted by the symbol O. Addition and Subtraction: The usual rules of addition and subtraction of two tensors A and B apply only when the two tensors have the same order. It should be emphasized that tensors of different orders cannot be added or subtracted. A.3 Tensor products (special case) In order to define the operations of tensor product, dot product, and juxtaposition for general tensors, it is convenient to first consider the definitions of these properties for special tensors. Also, the operations of left and right transpose of the tensor product of a string of vectors will be defined. It will be seen later that the operation of dot product is defined to be consistent with the usual notion of the dot product as an inner product because it is a positive definite operation. Consequently, the dot product of a tensor with itself yields the square of the magnitude of the tensor. Also, the operation of juxtaposition is defined to be consistent with the usual procedures for matrix multiplication when two second order tensors are juxtaposed. TENSORS 431 Tensor Product (Special Case): The tensor product operation is denoted by the symbol ® and it is defined so that for an arbitrary vector v, the quantity (a I ®a2) is a second order tensor satisfying the relations (a l ®a2)v=al (a2 ·v) , v(al ®a2)=(v·a l )a2 . (A.3.1) Similarly, the quantity (al ®a2®a3) is a third order tensor satisfying the relations (a l ®a2®a3) v = (al®a2) (a3 • v) , v (a l ®a2®a3) = (v· a l ) (a2®a3) . (A.3.2) For convenience in generalizing these ideas, let A be a special tensor of order M which is formed by the tensor product of a string of M (M~2) vectors (al ,a2,a3, ... ,aM), and let B be a special tensor of order N which is formed by the tensor product of a string of N (N~2) vectors (b l ,b2,b3, ... ,bN) so that A = (a l ®a2®a3® ... ®aM), B = (b l ®b2®b3® ... ®bN) , (A.3.3) and for definiteness take M:.:;N. (A.3.4) It then follows that A satisfies the relations Av = (a l ®a2® ... ®aM) v = (al ®a2® ... ®aM_I ) (aM· v), vA = v (al ®a2® ... ®aM) = (v· a l ) (a2®a3® ... ®aM ) . (A.3.S) Notice that when v operates on either the right or left side of A, it has the effect of forming the scalar product with the vector in the string closest to it, and it causes the result A v or v A to have order one less than the order of A since one of the tensor products is removed. The remaining vectors in the string of tensor products are unaltered. Dot Product (Special Case): The dot product operation between two vectors can be generalized to an operation between two tensors of any orders. For example, the dot product between two second order tensors can be written as (al ®a2)· (bl ®b2) = (al • bl) (a2 • b2) , (b l ®b2)· (al®a2) = (b l • al) (b2 • a2) , (A.3.6) and the dot product between a second order tensor and a third order tensor can be written as (al ®a2)· (bl ®b2®b3) = (al • bl) (a2 • b2) b 3 ' (b l ®b2®b3)· (a l®a2) = b l (b2 • al) (b3 • a2) . (A.3.7) Also, the dot product between two third order tensors can be written as (a l ®a2®a3) • (bl ®b2®b3) = (a l • bl) (a2 • b2) (a3 • b3) , (b l ®b2®b3)· (al ®a2®a3) = (b l • a l ) (b2 • a2) (b3 • a3) , (A.3.8) and the dot product between a second order tensor and a fourth order tensor can be written as (al ®a2)· (b l ®b2®b3®b4) = (a l • bl) (a2 • b2) (b3®b4) , (bl ®b2®b3®b4)· (a l ®a2) = (bl ®b2) (b3 • al) (b4 • a2)· (A.3.9) This dot product operation can be generalized for special tensors of any orders like A and B by carefully examining examples (A.3.7) and (A.3.9). These examples indicate 432 APPENDIX A that the dot product between tensors of different orders does not necessarily commute, whereas the dot product of two tensors of the same order does commute. A • B :;t B • A for M<N , A· B = B • A for M=N . (A.3.1O) Moreover, it can be seen that the tensor of smallest order controls the outcome of the dot product operation. Specifically, in (A.3.7), the second order tensor (a,®a2) appears on the left of the third order tensor (b, ®b2®b3). Since (a, ®a2) is a second order tensor this causes only the first two vectors on the left of (b, ®b2®b3) to form inner products with a, and a2. Similarly, since (a,®a2) appears on the right side of (b,®b2®b3) in (A.3.7)2' this causes only the first two vectors on the right of (b, ®b2®b3) to form inner products with a, and a2. In general, the dot product A • B of the special tensors defined in (A.3.3) is a tensor of order IM-NI. Furthermore, in view of the restriction (A.3.4), only the first M vectors of B closest to A will form inner products with the vectors of A. It is also important to note that the order of the strings of vectors in the inner products remains the same as that in the tensors. Specifically, for the dot product A • B the first vector on the left side of A forms an inner product with the first vector on the left side of B and so forth.