Tensor Algebra

To work with tensors, we need to know their properties of addition and multi­plication. With your a priori knowledge of vectors and matrices, there will be little new ground to cover. Basically, tensors are manipulated like vectors and matrices. We only need to spend some time discussing the vector and dyadic products that take on a different flavor.

First-order tensors (vectors) can be added and subtracted, abiding by the com­mutative rule

Sbc = sba + Sac = Sac + sba (2.10)

and the associative rule

(®вс + Sca) + sab — Sbc + (sca+ sab) (2.11)

However, I do not recommend the exchange of the terms on the right-hand side of Eq. (2.10), because it will upset the rule of contraction of subscripts. Notice in Eq. (2.11)1 arranged the order of the subscripts to form the null vector s BB, which is characterized by zero displacement. You can verify this fact by contraction of the subscripts.

Multiplication of a vector sBC with scalars a and fi is commutative, associative, and distributive:

ci(Psbc) = P(usBC) = (ccP)sbc (a + P)sBc = otsBc + Psbc

Next I will deal with the multiplication of two vectors. There are three pos­sibilities, distinguishable by the results. The outcome could be a scalar, vector, or tensor. Therefore, we call them scalar, vector, or dyadic products. The word dyadic is borrowed from vector mechanics, which calls stress and inertia tensors dyadics. I will use the symbols x and у for the two vectors, but also write them in the bracketed form [jc] and [ y|, to emphasize their column matrix property. If they do not carry a superscript to mark the coordinate system, they abide in any

allowable coordinate system and are therefore first-order tensor symbols just like x andy.