A Basic Operations of Tensor Algebra - Springer

1 A Basic Operations of Tensor AlgebraThe Tensor calculus is a powerful tool for the description of the fundamentals in con-tinuum mechanics and the derivation of the governing equations for applied prob-lems. In general, there are two possibilities for the representation of the tensors andthe tensorial equations: the direct (symbolic, coordinate-free) notation and the index (component) notationThe direct notation operates with scalars, vectors and tensors as physical objectsdefined in the three-dimensional space (in this book we are limit ourselves to thiscase). A vector (first rank Tensor )aaais considered as a directed line segment ratherthan a triple of numbers (coordinates). A second rank tensorAAAis any finite sumof ordered vector pairsAAA=a ba ba b+.

2 +c dc dc d. The scalars, vectors and tensorsare handled as invariant (independent from the choice of the coordinate system)quantities. This is the reason for the use of the direct notation in the modern literatureof mechanics and rheology, [32, 36, 53, 126, 134, 205, 253, 321, 343] amongothers. The basics of the direct Tensor calculus are given in the classical textbooksof Wilson (founded upon the lecture notes of Gibbs) [331] and Lagally [183].The index notation deals with components or coordinates of vectors and a selected basis, ,i=1, 2, 3one can writeaaa=aigggi,AAA= aibj+..+cidj gggi gggjHere the Einstein s summation convention is used: in one expression the twice re-peated indices are summed up from 1 to 3, 3 k=1akgggk,Aikbk 3 k=1 AikbkIn the above exampleskis a so-called dummy index.

3 Within the index notationthe Basic Operations with tensors are defined with respect to their coordinates,e. g. the sum of two vectors is computed as the sum of their coordinatesci=ai+ introduced basis remains in the background. It must be noted that a change ofthe coordinate system leads to the change of the components of this book we prefer the direct Tensor notation over the index one. When solv-ing applied problems the Tensor equations can be translated into the language 168A Basic Operations of Tensor Algebraof matrices for a specified coordinate system. The purpose of this Appendix is togive a brief guide to notations and rules of the Tensor calculus applied through-out this book.

4 For more comprehensive overviews on Tensor calculus we recom-mend [58, 99, 126, 197, 205, 319, 343]. The calculus of matrices is presented in[44, 114, 350], for example. Section A provides a summary of Basic algebraic oper-ations with vectors and second rank tensors. Several rules from Tensor analysis aregiven in Sect. B. Basic sets of invariants for different groups of symmetry transfor-mation are presented in Sect. C, where a novel approach to find the functional basisis Polar and Axial VectorsA vector in the three-dimensional Euclidean space is defined as a directed linesegment with specified scalar-valued magnitude and direction. The magnitude (thelength) of a vectoraaais denoted by|aaa|.

5 Two vectorsaaaandbbbare equal if they have thesame direction and the same magnitude. The zero vector000has a magnitude equalto zero. In mechanics two types of vectors can be introduced. The vectors of thefirst type are directed line segments. These vectors are associated with translationsin the three-dimensional space. Examples for polar vectors include the force, thedisplacement, the velocity, the acceleration, the momentum, etc. The second typeis used to characterize spinor motions and related quantities, the moment, theangular velocity, the angular momentum, etc. Figure shows the so-called spinvectoraaa which represents a rotation about the given axis. The direction of rotationis specified by the circular arrow and the magnitude of rotation is the correspond-ing length.

6 For the given spin vectoraaa the directed line segmentaaais introducedaccording to the following rules [343]:1. the vectoraaais placed on the axis of the spin vector,2. the magnitude ofaaais equal to the magnitude ofaaa ,3. the vectoraaais directed according to the right-handed screw, Fig. , or theleft-handed screw, Fig. aaa aaa aaaaaaabcFig. vector and its representation by an axial vector,baxial vectorin the right-screw oriented reference frame,caxial vector in the left-screw oriented Operations with Vectors169 The selection of one of the two cases in of the third item corresponds to the con-vention of orientation of the reference frame [343] (it should be not confused withthe right- or left-handed triples of vectors or coordinate systems).

7 The directed linesegment is called a polar vector if it does not change by changing the orientationof the reference frame. The vector is called to be axial if it changes the sign bychanging the orientation of the reference frame. The above definitions are valid forscalars and tensors of any rank too. The axial vectors (and tensors) are widely usedin the rigid body dynamics, [342], in the theories of rods, plates and shells, [28], in the asymmetric theory of elasticity, [238], as well as in dynamics ofmicro-polar media, [111]. When dealing with polar and axial vectors it shouldbe remembered that they have different physical meanings. Therefore, a sum of apolar and an axial vector has no Operations with AdditionFor a given pair of vectorsaaaandbbbof the same type the sumccc=aaa+bbbis definedaccording to one of the rules in Fig.

8 The sum has the following properties aaa+bbb=bbb+aaa(commutativity), (aaa+bbb)+ccc=aaa+(bbb+ccc)(associativit y), aaa+000= Multiplication by a ScalarFor any vectoraaaand for any scalar a vectorbbb= aaais defined in such a way that |bbb|=| ||aaa|, for >0the direction ofbbbcoincides with that ofaaa, for <0the direction ofbbbis opposite to that =0the product yields the zero vector, It is easy to verify that (aaa+bbb)= aaa+ bbb,( + )aaa= aaa+ aaaaaaaaabbbbbbccccccabFig. of two rule,btriangle rule170A Basic Operations of Tensor Algebraaaaaaabbbbbb 2 nnnaaa=aaa|aaa|(bbb aaa)nnnaaaabFig. product of two between two vectors,bunit vector Scalar (Dot) Product of Two VectorsFor any pair of vectorsaaaandbbba scalar is defined by =aaa bbb=|aaa||bbb|cos ,where is the angle between the one can use any of the twoangles between the vectors, Fig.

9 The properties of the scalar product are aaa bbb=bbb aaa(commutativity), aaa (bbb+ccc)=aaa bbb+aaa ccc(distributivity)Two nonzero vectors are said to be orthogonal if their scalar product is zero. Theunit vector directed along the vectoraaais defined by (see Fig. )nnnaaa=aaa|aaa|The projection of the vectorbbbonto the vectoraaais the vector(bbb aaa)nnnaaa, length of the projection is|bbb cos |. Vector (Cross) Product of Two VectorsFor the ordered pair of vectorsaaaandbbbthe vectorccc=aaa bbbis defined in twofollowing steps [343]: the spin vectorccc is defined in such a way that the axis is orthogonal to the plane spanned onaaaandbbb, , the circular arrow shows the direction of the shortest rotation fromaaatobbb,Fig.

10 , the length is|aaa||bbb|sin , where is the angle of the shortest rotation fromaaatobbb, Bases171aaaaaaaaabbbbbbbbb ccc cccabcFig. product of two spanned on two vectors,bspin vector,caxialvector in the right-screw oriented reference frame from the resulting spin vector the directed line segmentcccis constructed accordingto one of the rules listed in Sect. properties of the vector product areaaa bbb= bbb aaa,aaa (bbb+ccc)=aaa bbb+aaa cccThe type of the vectorccc=aaa bbbcan be established for the known types of thevectorsaaaandbbb, [343]. Ifaaaandbbbare polar vectors the result of the cross productwill be the axial vector. An example is the moment of momentum for a mass pointmdefined byrrr (m vvv), whererrris the position of the mass point andvvvis the velocityof the mass point.