A Basic Operations of Tensor Algebra The 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 and the tensorial equations: – the direct (symbolic, coordinate-free) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three-dimensional space (in this book we are limit ourselves to this case). A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a ⊗ b + ...+ c ⊗ d. The scalars, vectors and tensors are 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 literature of mechanics and rheology, e.g. [32, 36, 53, 126, 134, 205, 253, 321, 343] among others. The basics of the direct tensor calculus are given in the classical textbooks of Wilson (founded upon the lecture notes of Gibbs) [331] and Lagally [183]. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write i i j i j ⊗ a = a gi, A = a b + ...+ c d gi gj Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k ≡ k ik ≡ ik a gk ∑ a gk, A bk ∑ A bk k=1 k=1 In the above examples k is a so-called dummy index. Within the index notation the 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 coordinates ci = ai + bi. The introduced basis remains in the background. It must be noted that a change of the coordinate system leads to the change of the components of tensors. In 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” 168 A Basic Operations of Tensor Algebra of matrices for a specified coordinate system. The purpose of this Appendix is to give a brief guide to notations and rules of the tensor calculus applied through- out this book. 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 are given 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 basis is discussed. A.1 Polar and Axial Vectors A vector in the three-dimensional Euclidean space is defined as a directed line segment with specified scalar-valued magnitude and direction. The magnitude (the length) of a vector a is denoted by |a|. Two vectors a and b are equal if they have the same direction and the same magnitude. The zero vector 0 has a magnitude equal to zero. In mechanics two types of vectors can be introduced. The vectors of the first type are directed line segments. These vectors are associated with translations in the three-dimensional space. Examples for polar vectors include the force, the displacement, the velocity, the acceleration, the momentum, etc. The second type is used to characterize spinor motions and related quantities, i.e. the moment, the angular velocity, the angular momentum, etc. Figure A.1a shows the so-called spin vector a∗ which represents a rotation about the given axis. The direction of rotation is specified by the circular arrow and the “magnitude” of rotation is the correspond- ing length. For the given spin vector a∗ the directed line segment a is introduced according to the following rules [343]: 1. the vector a is placed on the axis of the spin vector, 2. the magnitude of a is equal to the magnitude of a∗, 3. the vector a is directed according to the right-handed screw, Fig. A.1b, or the left-handed screw, Fig. A.1c a b c a a∗ a∗ a∗ a Fig. A.1 Spin vector and its representation by an axial vector. a Spin vector, b axial vector in the right-screw oriented reference frame, c axial vector in the left-screw oriented reference frame A.2 Operations with Vectors 169 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 with the right- or left-handed triples of vectors or coordinate systems). The directed line segment is called a polar vector if it does not change by changing the orientation of the reference frame. The vector is called to be axial if it changes the sign by changing the orientation of the reference frame. The above definitions are valid for scalars and tensors of any rank too. The axial vectors (and tensors) are widely used in the rigid body dynamics, e.g. [342], in the theories of rods, plates and shells, e.g. [28], in the asymmetric theory of elasticity, e.g. [238], as well as in dynamics of micro-polar media, e.g. [111]. When dealing with polar and axial vectors it should be remembered that they have different physical meanings. Therefore, a sum of a polar and an axial vector has no sense. A.2 Operations with Vectors A.2.1 Addition For a given pair of vectors a and b of the same type the sum c = a + b is defined according to one of the rules in Fig. A.2. The sum has the following properties • a + b = b + a (commutativity), • (a + b)+c = a +(b + c) (associativity), • a + 0 = a A.2.2 Multiplication by a Scalar For any vector a and for any scalar α a vector b = αa is defined in such a way that •|b| = |α||a|, • for α > 0 the direction of b coincides with that of a, • for α < 0 the direction of b is opposite to that of a. For α = 0 the product yields the zero vector, i.e. 0 = 0a. It is easy to verify that α(a + b)=αa + αb, (α + β)a = αa + βa a b b c c b a a Fig. A.2 Addition of two vectors. a Parallelogram rule, b triangle rule 170 A Basic Operations of Tensor Algebra a b b b ϕ ϕ a a 2π − ϕ a na = (b · a)n |a| a Fig. A.3 Scalar product of two vectors. a Angles between two vectors, b unit vector and projection A.2.3 Scalar (Dot) Product of Two Vectors For any pair of vectors a and b a scalar α is defined by α = a · b = |a||b| cos ϕ, where ϕ is the angle between the vectors a and b.Asϕ one can use any of the two angles between the vectors, Fig. A.3a. The properties of the scalar product are • a · b = b · a (commutativity), • a · (b + c)=a · b + a · c (distributivity) Two nonzero vectors are said to be orthogonal if their scalar product is zero. The unit vector directed along the vector a is defined by (see Fig. A.3b) a na = a |a| · The projection of the vector b onto the vector a is the vector (b · a)na,Fig.A.3b. The length of the projection is |bcos ϕ|. A.2.4 Vector (Cross) Product of Two Vectors For the ordered pair of vectors a and b the vector c = a × b is defined in two following steps [343]: • the spin vector c∗ is defined in such a way that the axis is orthogonal to the plane spanned on a and b,Fig.A.4a, the circular arrow shows the direction of the “shortest” rotation from a to b, Fig. A.4b, the length is |a||b| sin ϕ, where ϕ is the angle of the “shortest” rotation from a to b, A.3 Bases 171 a b c c c∗ ϕ ϕ ϕ a b a b a b Fig. A.4 Vector product of two vectors. a Plane spanned on two vectors, b spin vector, c axial vector in the right-screw oriented reference frame • from the resulting spin vector the directed line segment c is constructed according to one of the rules listed in Sect. A.1. The properties of the vector product are a × b = −b × a, a × (b + c)=a × b + a × c The type of the vector c = a × b can be established for the known types of the vectors a and b, [343]. If a and b are polar vectors the result of the cross product will be the axial vector. An example is the moment of momentum for a mass point m defined by r × (mv˙ ), where r is the position of the mass point and v is the velocity of the mass point. The next example is the formula for the distribution of velocities in a rigid body v = ω × r. Here the cross product of the axial vector ω (angular velocity) with the polar vector r (position vector) results in the polar vector v.