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Appendix a Basic Operations of Tensor Algebra Appendix A Basic Operations of Tensor Algebra The tensor calculus is a powerful tool for the description of the fundamentals in continuum mechanics and the derivation of the governing equations for applied problems. 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 ofordered 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. Altenbach (2012), Antman (1995), Besseling and van der Giessen (1994), Giesekus (1994), Haupt (2002), Lurie (1990), Palmov (1998), Truesdell and Noll (1992), Zhilin (2001) among others. The basics of the direct tensor calculus are given in the classical textbooks of Wilson (1901) (founded upon the lecture notes of Gibbs) and Lagally (1962). 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 ⊗ g j Here the Einstein’s summation convention is used: in one expression the twice repeated 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 © Springer International Publishing Switzerland 2016 323 K. Naumenko and H. Altenbach, Modeling High Temperature Materials Behavior for Structural Analysis, Advanced Structured Materials 28, DOI 10.1007/978-3-319-31629-1 324 Appendix A: Basic Operations of Tensor Algebra 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” 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 throughout this book. For more comprehensive overviews on tensor calculus we recommend Betten (1987), de Boer (1982), Giesekus (1994), Lebedev et al. (2010), Lippmann (1993), Lurie (1990), Trostel (1993), Zhilin (2001). The calculus of matrices is presented in Bellmann (1970), Faddejew and Faddejewa (1964), Zurmühl and Falk (1992), for example. AppendixA provides a summary of basic algebraic operations with vectors and second rank tensors. Several rules from tensor analysis are given in Appendix B. Basic sets of invariants for different groups of symmetry transformation are presented in Sect. B.5, 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 asso- ciated with translations in the three-dimensional space. Examples for polar vectors include the force, the displacement, the velocity, the acceleration, the momentum, (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 Appendix A: Basic Operations of Tensor Algebra 325 etc. The second type is used to characterize spinor motions and related quantities, i.e. the moment, the angular velocity, the angular momentum, etc., Fig.A.1ashowsthe so-called spin vector a ∗ which represents a rotation about the given axis. The direc- tion of rotation is specified by the circular arrow and the “magnitude” of rotation is the corresponding length. For the given spin vector a ∗ the directed line segment a is introduced according to the following rules (Zhilin 2001): 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 The selection of one of the two cases in of the third item corresponds to the convention of orientation of the reference frame (Zhilin 2001). 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. Altenbach et al. (2007), Zhilin (1996), in the theories of rods, plates and shells, e.g. Altenbach and Zhilin (1988), Naumenko and Eremeyev (2014), in the asymmetric theory of elasticity, e.g. Nowacki (1986), as well as in dynamics of micro-polar media, e.g. Altenbach et al. (2003), Eringen (1999). 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) (b) b c c b a a Fig. A.2 Addition of two vectors. a Parallelogram rule, b triangle rule 326 Appendix A: Basic Operations of Tensor Algebra 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.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 n = a |a| (a) (b) b b a a n b a na a a a Fig. A.3 Scalar product of two vectors. a Angles between two vectors, b unit vector and projection Appendix A: Basic Operations of Tensor Algebra 327 (a) (b) (c) c c 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 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 |b|| cos ϕ|. A.2.4 Vector (Cross) Product of Two Vectors For the ordered pair of vectorsa andb the vectorc = a ×b is defined in two following steps (Zhilin 2001): • 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, • 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 (Zhilin 2001). 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 × (mvv)˙ , where r is the position of the mass point and v is the velocity of the mass point.
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