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A Some Basic Rules of Tensor Calculus

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A Some Basic Rules of Tensor Calculus A Some Basic Rules of Tensor Calculus 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) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. 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) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. 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 a = aig , A = aibj + ... + cidj g g i i ⊗ j 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 a sum of their coordinates ci = ai + bi. The introduced basis remains in the background. It must be remembered that a change of the coordinate system leads to the change of the components of tensors. In this work 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 168 A SomeBasicRulesofTensorCalculus give a brief guide to notations and rules of the tensor calculus applied through- out this work. For more comprehensive overviews on tensor calculus we recom- mend [54, 96, 123, 191, 199, 311, 334]. The calculus of matrices is presented in [40, 111, 340], for example. Section A.1 provides a brief overview of basic alge- braic operations with vectors and second rank tensors. Several rules from tensor analysis are summarized in Sect. A.2. Basic sets of invariants for different groups of symmetry transformation are presented in Sect. A.3, where a novel approach to find the functional basis is discussed. A.1 Basic Operations of Tensor Algebra A.1.1 Polar and Axial Vectors A vector in the three-dimensional Euclidean space is defined as a directed line seg- ment with specified magnitude (scalar) 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 char- acterize 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 corresponding length. For the given spin vector a the directed line segment a is introduced according to the following rules [334]: ∗ 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 , ∗ a b c a a a a ∗ ∗ ∗ a Figure A.1 Axial vectors. a Spin vector, b axial vector in the right-screw oriented reference frame, c axial vector in the left-screw oriented reference frame A.1 Basic Operations of Tensor Algebra 169 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 3. corresponds to the convention of ori- entation of the reference frame [334] (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 refer- ence 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. [333], in the theories of rods, plates and shells, e.g. [25], in the asymmetric theory of elasticity, e.g. [231], as well as in dynamics of micro-polar media, e.g. [108]. By 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.1.2 Operations with Vectors 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 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 Figure A.2 Addition of two vectors. a Parallelogram rule, b triangle rule 170 A SomeBasicRulesofTensorCalculus a b b b ϕ ϕ a a 2π ϕ a − na = (b na )na a · Figure A.3 Scalar product of two vectors. a Angles| | between two vectors, b unit vector and projection 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. As ϕ 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 n )n , Fig. A.3b. · a a The length of the projection is b cos ϕ . · | || | 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 [334]: × – 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 Subsect. A.1.1. 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, [334]. If a and×b are polar vectors the result of the cross product A.1 Basic Operations of Tensor Algebra 171 a b c c c ∗ ϕ ϕ ϕ a b a b a b Figure 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 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. The mixed product of three vectors a, b and c is defined by (a b) c. The result is a scalar. For the mixed product the following identities are valid× · a (b c)= b (c a)= c (a b) (A.1.1) · × · × · × If the cross product is applied twice, the first operation must be set in parentheses, e.g., a (b c).
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