1.3 Cartesian Tensors a Second-Order Cartesian Tensor Is Defined As A

Total Page:16

File Type:pdf, Size:1020Kb

1.3 Cartesian Tensors a Second-Order Cartesian Tensor Is Defined As A 1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tijee i j . (1.3.1) The coefficients Tij are the components of T . A tensor exists independent of any coordinate system. The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to basis {ei} and components Tij with respect to basis {e i}, i.e., T T e e T e e . (1.3.2) pq p q ij i j From (1.3.2) and (1.2.4.6), Tpq ep eq TpqQipQjqei e j Tij e i e j . (1.3.3) Tij QipQjqTpq . (1.3.4) Similarly from (1.3.2) and (1.2.4.6) Tij e i e j Tij QipQjqep eq Tpqe p eq , (1.3.5) Tpq QipQjqTij . (1.3.6) Equations (1.3.4) and (1.3.6) are the transformation rules for changing second order tensor components under change of basis. In general Cartesian tensors of higher order can be expressed as T T e e ... e , (1.3.7) ij ...n i j n and the components transform according to Tijk ... QipQjqQkr ...Tpqr... , Tpqr ... QipQjqQkr ...Tijk ... (1.3.8) The tensor product S T of a CT(m) S and a CT(n) T is a CT(m+n) such that S T S T e e e e e e . i1i2 im j1j 2 jn i1 i2 im j1 j2 j n 1.3.1 Contraction T Consider the components i1i2 ip iq in of a CT(n). Set any two indices equal, iq ip say, and sum over ip from 1 to 3. Theses indices are contracted and the order of the tensor is reduced by two. Special cases: (a) Tensor product u v of two vectors become their dot product u v on contraction since uivj becomes uivi . April 15, 2016 1.3-1 (b) If T is a CT(2) with components Tij then this contracts to the scalar Tii . The scalar Tii is the trace of T and is denoted by trT. (It is a scalar invariant of T.) (c) IIf S and T are CT(2), then S T is a CT(4). Let SijTkl be the components of S T with respect to the basis {ei}. The indices can be contracted in many ways. For example SijTjl are the components of the CT(2) ST and this contracts to the scalar tr (ST) SijTji tr (TS). T T Similarly SijTkj are the components of ST where T denotes the transpose of T (i.e., TT T e e ). In addition tr (ST) tr (S TTT ) and tr (STT ) tr (S TT). ij j i If S T, then TT is denoted by T 2 , TT2 by T 3 etc. Consider the components vAuiijj= (where there is a repetition of only one index of the components of a tensor and a vector) and Aij= BC ik kj , (where there is a repetition of only one index of the components of two tensors). This will not be indicated by a dot when symbolic notation is used, i.e., for vu= A and ABC= , we do not write vu= A and A = BC , respectively. A double contraction of two tensors A and B will be denoted by A :B and is defined by AB: = tr()ABT = tr(BATTT ) = tr( AB ) = tr( B A ) , (1.3.1-a) = BA: See pgs. 13-14 and eqn (1.93) of Holzapfel (2001). The following properties can be deduced from eqn (1.3.1-a): A:(uvÄ= ) u(A v ), (1.3.2-b) ():()uvÄÄ= wx ()() uwvx, (1.3.3-c) See eqns (1.96) and (1.97) of Holzapfel (2001). Note: The symbol “:” denotes the contraction of a pair of repeated indices which appear in the same order, eg. AB: = AijB ij ; se= C : in component form is seij= C ijkl kl ; strain energy 1 ee::C in component form is 1 eeC . 2 2 ij ijkl kl (See pgs. 8-9 of Belytschko, Liu and Moran (2000)). April 15, 2016 1.3-2 .
Recommended publications
  • Arxiv:2012.13347V1 [Physics.Class-Ph] 15 Dec 2020
    KPOP E-2020-04 Bra-Ket Representation of the Inertia Tensor U-Rae Kim, Dohyun Kim, and Jungil Lee∗ KPOPE Collaboration, Department of Physics, Korea University, Seoul 02841, Korea (Dated: June 18, 2021) Abstract We employ Dirac's bra-ket notation to define the inertia tensor operator that is independent of the choice of bases or coordinate system. The principal axes and the corresponding principal values for the elliptic plate are determined only based on the geometry. By making use of a general symmetric tensor operator, we develop a method of diagonalization that is convenient and intuitive in determining the eigenvector. We demonstrate that the bra-ket approach greatly simplifies the computation of the inertia tensor with an example of an N-dimensional ellipsoid. The exploitation of the bra-ket notation to compute the inertia tensor in classical mechanics should provide undergraduate students with a strong background necessary to deal with abstract quantum mechanical problems. PACS numbers: 01.40.Fk, 01.55.+b, 45.20.dc, 45.40.Bb Keywords: Classical mechanics, Inertia tensor, Bra-ket notation, Diagonalization, Hyperellipsoid arXiv:2012.13347v1 [physics.class-ph] 15 Dec 2020 ∗Electronic address: [email protected]; Director of the Korea Pragmatist Organization for Physics Educa- tion (KPOP E) 1 I. INTRODUCTION The inertia tensor is one of the essential ingredients in classical mechanics with which one can investigate the rotational properties of rigid-body motion [1]. The symmetric nature of the rank-2 Cartesian tensor guarantees that it is described by three fundamental parameters called the principal moments of inertia Ii, each of which is the moment of inertia along a principal axis.
    [Show full text]
  • On Manifolds of Tensors of Fixed Tt-Rank
    ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest.
    [Show full text]
  • Matrices and Tensors
    APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM..
    [Show full text]
  • 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.
    [Show full text]
  • Tensor-Spinor Theory of Gravitation in General Even Space-Time Dimensions
    Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor.
    [Show full text]
  • Tensors (Draft Copy)
    TENSORS (DRAFT COPY) LARRY SUSANKA Abstract. The purpose of this note is to define tensors with respect to a fixed finite dimensional real vector space and indicate what is being done when one performs common operations on tensors, such as contraction and raising or lowering indices. We include discussion of relative tensors, inner products, symplectic forms, interior products, Hodge duality and the Hodge star operator and the Grassmann algebra. All of these concepts and constructions are extensions of ideas from linear algebra including certain facts about determinants and matrices, which we use freely. None of them requires additional structure, such as that provided by a differentiable manifold. Sections 2 through 11 provide an introduction to tensors. In sections 12 through 25 we show how to perform routine operations involving tensors. In sections 26 through 28 we explore additional structures related to spaces of alternating tensors. Our aim is modest. We attempt only to create a very structured develop- ment of tensor methods and vocabulary to help bridge the gap between linear algebra and its (relatively) benign notation and the vast world of tensor ap- plications. We (attempt to) define everything carefully and consistently, and this is a concise repository of proofs which otherwise may be spread out over a book (or merely referenced) in the study of an application area. Many of these applications occur in contexts such as solid-state physics or electrodynamics or relativity theory. Each subject area comes equipped with its own challenges: subject-specific vocabulary, traditional notation and other conventions. These often conflict with each other and with modern mathematical practice, and these conflicts are a source of much confusion.
    [Show full text]
  • Low-Level Image Processing with the Structure Multivector
    Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters.
    [Show full text]
  • 1 Vectors & Tensors
    1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.
    [Show full text]
  • PHYSICS 116A Homework 9 Solutions 1. Boas, Problem 3.12–4
    PHYSICS 116A Homework 9 Solutions 1. Boas, problem 3.12–4. Find the equations of the following conic, 2 2 3x + 8xy 3y = 8 , (1) − relative to the principal axes. In matrix form, Eq. (1) can be written as: 3 4 x (x y) = 8 . 4 3 y − I could work out the eigenvalues by solving the characteristic equation. But, in this case I can work them out by inspection by noting that for the matrix 3 4 M = , 4 3 − we have λ1 + λ2 = Tr M = 0 , λ1λ2 = det M = 25 . − It immediately follows that the two eigenvalues are λ1 = 5 and λ2 = 5. Next, we compute the − eigenvectors. 3 4 x x = 5 4 3 y y − yields one independent relation, x = 2y. Thus, the normalized eigenvector is x 1 2 = . y √5 1 λ=5 Since M is a real symmetric matrix, the two eigenvectors are orthogonal. It follows that the second normalized eigenvector is: x 1 1 = − . y − √5 2 λ= 5 The two eigenvectors form the columns of the diagonalizing matrix, 1 2 1 C = − . (2) √5 1 2 Since the eigenvectors making up the columns of C are real orthonormal vectors, it follows that C is a real orthogonal matrix, which satisfies C−1 = CT. As a check, we make sure that C−1MC is diagonal. −1 1 2 1 3 4 2 1 1 2 1 10 5 5 0 C MC = − = = . 5 1 2 4 3 1 2 5 1 2 5 10 0 5 − − − − − 1 Following eq. (12.3) on p.
    [Show full text]
  • Curvilinear Coordinates
    UNM SUPPLEMENTAL BOOK DRAFT June 2004 Curvilinear Analysis in a Euclidean Space Presented in a framework and notation customized for students and professionals who are already familiar with Cartesian analysis in ordinary 3D physical engineering space. Rebecca M. Brannon Written by Rebecca Moss Brannon of Albuquerque NM, USA, in connection with adjunct teaching at the University of New Mexico. This document is the intellectual property of Rebecca Brannon. Copyright is reserved. June 2004 Table of contents PREFACE ................................................................................................................. iv Introduction ............................................................................................................. 1 Vector and Tensor Notation ........................................................................................................... 5 Homogeneous coordinates ............................................................................................................. 10 Curvilinear coordinates .................................................................................................................. 10 Difference between Affine (non-metric) and Metric spaces .......................................................... 11 Dual bases for irregular bases .............................................................................. 11 Modified summation convention ................................................................................................... 15 Important notation
    [Show full text]
  • The Riemann Curvature Tensor
    The Riemann Curvature Tensor Jennifer Cox May 6, 2019 Project Advisor: Dr. Jonathan Walters Abstract A tensor is a mathematical object that has applications in areas including physics, psychology, and artificial intelligence. The Riemann curvature tensor is a tool used to describe the curvature of n-dimensional spaces such as Riemannian manifolds in the field of differential geometry. The Riemann tensor plays an important role in the theories of general relativity and gravity as well as the curvature of spacetime. This paper will provide an overview of tensors and tensor operations. In particular, properties of the Riemann tensor will be examined. Calculations of the Riemann tensor for several two and three dimensional surfaces such as that of the sphere and torus will be demonstrated. The relationship between the Riemann tensor for the 2-sphere and 3-sphere will be studied, and it will be shown that these tensors satisfy the general equation of the Riemann tensor for an n-dimensional sphere. The connection between the Gaussian curvature and the Riemann curvature tensor will also be shown using Gauss's Theorem Egregium. Keywords: tensor, tensors, Riemann tensor, Riemann curvature tensor, curvature 1 Introduction Coordinate systems are the basis of analytic geometry and are necessary to solve geomet- ric problems using algebraic methods. The introduction of coordinate systems allowed for the blending of algebraic and geometric methods that eventually led to the development of calculus. Reliance on coordinate systems, however, can result in a loss of geometric insight and an unnecessary increase in the complexity of relevant expressions. Tensor calculus is an effective framework that will avoid the cons of relying on coordinate systems.
    [Show full text]
  • Tensor Categorical Foundations of Algebraic Geometry
    Tensor categorical foundations of algebraic geometry Martin Brandenburg { 2014 { Abstract Tannaka duality and its extensions by Lurie, Sch¨appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocom- plete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl¨ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories. arXiv:1410.1716v1 [math.AG] 7 Oct 2014 This is a slightly expanded version of the author's PhD thesis. Contents 1 Introduction 1 1.1 Background . .1 1.2 Results . .3 1.3 Acknowledgements . 13 2 Preliminaries 14 2.1 Category theory . 14 2.2 Algebraic geometry . 17 2.3 Local Presentability . 21 2.4 Density and Adams stacks . 22 2.5 Extension result . 27 3 Introduction to cocomplete tensor categories 36 3.1 Definitions and examples . 36 3.2 Categorification . 43 3.3 Element notation . 46 3.4 Adjunction between stacks and cocomplete tensor categories . 49 4 Commutative algebra in a cocomplete tensor category 53 4.1 Algebras and modules . 53 4.2 Ideals and affine schemes .
    [Show full text]