On the Notion of Stress in Classical Continuum Mechanics
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All-Master-File-Problem-Set
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 THE CITY UNIVERSITY OF NEW YORK First Examination for PhD Candidates in Physics Analytical Dynamics Summer 2010 Do two of the following three problems. Start each problem on a new page. Indicate clearly which two problems you choose to solve. If you do not indicate which problems you wish to be graded, only the first two problems will be graded. Put your identification number on each page. 1. A particle moves without friction on the axially symmetric surface given by 1 z = br2, x = r cos φ, y = r sin φ 2 where b > 0 is constant and z is the vertical direction. The particle is subject to a homogeneous gravitational force in z-direction given by −mg where g is the gravitational acceleration. (a) (2 points) Write down the Lagrangian for the system in terms of the generalized coordinates r and φ. (b) (3 points) Write down the equations of motion. (c) (5 points) Find the Hamiltonian of the system. (d) (5 points) Assume that the particle is moving in a circular orbit at height z = a. Obtain its energy and angular momentum in terms of a, b, g. (e) (10 points) The particle in the horizontal orbit is poked downwards slightly. Obtain the frequency of oscillation about the unperturbed orbit for a very small oscillation amplitude. 2. Use relativistic dynamics to solve the following problem: A particle of rest mass m and initial velocity v0 along the x-axis is subject after t = 0 to a constant force F acting in the y-direction. -
1.2 Topological Tensor Calculus
PH211 Physical Mathematics Fall 2019 1.2 Topological tensor calculus 1.2.1 Tensor fields Finite displacements in Euclidean space can be represented by arrows and have a natural vector space structure, but finite displacements in more general curved spaces, such as on the surface of a sphere, do not. However, an infinitesimal neighborhood of a point in a smooth curved space1 looks like an infinitesimal neighborhood of Euclidean space, and infinitesimal displacements dx~ retain the vector space structure of displacements in Euclidean space. An infinitesimal neighborhood of a point can be infinitely rescaled to generate a finite vector space, called the tangent space, at the point. A vector lives in the tangent space of a point. Note that vectors do not stretch from one point to vector tangent space at p p space Figure 1.2.1: A vector in the tangent space of a point. another, and vectors at different points live in different tangent spaces and so cannot be added. For example, rescaling the infinitesimal displacement dx~ by dividing it by the in- finitesimal scalar dt gives the velocity dx~ ~v = (1.2.1) dt which is a vector. Similarly, we can picture the covector rφ as the infinitesimal contours of φ in a neighborhood of a point, infinitely rescaled to generate a finite covector in the point's cotangent space. More generally, infinitely rescaling the neighborhood of a point generates the tensor space and its algebra at the point. The tensor space contains the tangent and cotangent spaces as a vector subspaces. A tensor field is something that takes tensor values at every point in a space. -
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.. -
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. -
Cauchy Tetrahedron Argument and the Proofs of the Existence of Stress Tensor, a Comprehensive Review, Challenges, and Improvements
CAUCHY TETRAHEDRON ARGUMENT AND THE PROOFS OF THE EXISTENCE OF STRESS TENSOR, A COMPREHENSIVE REVIEW, CHALLENGES, AND IMPROVEMENTS EHSAN AZADI1 Abstract. In 1822, Cauchy presented the idea of traction vector that contains both the normal and tangential components of the internal surface forces per unit area and gave the tetrahedron argument to prove the existence of stress tensor. These great achievements form the main part of the foundation of continuum mechanics. For about two centuries, some versions of tetrahedron argument and a few other proofs of the existence of stress tensor are presented in every text on continuum mechanics, fluid mechanics, and the relevant subjects. In this article, we show the birth, importance, and location of these Cauchy's achievements, then by presenting the formal tetrahedron argument in detail, for the first time, we extract some fundamental challenges. These conceptual challenges are related to the result of applying the conservation of linear momentum to any mass element, the order of magnitude of the surface and volume terms, the definition of traction vectors on the surfaces that pass through the same point, the approximate processes in the derivation of stress tensor, and some others. In a comprehensive review, we present the different tetrahedron arguments and the proofs of the existence of stress tensor, discuss the challenges in each one, and classify them in two general approaches. In the first approach that is followed in most texts, the traction vectors do not exactly define on the surfaces that pass through the same point, so most of the challenges hold. But in the second approach, the traction vectors are defined on the surfaces that pass exactly through the same point, therefore some of the relevant challenges are removed. -
Lecture 1: Introduction
Lecture 1: Introduction E. J. Hinch Non-Newtonian fluids occur commonly in our world. These fluids, such as toothpaste, saliva, oils, mud and lava, exhibit a number of behaviors that are different from Newtonian fluids and have a number of additional material properties. In general, these differences arise because the fluid has a microstructure that influences the flow. In section 2, we will present a collection of some of the interesting phenomena arising from flow nonlinearities, the inhibition of stretching, elastic effects and normal stresses. In section 3 we will discuss a variety of devices for measuring material properties, a process known as rheometry. 1 Fluid Mechanical Preliminaries The equations of motion for an incompressible fluid of unit density are (for details and derivation see any text on fluid mechanics, e.g. [1]) @u + (u · r) u = r · S + F (1) @t r · u = 0 (2) where u is the velocity, S is the total stress tensor and F are the body forces. It is customary to divide the total stress into an isotropic part and a deviatoric part as in S = −pI + σ (3) where tr σ = 0. These equations are closed only if we can relate the deviatoric stress to the velocity field (the pressure field satisfies the incompressibility condition). It is common to look for local models where the stress depends only on the local gradients of the flow: σ = σ (E) where E is the rate of strain tensor 1 E = ru + ruT ; (4) 2 the symmetric part of the the velocity gradient tensor. The trace-free requirement on σ and the physical requirement of symmetry σ = σT means that there are only 5 independent components of the deviatoric stress: 3 shear stresses (the off-diagonal elements) and 2 normal stress differences (the diagonal elements constrained to sum to 0). -
General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ). -
Tensor Calculus and Differential Geometry
Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane . -
Hamilton's Principle in Continuum Mechanics
Hamilton’s Principle in Continuum Mechanics A. Bedford University of Texas at Austin This document contains the complete text of the monograph published in 1985 by Pitman Publishing, Ltd. Copyright by A. Bedford. 1 Contents Preface 4 1 Mechanics of Systems of Particles 8 1.1 The First Problem of the Calculus of Variations . 8 1.2 Conservative Systems . 12 1.2.1 Hamilton’s principle . 12 1.2.2 Constraints.......................... 15 1.3 Nonconservative Systems . 17 2 Foundations of Continuum Mechanics 20 2.1 Mathematical Preliminaries . 20 2.1.1 Inner Product Spaces . 20 2.1.2 Linear Transformations . 22 2.1.3 Functions, Continuity, and Differentiability . 24 2.1.4 Fields and the Divergence Theorem . 25 2.2 Motion and Deformation . 27 2.3 The Comparison Motion . 32 2.4 The Fundamental Lemmas . 36 3 Mechanics of Continuous Media 39 3.1 The Classical Theories . 40 3.1.1 IdealFluids.......................... 40 3.1.2 ElasticSolids......................... 46 3.1.3 Inelastic Materials . 50 3.2 Theories with Microstructure . 54 3.2.1 Granular Solids . 54 3.2.2 Elastic Solids with Microstructure . 59 2 4 Mechanics of Mixtures 65 4.1 Motions and Comparison Motions of a Mixture . 66 4.1.1 Motions............................ 66 4.1.2 Comparison Fields . 68 4.2 Mixtures of Ideal Fluids . 71 4.2.1 Compressible Fluids . 71 4.2.2 Incompressible Fluids . 73 4.2.3 Fluids with Microinertia . 75 4.3 Mixture of an Ideal Fluid and an Elastic Solid . 83 4.4 A Theory of Mixtures with Microstructure . 86 5 Discontinuous Fields 91 5.1 Singular Surfaces . -
Navier-Stokes-Equation
Math 613 * Fall 2018 * Victor Matveev Derivation of the Navier-Stokes Equation 1. Relationship between force (stress), stress tensor, and strain: Consider any sub-volume inside the fluid, with variable unit normal n to the surface of this sub-volume. Definition: Force per area at each point along the surface of this sub-volume is called the stress vector T. When fluid is not in motion, T is pointing parallel to the outward normal n, and its magnitude equals pressure p: T = p n. However, if there is shear flow, the two are not parallel to each other, so we need a marix (a tensor), called the stress-tensor , to express the force direction relative to the normal direction, defined as follows: T Tn or Tnkjjk As we will see below, σ is a symmetric matrix, so we can also write Tn or Tnkkjj The difference in directions of T and n is due to the non-diagonal “deviatoric” part of the stress tensor, jk, which makes the force deviate from the normal: jkp jk jk where p is the usual (scalar) pressure From general considerations, it is clear that the only source of such “skew” / ”deviatoric” force in fluid is the shear component of the flow, described by the shear (non-diagonal) part of the “strain rate” tensor e kj: 2 1 jk2ee jk mm jk where euujk j k k j (strain rate tensro) 3 2 Note: the funny construct 2/3 guarantees that the part of proportional to has a zero trace. The two terms above represent the most general (and the only possible) mathematical expression that depends on first-order velocity derivatives and is invariant under coordinate transformations like rotations. -
Stress Components Cauchy Stress Tensor
Mechanics and Design Chapter 2. Stresses and Strains Byeng D. Youn System Health & Risk Management Laboratory Department of Mechanical & Aerospace Engineering Seoul National University Seoul National University CONTENTS 1 Traction or Stress Vector 2 Coordinate Transformation of Stress Tensors 3 Principal Axis 4 Example 2019/1/4 Seoul National University - 2 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction Vector Consider a surface element, ∆ S , of either the bounding surface of the body or the fictitious internal surface of the body as shown in Fig. 2.1. Assume that ∆ S contains the point. The traction vector, t, is defined by Δf t = lim (2-1) ∆→S0∆S Fig. 2.1 Definition of surface traction 2019/1/4 Seoul National University - 3 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction Vector (Continued) It is assumed that Δ f and ∆ S approach zero but the fraction, in general, approaches a finite limit. An even stronger hypothesis is made about the limit approached at Q by the surface force per unit area. First, consider several different surfaces passing through Q all having the same normal n at Q as shown in Fig. 2.2. Fig. 2.2 Traction vector t and vectors at Q Then the tractions on S , S ′ and S ′′ are the same. That is, the traction is independent of the surface chosen so long as they all have the same normal. 2019/1/4 Seoul National University - 4 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Stress vectors on three coordinate plane Let the traction vectors on planes perpendicular to the coordinate axes be t(1), t(2), and t(3) as shown in Fig. -
Leonhard Euler Moriam Yarrow
Leonhard Euler Moriam Yarrow Euler's Life Leonhard Euler was one of the greatest mathematician and phsysicist of all time for his many contributions to mathematics. His works have inspired and are the foundation for modern mathe- matics. Euler was born in Basel, Switzerland on April 15, 1707 AD by Paul Euler and Marguerite Brucker. He is the oldest of five children. Once, Euler was born his family moved from Basel to Riehen, where most of his childhood took place. From a very young age Euler had a niche for math because his father taught him the subject. At the age of thirteen he was sent to live with his grandmother, where he attended the University of Basel to receive his Master of Philosphy in 1723. While he attended the Universirty of Basel, he studied greek in hebrew to satisfy his father. His father wanted to prepare him for a career in the field of theology in order to become a pastor, but his friend Johann Bernouilli convinced Euler's father to allow his son to pursue a career in mathematics. Bernoulli saw the potentional in Euler after giving him lessons. Euler received a position at the Academy at Saint Petersburg as a professor from his friend, Daniel Bernoulli. He rose through the ranks very quickly. Once Daniel Bernoulli decided to leave his position as the director of the mathmatical department, Euler was promoted. While in Russia, Euler was greeted/ introduced to Christian Goldbach, who sparked Euler's interest in number theory. Euler was a man of many talents because in Russia he was learning russian, executed studies on navigation and ship design, cartography, and an examiner for the military cadet corps.