Stress Energy Tensor

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  Dynamics of Classical and Quantum Fields

  An Introduction

  • Girish S. Setlur(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  Define ( T 1 a , T 2 a , T 3 a ) = T a . 1 c 2 ∂ S a ∂ t + ∇ · T a = 0 (3.135) Here u is the energy (density), S is the energy flux, and T a is the flux of the energy flux. This is because not only is the total energy in a volume conserved, the total momentum of the EM radiation ( R d 3 r S ) is also conserved. Figure 3.3: The meaning of the various components of the Stress Energy Tensor. There are other possible symmetries that one may consider. Some of them lead to trivial conservation laws. Consider a rotation in three dimensions so that r → r 0 ≡ M r , A ( r , t ) → A 0 ( r 0 , t ) ≡ M A ( r , t ) , and φ ( r , t ) → φ 0 ( r 0 , t ) = φ ( r , t ) , where M is an orthogonal matrix independent of position and time. Choosing an appropriate gauge such as φ ≡ 0 and ∇ · A = 0 (radiation gauge), it is easy to convince oneself that the conserved quantity has the expression P = R d 3 r ∂ t A × A . It is also easy to convince oneself that this quantity vanishes identically. Hence, this does not yield a constant of the motion. 74 Field Theory Find the energy momentum tensor of a particle of rest mass m moving with velocity v . The simplest way to do this is to use the tensor nature of this quantity. This tensor, being of rank two, transforms as the product of two coordinates under Lorentz transformation, T μ ν ( x ) = Λ μ ρ Λ ν σ T 0 ρσ ( x 0 ) , (3.136) where Λ μ ρ ≡ ∂ x 0 μ ∂ x ρ and summation over repeated indices is implied. Let us imagine that the reference frame of the label x 0 is one where the particle is at rest. In this case, the tensor has only one component viz. the time–time component equal to the energy density. T 0 ρσ ( x ) = δ ρ , 0 δ σ , 0 mc 2 δ ( r 0 ) (3.137) Now imagine that we view this particle moving with some velocity v in the positive x-direction, then the energy momentum tensor in this frame would be, T μ ν ( x 0 ) = Λ μ 0 Λ ν 0 mc 2 δ ( r 0 ) .

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  Concepts and Methods in Modern Theoretical Chemistry

  Electronic Structure and Reactivity

  • Swapan Kumar Ghosh, Pratim Kumar Chattaraj(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  235 12 Electronic Stress with Spin Vorticity Akitomo Tachibana 12.1 INTRODUCTION The study of the density functional theory by Hohenberg–Kohn has invoked the new idea of energy in terms of electron density. 1 A natural outcome is the concept of energy density, which has been published in a recent paper of stress tensor and the ref-erences cited therein. 2 Electronic stress tensor plays a very significant role as has been originally formulated and reviewed by Pauli 3 in his textbook of quantum mechanics. It has been shown in the preceding paper that the QED electronic stress ten-sor plays a fundamentally important role in order to understand the electron spin dynamics; the spin torque and zeta force originated from the chiral nature of electron that is intrinsic to the spin-1/2 Fermion. 4 The dynamics of electron spin with the realization of spin–orbit coupling has recently been of keen interest, particularly in the field of spin torque transfer in spintronics; see recent review and the references cited therein. 5 It shall be formulated here the energy density concept in terms of stress tensor in general relativity for the unified treatment of spin dynamics and chemical reaction dynamics. CONTENTS 12.1 Introduction .................................................................................................. 235 12.2 Spin Vorticity Principle and Energy Density by General Relativity ............ 236 12.2.1 Variation Principle of QED in Curved Spacetime ........................... 236 12.2.2 Quantum Electron Spin Vorticity Principle ..................................... 238 12.2.3 Energy Density ................................................................................. 239 12.3 Rigged Field Theory ..................................................................................... 241 12.3.1 Rigged QED ......................................................................................

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  Principles of Continuum Mechanics

  Conservation and Balance Laws with Applications

  • J. N. Reddy(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 STRESS VECTOR AND STRESS TENSOR A man may imagine things that are false, but he can only understand things that are true, for if the things be false, the apprehension of them is not understanding. Isaac Newton If a man is in too big a hurry to give up an error he is liable to give up some truth with it. Wilbur Wright 4.1 Introduction In the beginning of Chapter 3, we briefly discussed the need to study deformation in materials that we may design for engineering applications. All materials have a certain threshold to withstand forces, beyond which they “fail” to perform their intended function. The force per unit area, called stress , is a measure of the capacity of the material to carry loads, and all designs are based on the criterion that the materials used have the capacity to carry the working loads of the system. Thus, it is necessary to determine the state of stress in materials that are used in a system. In the present chapter we study the concept of stress and its various measures. For instance, stress can be measured as force (that occurs inside a deformed body) per unit deformed area or undeformed area. Stress at a point on the surface and at a point inside a three-dimensional continuum are measured using different entities. The stress at a point on the surface is measured in terms of force per unit area and depends on (magnitude and direction of) the force vector as well as the plane on which the force is acting. Therefore, stress defined at a point on the surface is a vector. As we shall see shortly, stress at a point inside the body can be measured in terms of nine quantities, three per plane, on three mutually perpendicular planes at the point. These nine quantities may be viewed as the components of a second-order tensor, called a stress tensor . One may suspect that the stress vector defined at a point on the surface of a continuum is related to the stress tensor defined inside the continuum at the point.

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  Advanced Mechanics of Materials and Applied Elasticity

  • Anthony E. Armenakas(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 2 Strain and Stress Tensors 2.1 The Continuum Model Bodies are composed of a large number of discrete particles (atoms, molecules) in constant motion. Solids differ from liquids and liquids from gases in the spacing of these particles and in the amplitude of their motion. In studying the behavior of bodies the assumption is usually made that the material is distributed in the space which it occupies without leaving gaps or empty spaces. In other words, it is assumed that at every instant of time, there is a particle at every point of the space occupied by the body at that time . This model is referred to in the literature as continuum and it is used in all engineering disciplines because it is mathematically convenient. It permits integration and differentiation of the quantities describing the behavior of a body which are functions of the space coordinates. An infinitesimal portion of a continuum is called a particle . In this text we study the behavior of deformable solid bodies subjected to external loads on the basis of the continuum model. 2.2 External Loads Consider a body initially in a reference undeformed and unstressed state of 0 mechanical and thermal equilibrium at the uniform temperature T . In this state the † †† body is not subjected to external loads and heat does not flow in or out of it because its temperature is uniform. Subsequently, the body is subjected to one or more of the following external loads: 1 . Body forces 2 . Surface forces 1 2 3 3 . A temperature field T ( x , x , x ) ††† 4. Specified components of displacements of some particles of the surface of the body † When a body is in a state of mechanical equilibrium, its particles do not accelerate; that is, the sum of the forces acting on any portion of the body and the sum of their moments about any point vanish. †† When a body is in a state of thermal equilibrium, heat does not flow in or out of it; that is, the temperature of all its particles is the same.

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  Elasticity

  Theory and Applications

  • Adel S. Saada, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  PART II THEORY OF STRESS This page intentionally left blank CHAPTER 7 ANALYSIS OF STRESS 7.1 Introduction When a body is subjected to external forces, its behavior depends upon the magnitude of the forces, upon their direction, and upon the inherent strength of the material of which it is made. Structural and mechanical construction units are usually subjected to various combi-nations of forces, some having more detrimental effects than others. It is therefore necessary to consider how forces are transmitted through the material constituting these units. In this chapter, the concepts of stress vector on a surface and state of stress at a point will be introduced. It will be shown that the compo-nents of the stress vector can be obtained through a linear symmetric transformation with a matrix whose elements are the components of a tensor of rank two called the stress tensor. All the properties of linear symmetric transformations will be applied to stress the same way they have been applied to linear strain. However, while the components of the linear strain tensor have to satisfy six compatibility relations of the second order, it will be shown that the components of the stress tensor must satisfy three partial differential equations of the first order, called the differential equations of equilibrium. These equations will be derived in both cartesian and orthogonal curvilinear coordinate sys-tems. 7.2 Stress on a Plane at a Point. Notation and Sign Convention Let us consider a body in equilibrium under a system of external forces Q x . . . Q n , and let us pass a fictitious plane P through a point O 147 148 Theory of Stress Fig. 7.1 in the interior of this body (Fig. 7.1). Part A of the body is in equilibrium under ~Q, Q 2 , Q 3 , and the effect of part B. We shall assume this effect is continuously distributed over the surface of intersection. Around the point 0 , let us consider a small surface AA and an outward unit normal vector n.

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  General Relativity

  Basics and Beyond

  • Ghanashyam Date(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Even though we can have very complicated physical stress tensor arising out of different species of matter, we expect certain general properties to be satisfied by any such tensor. These stipulations go under the name of energy conditions . Below are commonly used energy conditions [17]. Weak T μν v μ v ν ≥ 0 ∀ v · v < 0 Strong T μν v μ v ν ≥ -1 2 T αβ g αβ ∀ v · v = -1 Dominant -T μ ν v ν is future directed causal ∀ v future directed causal. There is also the Null energy condition which replaces the time-like v μ by a light-like k μ . To appreciate their implications, it is useful to have an algebraic classifi-cation of the stress tensor, which allows us to put the stress tensor in some canonical forms. The classification holds point-wise in the space-time. Math-ematically, one considers the eigenvalue equation: T μ ν X ν = λX μ or equiva-lently, T μν X ν = λg μν X ν . Note that T μ ν = g μα T αν is not a symmetric matrix, only T μν = T νμ . Hence, diagonalizability of T μ ν is not assured. Secondly, the metric is Lorentzian which means that eigenvectors X μ can be time-like and light-like as well. Nevertheless, symmetric nature of the stress tensor implies Dynamics of Space-Time 45 that eigenvectors corresponding to distinct eigenvalues are necessarily orthog-onal. This implies that among the eigenvectors, there can be at the most only one time-like eigenvector. Likewise, since no two distinct light-like vectors can be orthogonal, if they are eigenvectors, they must have the same eigenvalue. The classification is now arranged according to number of distinct eigenval-ues [18]. Type I: Four distinct eigenvalues. This implies 4 orthogonal eigenvectors. If one of these is light-like, then the remaining three must be space-like and this is impossible. Hence the eigenvectors must form a (pseudo-)orthonormal basis, E a μ , a = 0 , 1 , 2 , 3 , E a · E b = η ab . The T μ ν is diagonalizable and can be expressed as, T μν = -ρE 0 μ E 0 ν + 3 X i =1 p i E i μ E i ν .

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  Introduction to Continuum Mechanics

  • W Michael Lai, David Rubin, Erhard Krempl, David H. Rubin(Authors)
  • 2009(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  CHAPTER 4 Stress and Integral Formulations of General Principles

  In the previous chapter, we considered the purely kinematic description of the motion of a continuum without any consideration of the forces that cause the motion and deformation. In this chapter, we consider a means of describing the forces in the interior of a body idealized as a continuum. It is generally accepted that matter is formed of molecules, which in turn consist of atoms and subatomic particles. Therefore, the internal forces in real matter are those between these particles. In the classical continuum theory where matter is assumed to be continuously distributed, the forces acting at every point inside a body are introduced through the concept of body forces and surface forces. Body forces are those that act throughout a volume (e.g., gravity, electrostatic force) by a long-range interaction with matter or charges at a distance. Surface forces are those that act on a surface (real or imagined), separating parts of the body. We assume that it is adequate to describe the surface forces at a point on a surface through the definition of a stress vector , discussed in Section 4.1 , which pays no attention to the curvature of the surface at the point. Such an assumption is known as Cauchy’s stress principle and is one of the basic axioms of classical continuum mechanics.

  4.1 STRESS VECTOR

  Let us consider a body depicted in Figure 4.1-1 . Imagine a plane such as S , which passes through an arbitrary internal point P and which has aunit normal vector n . The plane cuts the body into two portions. One portion lies on the side of the arrow of n (designated by II in the figure) and the other portion on the tail of n (designated by I). Considering portion I as a free body, there will be on plane S a resultant force ΔF acting on a small area ΔA containing P . We define the stress vector (acting from II to I) at the point P on the plane S as the limit of the ratio ΔF /ΔA as ΔA → 0. That is, with

  tn

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  Geometry of Incompatible Deformations

  Differential Geometry in Continuum Mechanics

  • (Author)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  7 Stress Measures The conventional representations of stresses (Chapter 3) substantially use the Eu-clidean-affine structure of physical space, which contains both the reference and the actual shapes of the body. Below we list some characteristic features of this approach. ∙ The existence of the Cauchy stress tensor T is stated via the classical “Cauchy tetrahedron argument” that is based on the Euclidean structure. ∙ Integrals that represent the resulting contact forces essentially use the Euclidean parallel translation rule. Since contact force densities take values in V , one can sum these values taken at different points. Such summation is required by the definition of the integration in Euclidean space. ∙ Let S denote ∂ 𝜘 R ( P ) or ∂ 𝜘( P ) , where P ∈ Part ( B ) . For a point x ∈ S , one has the decomposition T x S ⊕ ( T x S ) ⊥ = V . Such a direct sum involves the Euclidean structure of V . Thus, n x denotes the unit normal vector, the element of ( T x S ) ⊥ . In this chapter, the notion of stresses is generalized on smooth manifolds. We follow the ideas set out in [124]. 7.1 Concentrated Forces and Force Densities Concentrated force. Let M = {𝜘 t } t ∈𝕋 be a smooth motion and X ∈ B be a material point. A concentrated force , which acts on the material point X at instant t ∈ 𝕋 , is a covector f t ∈ T ∗ 𝜘 t ( X ) P . Suppose that a family { f t } t ∈𝕋 of concentrated forces, which act on the material point X , corresponds to the given motion M . The motion M , in turn, generates the field { V M ; t } t ∈𝕋 of velocities V M ; t ∈ Sec (𝜘 ∗ t T P ) . For each t ∈ 𝕋 , the scalar f t ⌞ V M ; t ( X ) can be interpreted as the power that the force f expends at instant t . Finally, for each instant t ∈ 𝕋 , a concentrated force that acts on some material point at that instant is represented by an element of the pullback bundle 𝜘 ∗ t T P . Force density. As is usual in continuum mechanics, we assume that forces that act on the body B are divided into two groups.

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