DOCSLIB.ORG

Math 575-Lecture 2 1 Conservation of Momentum and Cauchy Stress Tensor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 575-Lecture 2 1 Conservation of Momentum and Cauchy Stress Tensor Math 575-Lecture 2 1 Conservation of momentum and Cauchy stress tensor Consider a fluid parcel with volume V . The force acting on it includes the body force and surface force F = F v + F S The body force usually comes from external field. For example, if there is only gravitational force, then Z F v = ρgdV: V The interesting part is the surface force, which is the forces exerted across the boundary of the volume by the neighboring material, called the stress. Let T be the stress vector (force per unit area, also called traction vector), the total surface force is given by Z F S = T dS @V Now, consider the material volume Rt. By conservation of momentum, we have d Z Z Z ρudV = ρbdV + T dS (1.1) dt Rt Rt @Rt where b is some general body force density. By the second convection identity (if you feel confused, consider each component) to the first term: d Z Z Z Du ρudV = (ρu)t + r · (u ⊗ (ρu))dV = ρ dV dt Rt Rt Rt Dt Du This actually makes perfect sense. Dt is the acceleration and thus the right hand side is the integrals of the forces of fluid elements, which should be total force. As a result, we have Z Du Z ρ − ρbdV = T dS: Rt Dt @Rt As long as we have this, it is clear that this equality should be true for any volume V , since any volume can be regarded as some material value at time t. Let V to be a tetrahedron with one vertex at x and the three edges origin from x are parallel to coordinate axes. The traction T should depend on x; t and the direction n. Consider we scale V by length L. Then, the equation is written as Du X (ρ − ρb)VL3 = T (x; t; n )S L2 + O(L3) Dt i i i 1 As L ! 0, we find that, we have X T (x; t; ni)Si = 0: i Hence, 3 X Sj T (x; t; n) = − T (x; t; −e ) i S i=1 4 It can be seen easily that Sj=S4 = nj. Hence, T (x; t; n) = n · σ = niσij where σij = −Tj(x; t; −ei). σ is called the Cauchy stress tensor. By an argument of conservation of angular momentum (which means the net torque acting on an infinitesimal fluid element is zero, and may not be true in some rare cases), σ is symmetric. In this course, we consider all fluids so that σ is symmetric. Now, we can rewrite the Equation (1.1) as d Z Z Z ρudV = ρbdV + n · σdS dt Rt Rt @Rt Applying the Reynolds' transport theorem and divergence theorem, we have Z Z Z Z Z (ρu)t + r · (ρu ⊗ u)dV = (ρu)tdV + n · (ρu ⊗ u)dS = ρbdV + n · σdS Rt Rt S Rt @Rt Hence, Z Z Z (ρu)tdV = ρbdV − n · (ρuu − σ)dS: (1.2) Rt Rt @Rt ρuu − σ is then called the total momentum flux. This relation yields, (ρu)t = ρb − r · (ρuu − σ): Meanwhile, as we have shown, Z Du Z Z ρ dV = ρbdV + n · σdS Rt Dt Rt @Rt Applying the divergence theorem, we have Du ρ = ρb + r · σ Dt This equation can be interpretated as F = ma. 2 Pressure and ideal fluid We can decompose the Cauchy stress tensor into the isotropic part and anisotropic part: σ = −pI + τ (1.3) where trace(τ) = 0. p is a scalar depending on x and t only. It is called the `pressure'. Usually, τ depends on the rate of strain, or in other words, the spatial derivative of velocity. In the case of Newtonian fluid, τ = µ(ru + ruT ) which we will come back later in detail, the equation is reduced to Du ρ = ρ(u + u · ru) = −∇p + µ∆u + ρb Dt t Here, µ is called the viscosity. Remark 1. In the compressible case, τ is not traceless. We allow τ = λ(r · u)I + µ(ru + ruT ). See C.M. book P32 but (Equation 1.3.1 has a sign error). An ideal fluid is that with τ = 0, which in the Newtonian fluid case corresponds to zero viscosity, or there is no energy dissipated in the fluid. Hence, we have the equations derived for ideal fluids: ρt + r · (ρu) = 0; (1.4) ρ(ut + u · ru) = −∇p + ρb You can see that we have unknowns ρ, u; p(5 unknowns) but only 4 equations. The system is not closed. There are two directions: • If the fluid is compressible, then, the pressure p depends on the density and entropy (and maybe other quantities like temperature). In this case, we may also consider conservation of energy and the equation of state. This is usually the case we do for ideal gas. The resulted equations will be called the compressible Euler equations. If pressure is a function of density only in the isentropic case, the two equations are closed, which we study in detail next lecture. 3 Dρ • If the fluid is incompressible, then Dt = 0 or r · u = 0. The first equation can be dropped and we have a closed equation ρ(ut + u · ru) = −∇p + ρb (1.5) r · u = 0: This is called the compressible Euler equations. For incompressible fluid, the pressure p can not be determined through the state of the system. You can imagine that for water, the pressure can change a lot with a little change of volume. In the ideal case, the liquid becomes incompressible, then the pressure is undetermined. It is now a Lagrangian multiplier of the constraint r · u = 0 and must be solved from the equations. Then, we will mention the gas dynamics briefly and then focus on the incompressible case in other lectures. Boundary conditions for ideal fluids? Usually, we impose that the condition so that the fluid particle initially at the wall will stay at the wall. The normal component of the fluid velocity should be the same as the normal component of a material point on the wall: u · n = w · n: (1.6) Alternatively, if the wall is given by an equation Σ(x; t) = 0. Then, Σtjx + u(x; t) · rΣ = 0: Mathematically, there is only first order spatial derivative in the PDEs, we can only impose the normal component. If you impose the full velocity at the boundary, you get too many boundary conditions. Conservation of energy for incompressible fluids Even though we do not need the conservation of energy for the system to be closed, we can still check that the energy satisfies the conservation property. For incompressible fluids, there is no exchange between internal energy and kinetic energy. We define the total energy to be Z 1 E = ρjuj2dx: Rt 2 4 By the second convection identity, we have d Z 1 Z 1 D Z D Z ρjuj2dV = ρ juj2dV = ρu · udV = u · (−∇p + ρb)dV dt Rt 2 Rt 2 Dt Rt Dt Rt Z Z Z = −∇ · (pu)dV + ρu · b)dV = − pu · ndS + ρu · bdV Rt @Rt Rt These two terms are work done by the stress and body force. 5.
Load more

Recommended publications
  • 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]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
    Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Euler and Chebyshev: from the Sphere to the Plane and Backwards Athanase Papadopoulos
    Euler and Chebyshev: From the sphere to the plane and backwards Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Euler and Chebyshev: From the sphere to the plane and backwards. 2016. hal-01352229 HAL Id: hal-01352229 https://hal.archives-ouvertes.fr/hal-01352229 Preprint submitted on 6 Aug 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EULER AND CHEBYSHEV: FROM THE SPHERE TO THE PLANE AND BACKWARDS ATHANASE PAPADOPOULOS Abstract. We report on the works of Euler and Chebyshev on the drawing of geographical maps. We point out relations with questions about the fitting of garments that were studied by Chebyshev. This paper will appear in the Proceedings in Cybernetics, a volume dedicated to the 70th anniversary of Academician Vladimir Betelin. Keywords: Chebyshev, Euler, surfaces, conformal mappings, cartography, fitting of garments, linkages. AMS classification: 30C20, 91D20, 01A55, 01A50, 53-03, 53-02, 53A05, 53C42, 53A25. 1. Introduction Euler and Chebyshev were both interested in almost all problems in pure and applied mathematics and in engineering, including the conception of industrial ma- chines and technological devices. In this paper, we report on the problem of drawing geographical maps on which they both worked.
    [Show full text]
  • On the Geometric Character of Stress in Continuum Mechanics
    Z. angew. Math. Phys. 58 (2007) 1–14 0044-2275/07/050001-14 DOI 10.1007/s00033-007-6141-8 Zeitschrift f¨ur angewandte c 2007 Birkh¨auser Verlag, Basel Mathematik und Physik ZAMP On the geometric character of stress in continuum mechanics Eva Kanso, Marino Arroyo, Yiying Tong, Arash Yavari, Jerrold E. Marsden1 and Mathieu Desbrun Abstract. This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other funda- mental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Mathematics Subject Classification (2000). Keywords. Continuum mechanics, elasticity, stress tensor, differential forms. 1. Motivation This paper proposes a reformulation of classical continuum mechanics in terms of bundle-valued exterior forms. Our motivation is to provide a geometric description of force in continuum mechanics, which leads to an elegant geometric theory and, at the same time, may enable the development of space-time integration algorithms that respect the underlying geometric structure at the discrete level. In classical mechanics the traditional approach is to define all the kinematic and kinetic quantities using vector and tensor fields. For example, velocity and traction are both viewed as vector fields and power is defined as their inner product, which is induced from an appropriately defined Riemannian metric. On the other hand, it has long been appreciated in geometric mechanics that force should not be viewed as a vector, but rather a one-form.
    [Show full text]
  • Stress and Strain
    Stress and strain Stress Lecture 2 – Loads, traction, stress Mechanical Engineering Design - N.Bonora 2018 Stress and strain Introduction • One important step in mechanical design is the determination of the internal stresses, once the external load are assigned, and to assess that they do not exceed material allowables. • Internal stress - that differ from surface or contact stresses that are generated where the load are applied - are those associated with the internal forces that are created by external loads for a body in equilibrium. • The same concept holds for complex geometry and loads. To evaluate these stresses is not a a slender block of material; (a) under the action of external straightforward matter, suffice to say here that forces F, (b) internal normal stress σ, (c) internal normal and they will invariably be non-uniform over a shear stress surface, that is, the stress at some particle will differ from the stress at a neighbouring particle Mechanical Engineering Design - N.Bonora 2018 Stress and strain Tractions and the Physical Meaning of Internal Stress • All materials have a complex molecular microstructure and each molecule exerts a force on each of its neighbors. The complex interaction of countless molecular forces maintains a body in equilibrium in its unstressed state. • When the body is disturbed and deformed into a new equilibrium position, net forces act. An imaginary plane can be drawn through the material • The force exerted by the molecules above the plane on the material below the plane and can be attractive or repulsive. Different planes can be taken through the same portion of material and, in general, a different force will act on the plane.
    [Show full text]
  • 8.09(F14) Chapter 6: Fluid Mechanics
    Chapter 6 Fluid Mechanics So far, our examples of mechanical systems have all been discrete, with some number of masses acted upon by forces. Even our study of rigid bodies (with a continuous mass distri- bution) treated those bodies as single objects. In this chapter, we will treat an important continuous system, which is that of fluids. Fluids include both liquids and gases. (It also includes plasmas, but for them a proper treatment requires the inclusion of electromagnetic effects, which will not be discussed here.) For our purposes, a fluid is a material which can be treated as continuous, which has the ability to flow, and which has very little resistance to deformation (that is, it has only a small support for shear stress, which refers to forces parallel to an applied area). Applica- tions include meteorology, oceanography, astrophysics, biophysics, condensed matter physics, geophysics, medicine, aerodynamics, plumbing, cosmology, heavy-ion collisions, and so on. The treatment of fluids is an example of classical field theory, with continuous field variables as the generalized coordinates, as opposed to the discrete set of variables qi that we have treated so far. Therefore the first step we have to take is understanding how to make the transition from discrete to continuum. 6.1 Transitioning from Discrete Particles to the Con- tinuum Rather than starting with fluids, lets instead consider a somewhat simpler system, that of an infinite one dimensional chain of masses m connected by springs with spring constant k, which we take as an approximation for an infinite continuous elastic one-dimensional rod.
    [Show full text]
  • Fundamental Governing Equations of Motion in Consistent Continuum Mechanics
    Fundamental governing equations of motion in consistent continuum mechanics Ali R. Hadjesfandiari, Gary F. Dargush Department of Mechanical and Aerospace Engineering University at Buffalo, The State University of New York, Buffalo, NY 14260 USA [email protected], [email protected] October 1, 2018 Abstract We investigate the consistency of the fundamental governing equations of motion in continuum mechanics. In the first step, we examine the governing equations for a system of particles, which can be considered as the discrete analog of the continuum. Based on Newton’s third law of action and reaction, there are two vectorial governing equations of motion for a system of particles, the force and moment equations. As is well known, these equations provide the governing equations of motion for infinitesimal elements of matter at each point, consisting of three force equations for translation, and three moment equations for rotation. We also examine the character of other first and second moment equations, which result in non-physical governing equations violating Newton’s third law of action and reaction. Finally, we derive the consistent governing equations of motion in continuum mechanics within the framework of couple stress theory. For completeness, the original couple stress theory and its evolution toward consistent couple stress theory are presented in true tensorial forms. Keywords: Governing equations of motion, Higher moment equations, Couple stress theory, Third order tensors, Newton’s third law of action and reaction 1 1. Introduction The governing equations of motion in continuum mechanics are based on the governing equations for systems of particles, in which the effect of internal forces are cancelled based on Newton’s third law of action and reaction.
    [Show full text]