Nonlinear Continuum Mechanics
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
Rotational Motion (The Dynamics of a Rigid Body)
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates. -
10. Collisions • Use Conservation of Momentum and Energy and The
10. Collisions • Use conservation of momentum and energy and the center of mass to understand collisions between two objects. • During a collision, two or more objects exert a force on one another for a short time: -F(t) F(t) Before During After • It is not necessary for the objects to touch during a collision, e.g. an asteroid flied by the earth is considered a collision because its path is changed due to the gravitational attraction of the earth. One can still use conservation of momentum and energy to analyze the collision. Impulse: During a collision, the objects exert a force on one another. This force may be complicated and change with time. However, from Newton's 3rd Law, the two objects must exert an equal and opposite force on one another. F(t) t ti tf Dt From Newton'sr 2nd Law: dp r = F (t) dt r r dp = F (t)dt r r r r tf p f - pi = Dp = ò F (t)dt ti The change in the momentum is defined as the impulse of the collision. • Impulse is a vector quantity. Impulse-Linear Momentum Theorem: In a collision, the impulse on an object is equal to the change in momentum: r r J = Dp Conservation of Linear Momentum: In a system of two or more particles that are colliding, the forces that these objects exert on one another are internal forces. These internal forces cannot change the momentum of the system. Only an external force can change the momentum. The linear momentum of a closed isolated system is conserved during a collision of objects within the system. -
Law of Conversation of Energy
Law of Conservation of Mass: "In any kind of physical or chemical process, mass is neither created nor destroyed - the mass before the process equals the mass after the process." - the total mass of the system does not change, the total mass of the products of a chemical reaction is always the same as the total mass of the original materials. "Physics for scientists and engineers," 4th edition, Vol.1, Raymond A. Serway, Saunders College Publishing, 1996. Ex. 1) When wood burns, mass seems to disappear because some of the products of reaction are gases; if the mass of the original wood is added to the mass of the oxygen that combined with it and if the mass of the resulting ash is added to the mass o the gaseous products, the two sums will turn out exactly equal. 2) Iron increases in weight on rusting because it combines with gases from the air, and the increase in weight is exactly equal to the weight of gas consumed. Out of thousands of reactions that have been tested with accurate chemical balances, no deviation from the law has ever been found. Law of Conversation of Energy: The total energy of a closed system is constant. Matter is neither created nor destroyed – total mass of reactants equals total mass of products You can calculate the change of temp by simply understanding that energy and the mass is conserved - it means that we added the two heat quantities together we can calculate the change of temperature by using the law or measure change of temp and show the conservation of energy E1 + E2 = E3 -> E(universe) = E(System) + E(Surroundings) M1 + M2 = M3 Is T1 + T2 = unknown (No, no law of conservation of temperature, so we have to use the concept of conservation of energy) Total amount of thermal energy in beaker of water in absolute terms as opposed to differential terms (reference point is 0 degrees Kelvin) Knowns: M1, M2, T1, T2 (Kelvin) When add the two together, want to know what T3 and M3 are going to be. -
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. -
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. -
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. -
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. -
Heat and Energy Conservation
1 Lecture notes in Fluid Dynamics (1.63J/2.01J) by Chiang C. Mei, MIT, Spring, 2007 CHAPTER 4. THERMAL EFFECTS IN FLUIDS 4-1-2energy.tex 4.1 Heat and energy conservation Recall the basic equations for a compressible fluid. Mass conservation requires that : ρt + ∇ · ρ~q = 0 (4.1.1) Momentum conservation requires that : = ρ (~qt + ~q∇ · ~q)= −∇p + ∇· τ +ρf~ (4.1.2) = where the viscous stress tensor τ has the components = ∂qi ∂qi ∂qk τ = τij = µ + + λ δij ij ∂xj ∂xi ! ∂xk There are 5 unknowns ρ, p, qi but only 4 equations. One more equation is needed. 4.1.1 Conservation of total energy Consider both mechanical ad thermal energy. Let e be the internal (thermal) energy per unit mass due to microscopic motion, and q2/2 be the kinetic energy per unit mass due to macroscopic motion. Conservation of energy requires D q2 ρ e + dV rate of incr. of energy in V (t) Dt ZZZV 2 ! = − Q~ · ~ndS rate of heat flux into V ZZS + ρf~ · ~qdV rate of work by body force ZZZV + Σ~ · ~qdS rate of work by surface force ZZX Use the kinematic transport theorm, the left hand side becomes D q2 ρ e + dV ZZZV Dt 2 ! 2 Using Gauss theorem the heat flux term becomes ∂Qi − QinidS = − dV ZZS ZZZV ∂xi The work done by surface stress becomes Σjqj dS = (σjini)qj dS ZZS ZZS ∂(σijqj) = (σijqj)ni dS = dV ZZS ZZZV ∂xi Now all terms are expressed as volume integrals over an arbitrary material volume, the following must be true at every point in space, 2 D q ∂Qi ∂(σijqi) ρ e + = − + ρfiqi + (4.1.3) Dt 2 ! ∂xi ∂xj As an alternative form, we differentiate the kinetic energy and get De -
Lecture 5: Magnetic Mirroring
!"#$%&'()%"*#%*+,-./-*+01.2(.*3+456789* !"#$%&"'()'*+,-".#'*/&&0&/-,' Dr. Peter T. Gallagher Astrophysics Research Group Trinity College Dublin :&2-;-)(*!"<-$2-"(=*%>*/-?"=)(*/%/="#* o Gyrating particle constitutes an electric current loop with a dipole moment: 1/2mv2 µ = " B o The dipole moment is conserved, i.e., is invariant. Called the first adiabatic invariant. ! o µ = constant even if B varies spatially or temporally. If B varies, then vperp varies to keep µ = constant => v|| also changes. o Gives rise to magnetic mirroring. Seen in planetary magnetospheres, magnetic bottles, coronal loops, etc. Bz o Right is geometry of mirror Br from Chen, Page 30. @-?"=)(*/2$$%$2"?* o Consider B-field pointed primarily in z-direction and whose magnitude varies in z- direction. If field is axisymmetric, B! = 0 and d/d! = 0. o This has cylindrical symmetry, so write B = Brrˆ + Bzzˆ o How does this configuration give rise to a force that can trap a charged particle? ! o Can obtain Br from " #B = 0 . In cylindrical polar coordinates: 1 " "Bz (rBr )+ = 0 r "r "z ! " "Bz => (rBr ) = #r "r "z o If " B z / " z is given at r = 0 and does not vary much with r, then r $Bz 1 2&$ Bz ) ! rBr = " #0 r dr % " r ( + $z 2 ' $z *r =0 ! 1 &$ Bz ) Br = " r( + (5.1) 2 ' $z *r =0 ! @-?"=)(*/2$$%$2"?* o Now have Br in terms of BZ, which we can use to find Lorentz force on particle. o The components of Lorentz force are: Fr = q(v" Bz # vzB" ) (1) F" = q(#vrBz + vzBr ) (2) (3) Fz = q(vrB" # v" Br ) (4) o As B! = 0, two terms vanish and terms (1) and (2) give rise to Larmor gyration. -
Rotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn that where the force is applied and how the force is applied is just as important as how much force is applied when we want to make something rotate. This tutorial discusses the dynamics of an object rotating about a fixed axis and introduces the concepts of torque and moment of inertia. These concepts allows us to get a better understanding of why pushing a door towards its hinges is not very a very effective way to make it open, why using a longer wrench makes it easier to loosen a tight bolt, etc. This module begins by looking at the kinetic energy of rotation and by defining a quantity known as the moment of inertia which is the rotational analog of mass. Then it proceeds to discuss the quantity called torque which is the rotational analog of force and is the physical quantity that is required to changed an object's state of rotational motion. Moment of Inertia Kinetic Energy of Rotation Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken. The total kinetic energy can be expressed as ..