Ductile Deformation - Concepts of Finite Strain
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10-1 CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams And
EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams and Material Behavior 10.2 Material Characteristics 10.3 Elastic-Plastic Response of Metals 10.4 True stress and strain measures 10.5 Yielding of a Ductile Metal under a General Stress State - Mises Yield Condition. 10.6 Maximum shear stress condition 10.7 Creep Consider the bar in figure 1 subjected to a simple tension loading F. Figure 1: Bar in Tension Engineering Stress () is the quotient of load (F) and area (A). The units of stress are normally pounds per square inch (psi). = F A where: is the stress (psi) F is the force that is loading the object (lb) A is the cross sectional area of the object (in2) When stress is applied to a material, the material will deform. Elongation is defined as the difference between loaded and unloaded length ∆푙 = L - Lo where: ∆푙 is the elongation (ft) L is the loaded length of the cable (ft) Lo is the unloaded (original) length of the cable (ft) 10-1 EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy Strain is the concept used to compare the elongation of a material to its original, undeformed length. Strain () is the quotient of elongation (e) and original length (L0). Engineering Strain has no units but is often given the units of in/in or ft/ft. ∆푙 휀 = 퐿 where: is the strain in the cable (ft/ft) ∆푙 is the elongation (ft) Lo is the unloaded (original) length of the cable (ft) Example Find the strain in a 75 foot cable experiencing an elongation of one inch. -
Energy, Vorticity and Enstrophy Conserving Mimetic Spectral Method for the Euler Equation
Master of Science Thesis Energy, vorticity and enstrophy conserving mimetic spectral method for the Euler equation D.J.D. de Ruijter, BSc 5 September 2013 Faculty of Aerospace Engineering · Delft University of Technology Energy, vorticity and enstrophy conserving mimetic spectral method for the Euler equation Master of Science Thesis For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology D.J.D. de Ruijter, BSc 5 September 2013 Faculty of Aerospace Engineering · Delft University of Technology Copyright ⃝c D.J.D. de Ruijter, BSc All rights reserved. Delft University Of Technology Department Of Aerodynamics, Wind Energy, Flight Performance & Propulsion The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled \Energy, vorticity and enstro- phy conserving mimetic spectral method for the Euler equation" by D.J.D. de Ruijter, BSc in partial fulfillment of the requirements for the degree of Master of Science. Dated: 5 September 2013 Head of department: prof. dr. F. Scarano Supervisor: dr. ir. M.I. Gerritsma Reader: dr. ir. A.H. van Zuijlen Reader: P.J. Pinto Rebelo, MSc Summary The behaviour of an inviscid, constant density fluid on which no body forces act, may be modelled by the two-dimensional incompressible Euler equations, a non-linear system of partial differential equations. If a fluid whose behaviour is described by these equations, is confined to a space where no fluid flows in or out, the kinetic energy, vorticity integral and enstrophy integral within that space remain constant in time. Solving the Euler equations accompanied by appropriate boundary and initial conditions may be done analytically, but more often than not, no analytical solution is available. -
Chapter 11. Three Dimensional Analytic Geometry and Vectors
Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1. -
Radar Back-Scattering from Non-Spherical Scatterers
REPORT OF INVESTIGATION NO. 28 STATE OF ILLINOIS WILLIAM G. STRATION, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M. BINKS, Director RADAR BACK-SCATTERING FROM NON-SPHERICAL SCATTERERS PART 1 CROSS-SECTIONS OF CONDUCTING PROLATES AND SPHEROIDAL FUNCTIONS PART 11 CROSS-SECTIONS FROM NON-SPHERICAL RAINDROPS BY Prem N. Mathur and Eugam A. Mueller STATE WATER SURVEY DIVISION A. M. BUSWELL, Chief URBANA. ILLINOIS (Printed by authority of State of Illinois} REPORT OF INVESTIGATION NO. 28 1955 STATE OF ILLINOIS WILLIAM G. STRATTON, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M. BINKS, Director RADAR BACK-SCATTERING FROM NON-SPHERICAL SCATTERERS PART 1 CROSS-SECTIONS OF CONDUCTING PROLATES AND SPHEROIDAL FUNCTIONS PART 11 CROSS-SECTIONS FROM NON-SPHERICAL RAINDROPS BY Prem N. Mathur and Eugene A. Mueller STATE WATER SURVEY DIVISION A. M. BUSWELL, Chief URBANA, ILLINOIS (Printed by authority of State of Illinois) Definitions of Terms Part I semi minor axis of spheroid semi major axis of spheroid wavelength of incident field = a measure of size of particle prolate spheroidal coordinates = eccentricity of ellipse = angular spheroidal functions = Legendse polynomials = expansion coefficients = radial spheroidal functions = spherical Bessel functions = electric field vector = magnetic field vector = back scattering cross section = geometric back scattering cross section Part II = semi minor axis of spheroid = semi major axis of spheroid = Poynting vector = measure of size of spheroid = wavelength of the radiation = back scattering -
Impulse and Momentum
Impulse and Momentum All particles with mass experience the effects of impulse and momentum. Momentum and inertia are similar concepts that describe an objects motion, however inertia describes an objects resistance to change in its velocity, and momentum refers to the magnitude and direction of it's motion. Momentum is an important parameter to consider in many situations such as braking in a car or playing a game of billiards. An object can experience both linear momentum and angular momentum. The nature of linear momentum will be explored in this module. This section will discuss momentum and impulse and the interconnection between them. We will explore how energy lost in an impact is accounted for and the relationship of momentum to collisions between two bodies. This section aims to provide a better understanding of the fundamental concept of momentum. Understanding Momentum Any body that is in motion has momentum. A force acting on a body will change its momentum. The momentum of a particle is defined as the product of the mass multiplied by the velocity of the motion. Let the variable represent momentum. ... Eq. (1) The Principle of Momentum Recall Newton's second law of motion. ... Eq. (2) This can be rewritten with accelleration as the derivate of velocity with respect to time. ... Eq. (3) If this is integrated from time to ... Eq. (4) Moving the initial momentum to the other side of the equation yields ... Eq. (5) Here, the integral in the equation is the impulse of the system; it is the force acting on the mass over a period of time to . -
John Ellipsoid 5.1 John Ellipsoid
CSE 599: Interplay between Convex Optimization and Geometry Winter 2018 Lecture 5: John Ellipsoid Lecturer: Yin Tat Lee Disclaimer: Please tell me any mistake you noticed. The algorithm at the end is unpublished. Feel free to contact me for collaboration. 5.1 John Ellipsoid In the last lecture, we discussed that any convex set is very close to anp ellipsoid in probabilistic sense. More precisely, after renormalization by covariance matrix, we have kxk2 = n ± Θ(1) with high probability. In this lecture, we will talk about how convex set is close to an ellipsoid in a strict sense. If the convex set is isotropic, it is close to a sphere as follows: Theorem 5.1.1. Let K be a convex body in Rn in isotropic position. Then, rn + 1 B ⊆ K ⊆ pn(n + 1)B : n n n Roughly speaking, this says that any convex set can be approximated by an ellipsoid by a n factor. This result has a lot of applications. Although the bound is tight, making a body isotropic is pretty time- consuming. In fact, making a body isotropic is the current bottleneck for obtaining faster algorithm for sampling in convex sets. Currently, it can only be done in O∗(n4) membership oracle plus O∗(n5) total time. Problem 5.1.2. Find a faster algorithm to approximate the covariance matrix of a convex set. In this lecture, we consider another popular position of a convex set called John position and its correspond- ing ellipsoid is called John ellipsoid. Definition 5.1.3. Given a convex set K. -
SMALL DEFORMATION RHEOLOGY for CHARACTERIZATION of ANHYDROUS MILK FAT/RAPESEED OIL SAMPLES STINE RØNHOLT1,3*, KELL MORTENSEN2 and JES C
bs_bs_banner A journal to advance the fundamental understanding of food texture and sensory perception Journal of Texture Studies ISSN 1745-4603 SMALL DEFORMATION RHEOLOGY FOR CHARACTERIZATION OF ANHYDROUS MILK FAT/RAPESEED OIL SAMPLES STINE RØNHOLT1,3*, KELL MORTENSEN2 and JES C. KNUDSEN1 1Department of Food Science, University of Copenhagen, Rolighedsvej 30, DK-1958 Frederiksberg C, Denmark 2Niels Bohr Institute, University of Copenhagen, Copenhagen Ø, Denmark KEYWORDS ABSTRACT Method optimization, milk fat, physical properties, rapeseed oil, rheology, structural Samples of anhydrous milk fat and rapeseed oil were characterized by small analysis, texture evaluation amplitude oscillatory shear rheology using nine different instrumental geometri- cal combinations to monitor elastic modulus (G′) and relative deformation 3 + Corresponding author. TEL: ( 45)-2398-3044; (strain) at fracture. First, G′ was continuously recorded during crystallization in a FAX: (+45)-3533-3190; EMAIL: fluted cup at 5C. Second, crystallization of the blends occurred for 24 h, at 5C, in [email protected] *Present Address: Department of Pharmacy, external containers. Samples were gently cut into disks or filled in the rheometer University of Copenhagen, Universitetsparken prior to analysis. Among the geometries tested, corrugated parallel plates with top 2, 2100 Copenhagen Ø, Denmark. and bottom temperature control are most suitable due to reproducibility and dependence on shear and strain. Similar levels for G′ were obtained for samples Received for Publication May 14, 2013 measured with parallel plate setup and identical samples crystallized in situ in the Accepted for Publication August 5, 2013 geometry. Samples measured with other geometries have G′ orders of magnitude lower than identical samples crystallized in situ. -
Equation of State for the Lennard-Jones Fluid
Equation of State for the Lennard-Jones Fluid Monika Thol1*, Gabor Rutkai2, Andreas Köster2, Rolf Lustig3, Roland Span1, Jadran Vrabec2 1Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum, Universitätsstraße 150, 44801 Bochum, Germany 2Lehrstuhl für Thermodynamik und Energietechnik, Universität Paderborn, Warburger Straße 100, 33098 Paderborn, Germany 3Department of Chemical and Biomedical Engineering, Cleveland State University, Cleveland, Ohio 44115, USA Abstract An empirical equation of state correlation is proposed for the Lennard-Jones model fluid. The equation in terms of the Helmholtz energy is based on a large molecular simulation data set and thermal virial coefficients. The underlying data set consists of directly simulated residual Helmholtz energy derivatives with respect to temperature and density in the canonical ensemble. Using these data introduces a new methodology for developing equations of state from molecular simulation data. The correlation is valid for temperatures 0.5 < T/Tc < 7 and pressures up to p/pc = 500. Extensive comparisons to simulation data from the literature are made. The accuracy and extrapolation behavior is better than for existing equations of state. Key words: equation of state, Helmholtz energy, Lennard-Jones model fluid, molecular simulation, thermodynamic properties _____________ *E-mail: [email protected] Content Content ....................................................................................................................................... 2 List of Tables ............................................................................................................................. -
Automatic Estimation of Sphere Centers from Images of Calibrated Cameras
Automatic Estimation of Sphere Centers from Images of Calibrated Cameras Levente Hajder 1 , Tekla Toth´ 1 and Zoltan´ Pusztai 1 1Department of Algorithms and Their Applications, Eotv¨ os¨ Lorand´ University, Pazm´ any´ Peter´ stny. 1/C, Budapest, Hungary, H-1117 fhajder,tekla.toth,[email protected] Keywords: Ellipse Detection, Spatial Estimation, Calibrated Camera, 3D Computer Vision Abstract: Calibration of devices with different modalities is a key problem in robotic vision. Regular spatial objects, such as planes, are frequently used for this task. This paper deals with the automatic detection of ellipses in camera images, as well as to estimate the 3D position of the spheres corresponding to the detected 2D ellipses. We propose two novel methods to (i) detect an ellipse in camera images and (ii) estimate the spatial location of the corresponding sphere if its size is known. The algorithms are tested both quantitatively and qualitatively. They are applied for calibrating the sensor system of autonomous cars equipped with digital cameras, depth sensors and LiDAR devices. 1 INTRODUCTION putation and memory costs. Probabilistic Hough Transform (PHT) is a variant of the classical HT: it Ellipse fitting in images has been a long researched randomly selects a small subset of the edge points problem in computer vision for many decades (Prof- which is used as input for HT (Kiryati et al., 1991). fitt, 1982). Ellipses can be used for camera calibra- The 5D parameter space can be divided into two tion (Ji and Hu, 2001; Heikkila, 2000), estimating the pieces. First, the ellipse center is estimated, then the position of parts in an assembly system (Shin et al., remaining three parameters are found in the second 2011) or for defect detection in printed circuit boards stage (Yuen et al., 1989; Tsuji and Matsumoto, 1978). -
Exploding the Ellipse Arnold Good
Exploding the Ellipse Arnold Good Mathematics Teacher, March 1999, Volume 92, Number 3, pp. 186–188 Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). More than 200 books, videos, software, posters, and research reports are available through NCTM’S publication program. Individual members receive a 20% reduction off the list price. For more information on membership in the NCTM, please call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright May 1991 by the National Council of Teachers of Mathematics. All rights reserved. Arnold Good, Framingham State College, Framingham, MA 01701, is experimenting with a new approach to teaching second-year calculus that stresses sequences and series over integration techniques. eaders are advised to proceed with caution. Those with a weak heart may wish to consult a physician first. What we are about to do is explode an ellipse. This Rrisky business is not often undertaken by the professional mathematician, whose polytechnic endeavors are usually limited to encounters with administrators. Ellipses of the standard form of x2 y2 1 5 1, a2 b2 where a > b, are not suitable for exploding because they just move out of view as they explode. Hence, before the ellipse explodes, we must secure it in the neighborhood of the origin by translating the left vertex to the origin and anchoring the left focus to a point on the x-axis. -
On Nonlinear Strain Theory for a Viscoelastic Material Model and Its Implications for Calving of Ice Shelves
Journal of Glaciology (2019), 65(250) 212–224 doi: 10.1017/jog.2018.107 © The Author(s) 2019. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re- use or in order to create a derivative work. On nonlinear strain theory for a viscoelastic material model and its implications for calving of ice shelves JULIA CHRISTMANN,1,2 RALF MÜLLER,2 ANGELIKA HUMBERT1,3 1Division of Geosciences/Glaciology, Alfred Wegener Institute Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany 2Institute of Applied Mechanics, University of Kaiserslautern, Kaiserslautern, Germany 3Division of Geosciences, University of Bremen, Bremen, Germany Correspondence: Julia Christmann <[email protected]> ABSTRACT. In the current ice-sheet models calving of ice shelves is based on phenomenological approaches. To obtain physics-based calving criteria, a viscoelastic Maxwell model is required account- ing for short-term elastic and long-term viscous deformation. On timescales of months to years between calving events, as well as on long timescales with several subsequent iceberg break-offs, deformations are no longer small and linearized strain measures cannot be used. We present a finite deformation framework of viscoelasticity and extend this model by a nonlinear Glen-type viscosity. A finite element implementation is used to compute stress and strain states in the vicinity of the ice-shelf calving front. -
1 Fluid Flow Outline Fundamentals of Rheology
Fluid Flow Outline • Fundamentals and applications of rheology • Shear stress and shear rate • Viscosity and types of viscometers • Rheological classification of fluids • Apparent viscosity • Effect of temperature on viscosity • Reynolds number and types of flow • Flow in a pipe • Volumetric and mass flow rate • Friction factor (in straight pipe), friction coefficient (for fittings, expansion, contraction), pressure drop, energy loss • Pumping requirements (overcoming friction, potential energy, kinetic energy, pressure energy differences) 2 Fundamentals of Rheology • Rheology is the science of deformation and flow – The forces involved could be tensile, compressive, shear or bulk (uniform external pressure) • Food rheology is the material science of food – This can involve fluid or semi-solid foods • A rheometer is used to determine rheological properties (how a material flows under different conditions) – Viscometers are a sub-set of rheometers 3 1 Applications of Rheology • Process engineering calculations – Pumping requirements, extrusion, mixing, heat transfer, homogenization, spray coating • Determination of ingredient functionality – Consistency, stickiness etc. • Quality control of ingredients or final product – By measurement of viscosity, compressive strength etc. • Determination of shelf life – By determining changes in texture • Correlations to sensory tests – Mouthfeel 4 Stress and Strain • Stress: Force per unit area (Units: N/m2 or Pa) • Strain: (Change in dimension)/(Original dimension) (Units: None) • Strain rate: Rate