DOCSLIB.ORG

Infinitesimal Strains and Their Accumulations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Infinitesimal Strains and Their Accumulations Chapter 2 Infinitesimal Strains and Their Accumulations In this chapter we will study infinitesimal deformation and the rate of deformation. Most geologic structures are the integral of infinitesimal deformations, so that it is important to understand how infinitesimal and finite deformations are related. 2.1 Infinitesimal strain The lifetime of a deformation zone or an orogenic belt can be over millions of years, during which time rock masses undergo large deformations. Tectonism is fairly slow and the incremantal defor- mation in our usual time-scale is very small. Therefore, the deformations detected by a geodetic survey are usually very small. Geology detects the result of long-term deformations. Geological measurement is much less accurate than geodetic one. Thus, it is usually difficult for geology to detect such small strains as geodesy detects. However, it is possible in some cases1 that geologic structures provide clues to quantify those strains. In addition, it is necessary to know the theories of infinitesimal strains to understand the following sections in which the dynamics of tectonic deformatoins is discussed. 2.1.1 Definition Infinitesimal strain is a very small strain. If a deformation gradient tensor is nearly equal to the iden- tity tensor (F ≈ I), it is called infinitesimal deformation. Let us first rewrite F with the displacement2, u = x − ξ. 1Note that strain is defined as the change of length compared to the original one. Given a tiny strain, the associated length change can be large enough for detection by geological measurement. 2The famous textbook by Turcotte and Schubert [245] defines the displacement u = ξ −x in order to deal with contraction and compression as possitive strain and stress, respectively. 31 32 CHAPTER 2. INFINITESIMAL STRAINS AND THEIR ACCUMULATIONS We assume that the deformation and associated displacement are very small. Namely, ξ ≈ x. Three components of ξ are independent of each other, consequently ∂ξi/∂ξj = δij. Therefore, we have the deformation gradient tensor ∂x ∂(ξ + u) ∂u ∂u F = = = I + ≈ I + . ∂ξ ∂ξ ∂ξ ∂x The last term is a tensor indicating the spatial gradient of displacement, so that we use the expression ∇u = ∂u/∂x for it. Taking the limit u → 0, we obtain F = I + ∇u, where the small departure from I is indicated by ∇u = ∂u/∂x. The equation F = I indicates no deformation. Let us write the difference as δF, with which we have the deformation gradient for infinitesimal deformation F = I + δF (δF ≈ O), (2.1) where δF = ∇u. In this case, 2 G + I = FT · F = (F + δF) T · (F + δF) ≈ I + δF + δFT, where the quadratic term δFT · δF was neglected. Geometrical linearity applies if this approxima- tion is satisfied. Rearranging the above equation, we have Green’s strain tensor 1 1 G = δF + δFT = ∇u + (∇u)T , (2.2) 2 2 the components of which are 1 ∂uj ∂ui 1 ∂uj ∂ui Gij = + = + . (2.3) 2 ∂ξi ∂ξj 2 ∂xi ∂xj Almansi’s strain tensor (Eq. (1.25)) is linearlized to be − −1 1 2 A + I = 2 I − F · FT = 2 I − I + δF · I + δFT −1 ≈ 2 I − I + δF + δFT ≈ I + δF + δFT. Therefore, Green’s and Almansi’s strain tensors are identical (G = A) for infinitesimal strains. We, accordingly, definethe infinitesimal strain tensor, 1 T 1 ∂uj ∂ui E = ∇u + (∇u) or Eij = + . (2.4) 2 2 ∂xi ∂xj 2.1. INFINITESIMAL STRAIN 33 This is often referred to as a strain tensor. Elongation and contraction are indicated by the pos- itive and negative components of this tensor (2.1.3). If we use the opposite sign convention, the infinitesimal strain tensor is written as 1 1 ∂u ∂uj e = − ∇u + (∇u) T or = − i + . 2 2 ∂xj ∂xi It is obvious that the infinitesimal strain tensors E and e satisfy the following equations, E = ET, e = eT, E = −e. (2.5) 2.1.2 Relationships among strain tensors for infinitesimal deformations We have seen that the infinitesimal strain tensor E was derived from G and A by geometrical lin- earization. Then, how is E related to other strain tensors? By the linealization, the right and left Cauchy-Green tensors become T 1 T 1 U = I + δF · I + δF 2 ≈ I + δF + δF 2 1 ≈ I + δF + δFT = I + G = I + E = V, (2.6) 2 indicating that both tensors coincide with each other for infinitesimal strains. The orthogonal tensor R, which is related to the deformation gradient F by the equation F = R · U, becomes a tensor approximately equal to the unit tensor: T −1 T − δF δF δF δF R = F · U 1 = I + δF · I + + ≈ I + δF · I − − 2 2 2 2 1 ≈ I + δF − δFT = I + X. (2.7) 2 The last term is called the rotation tensor defined by the equation 1 T 1 ∂ui ∂uj X ≡ ∇u − (∇u) or Ωij = − . (2.8) 2 2 ∂xj ∂xi E and X are the symmetric and antisymmetric parts of ∇u, respectively. Using Eqs. (2.6) and (2.7), we obtain U + R = 2 I + δF = I + F = 2 I + G + X. Thus F = U + R − I = V + R − I = I + G + X. (2.9) Since G approaches E for small strains, F = I + E + X. (2.10) For finite strains F = R · U = U · R, as matrix maltiplications are generally non-commutative. By contrast, matrix summations are commutative, so that Eq. (2.10) indicates that the order of rotation X and shape-change E are commutative for infinitesimal deformations. 34 CHAPTER 2. INFINITESIMAL STRAINS AND THEIR ACCUMULATIONS 2.1.3 Physical interpretation of infinitesimal strain and spin tensors To interpret the tensors E and X, we shall see how a small rectangle element deforms. Let us examine the diagonal components of E, first. If displacement is parallel to the coordinate axis O-1 (Fig. 2.1(a)), ∂ui u2 = u3 = 0, = 0(i = 1, 2, 3; j = 2, 3). ∂xj Therefore, all the components of E vanishes except for E11. In this case, neglecting higher-order terms, we have the length change ds ds0 ∂u1 ∂u1 ds − ds0 ≈ u1 + dx1 + dx1 − u1 − dx1 = dx1 = E11dx1. ∂x1 ∂x1 Therfore, we obtain (ds − ds0)/ds0 = E11. Accordingly, the diagonal terms of E represent the elongation along the coordinate axes. Using the angles θ and ψ shown in Figs. 2.1(b) and (c), we have ∂u1 ∂u2 2E12 = 2E21 = + = tan ψ + tan θ. ∂x2 ∂x1 For infinitesimal deformations, the last two terms of this equation become ψ + θ. These two com- ponents indicate the angular decrease of the corner at which two sides of the rectangle originally parallel to the axes O-1 and -2 meet. The indices of E12 and E21 correspond to the axes. Conse- quently, 2E12 and 2E21 represent the engineering shear strain of the corner. The term “shear strain” often refers to the diagonal components of E, which are half of the engineering shear strains. Com- paring Figs. 1.11 and 2.1(b), the parameter q in Eq. (1.13) is related to the displacement gradients as ∂u1/∂x2 = 2q and ∂u2/∂x1 = 0. Therefore, q equals shear strain for infinitesimal simple shears. The infinitesimal strain tensor is symmetric (Eq. (2.5)), so that it has three real eigenvalues, E1, E2, and E3, and three mutually perpendicular eigenvectors. Consider a rectangular parallelepiped whose edges are parallel to the eigenvectors. The length of the edge parallel to the ith eigenvector is elongated by (1 + Ei). Therefore the volume change is given by V − 1 = (E1 + 1)(E2 + 1)(E3 + 1) − 1 ≈ trace E = EI, (2.11) V0 where V0 and V stand for the volume before and after deformation. This quantity is called the dilatation, and is equal to the first basic invariant of the tensor E for infinitesimal strains. Figure 2.1(d) shows the counterclockwise rotation of a rectangle around the O-3 axis. Two sides at the corner were parallel to the coordinate axes before deformation. We are considering very small rotations associated with a infinitesimal deformation, so that the sides that were initially parallel to the O-1 and -2 axes are rotated by the angles ∂u2/∂x1 and −∂u1/∂x2, respectively. If it is a rigid-body rotation, the two quantities are equal. The average of these angles is 1 ∂u2 ∂u1 Ω12 = − . 2 ∂x1 ∂x2 2.1. INFINITESIMAL STRAIN 35 Figure 2.1: The gradient of u and the deformation of a rectangle (dotted line), the sides of which were originally parallel to the coordinate axes. Accordingly, the ijth component of X represents the mean rotation about the kth coordinate axis, where i = j = k. Generally an antisymmetric tensor has three independent components. The vector that is made up of these components is called the axial vector of the antisymmetric tensor. Rotation vector = T (1,2,3) is defined as the axial vector of the rotation tensor: ⎛ ⎞ 0 −3 2 ⎝ ⎠ X = 3 0 −1 . (2.12) −2 1 0 The above equation is alternatively written as Ωij = − ijkk. (2.13) Not only the correspondence of the components, but also the tensor and vector show an important property—for an arbitrary vector a the dot-product of the tensor and the cross-product of the vector 36 CHAPTER 2. INFINITESIMAL STRAINS AND THEIR ACCUMULATIONS are identical: X · a = × a. (2.14) This is a vector quantity, and the first component is Left-hand side = −3a2 + 2a3, Right-hand side = 2a3 − 3a2, therefore both sides are identical. The identity of Eq. (2.14) is readily demonstrated for other components. Combining Eqs. (2.8) and (2.12), we obtain 1 ∂u ∂u ∂u ∂u ∂u ∂u = 3 − 2 , 1 − 3 , 2 − 1 T, 2 ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 which is the expression of the rotation vector by means of displacement.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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 .............................................................................................................................
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow Through Porous Media
    Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Permalink https://escholarship.org/uc/item/6zv599hc Author Wu, Y.S. Publication Date 1990-02-01 eScholarship.org Powered by the California Digital Library University of California Lawrence Berkeley Laboratory e UNIVERSITY OF CALIFORNIA EARTH SCIENCES DlVlSlON Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Y.-S. Wu (Ph.D. Thesis) February 1990 TWO-WEEK LOAN COPY This is a Library Circulating Copy which may be borrowed for two weeks. r- +. .zn Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098. :0 DISCLAIMER I I This document was prepared as an account of work sponsored ' : by the United States Government. Neither the United States : ,Government nor any agency thereof, nor The Regents of the , I Univers~tyof California, nor any of their employees, makes any I warranty, express or implied, or assumes any legal liability or ~ : responsibility for the accuracy, completeness, or usefulness of t any ~nformation, apparatus, product, or process disclosed, or I represents that its use would not infringe privately owned rights. : Reference herein to any specific commercial products process, or I service by its trade name, trademark, manufacturer, or other- I wise, does not necessarily constitute or imply its endorsement, ' recommendation, or favoring by the United States Government , or any agency thereof, or The Regents of the University of Cali- , forma. The views and opinions of authors expressed herein do ' not necessarily state or reflect those of the United States : Government or any agency thereof or The Regents of the , Univers~tyof California and shall not be used for advertismg or I product endorsement purposes.
    [Show full text]
  • Computational Rheology (4K430)
    Computational Rheology (4K430) dr.ir. M.A. Hulsen [email protected] Website: http://www.mate.tue.nl/~hulsen under link ‘Computational Rheology’. – Section Polymer Technology (PT) / Materials Technology (MaTe) – Introduction Computational Rheology important for: B Polymer processing B Rheology & Material science B Turbulent flow (drag reduction phenomena) B Food processing B Biological flows B ... Introduction (Polymer Processing) Analysis of viscoelastic phenomena essential for predicting B Flow induced crystallization kinetics B Flow instabilities during processing B Free surface flows (e.g.extrudate swell) B Secondary flows B Dimensional stability of injection moulded products B Prediction of mechanical and optical properties Introduction (Surface Defects on Injection Molded Parts) Alternating dull bands perpendicular to flow direction with high surface roughness (M. Bulters & A. Schepens, DSM-Research). Introduction (Flow Marks, Two Color Polypropylene) Flow Mark Side view Top view Bottom view M. Bulters & A. Schepens, DSM-Research Introduction (Simulation flow front) 1 0.5 Steady Perturbed H 2y 0 ___ −0.5 −1 0 0.5 1 ___2x H Introduction (Rheology & Material Science) Simulation essential for understanding and predicting material properties: B Polymer blends (morphology, viscosity, normal stresses) B Particle filled viscoelastic fluids (suspensions) B Polymer architecture macroscopic properties (Brownian dynamics (BD), molecular dynamics (MD),⇒ Monte Carlo, . ) Multi-scale. ⇒ Introduction (Solid particles in a viscoelastic fluid) B Microstructure (polymer, particles) B Bulk rheology B Flow induced crystallization Introduction (Multiple particles in a viscoelastic fluid) Introduction (Flow induced crystallization) Introduction (Multi-phase flows) Goal and contents of the course Goal: Introduction of the basic numerical techniques used in Computational Rheology using the Finite Element Method (FEM).
    [Show full text]
  • Chapter 3 Equations of State
    Chapter 3 Equations of State The simplest way to derive the Helmholtz function of a fluid is to directly integrate the equation of state with respect to volume (Sadus, 1992a, 1994). An equation of state can be applied to either vapour-liquid or supercritical phenomena without any conceptual difficulties. Therefore, in addition to liquid-liquid and vapour -liquid properties, it is also possible to determine transitions between these phenomena from the same inputs. All of the physical properties of the fluid except ideal gas are also simultaneously calculated. Many equations of state have been proposed in the literature with either an empirical, semi- empirical or theoretical basis. Comprehensive reviews can be found in the works of Martin (1979), Gubbins (1983), Anderko (1990), Sandler (1994), Economou and Donohue (1996), Wei and Sadus (2000) and Sengers et al. (2000). The van der Waals equation of state (1873) was the first equation to predict vapour-liquid coexistence. Later, the Redlich-Kwong equation of state (Redlich and Kwong, 1949) improved the accuracy of the van der Waals equation by proposing a temperature dependence for the attractive term. Soave (1972) and Peng and Robinson (1976) proposed additional modifications of the Redlich-Kwong equation to more accurately predict the vapour pressure, liquid density, and equilibria ratios. Guggenheim (1965) and Carnahan and Starling (1969) modified the repulsive term of van der Waals equation of state and obtained more accurate expressions for hard sphere systems. Christoforakos and Franck (1986) modified both the attractive and repulsive terms of van der Waals equation of state. Boublik (1981) extended the Carnahan-Starling hard sphere term to obtain an accurate equation for hard convex geometries.
    [Show full text]
  • What Is Hooke's Law? 16 February 2015, by Matt Williams
    What is Hooke's Law? 16 February 2015, by Matt Williams Like so many other devices invented over the centuries, a basic understanding of the mechanics is required before it can so widely used. In terms of springs, this means understanding the laws of elasticity, torsion and force that come into play – which together are known as Hooke's Law. Hooke's Law is a principle of physics that states that the that the force needed to extend or compress a spring by some distance is proportional to that distance. The law is named after 17th century British physicist Robert Hooke, who sought to demonstrate the relationship between the forces applied to a spring and its elasticity. He first stated the law in 1660 as a Latin anagram, and then published the solution in 1678 as ut tensio, sic vis – which translated, means "as the extension, so the force" or "the extension is proportional to the force"). This can be expressed mathematically as F= -kX, where F is the force applied to the spring (either in the form of strain or stress); X is the displacement A historical reconstruction of what Robert Hooke looked of the spring, with a negative value demonstrating like, painted in 2004 by Rita Greer. Credit: that the displacement of the spring once it is Wikipedia/Rita Greer/FAL stretched; and k is the spring constant and details just how stiff it is. Hooke's law is the first classical example of an The spring is a marvel of human engineering and explanation of elasticity – which is the property of creativity.
    [Show full text]
  • Fluid Inertia and End Effects in Rheometer Flows
    FLUID INERTIA AND END EFFECTS IN RHEOMETER FLOWS by JASON PETER HUGHES B.Sc. (Hons) A thesis submitted to the University of Plymouth in partial fulfilment for the degree of DOCTOR OF PHILOSOPHY School of Mathematics and Statistics Faculty of Technology University of Plymouth April 1998 REFERENCE ONLY ItorriNe. 9oo365d39i Data 2 h SEP 1998 Class No.- Corrtl.No. 90 0365439 1 ACKNOWLEDGEMENTS I would like to thank my supervisors Dr. J.M. Davies, Prof. T.E.R. Jones and Dr. K. Golden for their continued support and guidance throughout the course of my studies. I also gratefully acknowledge the receipt of a H.E.F.C.E research studentship during the period of my research. AUTHORS DECLARATION At no time during the registration for the degree of Doctor of Philosophy has the author been registered for any other University award. This study was financed with the aid of a H.E.F.C.E studentship and carried out in collaboration with T.A. Instruments Ltd. Publications: 1. J.P. Hughes, T.E.R Jones, J.M. Davies, *End effects in concentric cylinder rheometry', Proc. 12"^ Int. Congress on Rheology, (1996) 391. 2. J.P. Hughes, J.M. Davies, T.E.R. Jones, ^Concentric cylinder end effects and fluid inertia effects in controlled stress rheometry, Part I: Numerical simulation', accepted for publication in J.N.N.F.M. Signed ...^.^Ms>3.\^^. Date Ik.lp.^.m FLUH) INERTIA AND END EFFECTS IN RHEOMETER FLOWS Jason Peter Hughes Abstract This thesis is concerned with the characterisation of the flow behaviour of inelastic and viscoelastic fluids in steady shear and oscillatory shear flows on commercially available rheometers.
    [Show full text]
  • Engineering Viscoelasticity
    ENGINEERING VISCOELASTICITY David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 October 24, 2001 1 Introduction This document is intended to outline an important aspect of the mechanical response of polymers and polymer-matrix composites: the field of linear viscoelasticity. The topics included here are aimed at providing an instructional introduction to this large and elegant subject, and should not be taken as a thorough or comprehensive treatment. The references appearing either as footnotes to the text or listed separately at the end of the notes should be consulted for more thorough coverage. Viscoelastic response is often used as a probe in polymer science, since it is sensitive to the material’s chemistry and microstructure. The concepts and techniques presented here are important for this purpose, but the principal objective of this document is to demonstrate how linear viscoelasticity can be incorporated into the general theory of mechanics of materials, so that structures containing viscoelastic components can be designed and analyzed. While not all polymers are viscoelastic to any important practical extent, and even fewer are linearly viscoelastic1, this theory provides a usable engineering approximation for many applications in polymer and composites engineering. Even in instances requiring more elaborate treatments, the linear viscoelastic theory is a useful starting point. 2 Molecular Mechanisms When subjected to an applied stress, polymers may deform by either or both of two fundamen- tally different atomistic mechanisms. The lengths and angles of the chemical bonds connecting the atoms may distort, moving the atoms to new positions of greater internal energy.
    [Show full text]