5 Key excerpts on "Elastic Forces"

• 

  eBook - ePub

  Science and Mathematics for Engineering

  • John Bird(Author)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  A good knowledge of some of the constants used in the study of the properties of materials is vital in most branches of engineering, especially in mechanical, manufacturing, aeronautical and civil and structural engineering. For example, most steels look the same, but steels used for the pressure hull of a submarine are about five times stronger than those used in the construction of a small building, and it is very important for the professional and chartered engineer to know what steel to use for what construction; this is because the cost of the high-tensile steel used to construct a submarine pressure hull is considerably higher than the cost of the mild steel, or similar material, used to construct a small building. The engineer must not only take into consideration the ability of the chosen material of construction to do the job, but also its cost. Similar arguments lie in manufacturing engineering, where the engineer must be able to estimate the ability of his/her machines to bend, cut or shape the artefact s/he is trying to produce, and at a competitive price! This chapter provides explanations of the different terms that are used in determining the properties of various materials. The importance of knowing about the effects of forces on materials is to aid the design and construction of structures in an efficient and trustworthy manner.

  At the end of this chapter, you should be able to:

  • define force and state its unit
  • recognise a tensile force and state relevant practical examples
  • recognise a compressive force and state relevant practical examples
  • recognise a shear force and state relevant practical examples
  • define stress and state its unit
  • calculate stress σ from

    σ =

    F A
  • define strain
  • calculate strain e from

    ε =

    x L
  • define elasticity, plasticity, limit of proportionality and elastic limit
  • state Hooke’s law
  • define Young’s modulus of elasticity E and stiffness
  • appreciate typical values for E
  • calculate Ε from

    E =

    σ ε
  • perform calculations using Hooke’s law
  • plot a load/extension graph from given data
  • define ductility, brittleness and malleability, with examples of each

  Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis. All rights reserved.

  26.1   Introduction

  A good knowledge of some of the constants used in the study of the properties of materials is vital in most branches of engineering, especially in mechanical, manufacturing, aeronautical and civil and structural engineering. For example, most steels look the same, but steels used for the pressure hull of a submarine are about 5 times stronger than those used in the construction of a small building, and it is very important for the professional and chartered engineer to know what steel to use for what construction; this is because the cost of the high-tensile steel used to construct a submarine pressure hull is considerably higher than the cost of the mild steel, or similar material, used to construct a small building. The engineer must not only take into consideration the ability of the chosen material of construction to do the job, but also its cost. Similar arguments lie in manufacturing engineering, where the engineer must be able to estimate the ability of his/her machines to bend, cut or shape the artefact s/he is trying to produce, and at a competitive price! This chapter provides explanations of the different terms that are used in determining the properties of various materials.

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  eBook - ePub

  Elasticity

  Tensor, Dyadic, and Engineering Approaches

  • Pei Chi Chou, Nicholas J. Pagano(Authors)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  It should be pointed out that engineering materials are seldom truly isotropic or homogeneous, because the crystalline or molecular structure of material is not continuous and may be randomly oriented. However, the assumptions of isotropy and homogeneity usually lead to results consistent with experiments (subject to exceptions as noted in the previous paragraph). This is because in experimental measurements the stresses and strains are averaged over dimensions, which although small, are still much larger than the dimensions of crystals and molecules. From a macroscopic point of view, therefore, the behavior of the real material can be treated (at least to a first approximation) as being isotropic and homogeneous.

  The equations relating stress, strain, stress-rate (increase of stress per unit time), and strain-rate are called the constitutive equations, since they depend upon the material properties of the medium under discussion. In the case of elastic solids, the constitutive equations take the form of generalized Hooke’s law, which involves only stress and strain and is independent of the stress-rate or strain-rate.

  3.2. Generalized Hooke’s Law

  As pointed out in the introduction, most engineering materials exhibit a well-defined elastic range under a condition of uniaxial normal stress. If the normal stress acts in the x direction, we have the relation known as Hooke’s law:

  (3.1)

  in the elastic range. The constant E is called the modulus of elasticity, or Young’s modulus.v

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  eBook - ePub

  Physical Properties of Textile Fibres

  • J. W. S. Hearle, W E Morton(Authors)
  • 2008(Publication Date)
  • Woodhead Publishing

    (Publisher)

  15 Elastic recovery 15.1 Introduction The extent to which a fibre becomes permanently deformed when it is stretched is of great technical importance. It may be just as serious a form of damage as actual breakage of the fibre. The values of stress and strain above which permanent deformation occurs may well be the limiting values in use. In some specialised applications, such as ropes used in rock-climbing, the fibres may safely be taken beyond their yield point once, but their properties will then be so altered that they are unfit for further use. Elastic recovery, that is, the behaviour on removal of stress, is only a special case of the general phenomenon of hysteresis. In a cyclic change of stress or strain, the results will not fall on a single line. After a few initial cycles, the fibre will become conditioned and the results will tend to fall on a loop, as in Fig. 15.1. This means that energy is used up by internal friction, and consequently the material will heat up and may tend to dry out. This is important where fibres are subject to repeated loading, as in tyres, and the heating will affect their properties

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  eBook - ePub

  Fundamental Physics of Ultrasound

  • Shutilov(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  §2. The stress tensor

  In a body that is not deformed, all parts of the body are in mechanical equilibrium with one another. This means that the resultant of all forces acting on a volume element singled out in the body vanishes. A deformation, however, takes the body out of the equilibrium state, as a result of which Elastic Forces due to intermolecular interactions appear in the body. The range of molecular forces is of the order of the distance between molecules, so that in the theory of elasticity of a continuous medium this range is assumed to equal zero. Thus, when a body is deformed, the internal forces exerted directly on an individual volume element of the body act only from the surrounding parts of the body through the surface of the element, i.e. , they are surface forces, which we shall examine in what follows neglecting body forces such as gravity. Surface forces are proportional to the surface area on which they act. A force referred to unit surface area is called a mechanical stress.

  Let dS denote an element of the surface of an arbitrary volume ΔV of a deformed body (Fig. 5 ) which is small enough so that the mechanical stress acting through it* may be assumed to be uniform. Let us draw the outer normal n to this surface. The stress acting on the surface dS is a vector whose orientation may not, in general, coincide with the normal to the surface. The sign of the stress is chosen arbitrarily. It is customary to consider a stress making an acute angle with the normal n , i.e. , a tensile stress, to be positive. The stress depends on the position and orientation of the surface element dS , so that the stress vector corresponding to a given surface element with the outer normal n is denoted by an index referring to this area

  σ n

  . The vector

  σ n

  can be decomposed into the components

  σn x ,

  σn y

  ,

  σn z

  along the coordinate axes. In general, the stress

  σ n

  and its components are functions of the coordinates and time.

  Fig. 5.

    *

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  eBook - ePub

  Mechanical Vibration

  Analysis, Uncertainties, and Control, Fourth Edition

  • Haym Benaroya, Mark Nagurka, Seon Han(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  This chapter introduces the concept of energy dissipation, or damping, for vibrating systems. Damping is complex physical phenomena (it is temperature and pressure dependent) and there are various sources of dissipation in a system. We will develop mathematical models of damping that are commonly used in equations governing oscillatory motion. Our focus is on linear damping models.

  As with any study of real systems, we are forced to idealize and approximate in order to begin to understand complex behavior. When considering vibrating systems, we first idealize by discretizing real continuous systems into discrete approximate systems. Further, we assume small motions so that linear theory applies, and assume some components to be rigid and some to be elastic. Just as linearity is always an approximation, in many instances leading to almost exact results, elasticity is also always an approximation.

  We will see that damping introduces a phase lag in system response. Damping, in addition to accounting for energy dissipation, implies that structural displacement lags forcing and strain lags stress. We also know from material behavior experiments that strain is not a single‐valued function of stress. “Solids creep when a sufficiently high stress is applied, and the strain is a function of time.”

  3

  Anelasticity is the term used to denote such creep and time‐dependence. The response of a solid to a force can be viewed as having two components, one part that is (ideally) elastic and instantaneous, and one part that is anelastic and time‐dependent, or delayed. As per Anderson, “The anelastic part contains information about temperature, stress and the defect nature of the solid

  Energy is lost either externally to the system or internally within the system. There is always energy dissipation in any real system. We sometimes ignore damping in order to study idealized (undamped) systems because they exhibit key oscillatory characteristics that are useful to our fundamental understanding.

  Damping mechanisms are not as well understood as are other aspects of vibratory systems. There is no universal model for damping. “There are excellent reasons why the stiffness and inertia properties of a general discrete system, executing small vibration around a position of stable equilibrium, may be approximated via the familiar stiffness and mass matrices. These simply represent the first non-trivial terms which do not vanish when the potential and kinetic energy functions are Taylor-expanded for small amplitudes of motion. Nothing so simple can be done to represent damping, because it is not in general clear which state variables the damping forces will depend on.”

  4

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