7 Key excerpts on "Elastic Strain"

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  eBook - ePub

  Structure for Architects

  A Case Study in Steel, Wood, and Reinforced Concrete Design

  • Ashwani Bedi, Ramsey Dabby(Authors)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  ductility, a unique property that enables it to absorb large deformations beyond the elastic range. For this reason, steel’s stress-strain curve in tension is the one of usual concern and the one we will look at in detail.

  Steel’s tensile stress-strain curve is based on laboratory tests of a cylindrical specimen pulled apart to failure.

  An entire generalized curve for a typical steel specimen is shown in Figure 3.5 .

  Figure 3.5 Steel’s Stress-Strain Curve in Tension

  Let’s take a closer look.

  When stress is initially applied, strain is directly proportional to stress, up to the proportional limit (Figure 3.5a ).

  Figure 3.5a

  Beyond the proportional limit, the relationship of stress to strain becomes non-linear. For every unit of stress, the increase in strain becomes greater. The upper limit of this behavior is termed the elastic limit.

  Within the elastic limit, the strain disappears when the stress is removed, and the material returns to its original position without any permanent deformation (Figure 3.5b ). Beyond the elastic limit, when the stress is removed, the material no longer returns to its original position and the material becomes permanently deformed. This deformation (i.e., strain) is termed permanent set.

  For steel, the strain up to the elastic limit is termed the elastic range wherein the material is said to behave elastically.

  Figure 3.5b

  With additional stress, the disproportionality of stress to strain continues to the yield point, the point at which the material is said to ‘yield’—i.e., exhibit large increases in strain with little or no increase in stress (Figure 3.5c ).

  Note that steel’s proportional limit, elastic limit, and yield point occur relatively close to each other, but they are distinct points for steel.

  Figure 3.5c

  Yielding continues throughout the plastic range until the start of strain hardening. For steel, the strain from the elastic limit to the start of strain hardening is termed the plastic range, wherein the material is said to behave plastically (Figure 3.5d

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  eBook - ePub

  Structure for Architects

  A Primer

  • Ramsey Dabby, Ashwani Bedi(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  When any particular material is stressed, its resulting strain behavior (i.e., deformational behavior) is unique, not only for that material but also for the type of stress (compression or tension) to which the material is subjected. Steel, for example, behaves similarly in compression as in tension, while concrete behaves quite differently in compression than in tension.

  A material's stress-strain relationship is graphically depicted in a stress-strain curve with stress plotted along the y axis and strain along the x axis. A stress-strain curve is, in a sense, a material's signature behavior under stress. Although any particular material's stress-strain curve is unique, materials generally exhibit some (or all) of the strain behavior patterns broadly characterized as elastic range, plastic range, and strain-hardening range prior to rupture, the point at which the material actually breaks. The stress-strain curve in Figure 13.3 is for a hypothetical ductile material, such as steel, in tension. Table 13.1 describes the general characteristics of the three strain ranges.

  Figure 13.3 Stress-Strain Curve for a Hypothetical Ductile Material in Tension

  Table 13.1 Elastic, Plastic, and Strain-Hardening Range Characteristics

  Range	Characteristics
  Elastic Range	In this range, a force applied to a material causes it to become stressed, resulting in deformation (strain). When the force is removed, the material returns to its original unstressed shape.
  Plastic Range	In this range, with relatively little increase in stress, the material behaves somewhat taffy-like and becomes permanently deformed.
  Strain-Hardening Range	In this range, the material stabilizes somewhat and is able to take on additional stress with a corresponding increase in deformation (strain) until it reaches its ultimate strength and ruptures.

  Stress-strain behavior for the variety of structural materials available is a complex field of study. The reader is referred to more technical sources for a detailed review of the topic. For a simplified comparison, however, the generalized stress-strain curves for steel, concrete (in compression), and wood are shown in Figure 13.4

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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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  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

  Introduction to Engineering Mechanics

  A Continuum Approach, Second Edition

  • Jenn Stroud Rossmann, Clive L. Dym, Lori Bassman(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 5 .

  Strain is a local property, and the values of each strain component may change dramatically within a material. And so we come to our mathematical definition of strain, which relates to relative deformations of an infinitesimal element.

  If we consider the extensional strain in one direction of an original element AB with length Δx, as shown in Figure 4.2 , we see that point A experiences a displacement u. This displacement is common to the whole element, a kind of “rigid-body displacement.” A stretching Δu also takes place within the element, so that point B experiences a total displacement u + Δu.

  FIGURE 4.2One-dimensional extensional strain.

  Based on this situation, we define the extensional normal strain of this element as

  ε  = lim Δ x → 0 Δ u Δ x = d u d x ,

  (4.3)

  taking the limit as Δx → 0 so that the expression will apply to any length Δx of the element. We see that this definition is independent of whatever “rigid body displacement” occurred and recalls our third definition of normal strain, Equation 2.4, in Section 2.1.1

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  eBook - ePub

  Mechanics of Sheet Metal Forming

  • Jack Hu, Zdzislaw Marciniak, John Duncan(Authors)
  • 2002(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Figure 10.4(c) .

  For a tension of 381 kN/m, the moment in the sheet is approximately 4.8 Nm/m. For elastic unloading, the change in curvature is proportional to the change in moment, and from Equation 6.30 ,

  The curvature of the form block is 1/2.39 = 0.418 m−1 , hence the final curvature is

  i.e. the final radius of curvature is 1/0.340 = 2.94, or a change in radius from that of the form block of ((2.94–2.39)/2.39)/100 = 23%. This illustrates that the curvature of the sheet is very sensitive to tension if the process is in the elastic, plastic region. For this reason, the sheet is usually overstretched, as mentioned above, to ensure that changes in the strength or thickness of the incoming sheet will not result in springback.

  10.3 Bending and stretching a strain-hardening sheet

  The previous section assumed that the material did not strain-harden. In practice, sheet having some strain-hardening potential will be used to ensure that it can be stretched beyond the elastic limit in a stable manner. The strain distribution will be similar to that shown in Figure 10.3 , but once the whole section has yielded, the stress distribution will not be uniform; the stress will increase from the inner to the outer surface as shown in Figure 10.5 . As this stress is not constant, there will be some moment in equilibrium with the section as well as the tension.

  Figure 10.5 (a) Stress–strain curve for a strain-hardening material showing the range of stress in a sheet bent and stretched to a mid-surface strain of ε1a . (b) The strain distribution when bent over a former of radius ρ0 , and (c) the stress distribution.

  The stress–strain curve for the material is shown in Figure 10.5(a) . The range of strain across the section is from just below the mid-surface strain ε1a , to just above it; the stress–strain relation is assumed to be linear in this range with a slope of |dσ/dε|ε1a

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  eBook - ePub

  Fundamental Physics of Ultrasound

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

    (Publisher)

  Since the propagation of ultrasound is a nearly adiabatic process, in what follows we shall have in mind the adiabatic values of the moduli. Since Eqs. (I.13) are linear homogeneous equations, they can be solved for the components of the strain e. This gives a system of equations ϵ m = k mn σ n, relating the strains to the stresses. The coefficients of proportionality k mn can be called the elastic susceptibilities or elastic compliances. They likewise form a tensor of rank four, for which the same remarks hold true as for the elastic modulus tensor. The dimensions of the compliance coefficients are the inverse of the dimensions of mechanical stress. § 5. Energy of elastic deformation Let us calculate the energy of an elastically deformed body. Let the displacement vector u due to the deformation of the body vary by a small amount du i. The elementary work performed in this case by the internal stresses is the product of the force F i = ∂σ ik / ∂x k and the displacement du i, integrated over the entire volume of the body V : dA = ∫ V (∂σ ik / ∂x k) (du i) dV. Integrating by parts, we obtain d A = ∮ σ i k (d u i) d S - ∫ V σ i ⁢ k ⁢ ∂ ∂ x k (d u i) d V. The first (surface). integral vanishes for an unbounded medium which is not deformed at infinity, since σ ik = 0. The second integral, by virtue of the fact that (∂ / ∂x k) (du i) = d (∂u i / ∂x k), can be written in the form ∫ σ ik d (∂u i / ∂x k) dV. The integrand here represents the work performed by internal stresses per unit volume of the body: d A ′ = − σ i k d (∂ u i ∂ x k), (I.16) In the case of a linear-elastic deformation, taking into account the symmetry of the stress tensor σ ik, we have σ i k d (∂ u i ∂ x k) = σ i k d [ 1 2 (∂ u i ∂ x k + ∂ u k ∂ x i) ] = σ i k d ϵ i k, where ϵ ik is the. strain tensor

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