Elasticity of Materials

10 Key excerpts on "Elasticity of Materials"

• 

  eBook - ePub

  Introduction to Mechanical Engineering

  Part 1

  • Michael Clifford(Author)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  The elastic modulus enables the extension or deflection of a material under load to be calculated. It is applicable to situations where the applied load produces reversible elastic rather than permanent plastic deformation. In most engineering situations, this is the case, as permanent plastic deformation would lead to unacceptable distortion of the component. For example, the elastic modulus would be used to predict the amount a wire will elastically extend under tension and the deflection of structures such as beams under loading. The elastic modulus can also be used to determine the energy stored in a structure that has been elastically deformed (and hence can be recovered to do work), such as a spring.

  Learning Summary

  • Elastic modulus, modulus of elasticity or stiffness, is defined as an object’s resistance to elastic deformation;
  • Units are the pascal (Pa); most materials have elastic moduli ~109 Pa (GPa);
  • Young’s modulus can be determined from the slope of the straight line portion of stress–strain curves produced by tensile testing;
  • The origin of the elastic modulus of a material lies predominantly in the stiffness of the bonding between the atoms that comprise the material. Stiff bonding gives a high modulus. As little can be done to change the stiffness of the bonds, the elastic modulus varies very little with alloying, heat treatment and mechanical working;
  • The elastic modulus enables the extension, compression, deflection and energy stored in structures under load to be determined (as long as the load produces reversible elastic deformation).

  Table 2.1

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  eBook - ePub

  Mechanical Engineering Principles

  • John Bird, Carl Ross(Authors)
  • 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 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 take into consideration not only 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 ε 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 E 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
  • define rigidity or shear modulus
  • understand thermal stresses and strains
  • calculates stresses in compound bars

  3.1  

  Introduction

  A force exerted on a body can cause a change in either the shape or the motion of the body. The unit of force is the newton*, N.

  No solid body is perfectly rigid and when forces are applied to it, changes in dimensions occur. Such changes are not always perceptible to the human eye since they are so small. For example, the span of a bridge will sag under the weight of a vehicle and a spanner will bend slightly when tightening a nut. It is important for engineers and designers to appreciate the effects of forces on materials, together with their mechanical properties. The three main types of mechanical force that can act on a body are:

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  eBook - PDF

  Fundamentals of Materials Science and Engineering

  An Integrated Approach

  • William D. Callister, Jr., David G. Rethwisch(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Second, it specifies the degree of allowable deformation during fabrication operations. We sometimes refer to relatively ductile materials as being “forgiving,” in the sense that they may experience local deformation without fracture, should there be an error in the magnitude of the design stress calculation. Brittle materials are approximately considered to be those having a fracture strain of less than about 5%. Thus, several important mechanical properties of metals may be determined from tensile stress–strain tests. Table 8.2 presents some typical room-temperature values of yield strength, tensile strength, and ductility for several common metals (and also for a number of polymers and ceramics). These properties are sensitive to any prior deforma- tion, the presence of impurities, and/or any heat treatment to which the metal has been subjected. The modulus of elasticity is one mechanical parameter that is insensitive to these treatments. As with modulus of elasticity, the magnitudes of both yield and tensile strengths decline with increasing temperature; just the reverse holds for ductility—it usually increases with temperature. Figure 8.14 shows how the stress–strain behavior of iron varies with temperature. Resilience Resilience is the capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered. The associated property is the modulus of resilience, U r , which is the strain energy per unit volume required to stress a material from an unloaded state up to the point of yielding. Computationally, the modulus of resilience for a specimen subjected to a uniaxial tension test is just the area under the engineering stress–strain curve taken to yielding (Figure 8.15), or U r =  0 ε y σ dε (8.13a) Ductility, as percent reduction in area Tutorial Video: How Do I Determine Ductility in Percent Elongation and Percent Reduction in Area? resilience Definition of modulus of resilience

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  eBook - ePub

  Fundamentals of Materials Engineering - A Basic Guide

  • Shashanka Rajendrachari, Orhan Uzun(Authors)
  • 2021(Publication Date)
  • Bentham Science Publishers

    (Publisher)

  4 ]. Toughness is measured by calculating the area under the stress-strain curve from a tensile test and it is the unit of energy per volume. One of the metals having maximum toughness is Tungsten, and its toughness can be significantly increased by alloying with suitable metals.

  Fig. (7)) The stress-strain curve showing the toughness [4 ].

  5.8. Resilience

  It is the ability of a material to rebound elastically into the original shape. In other words, it is the total energy absorbed by the materials during elastic deformation. Fig. (

  8

  ) shows the resilience and modulus of resilience in load-extension and stress-strain diagrams, respectively [4 ]. It is the area up to the elastic limit, as shown in the figure.

  Fig. (8)) (a) Load-extension diagram showing resilience. (b) Stress-strain diagram showing modulus of resilience [4 ].

  The maximum amount of energy absorbed up to the elastic limit, without permanent distortion, is called proof resilience. Similarly, the modulus of resilience is defined as the maximum energy that can be absorbed per unit volume without permanent distortion. The unit of resilience is joule per cubic meter (J/m3 ).

  5.9. Stiffness

  Stiffness is expressed by Young’s modulus, and it is defined as the resistance of material for elastic deformation. The values of both true stress and true strain are from the elastic region, and it does not affect by alloying or change in the microstructure.

  5.10. Ductility

  Ductility is an ability of a material to stretch under tensile stress before fracture. In other words, it can also be defined as an ability of a material to drawn in to thin wire. Generally, ductility can be expressed in terms of percentage elongation, and it is given as below;

  Ductility of a material can be graphically represented as below:

  Fig. (

  9

  ) represents the load-extension diagram showing the ductility of materials [4 ]. Examples of ductile materials are copper, aluminum, steel, and some more metals. Ductility is a physical property, and it is not having any unit.

  5.11. Malleability

  Malleability is just opposite to that of ductility. In the case of the ductility test, tensile forces are used, but here compressive forces are used. It is defined as the ability of a material to undergo plastic deformation before fracturing under compressive stress. It can also be defined as the ability of a material to undergo rolling or flattening into thin sheets. Most of the metals with high ductility also possess greater malleability, and it can be measured by % reduction in the cross-sectional area of the material under study.

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  eBook - PDF

  Callister's Materials Science and Engineering

  • William D. Callister, Jr., David G. Rethwisch(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  However, for some polymeric materials, its magnitude is significant; in this case it is termed viscoelastic behavior, which is the discussion topic of Section 15.4. anelasticity 6.4 ANELASTICITY EXAMPLE PROBLEM 6.1 Elongation (Elastic) Computation A piece of copper originally 305 mm (12 in.) long is pulled in tension with a stress of 276 MPa (40,000 psi). If the deformation is entirely elastic, what will be the resultant elongation? Solution Because the deformation is elastic, strain is dependent on stress according to Equation 6.5. Furthermore, the elongation Δl is related to the original length l 0 through Equation 6.2. Combining these two expressions and solving for Δl yields  = E = ( Δl l 0 ) E Δl = l 0 E The values of  and l 0 are given as 276 MPa and 305 mm, respectively, and the magnitude of E for copper from Table 6.1 is 110 GPa (16 × 10 6 psi). Elongation is obtained by substitution into the preceding expression as Δl = (276 MPa)(305 mm) 110 × 10 3 MPa = 0.77 mm (0.03 in.) When a tensile stress is imposed on a metal specimen, an elastic elongation and ac- companying strain  z result in the direction of the applied stress (arbitrarily taken to be the z direction), as indicated in Figure 6.9. As a result of this elongation, there will be constrictions in the lateral (x and y) directions perpendicular to the applied stress; from these contractions, the compressive strains  x and  y may be determined. If the applied stress is uniaxial (only in the z direction) and the material is isotropic, then  x =  y . A parameter termed Poisson’s ratio  is defined as the ratio of the lateral and axial strains, or  = −  x  z = −  y  z (6.8) Poisson’s ratio Definition of Poisson’s ratio in terms of lateral and axial strains 6.5 ELASTIC PROPERTIES OF MATERIALS

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  eBook - PDF

  Fundamentals of Modern Manufacturing

  Materials, Processes, and Systems

  • Mikell P. Groover(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  If the stress had been applied and then immediately removed, the material would have returned immediately to its starting shape. However, time has entered the picture and played a role in affecting the behavior of the material. A simple model of viscoelasticity can be developed using the definition of elasticity as a starting point. Elasticity is concisely expressed by Hooke’s law, σ = Eϵ, which simply relates stress to strain through a constant of proportionality. In a viscoelastic solid, the relationship between stress and strain is time dependent; it can be expressed as = ( ) ( ) t f t σ є (3.27) The time function f(t) can be conceptualized as a modulus of elasticity that depends on time. It might be written E(t) and referred to as a viscoelastic modulus. The form of this time function can be Plastic solid Pseudoplastic fluid Newtonian fluid Shear rate, γ Shear stress, τ Yield stress · ■ Figure 3.18 Viscous behaviors of Newtonian and pseudoplastic fluids. Polymer melts exhibit pseudoplastic behavior. For comparison, the behavior of a plastic solid material is shown. 62 | Chapter 3 | Mechanical Properties of Materials complex, sometimes including strain as a factor. Without getting into the mathematical expressions for it, the effect of the time dependency can nevertheless be explored. One common effect can be seen in Figure 3.20, which shows the stress–strain behavior of a thermoplastic polymer under differ- ent strain rates. At low strain rate, the material exhibits significant viscous flow; but at high strain rate, it behaves in a much more brittle fashion. Temperature is a factor in viscoelasticity. As temperature increases, the viscous behavior becomes more and more prominent relative to elastic behavior. The material becomes more like a fluid. Figure 3.21 illustrates this temperature dependence for a thermoplastic polymer. At low tempera- tures, the polymer shows elastic behavior.

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  eBook - ePub

  Micro- and Macromechanical Properties of Materials

  • Yichun Zhou, Li Yang, Yongli Huang(Authors)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  12 Mechanical Properties of Polymer Materials

  Polymers are an important category of engineering materials, and are widely used in a variety of areas such as mechanical, construction, automotive, electrical appliances, light industry, electronics, and aerospace. Because of their unique molecular chain structures and aggregation structures, polymers have different physical properties compared with other materials, especially their mechanical properties. The mechanical properties of a polymer strongly depend on the conditions of temperature, loading time, deformation rate, and load frequency. Usually—depending on different temperatures and the time of observation—a polymer shows different mechanical behaviors including the glassy state, the viscoelastic state, the high-elastic state (hyperelastic or rubbery state), and the viscous flow state. As a structural material, the main features of a polymer’s mechanical properties are its viscoelasticity and its hyperelasticity. This chapter will introduce viscoelasticity and its mechanical models, the hyperelasticity of polymer materials, and the yield and pattern of the brittle–ductile transition of polymer materials.

  12.1 Viscoelasticity of High Polymer

  In understanding substances and materials, human beings first became familiar with two types of materials: elastic solids and viscous fluids. Elastic solids have specific volumes and configurations, and their stress state and deformation are time-independent when subject to static loads. Elastic solids can be fully restored after the external force is removed. The work done by external forces in the elastic deformation process is not only stored in the form of elastic potential energy, but is also completely released in the process of unloading. On the contrary, viscous fluids do not have determined volumes and configurations, and their shapes depend on the container. Under external forces, they continually deform over time, resulting in an irreversible flow. When deformation movement occurs, the adjacent fluid layers will cause internal friction that consumes energy. In general, polymer materials are neither attributable to the elastic solid, nor attributable to the viscous fluid. They often have the characteristics of both elastic solids and viscous fluids, and comprehensively show the two different mechanisms of elastic and viscous deformations. This property of substances is called viscoelasticity [1

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  eBook - PDF

  Fundamentals of Materials Science and Engineering

  An Integrated Approach

  • William D. Callister, Jr., David G. Rethwisch(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  The slope of this linear segment corresponds to the modulus of elasticity E. This modulus may be thought of as stiffness, or a material’s resistance to elastic deformation. The greater the modulus, the stiffer is the material, or the smaller is the elastic strain that results from the application of a given stress. The modulus is an important design parameter for computing elastic deflections. Elastic deformation is nonpermanent, which means that when the applied load is released, the piece returns to its original shape. As shown in the stress–strain plot (Figure 7.5), application of the load corresponds to moving from the origin up and along the straight line. Upon release of the load, the line is traversed in the opposite direction, back to the origin. modulus of elasticity elastic deformation 7.3 STRESS–STRAIN BEHAVIOR : VMSE Metal Alloys Hooke’s law— relationship between engineering stress and engineering strain for elastic deformation (tension and compression) Tutorial Video: Calculating Elastic Modulus Using a Stress vs. Strain Curve

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  eBook - PDF

  Materials Science and Engineering

  An Introduction

  • William D. Callister, Jr., David G. Rethwisch(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Such treatments are beyond the scope of the present discussion. Elastic Deformation The degree to which a structure deforms or strains depends on the magnitude of an imposed stress. For most metals that are stressed in tension and at relatively low levels, stress and strain are proportional to each other through the relationship  = E (6.5) This is known as Hooke’s law, and the constant of proportionality E (GPa or psi) 6 is the modulus of elasticity, or Young’s modulus. For most typical metals, the magnitude of this modulus ranges between 45 GPa (6.5 × 10 6 psi), for magnesium, and 407 GPa (59 × 10 6 psi), for tungsten. Modulus of elasticity values for several metals at room temperature are presented in Table 6.1. Deformation in which stress and strain are proportional is called elastic deformation; a plot of stress (ordinate) versus strain (abscissa) results in a linear relationship, as shown in Figure 6.5. The slope of this linear segment corresponds to the modulus of elasticity E. This modulus may be thought of as stiffness, or a material’s resistance to elastic deformation. The greater the modulus, the stiffer the material, or the smaller the elastic strain that results Hooke’s law— relationship between engineering stress and engineering strain for elastic deformation (tension and compression) modulus of elasticity elastic deformation 6.3 STRESS–STRAIN BEHAVIOR 5 See, for example, W. F. Riley, L. D. Sturges, and D. H. Morris, Mechanics of Materials, 6th edition, John Wiley & Sons, Hoboken, NJ, 2006. 6 The SI unit for the modulus of elasticity is gigapascal (GPa), where 1 GPa = 10 9 N/m 2 = 10 3 MPa.

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  eBook - PDF

  Foundations of the Theory of Elasticity, Plasticity, and Viscoelasticity

  • Eduard Starovoitov, Faig Bakhman Ogli Naghiyev(Authors)
  • 2012(Publication Date)
  • Apple Academic Press

    (Publisher)

  Chapter 8 Foundations of the Theory of Plasticity Deformability of solids under the effect of external forces and their capability of per-ceiving constant or residual ( plastic ) strains at unloading is called plasticity . There is, however, no unequivocal dependence between stresses and strains arising in the body, which means that it is impossible to find strains in terms of stresses, and vice versa, one cannot determine stresses proceeding from known strains. The theory of plasticity deals with the laws interrelating stresses with elastoplastic deformations and the development of problem-solving methods on equilibrium and motion of deformed solid bodies. The theory of plasticity, forms the basis of today’s calculations of different structures, forging processes, rolling, punching, and so forth, as well as natural processes (e.g., orogenesis). This allows to reveal strength and de-formation potential of materials. Plastic deformation till fracture may reach 10–20%, while elastic deformations––only 0.3–0.5%. That is why strength calculations based on the assumption of only elastic deformations are often inexpedient both technically and economically. By taking plastic deformations into account, we may reduce stress concentration in structures, increase resistance of bodies to impact loads, de fi ne safety margins, rigidity and stability, ensuring thereby most ef fi cient functioning, reliability and safety of structures. PLASTICITY OF MATERIALS AT TENSION AND COMPRESSION The phenomena of elasticity and plasticity are displayed at sufficiently slow so-called static or quasi-static application of external forces. In this case, the phenomenon of deformability does not in fact depend on time, loading rate, and duration of external forces. Let us consider the basic phenomena of plasticity by a simplest example of tension and compression of a cylindrical sample.

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