3 Key excerpts on "Elastic Potential Energy"

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  eBook - ePub

  Energy Storage

  A New Approach

  • Ralph Zito, Haleh Ardebili(Authors)
  • 2019(Publication Date)
  • Wiley-Scrivener

    (Publisher)

  Humans have been using uncoiled mechanical springs in bows and arrows for thousands of years, for hunting and survival. When using bows and arrows, we are in fact harnessing the Elastic Potential Energy that was stored in the unoiled springs. As time passed, upon the emergence of coiled springs, the usage of mechanical springs expanded to many common applications including watches.

  Imagine, as you twist the knob on a watch, you are causing the spring to coil, and thus, you are powering the watch with Elastic Potential Energy. The watch would then uncoil, gradually in time, during the operation and thus, release the stored energy. Figure 2.4 shows the schematics of the coiled springs in an Elgin watch that was commonly used from 1864 to 1968.

  Figure 2.4

  Illustration of the coiled mainsprings used in Elgin pocket watch

  (Courtesy of Elgin National Watch Co.)

  To quantify the potential energy stored in elastic coiled structures such as a spring, we express the elastic energy U as

  (2.14)

  where k is the spring constant, and x is the displacement of the spring.

  For deformable bodies that can undergo deformation d or strain ε (the derivative of displacement with respect to position), the linear elastic strain energy (U) and strain energy density (u) (i.e. energy per volume), can be calculated as

  (2.15)

  (2.16)

  where, F is the force on the body, k is the material stiffness (or spring constant), σ is the stress in the body, and E is the elastic modulus.

  The strain energy density in one dimension can be converted to 2D or 3D based on the actual state of stresses and strains in the body. For example, if a body is subjected to plane stress (i.e., the normal and shear stresses in a plane are non-zero), the strain energy density (u) is expressed as

  (2.17)

  where x and y refer to the directions of the normal stresses and strains, and τxy is the γxy are the shear stress and strain, respectively, in the xy plane.

  2.2 Electrical Energy

  Imagine a charged particle at a certain voltage. The electrical or electrostatic energy (or work) associated with this particle can be calculated by multiplying its charge (in coulomb) by the voltage (in volt):

  (2.18)

  Now, imagine two particles with respective charges q1 and q2

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  eBook - ePub

  Integrated Reservoir Asset Management

  Principles and Best Practices

  • John Fanchi(Author)
  • 2010(Publication Date)
  • Gulf Professional Publishing

    (Publisher)

  6

  Petrophysics

  The study of the mechanical and acoustical properties of reservoir rocks and fluids is the focus of petrophysics. Petrophysical information is valuable in well logging, time-lapse seismology, and reservoir modeling. This chapter defines petroelastic parameters, introduces elasticity theory, presents models that can be used to estimate petroelastic and geomechanical quantities, and discusses the importance of petrophysics in time-lapse seismology.

  6.1 Elastic Constants

  The behavior of an object when it is subjected to deforming forces is described by the theory of elasticity. Elasticity is the property of the object that causes it to resist deformation. The deforming force applied to one or more surfaces of the object is called “stress.” Stress has the unit of pressure, or force per unit area, and is proportional to the force causing the deformation. The deformation of the object in response to the stress is called “strain.” Strain is a dimensionless quantity that reflects the relative change in the shape of the object as a result of the applied stress. Figure 6.1 illustrates the relationship between stress and strain.

  Figure 6.1 Stress–strain curve for an elastic solid.

  If the stress is not too great, object can return to its original shape when the stress is removed. In this case, Hooke’s law states that stress is proportional to strain. The proportionality constant is called the elastic modulus, which is the ratio of stress to strain:

  (6.1.1)

  Dimensional analysis shows that the elastic modulus has the unit of pressure. The elasticity of a substance determines how effective the object is in regaining its original form.

  Stresses are either one or a combination of three basic stresses: compressional stress, tensile stress, and shear stress. The corresponding strains are compressional strain, tensile strain, and shear strain, respectively. The corresponding elastic moduli are bulk modulus, Young’s modulus, and shear modulus. The following discusses each of these moduli.

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  Polymers for Vibration Damping Applications

  • Bikash C. Chakraborty, Debdatta Ratna(Authors)
  • 2020(Publication Date)
  • Elsevier

    (Publisher)

  In general, there are three states of maters, for example, solid, liquid, and gas which are relevant as media for vibration and sound propagation. The intermolecular forces in solids are highest, followed by liquid and gas. The forces in organic solids can be due to Van der Waal’s, apart from polar interaction and hydrogen bonding in some cases. In case of large molecules like polymers, these secondary valence forces assume much more importance than primary valence bonds, namely covalent and coordinate covalent bonds. This can be appreciated by the fact that polymers chemically decompose but do not evaporate (separation of molecules) upon heating. The strong intermolecular attraction in polymers results in considerable mechanical integrity and elastic modulus. Liquid molecules, on the other hand, are physically united by same forces but the intensity of the bond energies is far less, hence they can be vaporised upon heating prior to decomposition. In gases, these forces are negligible and individual molecules are free to move independently. Therefore, the response to external force on each of these is different and is the basis of complex behaviour of matters. Among the solid substances, we can have subdivisions as perfectly rigid bodies and elastic bodies. A rigid body is defined as which do not deform under any load, in any direction or mode. Such bodies are ideal and do not exist in the real world. A solid is defined as ‘elastic’, if its strain (deformation per unit dimension) linearly varies with the stress. Again, in the real world, there is no material which is perfectly elastic for all strain levels or all thermodynamic conditions. Hooke’s law states that within the elastic limit, if the temperature is kept constant, the stress varies linearly with strain and the ratio of stress to strain in the longitudinal direction of stretching is termed as ‘Young’s modulus’. This is a material property and does not depend on the values of implied force or extent of deformation, within the elastic limit. A mathematical representation of Hooke’s Law is

  σ = Eɛ

    (3.1)

  where σ is the stress in tension, measured as the force per unit area upon which it is acting, ɛ is the longitudinal strain, measured as the extent of deformation per unit length and E is the modulus of elasticity or Young’s modulus.

  Strain is defined as the deformation of a body per unit dimension. In case of longitudinal stretching, it is extension divided by the original length. This is mathematically expressed as

  ɛ =

  δ

  l 0

    (3.2)

  Strain is a dimensionless quantity. Stress in an elastic body is the resistance offered by the body against any attempt to deform it. Therefore, stress is exactly the force applied divided by the area upon which it acts for a perfect elastic body. It is mathematically expressed as

  σ =

  F A

    (3.3)

  Fig. 3.1 shows the stress–strain relationship of the elastic body. SI unit of stress is N/m2 or Pascal (Pa). Since strain is dimensionless, the unit of Young’s modulus is N/m2 (Pa). It is customary to use MPa (Mega Pascal), which is 106  Pa (or 1 N/mm2 ). The modulus is the stress of the body per unit strain. Therefore, it is a property of the material. It depends on the chemical structure and thermodynamic state of the material.

  Fig. 3.1 Stress–strain plot of a typical elastic solid.

  Eq. (3.1) reveals that if a stress is kept constant, the deformation instantly stops at that point. Also, the definition of elastic solid does not include any time factor, which means, the modulus is independent of strain rate.

  Elasticity is also related to the stiffness of bodies, measured as the load per unit deflection. The stiffness is analogous to spring constant k

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