Elasticity

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Elasticity
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• All objects are deformable. It is possible to
  change the shape or the size of an object by
  applying external forces. However, internal
  forces in the object resist deformation.

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• Stress and Strain
• Stress is a quantity that is proportional to the
  force causing a deformation. Stress is the
  external force acting on an object per unit cross
  sectional area.
• Strain is a measure of the degree of
  deformation. It is found that for sufficiently
  small stresses strain is proportional to stress.

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• The constant of the proportionality depends on
  the material being deformed and on the nature of
  deformation
• We call this proportionality constant the elastic
  modulus.

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• The elastic modulus is therefore the ratio of
  stress to the resulting strain.
• Elastic ModulusStress/Strain
• In a very real sense it is a comparison of what
  is done to a solid object (a force is applied)
  and how that object responds (it deforms to some
  extent)

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We consider three types of deformation and
define an elastic modulus for each

• Youngs Modulus which measures the resistance of
  a solid to a change in its length
• Shear Modulus which measures the resistance to
  motion of the planes of a solid sliding past each
  other
• Bulk Modulus which measures the resistance of
  solids or liquids to changes in their volume

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Youngs Modulus

• Consider a long bar of cross sectional area A and
  initial length Li that is clamped at one end.
  When an external force is applied perpendicular
  to the cross section internal forces in the bar
  resist distortion stretching but the bar
  attains an equilibrium in which its length Lf is
  greater than Li and in which the external force
  is exactly balanced by internal forces.

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• In such a situation the bar is said to be
  stressed. We define the tensile stress as the
  ratio of the magnitude of the external force F to
  the cross sectional area A. the tensile strain in
  this case is defines as the ratio of the change
  in length ?L to the original length Li.
• Ytensile stress/ tensile strain
• Y(F/A)/(?L/Li)

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The Elastic Limit

• The elastic limit of a substance is defined as
  the maximum stress that can be applied to the
  substance before it becomes permanently deformed.
  It is possible to exceed the elastic limit of a
  substance by applying sufficiently large stress,
  as seen in in the figure

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• Initially a stress strain curve is a straight
  line. As the stress increases, however the curve
  is no longer a straight line.
• When the stress exceeds the elastic limit the
  object is permanently distorted and it does not
  return to its original shape after the stress is
  removed.

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• What is Youngs modulus for the elastic solid
  whose stress strain curve is depicted in the
  figure ??
• Youngs modulus is given by the ratio of stress
  to strain which is the slope of the elastic
  behavior section of the graph in slide 9 reading
  from the graph we note that a stress of
  approximately 3x108N/m² results in a strain of
  0.003. The slope, and hence Youngs modulus are
  therefore 10x10¹ºN/m².

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Shear Modulus

• Another type of deformation occurs when an object
  is subjected to a force tangential to one of its
  faces while the opposite face is held fixed by
  another force. The stress in this case is called
  a shear stress.

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• If the object is originally a rectangular block a
  shear stress results in a shape whose cross
  section is a parallelogram. To a first
  approximation (for small distortions) no change
  in volume occurs with this deformation.
• We define the shear stress as F/A, the ratio of
  the tangential to the area of A of the force
  being sheared.

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• The shear strain is defined as the ratio
  ?X/H where ?X is the horizontal distance
  that the sheared force moves and H is the height
  of the object.
• In terms of these quantities the shear modulus is
  
• S shear stress/ shear strain
• S (F/A)/ (?X/H)

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Bulk Modulus

• Bulk modulus characterizes the response of a
  substance to uniform squeezing or to a reduction
  in pressure when the object is placed in a
  partial vacuum. Suppose that the external forces
  acting on an object are at right angles to all
  its faces, and that they are distributed
  uniformly over all the faces.

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• A uniform distribution of forces occur when an
  object is immersed in a fluid. An object subject
  to this type of deformation undergoes a change in
  volume but no change in shape. The volume stress
  is defined as the ratio of the magnitude of the
  normal force F to the area A.
• The quantity PF/A is called the pressure. If the
  pressure on an object changes by an amount ?P
  ?F/A the object will experience a volume change
  ?V.

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• The volume strain is equal to the change in
  volume ?V divided by the initial volume Vi
• B volume stress/volume strain
• B-(?F/A)/(? V/Vi)
• B- ? P/(?V/Vi)

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When a solid is under uniform pressure it
undergoes a change in volume but no change in
shape. This cube is compressed on all sides by
forces normal to its 6 faces.
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Prestressed Concrete

• If the stress on a solid object exceeds a certain
  value, the object fractures. The maximum stress
  that can be applied before fracture occurs
  depends on the nature of the material and on the
  type of applied stress.

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• For example concrete has a tensile strength of
  about 2 x 106 N/m², a compressive strength of 20
  x 106 N/m², and a shear strength of 2 x 106
  N/m.²
• It is common practice to use large safety factors
  to prevent failure in concrete structures.

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• Concrete is normally very brittle when it is cast
  in thin sections. Thus concrete slabs tend to
  slab and crack at unsupported areas as shown in
  figure A.

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• The slab can be strengthened by the use of steal
  rods to reinforce the concrete as illustrated in
  figure B. Because concrete is much stronger under
  compression squeezing than under tension
  stretching or shear, vertical columns of
  concrete can support very heavy loads, whereas
  horizontal beams of concrete tend to sag and
  crack.

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• However, a significant increase in shear strength
  is achieved if the reinforced concrete is
  prestressed as shown in figure C. As the concrete
  is being poured the steal rods are held under
  tension by external forces.

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• The external forces are released after the
  concrete cures this results in a permanent
  tension in the steel and hence a compressive
  stress on the concrete. This enables the concrete
  slab to support a much heavier load.

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Viscosity

• The term viscosity is commonly used in the
  description of fluid flow to characterize the
  degree of internal friction, or viscous force is
  associated with the resistance that two adjacent
  layers of fluid have to moving relative to each
  other. Viscosity causes part of the kinetic
  energy of a fluid to be converted to internal
  energy.

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Units of Measure

• Dynamic viscosity and absolute viscosity are
  synonymous. The IUPAC symbol for viscosity is the
  Greek symbol eta (?), and dynamic viscosity is
  also commonly referred to using the Greek symbol
  mu (µ). The SI physical unit of dynamic viscosity
  is the Pascal-second (Pas), which is identical
  to 1 kgm-1s-1. If a fluid with a viscosity of
  one Pas is placed between two plates, and one
  plate is pushed sideways with a shear stress of
  one Pascal, it moves a distance equal to the
  thickness of the layer between the plates in one
  second.

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• The name Poiseuille (Pl) was proposed for this
  unit (after Jean Louis Marie Poiseuille who
  formulated Poiseuille's law of viscous flow), but
  not accepted internationally. Care must be taken
  in not confusing the Poiseuille with the poise
  named after the same person.

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• The cgs physical unit for dynamic viscosity is
  the poise (P), named after Jean Louis Marie
  Poiseuille. It is more commonly expressed,
  particularly in ASTM standards, as centipoise
  (cP). The centipoise is commonly used because
  water has a viscosity of 1.0020 cP (at 20 C the
  closeness to one is a convenient coincidence).
• 1 P 1 gcm-1s-1
• The relation between poise and Pascal-seconds is
• 10 P 1 kgm-1s-1 1 Pas
• 1 cP 0.001 Pas 1 mPas

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Example 1

• A solid brass sphere is initially surrounded by
  air, the air pressure exerted on it is 1.0 x 105
  N/m² (normal atmospheric pressure) the sphere is
  lowered into the ocean to a depth which the
  pressure is 2.0 x 107 N/m² . The volume of the
  sphere in air is 0.50 m³. By how much does this
  volume change once the sphere is submerged ?

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• From the definition of bulk modulus, we have B
  - ?P / (?V/Vi)
• ?V - (Vi ?P) / B
• Because the final pressure is so much greater
  than the initial pressure, we neglect the initial
  pressure and state that
• ?P Pf Pi Pf 2.0 x 107 N/m²
• Therefore
• ?V - (0.5 m³) (2.0 x 107 N/m²)
• 6.1 x 10¹º N/ m²
• 1.6 x 104 m³
• The negative sign indicates a decrease in volume.

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Example 2

• We analyzed a cable used to support an actor as
  he swung onto the stage. The tension in the cable
  was 940 N. what diameter should a 10-m-long steel
  wire have if we do not want it to stretch more
  than 0.5cm under these conditions?

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• From the definition of Youngs modulus, we can
  solve for the required cross-sectional area.
  Assuming that the cross-sectional is circular, we
  can determine the diameter of the wire.
• Y(F/A)/(?L/Li)
• A(F Li)/(Y ?L)
• (940N)(10m) .
• (20 x 10¹ºN/m²)(0.005m)
• 9.4x10(-6) m²

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• The radius of the wire can be found from A?r²
• r v(A/ ?)
• v(9.4x10(-6) m²/ ?)
• 1.7x10³m 1.7mm
• D2r 2(1.7mm) 3.4mm
• To provide a large margin of safety, we should
  probably use a flexible cable made up of many
  smaller wires having a total cross-sectional area
  substantially greater than our calculated value.

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References

• Physics For Scientists and Engineers
• (Serway . Beichner).