Elasticity, Hooke's law, Poisson's ratio

 Elasticity, Hooke's law, Poisson's ratio

Introduction:

     The universe is made up of matter and energy. The universe is made up of either matter or energy.

 Matter exists in different states. There are three states of matter. These are

1.     Solid   2. Liquid   3.  Gas   4.  Plasma   5.  Bose Einstein Condensate

      Out of these, the first 3 states exist at normal conditions, such as pressure and temperature. The 4^th state exists above 6000⁰ C. The 5^th state exists at nearly 0⁰ K.

All these matters can be studied on the basis of Kinetic theory. According to this--

1.     All the matter is made up of atoms or molecules.

2.     These molecules are in some are the motion.

3.     There is attraction between the molecules depends on distance between the molecules.

4.     The molecules are in spherical shape.

Let us consider the first three states of matters on the basis of kinetic theory.

Solid:

     In solid, molecules are closely packed because solid is made by taking

10²²   molecules / cm³ .Due to this there is a strong attraction between the molecules and stay in a stable equilibrium position. Hence molecules arrange in a particular shape. Thus solid have shape and size. The molecules in the solid are in vibrational motion. The vibration depends on temperature.

   When a solid is deformed, the atoms or molecules are displaced from their equilibrium positions causing a change in inter atomic (or inter-molecular) distances. When the deforming force is removed, inter atomic forces tend to bring them back to their original positions. Thus the body regains its original shape and size. The restoring mechanism can be visualized by taking a model of Ball- Spring model. 

 Fig A

   Fig A represents Ball – Spring model of solid. The ball represents atom or molecule of solid and the spring represents inter atomic force. If any ball is displaced from its equilibrium position, the spring system tries to restore the ball back to its original position. Thus elastic behavior of solids can be explained by this Ball- Spring model.

Elasticity:

    In our daily life, there are some objects whose shapes, size may be changed permanently or temporarily by applying external unbalanced force. After removal of this applied force, the objects may recover completely or partially its original shape, size. 

    As for example, if the spring is pressed, the length of the spring becomes shorter than previous. Again, if the applied pressure is removed then the spring almost recovers its original length. Similarly, if the rubber band is stretched its length increases. This property of the material is termed as Elasticity.

Some of the terms related to Elasticity:

Inter molecular force or Elastic force:

    The strong attractive force between the molecules in solid is called inter molecular force. This is short range force.

Deforming Force:

   The force which produces a change in shape and size of the body on applying it is called deforming force.

Restoring Force:

    The force developed within the elastic body by virtue of relative molecular displacement is called a restoring force. After the removal of deforming force, the elastic body regains its original shape and size due to restoring force.

Elasticity: 

      The property of a body by virtue of which the body tries to regain its original shape and size after the removal of deforming force is called elasticity.

Elastic Body:

   The body which tries to regain its original shape and size after the removal of deforming force is called elastic body.

    Examples of elastic body: Wire, Spring, Rubber band, Cartilage tissue in human body which connects bones.

Perfectly Elastic Body:

  The body which tries to regain its original shape and size immediately and completely after the removal of deforming force is called perfectly elastic body.

Examples of perfectly elastic body: Quartz and Phosphor bronze etc.

Plasticity: 

      The property of a body by virtue of which the body tries remain deformed after the removal of deforming force is called plasticity.

Plastic Body:

     The body which tries remains deformed after the removal of deforming force is called plastic body.

Examples of plastic body: Soil, Concrete, Foam, Mud, Clay.

Perfectly Plastic Body:

 The body which tries remains permanently deformed after the removal of deforming force is called perfectly plastic body.

Examples of perfectly plastic body: Wet clay, Butter, Wax, Putty.

Perfectly Rigid Body: 

  The body which is not deformed under the action of applied force is called perfectly rigid body. The distance between the two points of rigid body remains constant under the action of applied force. There is no rigid body in nature. The point masses like electron, proton, atom, molecule and also thick metal pole are considered as perfectly rigid bodies.

Elastic Limit:

    The upper limit of deforming force up to which, if deforming force is removed, the body regains its original form completely and beyond which if deforming force is increased the body loses its property of elasticity is called elastic limit.

Deformation:

The change in shape and size or both of the body on applying external force is called deformation.

Deformation α Deforming force

Deformation may take place in three ways:

1.     Deformation is due to change in length in case of wire.

2.     Deformation is due to change in volume of body.

3.     Deformation is due to change in shape of body.

  When deformation takes place then the body is in deformed or in strained state. The force developed within the elastic body by virtue of relative molecular displacement is called a restoring force. After the removal of deforming force, the elastic body regains its original shape and size due to restoring force. This internal force is equals to the applied force but apposite in direction.

  The following terms to be considered, corresponding to these points.

1.   Stress          2. Strain         3. Hooke’s law   and

    Modulii of elasticity

1. Stress: The internal restoring force acting per unit area of a deformed           body is called stress.

Stress = Restoring force / Area = Applied force / Area

           The SI unit of stress is N/m² or pascal (Pa) and

CGS unit is dynes/cm². The dimensional formula is [M¹L^-1T^-2].

Three types of stresses corresponds three types of deformation (change in length, volume and shape). These are:

a. Longitudinal or tensile stress   b. Volume stress    c. Shearing stress

2. Strain: Strain is the ratio of change in dimension to the original dimension of the deformed body.

  Strain = Change in dimension / Original dimension

 Strain has neither unit nor dimensions.

Three types of strains corresponds three types of deformation (change in length, volume and shape). These are:

a. Longitudinal or tensile strain   b. Volume strain    c. Shearing strain

3. Hooke’s law: Robert Hooke (1635-1703) an English physicist studied stress and strain relationship and given fundamental law of elasticity know as Hooke’s law elasticity.   

“Within the limit of elasticity, the stress is directly proportional to the strain.”

                                Stress ∝ Strain 

Stress = M x Strain or   Stress / Strain = M = Modulus of elasticity

Where M is proportionality constant and is called as modulus of elasticity.           The SI unit of modulus of elasticity is N/m² or pascal (Pa) and

CGS unit is dynes/cm². The dimensional formula is [M¹L^-1T^-2].

These are same as that of stress.

Three types of modulii of elasticity corresponds three types of deformation (change in length, volume and shape). These are:

a. Young’s modulus     b. Bulk modulus     c. Modulus of rigidity 

a. The change in length of wire:

 Fig B

  Consider the wire of length L and cross-sectional area A is suspended at support (Fig B). A force (load) F is applied at bottom end of the wire along the length. This produces elongation in the wire of extension l.

The applied force is called Longitudinal or tensile stress.

Let the mass of load be M and radius of the wire be r. Then F = Mg ,where g is acceleration due to gravity and A = πr².

Longitudinal or tensile Stress = Applied force / Area

                                                   = F / A  = Mg / πr²                  ….  1

The corresponding strain is Longitudinal or tensile strain.

Longitudinal or tensile strain = Change in dimension / Original dimension

                                                   =   l /L                                  …. 2

Now related to change in length of the wire according to Hooke’s law modulus of elasticity is Young’s modulus.

 Young’s modulus

                = Longitudinal or tensile Stress/ Longitudinal or tensile strain         

 From equation 1 and 2, we get

    Young’s modulus Y  = Mg L / πr²  l                                … 3

Young’s modulus is property of solid only, as solid only can be stretched in length.  e.g. Young’s modulus  Y of steel is 2.1x10¹¹ N/m² .

b. The change in volume of body:

 Fig C

Consider a ball of volume V. When pressure P is applied to entire surface of ball, then the ball will compress and there is decrease in volume by dV.  The new volume will be V – dV (Fig C).

The pressure P applied on the ball is nothing but applied force per unit area.

This pressure is called volume stress.

Volume stress = Applied force / area = pressure =P                  …4

Volume strain = Change in volume / Original volume

                         = dV / V                                                                 … 5

Now related to change in volume of the wire according to Hooke’s law modulus of elasticity is Bulk modulus.

Bulk modulus = Volume stress / Volume strain

From equation 4 and 5, we get

Bulk modulus   K =  P .dV / V                                                            …. 6

Bulk modulus is property of solid, liquid and gas. .  e.g. Bulk modulus  K of steel is 1.6 x10¹¹ N/m² .

The reciprocal of Bulk modulus is called Compressibility.

        Compressibility = 1/   Bulk modulus

c. Modulus of rigidity:    

 Fig D

Consider a cube MNOPQRST whose bottom face MNOP is fixed and the tangential force F is applied on top face QRST (Fig D). Due to this the top face QRST is shifted as Q’R’S’T’. The shift is dl. Let the distance between fixed face and the shifted face be l and the area of each face of cube be A.

Shearing stress = Tangential force / Area of face

                         = F / A                                                                  …. 7

Shearing strain = Relative displacement of layer / Its distance from fixed layer

                           = QQ’ / MQ = dl / l = tan Ө ≈ Ө                    …. 8

Now related to change in shape of the body, according to Hooke’s law modulus of elasticity is modulus of rigidity.

Modulus of rigidity = Shearing stress / Shearing strain

      From equations 7 and 8, we get

Modulus of rigidity   ɳ = F / A Ө                                                   ….9

Modulus of rigidity is property of solid only, as solid only can be stretched to change in shape. .  e.g. Modulus of rigidity ɳ of steel is 0.83x10¹¹ N/m² .

Behavior of metal wire under increasing load:

A metal wire is suspended vertically from rigid support and free end is loaded with step by step increasing load until the wire breaks. The extension produced in the wire is measured every time. Then corresponding values of stress and strain are calculated. The graph of stress against strain is plotted, and curve is obtained. This curve is called Stress-Strain curve (Fig E).

 Fig E

1.     From the origin O to the point called proportional limit P, the stress-strain curve is a straight line. If the wire is unloaded anywhere between points O and P, then it will return to its original shape. This type of behavior is termed elastic and the region between points O and P is the proportional region. In this region Hooke’s law is obeyed. The stress corresponding to point P is called elastic limit of material of wire.

2.     If the stress increases beyond the elastic limit, then the extension starts increasing faster up to point E. At this point E if the load is removed, the wire is not able to recover original length. But the wire will still retain its elastic properties. Now reloading produce the path OE and the wire undergo permanent deformation of OS. This permanent strain OE is called set.

3.     If the load is increased a point Y will reached, then the extension begins to increase even without any increase in load. The wire is said to be plastic flow. The point Y is called Yield point. The value of stress corresponding to Y is called Yield stress.

4.     Next to yield point the curve again begin to go upwards in the portion YU. The cross section of the wire decreases up to point U and volume remains constant.  The point U represents the maximum stress which wire can bear and is called breaking stress or ultimate stress.

5.     The extension of the wire goes on increasing beyond U without any increase in load. The wire will flow and finally breaks at point B.  This point is called breaking point.

Poisson’s ratio:

  Fig F

Consider a tube of diameter D and length L, one end of which is suspended at support and other end is loaded by a load F (Fig F). Due to this the length of tube increases and diameter of tube decreases. The decrease in diameter of stretched tube is d and increase in length is l. The strain produced along the direction of force is called longitudinal strain which is considered as positive and that developed in perpendicular direction to force is called lateral strain, and which is considered as negative.

“Longitudinal strain is defined as ratio of increase in the length of the tube in the direction of applied load to that of the original length”.

 Longitudinal strain = Change in length / Original length

                                     = l / L                                               …. 10

“Lateral strain is defined as ratio of decrease in the diameter of the tube to that of the original diameter of tube”.

Lateral strain = Change in diameter / Original diameter

                        =  d / D                                                       …. 11

Poisson’s ratio: 

 Within elastic limit the ratio of lateral strain to that of the longitudinal strain is termed as Poisson’s ratio.

Poisson’s ratio σ = -  Lateral strain / Longitudinal strain

                             = -  d L / D l          (From equation 10 and 11)       ….12

Poisson’s ratio has no unit and it is dimensionless. The value is -1 ≤ σ ≤ 0.5.