Physics > Mechanical Properties of Solids

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  • The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed, is known as elasticity and the deformation caused is known as elastic deformation.
  • However, if you apply force to a lump of putty or mud, they have no gross tendency to regain their previous shape, and they get permanently deformed. Such substances are called plastic and this property is called plasticity.
  • When a solid is deformed, the atoms or molecules are displaced from their equilibrium positions causing a change in the interatomic (or intermolecular) distances. When the deforming force is removed, the interatomic forces tend to drive them back to their original positions. Thus the body regains its original shape and size.
  • When a body is subjected to a deforming force, a restoring force is developed in the body. This restoring force is equal in magnitude but opposite in direction to the applied force. The restoring force per unit area is known as stress. If is the force applied and  is the area of cross section of the body,

    Magnitude of the stress

    The unit of stress is

  • The restoring force per unit area is called tensile stress. If the cylinder is compressed under the action of applied forces, the restoring force per unit area is known as compressive stress. Tensile or compressive stress can also be termed as longitudinal stress.
  • The change in the length  to the original length  of the body (cylinder in this case) is known as longitudinal strain.
  • The restoring force per unit area developed due to the applied tangential force is known as tangential or shearing stress.
  • Shearing strain is defined as the ratio of relative displacement of the faces  to the length of the cylinder .

  • A solid sphere placed in the fluid under high pressure is compressed uniformly on all sides. The force applied by the fluid acts in perpendicular direction at each point of the surface and the body is said to be under hydraulic compression. This leads to decrease in its volume without any change of its geometrical shape.
  • The strain produced by a hydraulic pressure is called volume strain and is defined as the ratio of change in volume  to the original volume
  • For small deformations the stress and strain are proportional to each other. This is known as Hooke's law.
  • Thus,

    stress  strain

    where  is the proportionality constant and is known as modulus of elasticity.

  • The relation between the stress and the strain for a given material under tensile stress can be found experimentally. The fractional change in length (the strain) and the applied force needed to cause the strain are recorded. The applied force is gradually increased in steps and the change in length is noted. A graph is plotted between the stress (which is equal in magnitude to the applied force per unit area) and the strain produced.

  • Substances like tissue of aorta, rubber etc. which can be stretched to cause large strains are called elastomers.
  • The ratio of stress and strain is called modulus of elasticity.
  • The ratio of tensile (or compressive) stress  to the longitudinal strain  is defined as Young's modulus and is denoted by the symbol Y.

  • The ratio of shearing stress to the corresponding shearing strain is called the shear modulus of the material and is represented by G. It is also called the modulus of rigidity.

    The shearing stress  can also be expressed as

  • The ratio of hydraulic stress to the corresponding hydraulic strain is called bulk modulus. It is denoted by symbol .

    The negative sign indicates the fact that with an increase in pressure, a decrease in volume occurs. That is, if is positive,  is negative. Thus for a system in equilibrium, the value of bulk modulus B is always positive.

  • The reciprocal of the bulk modulus is called compressibility and is denoted by . It is defined as the fractional change in volume per unit increase in pressure.

  • A square lead slab of side  and thickness  is subject to a shearing force (on its narrow face) of

   

Sample Examples

 

Question

 The lower edge is riveted to the floor. How much will the upper edge be displaced?

Solution

The lead slab is fixed and the force is applied parallel to the narrow face. The area of the face parallel to which this force is applied is

Therefore, the stress applied is

 

We know that shearing strain

Therefore the displacement

   

Question

A structural steel rod has a radius of  and a length of  force stretches it along its length. Calculate (a) stress, (b) elongation, and (c) strain on the rod. Young's modulus, of structural steel is

Solution

We assume that the rod is held by a clamp at one end, and the force  is applied at the other end, parallel to the length of the rod. Then the stress on the rod is given by

The elongation,

The strain is given by

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