RELATION BETWEEN STRESS AND STRAIN

FOR ONE-DIMENSIONAL STRESS SYSTEM:

          The relationship between stress and strain for one directional stress (i.e., Normal stress in one direction only) is given by Hook's Law. This law states that when a material is loaded within elastic limit, the developed normal stress is proportional to the strain produced. This means that the ratio of the normal stress to the corresponding strain is a constant within the elastic limit. This constant is represented by ‘E’ and is known as Modulus of Elasticity or Young’s modulus of Elasticity.

FOR TWO-DIMENSIONAL STRESS SYSTEM:

          Before going to learn about Two dimensional stress system, we have to know about Longitudinal strain, Lateral strain and Poison's ratio. 
1. Longitudinal Strain:
          When a body is subjected to an axial tensile load, then there is an increase in length of the body. But at the same time there is a decrease in other dimensions of the body at right angles to the line of action of the load applied. Thus the body having axial deformation and also deformation at right angles to the line of action of the applied load (i.e., Lateral deformation).
          The ratio of axial deformation to the original length of the body is known as Longitudinal strain (Linear). The Longitudinal strain is also defined as the deformation of the body per unit length in the direction of applied load.    
2. Lateral Strain:
          The strain which is produced right angles to the direction of applied load is known as Lateral strain.   

          Let us take an example, A rectangular bar of length 'L', breadth 'B' and depth 'D' is subjected to an axial tensile load 'P' as shown in below fig., . The length of the bar will increase while applying load, the breadth and the depth will decrease.

Let     δl = Increase in length
          δb = Decrease in breadth and, 
          δd = Decrease in depth.

Then Longitudinal strain = 
and Lateral strain =

Note: 
1. If longitudinal stain is tensile, the lateral strain will be compressive, vice-versa,
2. Hence for every longitudinal strain in the direction of the load applied is accompanied by lateral strain of the opposite kind in all directions perpendicular to the load.

3. Poisson's Ratio:
          When the material is subjected to stress within the elastic limit, The ratio of lateral strain to longitudinal strain is a constant. This ratio is called Poisson's ratio and it is denoted by a symbol 'μ' (MU).

Mathematically, 

as lateral strain is opposite in sign to longitudinal strain, hence algebraically, lateral strain is written as,  

4. Relation between Stress and Strain:
          Consider a two dimensional figure ABCD, subjected to two mutually perpendicular stresses 


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TYPES OF STRESSES

Stresses are two types they are

               1. Normal stresses, and 

               2. Shear stresses

          Normal stress is the stress which acts in a way perpendicular to he area. It is represented by 'σ' (SIGMA). Normal stress is further divided into Tensile stress and Compressive stress.

Let us discuss each of them briefly

1. Tensile stress:

          The stress induced in a body, when subjected to two equal and opposite pull forces as shown fig., as a result of which there is an increase in length, is known as Tensile stress. The ratio of increase in length to the initial (or) original length is known as Tensile strain. The tensile stress acts normal to the area and it pulls on the area. 

              Let P = Pull force acting on the body,

                     A = Cross-sectional area of the body, 

                     L = Original length of the body, 

                   dL = Change in length 

                    σ  = Stress induced in the body, and 

                     e = Strain 

          Below fig., shows a bar subjected to a tensile force 'P' at its ends. Consider a cross section x-x which divides the bar into two parts. The part left to the section x-x, will be in equilibrium if Tensile force (P)=Resisting force (R). Similarly the part which is right to the cross section x-x, will be in equilibrium if P= Resisting force. This resisting force per unit area is known as Tensile stress or Intensity of stress.

2. Compressive stress: The stress induced in a body, when it is subjected to two equal and opposite push forces as shown in fig., as a result of which there is a decrease in length of the body, is known as Compressive Stress. The ratio of decrease in length to the initial (or) original length is known as Compressive strain. The compressive stress acts normal to the area and it pushes on the area. 

  

3. Shear stress: The stress induced in a body, when it is subjected to two equal and opposite push forces which are acting tangentially across the resisting section as shown in fig., as a result of which the body tends to shear off across the section, is known as Shear Stress. The corresponding strain is known as Shear strain. The shear stress which acts tangential to the area. Which is represented by 'τ' (Tau).

Credits: Mechanical engineers

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