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:
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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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
Credits: books.google.co.in
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