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Strain energy - Resilience

ork is the product of the component of the force in the direction of the displacement and the magnitude of this displacement.
Energy is defined as the capacity to do work.

Strain energy

Strain energy is the ability of a body to do work because of its deformation and its tendency to return to its original shape. This internal energy stored in a in the body due to its deformations.
Strain energy (U) = Work done (W)
Strain Energy is Always a positive quantity and is expressed in units of work (N-m,Joule).
This may be calculated as the area under the stress-strain curve from the origin O to up to the elastic limit (the shaded area in the above figure). 

Resilience

The Strain Energy stored within elastic limit without creating a permanent distortion is called Resilience.
The maximum energy stored at elastic limit is known as Proof Resilience. 
Note: In Proof Resilience, stress is the value at the elastic limit or for non-ferrous materials,the stress is the Proof Stress.
Proof Resilience for unit volume is called as Modulus of Resilience.

Strain Energy for Uniaxial stress: 

1. Strain energy due to gradually applied load

For a bar of uniform cross section A and length L is subjected to a Tensile or Compressive Load of P gradually within the proportional limit.
The change in length of the bar will be                                             dx=PL/AE  
The Strain Energy (U) of the bar is defined as the work done by the gradually applied load due to this deformation. (i.e. Producing Strain) and it will be
U =  ( Average Load x Deformation) =1/2 .P.dx
    = 1/2 ( P x PL/AE) = 1/2 ( P/A X P/A X AL/E )
    = 1/2 ( Stress x Stress/E x AL)
But (A L) is the volume of the bar V,
P/A is stress (s) and strain is e = Stress/E
The Strain Energy can be expressed as
U = 1/2 (Stress)x(Strain)x volume of the bar 
U = 1/2 (s).(e).V 
U = 1/2 (s).(s/E).V
Strain energy per unit volume
 U = 1/2 (s).(e) 
The stress induced in the bar due to gradually applied load s = P/A

2. Strain energy due to suddenly applied load:
When the load ‘P’ is suddenly applying to the body, the load P is constant during the extension of the bar takes place. 
Hence the work done is equal to the maxi. energy storing in the body.
Work Done by the load W = Maximum Strain Energy stored in the bar U
W= Load × Extension = P.dx = P.(PL/AE)
U = 1/2 (s).(e).AL = 1/2 (s).(P/AE).AL
Therefore U =W 
1/2 (s).(s/E).AL = P.(PL/AE)
s = 2P/A
The stress induced in the bar due to suddenly applied load s = 2P/A
Note:
Instantaneous Stress developed in a bar subjected to suddenly applied load is equal to the twice of stress produced by the same load applied gradually. 

3.Strain Energy due to Impact Load:

Consider a vertical rod is fixing at the top or upper end and attached with the collar at the other end of the rod.The load P drop from a certain height to stretch the bar as shown in figure.


P = Load Apply with impact
L = Length of the rod
A = cross-sectional area of the rod
V = Volume of the rod is A x L
h = Height through which load is drop.
δL = Extension of the rod
E = Modulus of Elasticity and also
σ = Stress-inducing in the rod

Work done by the load = Load×Distance 
  W = P (h+δL)
Work Done by the load W = Maximum Strain Energy stored in the bar U
         P (h+δL) = 1/2 (s).(s/E).AL  

The stress induced in the bar due to Impact load 


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