STRAIN ENERGY

 When a body is in a state of stress loaded externally deformation takes place a simple tensile member tends to elongate and a compressive member shortens the ratio of change in length to the original length is called strain since strain is a ratio of two lengths it is a dimensionless quantity.

Thus the tensile or compressive strain is given by
Strain                   = change in length/original length
e = δl/l
Shear strain:Consider a block is rigidly fixed to a vertical surface and force ‘p’ acts along its top fact tangentially to the top face under the action of shear force the block.
Shear strain Φ      = BB’/AB = x/y
Since ‘x’ will be small therefore Φ is expressed in radians thus the measure of shear strain is the rotation of the planes perpendicular to the applied shear load.
Volumetric strain:Consider each face of a cube is subjected to equal compressive force. The result of this force will be reduction in volume the ratio of change in volume to the original volume is called volumetric strain.
Volumetric strain           = change in volume/ original volume
e        =δv/v
Suppose the original length of each side of the cube is L and δl is the decrease in length of each side due to stress then
Volumetric strain           = L³ – (L – δl) ³/L³
Neglecting small quantities,
Volumetric strain           = 3³ l.δl/L³ =3.δl/L
 ev =3.e
Volumetric strain is equal to three times the linear strain.

STRAINS ENERGY:When a force is applied to an elastic body it causes a linear deformation and as a result of this deformation the point of application of a force undergoes a displacement. Thus the work is done by a force this work is stored in the elastic body as strain energy. When external force is removed this stored energy takes back the material to original position according to the principle of conservation of energy the work done by external force is equal to strain energy.
Resilience:Strain energy stored within the elastic limit is called resilience.
Proof resilience:The maximum energy stored within the elastic limit is called proof resilience it is the strain energy corresponding to stress at elastic limit.
Modulus of resilience:The proof resilience per unit volume of a body is called modulus of resilience.

Strain energy:

Consider a body of cross-sectional area a and length l is subjected to a tensile load p. let ‘δl’ be the extension of the bar the work done by the load is represented by shaded area of load extension diagram.

Thus strain energy U     = Work done

                                      = ½ P.δl

                                      = ½ f.A. f.l/E

                                      = f²/2E.A.l

                                   U    = f²/2E.V Nmm

Where         f = stress induced in the body, N/mm²

V = Volume of the body, mm³

E = modulus of elasticity, N/mm²

If ‘fe’ is the stress induced at elastic point.

Then proof resilience    =f²e/2E V N.mm

And modulus of resilience       = f²e/2E N/mm³.

Alternate expression:

U =½ P.δl

δl = Pl/AE

U = P²l/2AE

Strain energy for different modes of loads:

The load P can be applied to the body in three different ways

1.    Gradually applied load

2.    Suddenly applied load

3.    Falling or impact load

Gradually applied load:

In this case load increases gradually from zero to final value therefore the average load is considered to calculate the strain energy consider an elastic body of uniform cross-section of length l is subjected to gradually apply tensile load.

Let     δl = extension of the bar

Work done  = average load x extension

=(o+p/2)δl =pδl/2

From principle of conservation of energy.

Work done  = strain energy

p δl/2 = σ²/2E Al

p σl/2E = σ²/2E Al

Thus           σ = P/A N/mm²

Where         P = load applied N

A = cross-sectional area, mm²

U = σ²/2E Al Nmm²

Where         σ = stress induced N/mm².

E = modulus of elasticity N/mm².

A =cross-sectional area mm².

l = length of bar mm.

Suddenly applied load:

In this case load acts instantly on a body and the extension takes place at a constant load. P

Work done by load        = load x extension

=p x δl

Strain energy: σ². Al/2E

Work done  = strain energy

 p. δl = σAl²/2E

p .σ.l/E = σ²Al/2E

(OR) σ – 2P/A

Falling or impact load:

In this case load dropped from a certain height before it stretch the bar consider a load p falls from a height h on a collar which is attached to the lower end of the bar let δl be the extension of the bar under impact load.

Work done  = P(h+δl)

strain energy stored   = σ².Al/2E

Since W.D.is equal to strain energy stored therefore

                   p(h++δl)      = σ².Al/2E

ph+p.σl/E    = σ².Al/2E

ph= σ².Al/2E- p.σl/E

Multiplying both sides by 2E/Al.

2E ph/Al = σ²-2pσ/A

Solving the above quadratic equation σ = 2p/A±√4p²/A²+4.2Eph/Al

σ = p/A±√p²/A²+2Eph/Al

σ = p/A+√(p/A)²+2Eph/Al

The above equation gives the stress induced due to falling load a after calculating the value of stress the corresponding strain energy may be calculated.

Salience: strain energy stored within elastic limiter.

Proof resilience: maximum strain energy stored within elastic limit.

Modulus of resilience: proof resilience per unit of volume of bar.

1.    General expression for strain energy.

Strain energy, u = σ². Al/2E

2.    Stress induced in gradually applied load.

σ = 2p/A

3.    Stress induced in suddenly applied load.

σ = 2p/A

4.    Stress induced due to impact load.

σ = p/A+√(P/A)²+2Eph/Al

If maximum extension is negligible then.

σ = √2Eph/Al

5.    Stress induced due to sudden stopping of mss m moving with a velocity V

σ = √E.m.V²/Al