Question

A cylindrical specimen of a hypothetical metal alloy is stressed in compression. If its original and final diameters are 30.00 and 30.04 mm, respectively, and its final length is 105.20 mm, compute its original length if the deformation is totally elastic. The elastic and shear moduli for this alloy are 65.5 and 25.4 GPa, respectively.

Answer 1

The original length of the specimen
`l_(o) = 104.7 mm`

Step-by-step explanation:

Original diameter
`d_(o)` = 30 mm

Final diameter
`d_(1)` = 30.04 mm

Change in diameter Δd = 0.04 mm

Final length
`l_(1)` = 105.20 mm

Elastic modulus E = 65.5 G pa = 65.5 ×
`10^(3)` M pa

Shear modulus G = 25.4 G pa = 25.4 ×
`10^(3)` M pa

We know that the relation between the shear modulus & elastic modulus is given by

`G = (E)/(2(1 + \mu))`

`25.5 = (65.5)/(2 (1 + \mu))`

`\mu = 0.28`

This is the value of possion's ratio.

We know that the possion's ratio is given by

`\mu = ((0.04)/(30) )/((change \ in \ length)/(l_(o) ) )`

`{(change \ in \ length)/(l_(o) ) = ((0.04)/(30) )/(0.28)`

`{(change \ in \ length)/(l_(o) ) =` 0.00476

`(l_(1) - l_(o) )/(l_(o) ) = 0.00476`

`(l_(1) )/(l_(o) ) = 1.00476`

Final length
`l_(o)` = 105.2 m

Original length

`l_(o) = (105.2)/(1.00476)`

`l_(o) = 104.7 mm`

This is the original length of the specimen.