66 APPLIED MECHANICS 
If 7 is the length of the edge of a cube, which, when placed under 
pressure all over its surface, becomes 1—z, then the new volume becomes 
B — 32x + 3lx2 — «8, and the change in volume is 73 — (7° — 3a + 3/a2 — x?) 
or 32x —3lz? +23, but since x is always a very small quantity, the second 
and third terms of this latter expression may usually be neglected 5 
hence the change in volume is very approximately 3/’, and the volume 
strain is 3/2x/l3 or 3x/, which is three times the linear strain. 
82. Elasticity.— Strain is produced in a body by the action of a load 
on it, and if, when the load is removed, the strain disappears, the body 
is said to be perfectly elastic up to that particular load, or up to the 
particular stress corresponding to that load. If when the load is removed 
the strain does not entirely disappear a permanent set has been produced, 
and the elastic limit is reached when the load is the largest which will 
not cause a permanent set. 
It was discovered by Robert Hooke that so long as the elastic limit is 
not passed the strain produced is directly proportional to the load pro- 
ducing it, and since the stress is directly proportional to the load causing 
it, it follows that stress+strain is a constant ratio up to the elastic 
limit for a given material, or more correctly for a given piece of material. 
This is known as Hooke’s law. 
The value of the constant ratio stress+ strain is called the modulus 
of elasticity or the coefficient of elasticity. 
When a body is subjected to simple tension or simple compression, 
there being no external forces acting to prevent the exceedingly small 
lateral contraction or lateral expansion of the body, the coefficient of 
elasticity is the coefficient of direct elasticity, and is called Young's 
modulus. The letter E is generally used to denote Young’s modulus. 
When the strain is a shearing strain, and the stress of course a 
shearing stress, the ratio stress + strain is called the coefficient of trans- 
verse elasticity or the coefficient of rigidity. In this work the coefficient 
of rigidity will be denoted by the letter C. 
When the strain is a volume strain the ratio stress + strain is called 
the coefficient of elasticity of volume or the coefficvent of cubical elasticity. 
In this work the coefficient of elasticity of volume will be denoted b 
the letter K. “ 
83. Applications of Young’s Modulus.—If a bar of length /, whose 
area of cross section is a, suffers an alteration of length amounting to a, 
under the application of a load W acting in the line of the axis of 
the bar, and if f is the stress produced, then by Art. 78 or Art. 79 
W=af. Also by Art, 82, E=°S_/” From these two equations 
stram @# 
the following results are easily obtained :— 
__ ft _ Wl _aEr _ Ex 
t= = oR? W= j sand f= —. 
If the bar mentioned above be heated or cooled so that if free to 
expand or contract it would expand or contract by an amount #, then 
the forces which must be applied at each end of the bar, to prevent the © 
expansion or contraction, will be each equal to W, and the equations 
above will apply to this case if /+2 be substituted for 7. But since is 
very small compared with /, the error introduced by using / instead. of 
