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When a straight bar is subjected to a tensile load, the bar becomes longer. The amount of stretch, or elongation, is called strain. The elongation per unit length of the bar is called unit strain. In spite of these definitions, however, it is customary to use the word "strain" to mean "unit strain" and the expression "total strain" to mean total elongation, or deformation, of a member. Using this custom here, the expression for strain is
epsilon = delta / l [2-13]
where delta is the total elongation (total strain) of the
bar within the length l.
Shear strain, gamma, is the change in a right angle of a stress element subjected to pure shear.
Elasticity is that property of a material which enables it to regain its original shape and dimensions when the load is removed. Hooke's law states that, within certain limits, the stress in a material is proportional to the strain which produced it. An elastic material does not necessarily obey Hooke's law, since it is possible for some materials to regain their original shape without the limiting condition that stress be proportional to strain. On the other hand, a material which obeys Hooke's law is elastic. For the condition that stress is proportional to strain, we can write
sigma = E * epsilon [2-14]
tau = G * gamma [2-15]
where E and G are the constants of proportionality. Since the strains
are dimensionless numbers, the units of E and G are the same as the
units of stress. The constant E is called the modulus of
elasticity. The constant G is called the shear modulus of
elasticity, or sometimes, the modulus of rigidity.
Both E and G, however, are numbers which are indicative of the
stiffness or rigidity of the materials. These two constants
represent fundamental properties.
By substituting sigma = F / A and epsilon = gamma / l into equation 2-14 and rearranging, we obtain the equation for the total deformation of a bar loaded in axial tension or compression.
delta = (F*l)/(A*E) [a]
Experiments demonstrate that when a material is placed in tension, there exists not only an axial strain, but also a laterial strain. Poisson demonstrated that these two strains were proportional to each other within the range of Hooke's law. This constant is expressed as
mu = -(lateral strain)/(axial strain) [2-16]
and is known as Poisson's ratio. These same relations
apply for compression, except that a lateral expansion takes
place instead.
The three elastic constants are related to each other as follows:
E = 2*G*(1+mu) [2-17]