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Between longitudinal and cross deformation of the rod there is a
clear experimentally established relationship.
The absolute value of the ratio of relative cross deformation to
relative longitudinal deformation in tension or compression is the
constant value for a given material and is called Poisson’s ratio:

  
  cross  z  y . (2.16)
  
long x x
This is a dimensionless coefficient characterizing the elastic
properties of the material and is determined experimentally. For
different materials the value of Poisson’s coefficient is determined
0
within 0   0,5. For example, for cork   , for wax and
rubber   0,5, for most metals 0,25   0,35 .

Generally speaking, the Poisson's ratio for isotropic materials
should be within  1   0,5. This negative lower limit is
obtained theoretically on the basis of energy considerations.
However, at present we do not know any structural materials with
the negative Poisson's ratio. Obviously, they will appear in future.
The cross deformation in the direction of the axes and z y we

obtain from the equation (2.16) taking into account Hooke's law
(2.10):

         x . (2.17)
z y x
E

2.5 Determination of displacements. Differential
equations of motion

Consider a rod that is loaded by the tensile force P . Let
distinguish two arbitrary cross-sections a a and b b at the
distance dx from each other (fig.2.7). Under the load the section

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