Problem 2.1a
Show that when the only nonzero applied stress is
, Hooke’s law requires that the normal strains
, and that Poisson’s ratio
, defined as
, satisfies the equation
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(2.1a)
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Background
Stress is force/unit area and is denoted by
, etc., where a force in the
-direction acts upon a surface perpendicular to the
-axis. The stresses
and
are, respectively, a normal stress and a shearing stress.
Stresses produce strains (changes in size and/or shape). If the stresses cause a point
to have displacements
along the coordinate axes, the basic strains are given by derivatives of these displacements as follows:
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(2.1b)
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(2.1c)
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The vector displacement
is
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(2.1d)
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where
are unit vectors in the
-directions (see Sheriff and Geldart, 1995, problem 15.3). The dilatation
is the change in volume per unit volume, i.e.,
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(2.1e)
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A pressure
produces a decrease in volume, the proportionality constant being the bulk modulus
:
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(2.1f)
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Sometimes the compressibility,
, is used instead of
.
In addition to creating strains, stresses cause rotation of the medium, the vector rotation
being equal to
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(2.1g)
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where
,
,
For small strains and an isotropic medium (where properties are the same regard-less of the direction of measurement), Hooke’s law relates the stresses to the strains as follows:
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(2.1h)
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(2.1i)
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where
and
are Lamé’s constants (
is usually called the modulus of rigidity or the shear modulus).
Solution
Subtracting equation (2.1h) for
from the same equation for
gives
.
Dividing equation (2.1h) for
by
gives
so
Problem 2.1b
2.1b Show that Young’s modulus
, defined as
, is given by the equation
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(2.1j)
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Solution
Adding the three equations (2.1h) for
,
, and
, and recalling that
, we get
Dividing both sides by
gives
Using equation (2.1a) we get
Problem 2.1c
A pressure
is equivalent to stresses
. Derive the following result for the bulk modulus
:
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(2.1k)
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Solution
Since
, we add equation (2.1h) for each of the three values
, obtaining
, so from equation (2.1f),
Continue reading
Also in this chapter
External links
find literature about The basic elastic constants
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