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Elastic Constants
Describes the realtionship between the Elastic Constants, and introduces Bulk Modulus and Young's Modulus.
Introduction - Elastic Constants
In the science of materials, numbers that quantify the response of a particular material to elastic or non-elastic deformation when a stress load is applied to that material, are known as Elastic Constants. They are the relationships that determine the deformations produced by a given Stress system acting on a particular Material, and within the limits for which Hooke's Law is obeyed, these factors are constant:- The Modulus of Elasticity,
- The Modulus of rigidity,
- The Bulk Modulus,
- Poisson's Ratio
or
An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i.e. non-permanently) when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region.
Shear modulus or modulus of rigidity is defined as the ratio of shear stress to the shear strain.
Poisson's ratio is the ratio, when a sample object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).
Hooke's Law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it.
Shear modulus or modulus of rigidity is defined as the ratio of shear stress to the shear strain.
Poisson's ratio is the ratio, when a sample object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).
Hooke's Law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it.
Bulk Modulus
If a "hydrostatic" pressure
(i.e. one which is equal in all directions) acting on a body of initial volume
, produces a reduction in the Volume equal numerically to
, then the Bulk Modulus
is defined as the ratio between the fluid pressure and the Volumetric Strain, i.e.
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Or, Strain Energy per unit volume
Example:
[imperial]
Example - Example 1
Problem
A frictionless plunger
in diameter and weighing
, compresses oil in steel container. A weight of
is dropped from a height of
onto the plunger.
Calculate the maximum pressure set up in the oil if its volume is
and the container is assumed to be rigid.
for Water
Workings
Let
be the additional momentary maximum pressure produced by the falling weight, if the loss of energy at impact is neglected.
The loss of the Potential energy of the falling weight = The gain in Strain energy of the water
The Volumetric Strain produced
is
and hence the decrease in the volume of water is
and this is taken up by the Plunger which will therefore sink a further distance equal to :
Therefore, Loss of potential energy =
And gain in Strain energy =
Equating these last two quantities and multiplying through by
produces the quadratic
Or
Solving and taking the positive root gives:
Adding the pressure due to the
weight gives a final maximum pressure of:
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Solution
- The maximum pressure is
The Relationship Between E And C
It is necessary to establish , first of all, the relationship between Pure Shear Stress and a pure normal Stress system at a point in an elastic material.MISSING IMAGE!
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Resolving along and at right angles to
And,
i.e., there is pure shear on planes at
to
and
of magnitude equal to the applied normal Stresses.
The square element
has sides of unstrained length 2 units which are under the equal normal Stresses
both tension and compression. It has been shown the element
is in pure shear of equal magnitude
.
The linear Strain
in the direction
The linear Strain in the direction
Hence the Strained lengths of
and
are
and
respectively.
The Shear Strain,
( See "Modulus of Rigidity" in pages on Shear Stress)
This distorts the element
and the angle
increase to
. Angle
is half this i.e.
Consider the triangle
.
Expanding this equation gives:
Note
and for small angles it is permissible to write
By inspection
and by substituting for
and
from equations (3) and (4)
Or re-arranged into a more normal form:
By using equation (2) it is possible to eliminate Poisson's Ratio from equation (5) and hence it can be shown that:
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In fact if any two elastic constants are known, the other two may be calculated. Experimentally however, it is not satisfactory to calculate Poisson's Ratio by determining MISSING IMAGE!
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Example:
[imperial]
Example - Example 2
Problem
Show that if
is assumed correct then an error of
in the determination of
will involve an error of
in the calculation of Poisson's Ratio.
Workings