elaticity basics....
- From: vinay.kumar.katiriya@xxxxxxxxx
- Date: Wed, 22 Oct 2008 10:12:06 -0700 (PDT)
The property whereby a solid material changes its shape and size under
the action of opposing forces, but recovers its original configuration
when the forces are removed. The theory of elasticity deals with the
relations between the forces acting on a body and the resulting
changes in configuration, and is important in many branches of science
and technology, for instance, in the design of structures, in the
theory of vibration and sound, and in the study of the forces between
atoms in crystal lattices.
The forces acting on a body are expressed as stresses and measured as
force per unit area. Thus if a bar ABCD of square cross section
(illus. a) is fixed at one end and subjected to a force F uniformly
distributed over the other end DC, the stress is F/(DC)2. This stress
causes the bar to become longer and thinner and to assume the shape A′B
′C′D′. The strain is measured by the ratio (change in length)/
(original length), that is, by (B′C′ − BC)/(BC).
According to Hooke's law, stress is proportional to strain, and the
ratio of stress to strain is therefore a constant, in this case the
Young's modulus, denoted by E, so that
E = F(BC)/(DC)2 (B′C′ − BC).
See also Hooke's law; Stress and strain; Young's modulus.
shear stress. (c) Change in volume with no change in shape. (All
deformations are exaggerated.)">
Stresses on a bar. (a) Direct or normal stress. (b) Tangential or
shear stress. (c) Change in volume with no change in shape. (All
deformations are exaggerated.)
Poisson's ratio σ is the ratio of lateral strain to longitudinal
strain so that
σ = BC(DC − D′C′)/DC(B′C′ − BC).
The bar of illustration a is in a state of tension, and the stress is
tensile; if the force F were reversed in direction, the stress would
be compressive. Stresses of this type are called direct or normal
stresses; a second type of stress, known as tangential or shear
stress, is shown in illus. b. In this case, the configuration ABCD
becomes ABC′D′, with the shear forces F acting in the directions AB
and CD. The shear strain is measured by the angle θ, and if the body
is originally a cube, the shear stress is F/(DC). The ratio of stress
to strain, F/(DC)2 θ, is the shear or rigidity modulus G, which
measures the resistance of the material to change in shape without
change in volume.
A further elastic constant, the bulk modulus k, measures the
resistance to change in volume without changes in shape, and is shown
in illus. c. The original configuration is represented by the circle
AB, and under a hydrostatic (uniform) pressure P, the circle AB
becomes the circle A′B′. The bulk modulus is then k = Pv/Δv, where Δv/
v is the volumetric strain. The reciprocal of the bulk modulus is the
compressibility.
The elastic constants may be determined directly in the way suggested
by their definitions; for instance, Young's modulus can be determined
by measuring the relative extension of a rod or wire subjected to a
known tensile stress. Less direct methods are, however, usually more
convenient and accurate. Prominent among these are the dynamic methods
involving frequency of vibration and velocity of sound propagation.
The elastic constants can be expressed in terms of frequency of (or
velocity in) regularly shaped specimens, together with the dimensions
and density, and by measuring these quantities, the elastic constants
can be found. The elastic constants can also be determined from the
flexure and torsion of bars. See also Ultrasonics.
In practice, stress is only proportional to strain, and the strain is
only completely recoverable within certain limits called the elastic
limits of the material. Above the elastic limits, the material is
subject to time-dependent effects, and as the stress is further
increased, the ultimate strength of the material is approached. See
also Plasticity; Strength of materials.
Elasticity
A measure of sensitivity of one variable to another. More
specifically, the degree to which consumers respond to price changes.
Elasticity is a measure of the responsiveness of one variable to
changes in some other variable. For example, advertising elasticity is
the relationship between a change in a firm's advertising budget and
the resulting change in product sales. Economists are often interested
in the price elasticity of demand, which measures the response of the
quantity of an item purchased to a change in the item's price.
Elasticity measures are reported as a proportional or percent change
in the variable being studied. The general formula for elasticity,
represented by the letter "E" in the equation below, is:
E percent change in x / percent change in y.
Elasticity can be zero, one, greater than one, less than one, or
infinite. When elasticity is equal to one, there is unit elasticity.
This means the proportional change in one variable is equal to the
proportional change in another variable, or in other words, the two
variables are directly related and move together. When elasticity is
greater than one, the proportional change in x is greater than the
proportional change in y and the situation is said to be elastic.
Inelastic situations result when the proportional change in x is less
than the proportional change in y. Perfectly inelastic situations
result when any change in y will have an infinite effect in x.
Finally, perfectly elastic situations result when any change in y will
result in no change in x. A special case known as unitary elasticity
of demand occurs if total revenue stays the same when prices change.
Elasticity for Managerial Decision Making
Economists compute several different elasticity measures, including
the price elasticity of demand, the price elasticity of supply, and
the income elasticity of demand. Elasticity is typically defined in
terms of changes in total revenue since that is of primary importance
to managers, CEOs, and marketers. For managers, a key point in the
discussions of demand is what happens when they raise prices for their
products and services. It is important to know the extent to which a
percentage increase in unit price will affect the demand for a
product. With elastic demand, total revenue will decrease if the price
is raised. With inelastic demand, however, total revenue will increase
if the price is raised.
The possibility of raising prices and increasing dollar sales (total
revenue) at the same time is very attractive to managers. This occurs
only if the demand curve is inelastic. Here total revenue will
increase if the price is raised, but total costs probably will not
increase and, in fact, could go down. Since profit is equal to total
revenue minus total costs, profit will increase as price is increased
when demand for a product is inelastic. It is important to note that
an entire demand cure is neither elastic or inelastic; it only has the
particular condition for a change in total revenue between two points
on the curve (and not along the whole curve).
Demand elasticity is affected by the availability of substitutes, the
urgency of need, and the importance of the item in the customer's
budget. Substitutes are products that offer the buyer a choice. For
example, many consumers see corn chips as a good or homogeneous
substitute for potato chips, or see sliced ham as a substitute for
sliced turkey. The more substitutes available, the greater will be the
elasticity of demand. If consumers see products as extremely different
or heterogeneous, however, then a particular need cannot easily be
satisfied by substitutes. In contrast to a product with many
substitutes, a product with few or no substitutes―like gasoline―will
have an inelastic demand curve. Similarly, demand for products that
are urgently needed or are very important to a person's budget will
tend to be inelastic. It is important for managers to understand the
price elasticity of their products and services in order to set prices
appropriately to maximize firm profits and revenues.
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