Thermal Expansion
One of the main sources of stresses in an object is heat. Here we begin
to study some of the properties of heat. We are all familiar with the ideas that an
object is hot or cold. In order to quantify this, we introduce the notion of
temperature. Thus, for a certain amount of "hotness" we can associate a
number. This leads to the concept of a temperature scale.
Temperature Scales
In order to define a temperature scale, we need to make certain
assumptions. The first one is that there will be a linear relationship between how
hot an object is and its temperature. This allows us to specify two points on the
scale and divide the rest of the scale evenly. There are two main scales used in
the U.S. today. The first one is the Fahrenheit scale. It takes the freezing point
of water and associates the number 32 with it, and it associates the number 212
with the boiling point of water. Thus, there are 180 degrees between the freezing
and boiling points of water. The other main scale is the Celsius, or Centigrade,
scale. It is the main temperature scale used in the rest of the world, and is the
main temperature scale used in technological work in the U.S. In this scale the
freezing point of water is defined to be at 0 degrees, and the boiling point is
defined to be 100 degrees. Notice however, that neither of these scales actually
refers to any physical properties of water. Instead, two arbitrary points for an
arbitrary material were chosen and used to define the scale. We can define a
temperature scale that uses the physical properties of matter, the absolute
temperature scale. The Celsius version of the absolute scale is the Kelvin
scale. Similarly, the absolute version of the Fahrenheit scale is the Rankine
scale. All of these scales can be related to one another. The relations between
the scales are
5
TF  32
9
9
TF  TC  32
5
TK  TC  273.15
TC 
TR  TF  459.7
Thermal Equilibrium
Once we have defined a temperature scale, we are ready to begin to
study some of the thermal properties of matter. This study is called
thermodynamics. One of the most useful concepts in thermodynamics is the
idea of thermal equilibrium. Imagine buying a soda from a machine. Originally
the can is much colder than the outside air. After a certain amount of time has
gone by however, we would notice that the can has warmed up so that it felt to
be as warm as the outside air. If we now quantify this, we say that the soda was
originally at a lower temperature than the air, and at the later time we would find
that the temperature is the same as the air. During the time that the soda was
not at the same temperature as the air, the soda's temperature would be
constantly changing. Not until the two temperatures were equal would the
temperature stabilize. It is this stable condition that is what we refer to thermal
equilibrium. In order to prove that they are in thermal equilibrium, we must
conduct an experiment. Let us define an ideal insulator as one which permits
no interactions at all between two systems. Intuitively, we expect that an
insulator does not allow the systems to attain thermal equilibrium. Now consider
three systems, A, B and C, surrounded by an insulator so as to isolate them from
the rest of the world. Let A and B be separated by another insulator, but let C be
in contact with A and B via a thermal conductor. A thermal conductor is a
material which allows thermal interactions between two systems to occur.
C
A
B
Let the three systems be at different temperatures initially. Wait until there are
no further changes in any of the systems. Now replace the thermal conductor
with an insulator, and the insulator with a thermal conductor. What happens?
Our experiment shows that nothing happens; the states of A and B do not
change! Thus the experiment shows that if C were initially in thermal equilibrium
with both A and B, then when A and B are placed in contact with each other they
are also in thermal equilibrium with each other. This can be restated as two
systems in thermal equilibrium with a third system are in thermal
equilibrium with each other. This principle is called the zeroth law of
thermodynamics.
Thermal Expansion
Let’s consider a solid bar that is initially at thermal equilibrium with the
outside air. Now place a burner under the solid and see what happens. We
would observe that the length would change in a manner proportional to the
change in temperature and to the initial length. This can be written
mathematically as
dL = L0dT
(19.1)
The constant , which characterizes the thermal expansion properties of a
specific material, is called the coefficient of linear expansion. It should be
noted that (19.1) is only an approximate relationship. It varies somewhat
depending on the initial temperature and the size of the temperature change.
Volume Expansion
If the material does not have any preferential directions (for instance wood
has specific directions, across the grain and with the grain) every linear
dimension varies according to (19.1). Consider an object with an original volume
given by
V0 = L 1 L 2 L 3
Then if the temperature increased by dT, each linear dimension would increase
according to (19.1), and the new volume would be
V0  dV  L1 1  dT L2 1  dT L3 (1  dT 
 L1 L2 L3 1  dT 
3
 V0 1  dT 
3

 V0 1  3dT  3 2 dT 2   3 dT 3

But if dT is small, then we can ignore the higher powers, and we find
dV  3V0 dT
 V0 dT
(19.2)
where  is called the coefficient of volume expansion. Notice that if there
were a hole in the material, the volume of the hole would also increase as the
body expands, just as if the hole were filled with the same material as the rest of
the object.
Example:
A glass flask of volume 200 cm3 is just filled with mercury at 20°C. How
much mercury will overflow when the temperature of the system is raised to
100°C? Assume that the coefficient of volume expansion for glass is 1.2x10-5/C°,
and the coefficient of volume expansion for mercury is 18x10-5/C°.
The increase in volume of the flask is
dV  V0 dT





 1.2 10 -5 /C 200 cm 3 80C
 0.192 cm 3
The increase in volume of the mercury is
dV '   'V0 dT

 18 10 -5 /C 200 cm 3 80C
 2.88 cm
3
So 2.88 cm3 - 0.19 cm3 = 2.69 cm3 overflows.
Thermal Stress
Now consider what happens if we take an object and clamp it so that its
length is fixed. As it heats up, it will generate a tensile or compressive stress in
the material. In order to determine the amount of stress created, notice that
(19.1) can be rearranged to read
dL/L0 = dT
This would be the fractional change in length if the object were allowed to
change. Recall that the Young's modulus was defined to be
Y
F
dL
A
L0
or
dL
F

L0 AY
Since the object is not being allowed to expand, the sum of the thermal
expansion and the tensile strain must be zero
dT + F/AY = 0
or
F/A = YdT
(19.3)
Example:
What is the magnitude of the stress generated in a piece of steel for a
change of 1C°?
F
 YdT
A
 1.2 10 -5 / C 2.0 10 -11 Pa 1C 


 2.4 10 16 Pa
