THE COEFFICIENT OF LINEAR

THE COEFFICIENT OF LINEAR 

thermal
expansion (CTE, α, or α1) is a material property
that is indicative of the extent to which a mate-
rial expands upon heating. Different substances
expand by different amounts. Over small tem-
perature ranges, the thermal expansion of uni-
form linear objects is proportional to tempera-
ture change. Thermal expansion finds useful
application in bimetallic strips for the construc-
tion of thermometers but can generate detrimen-
tal internal stress when a structural part is heated
and kept at constant length.
For a more detailed discussion of thermal
expansion including theory and the effect of
crystal symmetry, the reader is referred to the
CINDAS Data Series on Material Properties,
Volumes 1 to 4, Thermal Expansion of Solids
(Ref 1).
Definitions
Most solid materials expand upon heating and
contract when cooled. The change in length with
temperature for a solid material can be ex-
pressed as:
(l
f – l
0)/l
0 = α1 (Tf – T0) ∆l/l
0 = α1∆T
α1 = 1/l(dl/dT)
where l
0 and l
f represent, respectively, the origi-
nal and final lengths with the temperature
change from T0 to Tf
. The parameter α1 CTE and
has units of reciprocal temperature (K–1) such as
µm/m · K or 10–6/K. Conversion factors are:
To convert To Multiply by
10–6/K 10–6/°F 0.55556
10–6/°F 10–6/K 1.8
ppm/°C 10–6/K 1
10–6/°C 10–6/K 1
(µm/m)/°F 10–6/K 1.8
(µm/m)/°C 10–6/K 1
10–6/R 10–6/K 1.8
The coefficient of thermal expansion is also
often defined as the fractional increase in length
per unit rise in temperature. The exact definition
varies, depending on whether it is specified at a
precise temperature (true coefficient of thermal
expansion or α
− or over a temperature range
(mean coefficient of thermal expansion or α).
The true coefficient is related to the slope of the
tangent of the length versus temperature plot,
while the mean coefficient is governed by the
slope of the chord between two points on the
curve. Variation in CTE values can occur ac-
cording to the definition used. When α is con-
stant over the temperature range then α = α
−.
Finite-element analysis (FEA) software such as
NASTRAN (MSC Software) requires that α be
input, not α
−.
Heating or cooling affects all the dimensions
of a body of material, with a resultant change in
volume. Volume changes may be determined
from:
∆V/V0 = αV∆T
where ∆V and V0 are the volume change and
original volume, respectively, and αV represents
the volume coefficient of thermal expansion. In
many materials, the value of αV is anisotropic;
that is, it depends on the crystallographic direc-
tion along which it is measured. For materials in
which the thermal expansion is isotropic, αV is
approximately 3α1.
Measurement
To determine the thermal expansion coeffi-
cient, two physical quantities (displacement and
temperature) must be measured on a sample that
is undergoing a thermal cycle. Three of the main
techniques used for CTE measurement are
dilatometry, interferometry, and thermomechani-
cal analysis. Optical imaging can also be used at
extreme temperatures. X-ray diffraction can be
used to study changes in the lattice parameter but
may not correspond to bulk thermal expansion.
Dilatometry. Mechanical dilatometry tech-
niques are widely used. With this technique, a
specimen is heated in a furnace and displace-
ment of the ends of the specimen are transmitted
to a sensor by means of push rods. The precision
of the test is lower than that of interferometry,
and the test is generally applicable to materials
with CTE above 5 × 10–6/K (2.8 × 10–6/°F) over
the temperature range of –180 to 900 °C (–290
to 1650 °F). Push rods may be of the vitreous sil-
ica type, the high-purity alumina type, or the
isotropic graphite type. Alumina systems can ex-
tend the temperature range up to 1600 °C
(2900 °F) and graphite systems up to 2500 °C
(4500 °F). ASTM Test Method E 228 (Ref 2)
cove the determination of linear thermal expan-
sion of rigid solid materials using vitreous silica
push rod or tube dilatometers.