E1C11 09/14/2010
13:14:1 Page 470
Metallic Gauges
To understand how metallic strain gauges work, consider a conductor having a uniform crosssectional area A c and length L made of a material having an electrical resistivity, r e . For this
electrical conductor, the resistance, R, is given by
R ¼ r e L=A c
ð11:6Þ
If the conductor is subjected to a normal stress along the axis of the wire, the cross-sectional
area and the length change resulting in a change in the total electrical resistance, R. The total change
in R is due to several effects, as illustrated in the total differential:
dR ¼
A c r e dL þ Ldr e
ð
ÞÀr e LdA c
A
2
c
ð11:7Þ
which may be expressed in terms of Poisson’s ratio as
dR
R
¼
dL
L
1 þ 2y p
À
Á þ
dr e
r e
ð11:8Þ
Hence, the changes in resistance are caused by two basic effects: the change in geometry as the
length and cross-sectional area change, and the change in the value of the resistivity, r e . The
dependence of resistivity on mechanical strain is called piezoresistance, and may be expressed in
terms of a piezoresistance coefficient, p 1 defined by
p 1 ¼
1
E m
dr e =r e
dL=L
ð11:9Þ
With this definition, the change in resistance may be expressed
dR=R ¼ dL=L 1 þ 2y p þ p 1 E m
À
Á
ð11:10Þ
Example 11.1
Determine the total resistance of a copper wire having a diameter of 1 mm and a length of 5 cm. The
resistivity of copper is 1:7 Â 10
À8
V m.
KNOWN
D ¼ 1 mm
L ¼ 5 cm
r e ¼ 1:7 Â 10
À8
V m
FIND The total electrical resistance
SOLUTION The resistance may be calculated from Equation 11.6 as
R ¼ r e L=A c
470 Chapter 11 Strain Measurement