Theoretical gage factor for resistive wire strain gages?

[Stephen Tashi]

What model correctly predicts the sensitivity of a resisitive wire straing gage to applied strain?

This question is motivated by an an old pamphlet found in a used book store, "Theory Of Strain Gage Flight Test Instrumentation" by Eugene Frank, 27 May 1946.

The gage factor of a strain gage made from resistive wire is defined as
$K = \frac{\triangle R/ R}{\triangle L/L}$
where R is the total resistance of the un-strained gage, L is the total length of un-strained "the specimen" being measured and the deltas are the respective changes in these quantities when a load is applied.

Frank says:

Theoretically, the stretched wire length under discussion should have a strain sensitivity or gage factor of 1.7 due to the geometric change during the stretching process as mentione above. Actually, for reasons unknown, measured strain sensitivity factors differ from the theoretical. For Advance wire, which is most commonly used for strain gauge wire ( 45% Ni & 55% Cu) the gage factor is approximately 20.

Frank's model assumes the change in resistance of the wire is entirely due to the change in the length and diameter of the wire as it is stretched.

When a wire is stretched elastically, its length changes and so does its diameter. To keep the wire's volume constant, its diameter will diminish by a factor of 0.3 as its length increases, corresponding to Poisson's ratio. The electrical resistance of a wire is affected by both the changes in length and in diameter.

 

[Valerie Park]

The model used to predict the sensitivity of a resistive wire strain gage is then:K = \frac{\triangle R/ R}{\triangle L/L + 0.3 \triangle L/L}