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As I am reading the document on physics of strain gauge, I encounter the following paragraph that I do not understand. How could resistivity change (delta rho) under strain/elongation? I always thought resistivity is a constant.

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  • $\begingroup$ Under strain or elongation, the inter-atomic distances change, and this potentially could change the "effective-mass" of the conduction electrons. $\endgroup$
    – NaOH
    Nov 4, 2016 at 8:47
  • $\begingroup$ All well explained here in terms of changes in the dimensions of the strain gauge. en.m.wikipedia.org/wiki/Strain_gauge $\endgroup$
    – Farcher
    Nov 4, 2016 at 8:53

1 Answer 1

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I always thought resistivity is a constant.

No, resistivity does change under strain. This phenomenon is called piezoresistivity, and it is, in fact, the main physical phenomenon on which strain gauges are based.

For metallic conductors the relative change in resistivity is approximately proportional to the change in volume of the conductors [1],

$$\frac{\Delta\rho}{\rho} = C\frac{\Delta V}{V},$$

where $C$ is called Bridgman's constant. For many metals its value is around unity.

A (too) simplified way to understand the above formula is to refer to the Drude's model of conductivity. According to this model the conductivity is given by

$$\sigma = \frac {ne^{2}\tau }{m},$$

where $n$ is the electron density, $e$ the elementary charge and $\tau$ is the relaxation time, a time constant related to the dynamics of the electrons within the crystal. Roughly assuming that the electron density is given by a constant number of electrons $N$ divided by the sample volume $V$ yields

$$\sigma = \frac {Ne^{2}\tau }{mV},$$

from which one obtains

$$\frac{\Delta\rho}{\rho} = \frac{\Delta\sigma}{\sigma} = \frac{\Delta V}{V}.$$

As I wrote, this model is very simplified and the proportionality coefficient between the relative change of resistance and that of volume is not one, but close to one for many alloys.

For semiconductors things are more complicated, and the effect is anisotropic. The explanation lies in the change of the energy bands caused by the strain. The relative size of the effect is much greater than the case of metallic conductors. Thus, semiconductor strain gauges have a much greater sensitivity with respect to the metallic ones, but are less stable and accurate.

It is worth noting that the piezoresistivity effect causes a change in the temperature coefficient of conductors when they are mechanically constrained, because the resistivity becomes affected by the thermally-induced strain of the supporting element. Interestingly, this effect can be exploited to build resistors with very low temperature coefficients, of the order or below $10^{-6}/\mathrm{K}$.

[1] R. Pallàs-Areny, Sensors and signal conditioning, John G. Webster, 2001.

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