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86 Mechanical Transduction Techniques
The sensitivity of a strain gauge is generally termed the gauge factor. This is a
dimensionless quantity and is given by
relative change in resistance ∆ / ∆ RR
RR
/
GF = = = (5.1)
applied strain ∆ R / L ε
where R is the initial resistance of the strain gauge and ∆R is the change in resistance.
The term ∆L/L is, by definition, the applied strain and is denoted as ε (dimension-
2
less). For all elastic materials, there is a relationship between the stress σ(N/m ) and
the strain ε; that is, they obey Hooke’s law and thus deform linearly with applied
force. The constant of proportionality is the elastic modulus or Young’s modulus of
the material and is given by
Stress σ
Young’s modulus, E = = (Nm 2 ) (5.2)
Strain ε
2
The Young’s modulus of silicon is 190 GPa (1 Pa=1N/m ), which is close to
that of typical stainless steel (around 200 GPa). For a given material, the higher the
value of Young’s modulus, the less it deforms for a given applied stress (i.e., it is
stiffer).
When an elastic material is subjected to a force along its axis, it will also deform
along the orthogonal axes. For example, if a rectangular block of material is
stretched along its length, its width and thickness will decrease. In other words, a
tensile strain along the length will result in compressive strains in the orthogonal
directions. Typically, the axial and transverse strains will differ and the ratio
between the two is known as Poisson’s ratio, ν. Most elastic materials have a Pois-
son’s ratio of around 0.3 (silicon is 0.22). The effect on a rectangular block is
depicted in Figure 5.2. The strains along the length, width, and thickness are
denoted by ε, ε , and ε , respectively.
l w t
Initial shape
Final shape
t−∆ t
w−∆ w
l + ∆ l
ε t
Note: The original length, width, and
thickness of the block is ,lw, and t,
ε w respectively.
ε l
Figure 5.2 Illustration of Poisson’s ratio on a rectangular, isotropic, elastic block. A longitudinal
tensile strain results in deformation in the two orthogonal axes.