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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.
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