An FBG is sensitive to both strain (ε) and temperature (T) changes. In a simplified approximation, one can write:
Wwere λ is the FBG-wavelength; λ0 the wavelength at an arbitrary reference strain ε0 and a reference temperature T0 ; Sε the strain sensitivity or gage factor; ST the temperature sensitivity; Δε = ε – ε0 the strain change with respect to the reference strain and ΔT = T – T0 the temperature shift with respect to the reference temperature.
The above formula holds for free FBGs and not for FBGs which are fixed to a structure, like for strain gages. In this case, formula (1) should be modified slightly due to the influence of the thermal expansion of the structure. If the structure has a linear expansion coefficient αt , then the thermal expansion of the structure is given by αt . ΔT. If this expansion exceeds the thermal expansion of the fiber glass, which is αf ΔT , the structure exercises an extra strain on the fiber which is given by (αt – αf) ΔT. Notice that the difference of the thermal expansion coefficients has to be taken because only the excess expansion will be transferred into strain. This strain gives an additional wavelength shift corresponding to Sε (αt – αf) ΔT. The modified formula thus is given by:
It needs to be noticed that a more accurate approximation of the DTG® response is given by the following function:
where λ is the DTG® wavelength; λ0 the nominal wavelength i.e. the wavelength at zero strain and at a fixed reference temperatureT0 ; S1 and S2 the temperature sensitivities and k the strain sensitivity or gage factor. The strain ε is the mechanical strain that is experienced by the fiber. Due to the quadratic nature, the calibration parameters S1 and S2 are depending on the used reference temperature. For strain gauges supplied from FBGS, the parameters S1 and S2 are determined for a reference temperature of 22.5°C.
This more complex formula is important to use when large temperature variations are expected and is standard implemented for strain gauges in the Illumisense software of FBGS.
Typical values for S1, S2 and k are:
S1 = 6.37E-06
S2 = 7.46E-09
k = 0.772
assuming T expressed in °C and ε expressed in strain.
The thermal expansion coefficient of te fiber αf is typical around 0.5 μstrain/°C
Notice: Parameters S1, S2 and αf do not take into account the influence of the coating.