Introduction to Bourdon Tubes

2.1 Hysteresis loop area

The hysteresis-loop area is equal to deformation work produced during one loading cycle. This deformation work is mainly transformed into heat energy, and a very small part of it into stored energy. The total hysteresis energy is then equal to the sum of areas of all hysteresis loops. If we neglect the (usually short) hardening/softening period during which the loop area undergoes changes, the total energy can be expressed as the product of the area of the saturation hysteresis loop and the number of cycles to fracture. This has been utilized in some energy-based theories of the fatigue process, and thence the area of the stable hysteresis loop becomes an important quantity. The hysteresis-loop area can be generally expressed as

$$\begin{array}{}\text{(2.1)}& \mathit{\Delta W}=\iint \mathit{d}\theta {\mathit{d}\epsilon }_p=\oint \sigma \left({\epsilon }_p\right){\mathit{d}\epsilon }_p\end{array}$$

This area depends on the hysteresis-loop shape (via the function σ=σ(ε_p)) and on the stress—and plastic—strain amplitudes, which are here limits of integration. Specific expressions for the loop area are then given by the choice of the stress-strain relation along the hysteresis loop.

Feltner and Morrow approximated the loop as shown in Fig. 2.16. This approximation assumes that (i) the half width of the loop is equal to the plastic-strain amplitude and (ii) the tensile part of the loop can be described by a power-law dependence of the type

$$\begin{array}{}\text{(2.2)}& {\mathit{\epsilon }}_p={\mathit{k\sigma }}_a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}\end{array}$$

The constant k in this equation can be expressed using the boundary condition σ=σ_a, for ε=2ε_ap  as k=2ε_ap/${\mathit{\sigma }}_a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}$. Equation 2.2 can then be written in the form

$$\begin{array}{}\text{(2.3)}& \mathit{\epsilon }=\frac{{\mathit{\sigma }}_{\mathit{a}}}{{\left({2\epsilon }_{\mathit{ap}}\right)}^{\mathit{m}}}{\mathit{\sigma }}_{\mathit{p}}^{\mathit{m}}\end{array}$$

The hysteresis-loop area is then (Fig. 2.16)

$$\begin{array}{}\text{(2.4)}& \mathit{\Delta W}=2{\int }_0^{2{\mathit{\epsilon }}_{\mathit{ap}}}\sigma \left({\epsilon }_p\right){\mathit{d\epsilon }}_p=2{\int }_0^{2{\mathit{\epsilon }}_{\mathit{ap}}}\frac{\mathit{\sigma }}{{\left({2\epsilon }_{\mathit{ap}}\right)}^{\mathit{m}}}{\mathit{\epsilon }}_{\mathit{p}}^{\mathit{m}}\mathit{d}{\mathit{\epsilon }}_{\mathit{p}}\end{array}$$

After integration we obtain

$$\begin{array}{}\text{(2.5)}& \mathit{\Delta W}=4{\mathit{\sigma }}_{\mathit{p}}{\epsilon }_{\mathit{ap}}\frac{1}{\mathit{m}+1}\end{array}$$

This hysteresis-loop area can be expressed alternatively as a function of stress amplitude (by applying the equation for constant k) in the form

$$\begin{array}{}\text{(2.6)}& \mathit{\Delta W}=\frac{2\mathit{k}}{\mathit{m}+1}{\mathit{\sigma }}_{\mathit{a}}^{\raisebox{1ex}{$\left(\mathit{m}+1\right)$}\!\left/ \!\raisebox{-1ex}{$\mathit{m}$}\right.}\end{array}$$

The assumptions used in this derivation are only rough approximations of experimental findings. The half width of the hysteresis loop is always lower than the plastic-strain amplitude. This difference is not important for high-amplitude cycling, but may be dominant for low-amplitude cycling. Figure 2.17 shows how far the power-law relation between stress and plastic strain along the hysteresis loop is justified. Figure 2.17a is a typical example from the high-amplitude region. In log-log coordinates, the relation is a straight line, i. e. here the relation is a power law. Figure 2.17b presents a result in the same coordinates for the low-amplitude region. It is obvious that the power-law approximation is no longer valid.
Morrow also assumes the power-law stress-strain relation, expressed in his paper as:

  $$\begin{array}{}\text{(2.7)}& \mathit{\sigma }=const.{\mathit{\epsilon }}_{\mathit{p}}^{{\mathit{n}}^{\mathit{\text{'}}}}\end{array}$$

… but does not identify a priori half width of the loop with the plastic-strain amplitude. Applying the boundary conditions according to Fig. 2.18, σ=σ_a, for ε_p=2ε_ap we get from eqn. 2.7

$$\begin{array}{}\text{(2.8)}& \mathit{\sigma }=\frac{2{\mathit{\sigma }}_{\mathit{a}}}{{\left(2{\mathit{\epsilon }}_{\mathit{ap}}\right)}^{{\mathit{n}}^{\mathit{\text{'}}}}}{\mathit{\epsilon }}_{\mathit{p}}^{{\mathit{n}}^{\mathit{\text{'}}}}\end{array}$$

According to Fig. 2.18, the loop area can be then expressed in the form

$$\begin{array}{}\text{(2.9)}& \mathit{\Delta W}=4{\mathit{\sigma }}_{\mathit{a}}{\mathit{\epsilon }}_{\mathit{ap}}-2{\int }_0^{2{\mathit{\sigma }}_{\mathit{a}}}{\mathit{\epsilon }}_{\mathit{p}}d\mathit{\sigma }\end{array}$$

Substituting eqn. 2.8 in eqn. 2.9, and integrating we obtain

$$\begin{array}{}\text{(2.10)}& \mathit{\Delta W}=4{\mathit{\sigma }}_{\mathit{a}}{\mathit{\epsilon }}_{\mathit{ap}}\frac{1-{\mathit{n}}^{\mathit{"}}}{1+{\mathit{n}}^{\mathit{\text{'}}}}\end{array}$$

As can be seen in Fig. 2.17b, the value of n" in the low-amplitude region—if the power-law representation is justified at all—lies in the range 0.2 to 0.3. The loop area lies then in the range 2.2 to 2.7.

Some materials exhibit so-called Masing behaviour. In this case, the ascending branches of a set of hysteresis loops of different ε_ap, translated in such a way that their tips coincide at the positions of minimum stress, follow a common curve. In other words, the exponent n" in Morrow’s expression (eqn. 2.10) is identical with the cyclic strain-hardening exponent n' (eqn. 2.1). The later modifications of the Morrow’s expression are based on the difference between the exponents n' and n". Nevertheless, the experimental data obtained both in the low-amplitude region and in the high-amplitude region show a good applicability of the original Morrow's expression (eqn. 2.10), in which the exponent n' is replaced by the exponent of the cyclic stress-strain curve n'.

Summarizing both the theoretical calculations and the experimental results, it can be said that the area of the saturation hysteresis loop can be expressed to a first approximation as

$$\begin{array}{}\text{(2.11)}& \mathit{\Delta W}=const.{\mathit{\sigma }}_{\mathit{a}}{\mathit{\epsilon }}_{\mathit{ap}}\end{array}$$

where the multiplicative constant, lying roughly between 2 and 3 depends on the material and slightly on the strain amplitude.

Author: Dave Waddell
Company: Computatest
Last edited date: June 20, 2020

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2. Giacobbe, J.B., Selecting and Working Bourdon-Tube Materials. Instrument Manufacturing, July 1952.

3. B. G. Lipták, J. E. Jamison, S. Edvi, Bourdon and Helical Pressure Sensors, Instrument Engineer's Handbook, chapter 5.4, pages 731-734.

4. Goitein. K., A Dimensional Analysis Approach to Bourdon Tube Design. Instrumentation Practice. September 1952.

5. Gillum. D., Industrial Pressure. Level, and Density Measurement. Research Triangle Park, NC. ISA Press, 1995.

6. Hall. J., Monitoring Pressure with Newer Technologies. Instruments and Control Systems. April 1979.

7. Hughes. T.A., Pressure Measurement. EMC series, downloadable PDF. Instrumentation, Systems, and Automation Society. Research Triangle Park, NC. 2002.

8. Feltner C.E., Morrow J.D. Microplastic strain hysteresis energy as a criterion for fatigue fracture. Trans. ASME D. 1961. Vol. 83. N 1. Pages 15-22.

9. Mirko Klesnil, Petr Lukáš Fatigue of Metallic Materials, Volume 71. Pages 30-35.

10. Georg Masing, Wilhelm Mauksch Eigenspannungen und Verfestigung des plastisch gedehnten und gestauchten Messings 1925 pp 244-256.

Appendix

Special Metals Ni-Span-C® Alloy 902
Categories:	Metal; Superalloy; Iron Base
Material Notes:	A nickel-iron-chromium alloy made precipitation hardenable by additions of aluminum and titanium. The titanium content also helps provide a controllable thermoelastic coefficient, which is the alloy's outstanding characteristic. The alloy can be processed to have a constant modulus of elasticity at temperatures from -50°F to 150°F (-45 to 65°C). Used for precision springs, mechanical resonators, and other precision elastic components. Standard product forms are round, strip, tube, pipe, and wire. Data provided by the manufacturer, Special Metals.
Key Words:	Ni Span C 902, Nispan C, UNS N09902; BS 3127; SAE AMS 5210, 5221, 5223, 5225
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Physical Properties	Metric	English	Comments
Density	8.05 g/cc	0.291 lb/in³
Mechanical Properties	Metric	English	Comments
Tensile Strength, Ultimate	1210 MPa	175000 psi	Precipitation Hardened
Tensile Strength, Yield	760 MPa @Strain 0.200 %	110000 psi @Strain 0.200 %	Precipitation Hardened
Elongation at Break	25 %	25 %	Precipitation Hardened
Modulus of Elasticity	165 200 GPa	23900 29000 ksi
Shear Modulus	62.0 69.0 GPa	8990 10000 ksi
Electrical Properties	Metric	English	Comments
Electrical Resistivity	0.000102 ohm-cm	0.000102 ohm-cm
Curie Temperature	190 °C	374 °F
Thermal Properties	Metric	English	Comments
CTE, linear	7.60 µm/m-°C @Temperature 20.0-100 °C	4.22 µin/in-°F @Temperature 68.0-212 °F
Specific Heat Capacity	0.500 J/g-°C	0.120 BTU/lb-°F
Thermal Conductivity	12.1 W/m-K	84.0 BTU-in/hr-ft²-°F
Melting Point	1450 1480 °C	2640 2700 °F
Solidus	1450 °C	2640 °F	Creep does not occur until the temperature ~>0.5 of the solidus i.e. 1320°F (715°C).
Liquidus	1480 °C	2700 °F
Component Elements Properties	Metric	English	Comments
Aluminum, Al	0.30 0.80 %	0.30 0.80 %
Carbon, C	<= 0.060 %	<= 0.060 %
Chromium, Cr	4.9 5.75 %	4.9 5.75 %
Iron, Fe	47 %	47 %	As remainder
Manganese, Mn	<= 0.80 %	<= 0.80 %
Nickel, Ni	41 43.5 %	41 43.5 %	Including Cobalt
Phosphorous, P	<= 0.040 %	<= 0.040 %
Silicon, Si	<= 1.0 %	<= 1.0 %
Sulfur, S	<= 0.040 %	<= 0.040 %
Titanium, Ti	2.2 2.75 %	2.2 2.75 %