# Critical Frequency of Self-Heating in a Superelastic Ni-Ti Belleville Spring: Experimental Characterization and Numerical Simulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Characterization

#### 2.1.1. Material and Spring Manufacturing

#### 2.1.2. Thermal Characterization of the NiTi Belleville Spring

#### 2.1.3. Mechanical Characterization of the NiTi Belleville Spring

#### 2.2. Numerical Simulation

## 3. Results and Discussion

#### 3.1. Experimental Analysis

#### 3.1.1. Transformation Temperatures

_{f}temperature is not determined due to an incomplete phase transformation between R-phase and monoclinic martensite B19′.

#### 3.1.2. Superelastic Response at Low Loading Frequency

#### 3.1.3. Mechanical Pre-Cycling

_{N}− F

_{N−1}/F

_{N−1}) is also shown in this figure (right axis). As typically observed in SMA cycling [10,25,26], the evolution of the superelastic behavior is not linear and the bulk of the force decrease occurred within the first cycles. Indeed, the force variation curve in Figure 6b shows a more accentuated decrease in the first cycles (303 N, ~18%), which becomes smaller as cycling progresses (1.1 N on average between the 40th and 50th cycles, or ~0.2%, as shown in detail in Figure 6c). Similar mechanical behavior was observed by [10,31]. The force decrease is due to the redistribution of internal stresses during stress-induced phase transformation, mainly caused by the accumulation of dislocation slip [32,33,34]. Diffraction patterns of NiTi wires under tensile loading show that this gradual plastic deformation generates a gradual accumulation of defects and residual martensite, and an increase of the volume fraction of non-transforming austenite, consequently decreasing the amount of stress-induced martensite with each new cycle [34].

#### 3.1.4. Superelastic Response and Thermomechanical Coupling in the Dynamic Regime

_{test}), measured as the initial value of the temperature signal (~26 °C). The mean temperature (T

_{mean}) decreases slightly at the end of cycling at 0.5 Hz. Peak temperature (${T}_{peak}^{128\mathrm{th}}$) and amplitude temperature (T

_{amp}) by the end of the 128th compressive cycle was 30.3 °C and 6.4 °C, respectively.

_{c}). If the SMA material or device is cyclically loaded at frequencies above f

_{c}, the SMA will accumulate latent heat and consequently become stiffer. The f

_{c}value for the NiTi Belleville spring analyzed in this work is 1.7 Hz. Note that this value is dependent on specific convective and conduction heat transfer conditions and amplitude levels. Changes in the geometrical parameters of the Belleville spring will change the stress-induced martensite fraction and consequently will influence the heat exchange by conduction between the spring and the compression plates and by convection, between the spring and the environment. If the heat exchanges are intensified, for example by increasing the convection coefficient, the self-heating frequency would increase, since the material would be capable of dissipating the generated heat.

_{mean}, which increases from 2 Hz; and the decrease in T

_{amp}with increasing frequency. For instance, at the 128th cycle, T

_{mean}reaches 29.3 °C at 2 Hz and 36.7 °C at 10 Hz. The number of cycles was not enough to stabilize T

_{mean}, and it is assumed that T

_{mean}would further increase before stabilizing, reaching thermal equilibrium.

_{amp}(for the 128th) decreases almost linearly with increasing frequency, decreasing by half between 2 Hz and 10 Hz (5.1 °C and 2.5 °C, respectively). We observed a linear decreasing relationship between the T

_{amp}and the frequency from 2 Hz, with an angular coefficient of −0.331 °C/Hz, being able to estimate with good precision (R

^{2}= 0.98) the T

_{amp}from a certain frequency. Although this coefficient is valid for the entire system (Belleville spring, compression plates, atmosphere), its value provides an order of magnitude for the relationship between temperature and loading frequency. Furthermore, the fact that the Belleville spring is not insulated is much closer to the environment found in industrial applications.

#### 3.1.5. Functional Properties

_{s}, in kN/m), dissipated energy per cycle (E

_{D}, in MJ/m

^{3}), and equivalent viscous damping factor (ξ, in %) can be obtained. These functional properties are useful for the design of devices manufactured from SMA, especially when the envisioned application works under a dynamic regime.

_{s}is calculated by the ratio between the difference of peak and valley forces (∆F) and the displacement (∆δ) related to deflection between 10% and 50%, as expressed by Equation (1).

_{s}vs. frequency. The cycles are plotted at each 2

^{n}cycle (n = 0, 1, 2, 4, …, n) to facilitate visualization. k

_{s}increases with cycling for all the analyzed frequencies. The rate of this increase, however, changes at different frequencies. Mainly for frequencies of 4 Hz and above, the rate of increase becomes stronger at the 8th cycle. This behavior is a consequence of the increase in the internal temperature after the 8th cycle (see Figure 10), which stiffens the NiTi Belleville spring. The percentage increase of k

_{s}during the performed cycling went from 4.8% at 0.5 Hz to 10% at 10 Hz.

_{D}with cycling for the analyzed frequencies, which is calculated using Equation (2) through the integration of the force vs. displacement superelastic loop divided by the volume of the Belleville spring (obtained from the CAD model). Overall, E

_{D}decreases linearly with both cycling and with the increase of frequency.

_{D}reached 24% for the cycling at 0.5 Hz (from 1.66 MJ/m

^{3}to 1.26 MJ/m

^{3}); it was only 0.2% at 10 Hz. As for the variation with the frequency, the E

_{D}variation between 2 Hz and 10 Hz at the last cycle reached 4.5% (1.12 MJ/m

^{3}and 1.07 MJ/m

^{3}, respectively). The energy dissipation capacity of SMA is mostly originated by the friction between transforming regions and the movement of defects in the crystal lattice [4]. In quasi-static loadings, a hypothesis for the decrease in energy dissipation capacity with increasing temperature is that higher temperatures lead to higher transformation stresses (due to the Clausius–Clapeyron relation), and consequently more favorably oriented martensite variants are formed in detriment of the less favorably oriented ones (due to the stronger stress field) [3]. If this is the case, less internal lattice movement occurs, causing less energy loss.

_{so}), as expressed by Equation (3) [24].

_{D}and, therefore, follows the same behavior. The highest values are observed for the frequency of 0.5 Hz, which are 4.49% and 4.03%, for the 1st and 128th cycles, respectively, while the lowest values are observed for the frequency of 10 Hz, which are 3.34% and 2.99% for the 1st and 128th cycles, respectively. According to [24], values around 5–10% are typical in structural engineering.

#### 3.2. Numerical Analysis

_{mean}from Figure 10, it is possible to make a simplification to estimate the force increase at 50% deflection, ΔF

_{50%}, using Equation (4) (ΔF

_{50%}= C(T − T

_{mean})). This estimated increase in force can be compared with the experimental observation in Figure 9 for the frequencies at which self-heating occurs (from 2 Hz). Figure 17 shows the results of this comparison.

_{50%}approaches the experimental value, up to 6 Hz. Between 7 Hz and 10 Hz, the estimated theoretical value is always lower than the experimental one, but this is because the mean temperature (T

_{mean}, Figure 10) is not stabilized, but still increasing. In other words, with the T

_{mean}value already stabilized, the estimated and experimental values will probably be very close, like those observed for 5 Hz and 6 Hz. This result means that by measuring the mechanical part’s temperature rise, it is possible to estimate both its stiffness increase (force to produce the same deflection) and the loading frequency.

## 4. Conclusions

_{c}) was determined by a proposed methodology using superelastic cycling at different frequencies. The temperature variation during cycling was analyzed, confirming the internal temperature increase of the NiTi Belleville spring. The proposed methodology consists in identifying the frequency at which there is no difference in the force at the end of loading between the first and last superelastic cycles (128 cycles were used). Keeping the same environmental conditions, cycling above this frequency will cause the self-heating of the SMA device. For the dimensions and NiTi alloy used in the Belleville spring of this study, this frequency was of the order of 1.7 Hz.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Isothermal superelastic response at low frequency (0.1 Hz) of the NiTi Belleville spring observed in force vs. deflection curves. The vertical lines indicate the deflection range selected for the cyclic compression tests.

**Figure 6.**Mechanical behavior during the pre-cycling (at 0.1 Hz) of the superelastic behavior. (

**a**) Force vs. deflection response. (

**b**) Force vs. cycles response (left axis) and percentage force variation in relation to the N-1 cycle (right axis). (

**c**) Detail of the percentage force variation.

**Figure 7.**Thermomechanical behavior at 0.5 Hz. (

**a**) Superelastic dynamic response in the 1st and 128th cycles. (

**b**) Temperature response during cycling.

**Figure 8.**Dynamic response of the superelastic behavior for loading frequencies from 1 Hz to 10 Hz. The horizontal line is used to compare the force levels at the beginning of each cycle.

**Figure 11.**Temperature amplitude (T

_{amp}) of the NiTi Belleville spring as a function of loading frequency.

**Figure 12.**Functional properties over cycles as a function of the load frequencies analyzed. (

**a**) Secant stiffness. (

**b**) Energy dissipated per cycle. (

**c**) Equivalent viscous damping factor.

**Figure 13.**FEA and experimental superelastic response at low loading frequency under controlled temperature.

Body (Material Model) | Material Parameter | Value | ||
---|---|---|---|---|

Compression plates (Linear elastic) | E (MPa) | 200,000 | ||

ν | 0.3 | |||

NiTi Belleville spring (SMA superelastic) | E (MPa) | 40,000 | ||

ν | 0.3 | |||

35 °C | 45 °C | 55 °C | ||

${\mathsf{\sigma}}_{\mathrm{s}}^{\mathrm{AM}}$ (MPa) | 350 | 450 | 525 | |

${\mathsf{\sigma}}_{\mathrm{f}}^{\mathrm{AM}}$ (MPa) | 575 | 625 | 650 | |

${\mathsf{\sigma}}_{\mathrm{s}}^{\mathrm{MA}}$ (MPa) | 300 | 350 | 375 | |

${\mathsf{\sigma}}_{\mathrm{f}}^{\mathrm{MA}}$ (MPa) | 150 | 175 | 200 | |

${\u03f5}_{\mathrm{L}}$ (mm/mm) | 0.06 | |||

α | 0 |

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**MDPI and ACS Style**

de Souza, E.F.; da Silva, P.C.S.; Grassi, E.N.D.; de Araújo, C.J.; de Lima, A.G.B.
Critical Frequency of Self-Heating in a Superelastic Ni-Ti Belleville Spring: Experimental Characterization and Numerical Simulation. *Sensors* **2021**, *21*, 7140.
https://doi.org/10.3390/s21217140

**AMA Style**

de Souza EF, da Silva PCS, Grassi END, de Araújo CJ, de Lima AGB.
Critical Frequency of Self-Heating in a Superelastic Ni-Ti Belleville Spring: Experimental Characterization and Numerical Simulation. *Sensors*. 2021; 21(21):7140.
https://doi.org/10.3390/s21217140

**Chicago/Turabian Style**

de Souza, Emmanuel Ferreira, Paulo César Sales da Silva, Estephanie Nobre Dantas Grassi, Carlos José de Araújo, and Antonio Gilson Barbosa de Lima.
2021. "Critical Frequency of Self-Heating in a Superelastic Ni-Ti Belleville Spring: Experimental Characterization and Numerical Simulation" *Sensors* 21, no. 21: 7140.
https://doi.org/10.3390/s21217140