# Finite Element Analysis for the Self-Loosening Behavior of the Bolted Joint with a Superelastic Shape Memory Alloy

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## Abstract

**:**

## 1. Introduction

## 2. Constitutive Modeling for NiTi SMA under Cyclic Loading

#### 2.1. Constitutive Equation and Internal Variables

- (i)
- Transition to the martensite phaseif $T>{M}_{S}$ and ${\sigma}_{s}^{cr}+{C}_{M}(T-{M}_{s})<\sigma <{\sigma}_{f}^{cr}+{C}_{M}(T-{M}_{s})$:$$\underset{A\to M}{{\xi}_{r}}=\frac{1-\underset{M\to A}{{\xi}_{ir}^{f}}}{2}\mathrm{cos}\left\{\frac{\pi}{{\sigma}_{s}^{cr}-{\sigma}_{f}^{cr}}[\sigma -{\sigma}_{f}^{cr}-{C}_{M}(T-{M}_{S})]\right\}+\frac{1+\underset{M\to A}{{\xi}_{ir}^{f}}}{2}$$
- (ii)
- Transition to the austenite phaseif $T>{A}_{S}$ and ${C}_{A}(T-{A}_{f})<\sigma <{C}_{A}(T-{A}_{s})$:$$\underset{M\to A}{{\xi}_{r}\text{}}=\frac{\underset{A\to M}{{\xi}_{r}^{f}}-\underset{M\to A}{{\xi}_{ir}}}{2}\left\{\mathrm{cos}[{a}_{A}(T-{A}_{s}-\frac{\sigma}{{C}_{A}})]+1\right\}$$

#### 2.2. Evolution Law of Parameters Governed by Accumulated Martensite Volume Fraction

- (i)
- Evolution equation for the residual martensite volume fraction:$${\dot{\xi}}_{ir}={\xi}_{ir\mathrm{max}}{c}_{AM}^{f}(\sigma ){e}^{-b{\xi}_{c}}{\dot{\xi}}_{c},$$$${c}_{AM}^{}(\sigma )={(\frac{<{Q}_{f}^{AM}-{Q}_{s}^{AM}-<{Q}_{f}^{AM}-\sigma >>}{{Q}_{f}^{AM}-{Q}_{s}^{AM}})}^{n}$$$${c}_{AM}^{f}(\sigma )=\mathrm{max}({c}_{AM}^{f}(\sigma ))$$
- (ii)
- Evolution law of transition stressDue to incomplete reverse transition during loading cycles, the superelastic NiTi SMA shows the mixture state of the austenite and residual martensite phases. The transition stresses decrease with the increasing of cycle numbers. Thus, according to the experimental observations in [32], the evolution equations in exponential formulation were proposed by [33] to describe the progressive evolution of the transition stresses with increasing cycle numbers from their initial values to stable ones, and are introduced here as$${\sigma}_{s}^{AM}={\sigma}_{{s}_{0}}^{AM}-({\sigma}_{{s}_{0}}^{AM}-{\sigma}_{{s}_{1}}^{AM})(1-{e}^{-{c}_{s}^{AM}{\xi}_{c}}),$$$${\sigma}_{f}^{AM}={\sigma}_{{f}_{0}}^{AM}-({\sigma}_{{f}_{0}}^{AM}-{\sigma}_{{f}_{1}}^{AM})(1-{e}^{-{c}_{f}^{MA}{\xi}_{c}}),$$$${\sigma}_{s}^{MA}={\sigma}_{{s}_{0}}^{MA}-({\sigma}_{{s}_{0}}^{MA}-{\sigma}_{{s}_{1}}^{MA})(1-{e}^{-{c}_{s}^{MA}{\xi}_{c}}),$$$${\sigma}_{f}^{MA}={\sigma}_{{f}_{0}}^{MA}-({\sigma}_{{f}_{0}}^{MA}-{\sigma}_{{f}_{1}}^{MA})(1-{e}^{-{c}_{f}^{MA}{\xi}_{c}}),$$

#### 2.3. Incremental Formulation of Constitutive Equations

#### 2.4. Finite Element Modeling for the One-Dimensional Bar Element of SMA Ratcheting Behavior

#### 2.5. Numerical Simulation and Model Verification

_{r}= 1.5%) remained. The high residual strain implies that there is an incomplete reverse transition from the stress-induced martensite to the original austenite after unloading, which leads to some amount of martensite strain remaining. It is further found that the amount of remained martensite increases progressively with the cyclic loadings. This accumulation phenomenon of deformation had been named as “phase transition ratcheting” and discussed in detail by [26,27,28,29,30,31,32], which is different from the ratcheting of ordinary metals without phase transition. It illustrates that after the first unloading, the pure austenite is replaced by the mixture of austenite and remaining martensite in the alloy. The stresses are certainly no longer the transition stresses of the pure austenite and martensite.

_{A}and martensite E

_{M}, the starting stress of the forward phase transition of austenite to martensite ${\sigma}_{s}^{AM}$ and the inverse transition of martensite to austenite ${\sigma}_{s}^{MA}$, the finishing stress of the forward phase transition of austenite to martensite ${\sigma}_{f}^{AM}$, and the inverse transition of martensite to austenite ${\sigma}_{f}^{MA}$. The subscript ‘0’ here indicates the first phase transition cycle of cyclic tension-unloading. It should be noted that the parameters of elastic modulus and transition stress are nominal variables due to the existence of residual martensite and its change during the cyclic loadings. Moreover, the dissipation energy W

_{d}is defined as the area around the stress-strain curve in each loading-unloading cycle:

_{d}decrease with an exponential function law during the cycle loadings. After certain numbers of cycles, both the ratcheting strains and dissipation energy show an apparent quick decrease to a stable value. These findings also agree with the conclusions presented by Kang [33].

## 3. Finite Element Modeling for Self-Loosening of the SMA Bolted Joint

#### 3.1. Finite Element Modeling under Cyclic External Force Load

- ①
- As the external load ${F}_{e}$ is smaller than the initial preloading force of bolt ${F}_{p0}$, the system will satisfy the force equilibrium relationship as ${F}_{b}={F}_{e}+{F}_{m}$.
- ②
- As the external load ${F}_{e}$ is larger than the initial preloading force of bolt ${F}_{p0}$, ${F}_{m}=0$ is satisfied at this time. Then, the system will satisfy the force equilibrium relationship as ${F}_{b}={F}_{e}$. Meanwhile, the constraint at the node 3 is removed and the element 2 is in the state of no stress.

#### 3.2. Finite Element Modeling under Cyclic External Displacement Load

## 4. Results and Discussion

^{8}N·m

^{−1}with the dimensional sizes of height 50 mm and diameter 30 mm, and the elastic modulus 70 × 10

^{9}Pa and Poisson ratio 0.28 of selected aluminum.

_{d}calculated across the area around the stress-strain curve of the SMA bolt bar in loading-unloading cycles decreases with an exponential function law during the stress cycle loading. Similar to the decrease of dissipation energy of a uniaxial cycle loading test, that of an SMA bolt also shows an obvious quick decrease to a stable value after a certain number of cycles. The remarkable decrease of dissipation energy means that the damping capability of superelastic NiTi SMA, which was also discussed in [46], degrades with the residual martensite accumulation and finally causes the functional failure of the NiTi SMA bolt.

_{0}= 7.9 kN and u

_{e}= 0.076 mm seems to be the demarcation line for the attenuation rate of the bolt clamping force. For the loading case of larger displacement loads than u

_{e}= 0.076 mm, the attenuation rate of the bolt clamping force shows an obvious increase than those of lower displacement loads. The change for the attenuation rate of the bolt clamping force is caused by the division of the contact surface of members under the loading cases of larger displacement loads than u

_{e}= 0.076 mm. This reason can also be used to explain why the larger preload case for P

_{0}= 8.3 kN produces a lower attenuation rate of bolt clamping force, to the contrary.

_{d}of the SMA bolt bar during the loading cycles. The dissipation energy of the SMA bolt also shows an obvious quick decrease to a stable value after a certain number of cycles. Obviously, the dissipation energy of the SMA bolt will increase with the increase of external displacement load that corresponds to the area around the stress-strain curve of the SMA bolt bar in loading-unloading cycles as shown in Figure 11. It can also be found that the dissipation energy of the SMA bolt increases with the increase of bolt preload. It may provide a beneficial reference for the engineer that a relatively higher preload force of the SMA bolt will increase the antivibration capability of the damping structure with the SMA bolted joint.

## 5. Conclusions

_{e}= 0.076 mm). The conclusion that the dissipation energy of the SMA bolt increases with a relatively higher preload force of the SMA bolt, which will be beneficial to increase the antivibration capability of the damping structure with the SMA bolted joint, could provide a beneficial reference for engineering design.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A.

**Figure A1.**Flow chart of analysis to determine the preload and self-loosening of a bolt under cyclic external force load.

**Figure A2.**Flow chart of analysis to determine the preload and self-loosening of a bolt under cyclic external displacement load.

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**Figure 1.**Typical tension-unloading stress-strain curve of materials [32].

**Figure 2.**Experiments and simulations for cyclic tension-unloading transition ratchetting with constant loading stress of 325 MPa: (

**a**) experimental results; (

**b**) simulated results; (

**c**) curves of peak and valley strains vs. cycle numbers; (

**d**) curves of dissipation energy vs. cycle numbers.

**Figure 3.**Experiments and simulations for cyclic tension-unloading transition ratchetting with constant loading stress of 365 MPa: (

**a**) experimental results; (

**b**) simulated results; (

**c**) curves of peak and valley strains vs. cycle numbers; (

**d**) curves of dissipation energy vs. cycle number.

**Figure 4.**Stress-strain response of SMA to cyclic loading up to a constant value of stress: curves for the first and 50th cycles.

**Figure 5.**Loading process and finite element modeling of the bolted joint. (

**a**) initial stage; (

**b**) after preloaded; (

**c**) cyclic loading stage.

**Figure 10.**Variation of displacement of nodes connecting the bolt and member during the loading cycles: (

**a**) change with the iterative substeps; (

**b**) change with the loading cycles.

**Figure 11.**Curves of stress-strain responses on bolt bar under different load cases. (

**a**) P

_{0}= 7.9 kN, u

_{e}= 0.036 mm; (

**b**) P

_{0}= 7.9 kN, u

_{e}= 0.046 mm; (

**c**) P

_{0}= 7.9 kN, u

_{e}= 0.056 mm; (

**d**) P

_{0}= 7.9 kN, u

_{e}= 0.066 mm; (

**e**) P

_{0}= 7.9 kN, u

_{e}= 0.076 mm; (

**f**) P

_{0}= 7.9 kN, u

_{e}= 0.086 mm; (

**g**) P

_{0}= 7.9 kN, u

_{e}= 0.096 mm; (

**h**) P

_{0}= 7.9 kN, u

_{e}= 0.106 mm; (

**i**) P

_{0}= 7.4 kN, u

_{e}= 0.086 mm; (

**j**) P

_{0}= 8.3 kN, u

_{e}= 0.086 mm.

**Figure 12.**Clamping force reduction of bolt with increasing loading cycles under different load cases.

**Figure 13.**Dissipation energy (W

_{d}) reduction of bolt for all loading cases with increase of loading cycles: (

**a**) for all cycles; (

**b**) for the initial two cycles.

${E}_{A}^{}$ = 48 GPa | ${E}_{M}^{}$ = 35 GPa | ${\nu}_{A}^{}$ = 0.3 | ${\nu}_{M}^{}$ = 0.3 | ${\epsilon}_{L}^{}$= 0.063 | $T$= 295 K |

${\sigma}_{s0,T}^{AM}$= 285 MPa | ${\sigma}_{f0,T}^{AM}$= 458 MPa | ${\sigma}_{s0,T}^{MA}$= 345 MPa | ${\sigma}_{f0,T}^{MA}$= 164 MPa | ||

${\sigma}_{s1,T}^{AM}$= 225 MPa | ${\sigma}_{f1,T}^{AM}$= 458 MPa | ${\sigma}_{s1,T}^{MA}$= 310 MPa | ${\sigma}_{f1,T}^{MA}$= 125 MPa; | ||

${c}_{s}^{AM}$= 0.05 | ${c}_{f}^{AM}$= 0.05 | ${c}_{s}^{MA}$= 0.05 | $n$ = 3 | ${\xi}_{\mathrm{max}}^{ir}$ = 0.84 | $b$ = 0.5 |

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**MDPI and ACS Style**

Jiang, X.; Huang, J.; Wang, Y.; Li, B.; Du, J.; Hao, P.
Finite Element Analysis for the Self-Loosening Behavior of the Bolted Joint with a Superelastic Shape Memory Alloy. *Materials* **2018**, *11*, 1592.
https://doi.org/10.3390/ma11091592

**AMA Style**

Jiang X, Huang J, Wang Y, Li B, Du J, Hao P.
Finite Element Analysis for the Self-Loosening Behavior of the Bolted Joint with a Superelastic Shape Memory Alloy. *Materials*. 2018; 11(9):1592.
https://doi.org/10.3390/ma11091592

**Chicago/Turabian Style**

Jiang, Xiangjun, Jin Huang, Yongkun Wang, Baotong Li, Jingli Du, and Peng Hao.
2018. "Finite Element Analysis for the Self-Loosening Behavior of the Bolted Joint with a Superelastic Shape Memory Alloy" *Materials* 11, no. 9: 1592.
https://doi.org/10.3390/ma11091592