# Influence of Structural Porosity and Martensite Evolution on Mechanical Characteristics of Nitinol via In-Silico Finite Element Approach

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Kinematics of Phase Transformations

## 2. Materials and Methods

#### 2.1. Constitutive Modelling of Nitinol SMAs

#### 2.2. Microscale Models

#### 2.3. Macroscale Models

- 1D and 3D constitutive model to reproduce superelasticity
- Time-discrete isothermal model
- Algorithmic implementation with a finite element (FE) framework.

- Conversion of austenite into single-variant martensite (A→S)
- Conversion of single-variant martensite into austenite (S→A)
- Reorientation of the single-variant martensite (S→S)

#### 2.3.1. Conversion of Austenite into Single-Variant Martensite (A→S)

#### 2.3.2. Conversion of Single-Variant Martensite into Austenite (S→A)

#### 2.3.3. Reorientation of the Single-Variant Martensite (S→S)

#### 2.4. Numerical Simulations

^{3}. For the macroscale study, a cube of geometry 5 × 5 × 5 mm was modelled with uniaxial mechanical loading/unloading (tension/compression) for a deformation of 0.5 mm (10% strain in fully dense model) at a constant strain rate of 0.1 min

^{−1}. The bottom of the cube was fully constrained (fixed).

#### 2.5. Mesh Convergence

## 3. Results

#### 3.1. Model Validation

#### 3.2. Response to Strain Levels

^{−1}using the material NT1. As seen in Figure 14, when the cube experienced a low strain of 3.5%, a partial phase transformation occurred, similar to what is found in an actual mechanical test. A 6% strain generated a complete martensitic transformation, while a 10% strain progressed with straining in the elastic region of detwinned martensite phase after completing the phase change.

#### 3.3. Asymmetry in Tension and Compression

#### 3.4. Compression of Porous Structures

^{3}accompanying a reduction of about 200 MPa in stress levels and a reduction of about 0.063 in strain levels. No particular trend could be interpreted between the levels of porosity and this reduction in dissipated energy. These factors play a major role in depicting the efficacy for heat pump applications.

#### 3.5. MVF vs. Elastic Modulus

## 4. Discussion

#### 4.1. Mechanical Strain

#### 4.2. Structural Porosity

_{out}) can be determined by calculating the area contained within the martensitic transformation region (enclosed by the loading and unloading curve) on the stress–strain curve. This is also referred to as the dissipated energy per cycle or enthalpy [103]. Q

_{out}is the latent heat energy required for the martensitic transformation (obtained from Differential Scanning Calorimetry analysis).

_{out}). As mentioned earlier, the simulations of these porous structures were conducted for 0.5 mm deformation, similar to that of the fully dense component. Structural porosity results in a lower resultant cross-sectional area and length compared to the fully dense part, and this has contributed towards an effective reduction in the stress and strain levels. No significant trend was noted for energy dissipation with respect to porosity. This could be attributed to the possibility of different resultant cross-sectional areas and lengths for the same levels of porosity. It should also be noted that a further reduction in mechanical properties takes place when the component is deployed in the high temperature ambient conditions prevalent in heat pumps. For example, a decline of about 10 GPa in ultimate strength was observed when temperature was increased by 200 K from room temperature [104].

_{out}and in turn affects the COP. However, a complex structure such as P6 and P7 might have lower stress limits that could affect the structural strength and operational integrity. Compared to most lattice structure topologies, the design of P7 (14-spoke lattice) possesses a higher mechanical strength (Poisson’s ratio and elastic modulus) [105]. To summarize, an all-embracing balance should be devised between the surface area (porosity) in contact, stress compensations, and maximum strain levels in order to achieve a high COP.

#### 4.3. Martensite Evolution vs. Stiffness

## 5. Conclusions

^{3}, along with a stress reduction of about 200 MPa and strain decrease of 0.063. The damping ratio increased from 1.8% to 4.0% with an increase in porosity. The apparent stiffness for damping operations showed no trend, however, exhibited a general rise in porous samples compared to the fully dense sample.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Elastic Modulus Estimation from Martensite Volume Fraction

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**Figure 1.**Stress–strain curve for shape memory and superelasticity of NiTi [10].

**Figure 2.**Schematic illustration showing the nomenclature used during phase transformations and mechanical loading. ${E}_{A}$—Young’s modulus of austenite (A) phase; ${E}_{M}$ —Young’s modulus of martensite (M) phase; $\epsilon $

_{L}—maximum longitudinal strain; ${\sigma}_{s}^{AS}$ —stress to start martensitic transformation (austenite to single-variant martensite); ${\sigma}_{f}^{AS}$ —stress at finish of martensitic transformation; ${\sigma}_{s}^{SA}$ —stress to start reverse transformation (single-variant martensite to austenite); ${\sigma}_{f}^{SA}$ —stress at finish of reverse transformation; ${\sigma}_{y}^{M}$ —martensite yield stress; ${h}_{p}^{M}$ —martensite hardening parameter.

**Figure 4.**The evolution of martensite during mechanical loading and unloading in the first cycle [39].

**Figure 5.**Schematic of the literature review content. Based on the gap in the literature in compressive response modelling of NiTi SMA, the scope of the current original work presented in this paper on the macroscale and microscale modelling of NiTi.

**Figure 7.**Porous structure designs showing the different types of cavities used; P1 to P5 varying degrees of cylindrical cavities; P6—enclosed honeycomb structure; and P7—lattice structure with 14-spokes per unit cell.

**Figure 10.**Stress values (red) at 4% strain and respective error % (black) for different mesh sizes.

**Figure 11.**Validation of superelasticity model with actual experiment by Jiang and Li [95].

**Figure 17.**Variation of stiffness and energy dissipated per unit volume per cycle, with the increasing levels of porosity.

**Figure 18.**(

**a**) Illustration of obtaining the dissipated and absorbed energy from stress–strain curves; (

**b**) variation of damping ratio ($\mathsf{\xi}\mathrm{R}$) and apparent stiffness with the increasing levels of porosity.

**Figure 20.**Stress–strain curve to illustrate the evolution of martensite phase during stress-induced transformation (SIMT) via stiffness slopes.

Material | ${\mathit{E}}_{\mathit{A}}\left(\mathbf{GPa}\right)$ | $\mathsf{\nu}$ | ${\mathit{\sigma}}_{\mathit{s}}^{\mathit{A}\mathit{S}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{f}}^{\mathit{A}\mathit{S}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{s}}^{\mathit{S}\mathit{A}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathit{f}}^{\mathit{S}\mathit{A}}\left(\mathbf{MPa}\right)$ | ${\mathit{\epsilon}}_{\mathit{L}}$ |
---|---|---|---|---|---|---|---|

NT1 | 71.1 | 0.3 | 500 | 700 | 400 | 200 | 0.044 |

NT2 | 41 | 0.33 | 380 | 390 | 145 | 110 | 0.040 |

NT3 | 50.3 | 0.3 | 556 | 643 | 315 | 246 | 0.075 |

Sample | Porosity (%) | Void Volume (mm^{3}) |
---|---|---|

Fully Dense (FD) | 0 | 0 |

P1 | 1.4 | 1.77 |

P2 | 2.8 | 3.53 |

P3 | 9.4 | 11.78 |

P4 | 15.7 | 19.63 |

P5 | 25.1 | 31.42 |

P6 | 72.7 | 90.85 |

P7 | 83.4 | 104.21 |

${\mathit{E}}_{\mathit{A}}\left(\mathbf{GPa}\right)$ | ${\mathit{E}}_{\mathit{M}}\left(\mathbf{GPa}\right)$ | $\mathsf{\nu}$ | |
---|---|---|---|

Upper bound (56 wt.% Ni) | 83 | 41 | 0.3 |

Lower bound (54 wt.% Ni) | 50.30 | 23.59 | 0.3 |

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## Share and Cite

**MDPI and ACS Style**

Chekotu, J.C.; Kinahan, D.; Goodall, R.; Brabazon, D.
Influence of Structural Porosity and Martensite Evolution on Mechanical Characteristics of Nitinol via In-Silico Finite Element Approach. *Materials* **2022**, *15*, 5365.
https://doi.org/10.3390/ma15155365

**AMA Style**

Chekotu JC, Kinahan D, Goodall R, Brabazon D.
Influence of Structural Porosity and Martensite Evolution on Mechanical Characteristics of Nitinol via In-Silico Finite Element Approach. *Materials*. 2022; 15(15):5365.
https://doi.org/10.3390/ma15155365

**Chicago/Turabian Style**

Chekotu, Josiah Cherian, David Kinahan, Russell Goodall, and Dermot Brabazon.
2022. "Influence of Structural Porosity and Martensite Evolution on Mechanical Characteristics of Nitinol via In-Silico Finite Element Approach" *Materials* 15, no. 15: 5365.
https://doi.org/10.3390/ma15155365