Development and Finite Element Implementation of a Simple Constitutive Model to Address Superelasticity and Hysteresis of Nitinol

Abstract

Nitinol shows superelasticity and clearly defined hysteresis that possesses close resemblance to biological components. This is attributed to stress-induced phase transformation of Nitinol. The present article proposes a new constitutive model based on a simple schematic arrangement of friction block, spring, and rigid walls to replicate this unique behavior of Nitinol. In addition to superelasticity, the strain hardening and viscoplasticity are thoroughly explored and also incorporated in the model. Results of simulation closely match with the experimental data obtained from uniaxial testing of Nitinol wire. This model can be readily used for any case of superelasticity either due to phase transformation or any other microstructural behavior.

Appendix

$$\varvec{A} := \frac{{{\text{d}}{\Delta }{\varvec{\upepsilon}}_{p}^{*} }}{{{\text{d}}\varvec{\sigma}_{\text{eff}} }} = \frac{3}{{2w_{1} x}}\left[ {\left( {x - 1} \right)\varvec{P} + \frac{3}{{2\varvec{x}^{2}\varvec{\sigma}_{\varvec{y}}^{0\,2} }}\varvec{P\sigma }_{\text{eff}}\varvec{\sigma}_{\text{eff}}^{\varvec{T}} \varvec{P}} \right];$$

$$\overline{\varvec{A}} = \frac{3}{{2x\sigma_{y}^{0} }}\left[ {\lambda \varvec{P} + \frac{1}{{x\left( {\sigma_{y}^{0} } \right)^{2} }}\left( {\frac{{\sigma_{y} }}{{w_{1} + \frac{{{\text{d}}\Delta \sigma_{v} }}{{{\text{d}}\lambda }}}} - \frac{\lambda }{x}} \right)\varvec{P\sigma }_{\text{eff}}\varvec{\sigma}_{\text{eff}}^{T} \varvec{P}} \right];$$

$$\varvec{B} := \frac{{{\text{d}}{\Delta }{\varvec{\upepsilon}}_{p} }}{{{\text{d}}{\Delta }{\varvec{\upepsilon}}_{p}^{*} }} = \frac{1}{y}\left[ {I - \frac{2}{{3y^{2} {\Delta }{\varvec{\upepsilon}}^{2} }}{\Delta }{\varvec{\upepsilon}}_{{p,{\text{eff}}}} {\Delta }{\varvec{\upepsilon}}_{{p,{\text{eff}}}}^{T} \overline{\varvec{P}} } \right];$$

$$\varvec{P} = \frac{1}{3}\left[ {\begin{array}{*{20}r} 2 \hfill & { - 1} \hfill & { - 1} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ { - 1} \hfill & 2 \hfill & { - 1} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ { - 1} \hfill & { - 1} \hfill & 2 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 6 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 6 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 6 \hfill \\ \end{array} } \right] ;$$

$$\overline{\varvec{P}} = \frac{1}{3}\left[ {\begin{array}{*{20}r} 2 \hfill & { - 1} \hfill & { - 1} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ { - 1} \hfill & 2 \hfill & { - 1} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ { - 1} \hfill & { - 1} \hfill & 2 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {1.5} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {1.5} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {1.5} \hfill \\ \end{array} } \right] .$$