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Thermomechanical model for NiTi-based shape memory alloys covering macroscopic localization of martensitic transformation

M. Frost, B. Benešová, H. Seiner, M. Kružík, P. Šittner, P. Sedlák

TL;DR

This work addresses macroscopic localization of stress-induced martensitic transformation in polycrystalline NiTi SMAs by extending a generalized standard solids model with a nonlocal austenite-martensite interaction term. The key method introduces a Mori-Tanaka–based interaction energy $E^{int}$, regularized through neighborhood averaging $(\bm{\varepsilon}^{\rm in})_\omega$ via a smooth kernel $\mathcal{G}_\omega$, and constrained by a 1-homogeneous convex transformation-strain surface; evolution is computed with a backward-Euler time discretization and FEM implementation in Abaqus. Numerical simulations on a NiTi ribbon under tension and a NiTi tube under bending demonstrate tension-localized martensite bands with an average inclination around $58^{\circ}$ and wedge-like bands on tensile regions in bending, while compression remains more homogeneous. The approach yields qualitative agreement with experimental observations, highlights tension-compression asymmetry due to the interaction energy, and provides a framework for exploring localization under arbitrary loading, with potential extensions to texture-induced anisotropy and non-spherical inclusions.

Abstract

The work presents a thermomechanical model for polycrystalline NiTi-based shape memory alloys developed within the framework of generalized standard solids, which is able to cover loading-mode dependent localization of the martensitic transformation. The key point is the introduction of a novel austenite-martensite interaction term responsible for strain-softening of the material. Mathematical properties of the model are analyzed and a suitable regularization and a time-discrete approximation for numerical implementation to the finite-element method are proposed. Model performance is illustrated on two numerical simulations: tension of a superelastic NiTi ribbon and bending of a superelastic NiTi tube.

Thermomechanical model for NiTi-based shape memory alloys covering macroscopic localization of martensitic transformation

TL;DR

This work addresses macroscopic localization of stress-induced martensitic transformation in polycrystalline NiTi SMAs by extending a generalized standard solids model with a nonlocal austenite-martensite interaction term. The key method introduces a Mori-Tanaka–based interaction energy , regularized through neighborhood averaging via a smooth kernel , and constrained by a 1-homogeneous convex transformation-strain surface; evolution is computed with a backward-Euler time discretization and FEM implementation in Abaqus. Numerical simulations on a NiTi ribbon under tension and a NiTi tube under bending demonstrate tension-localized martensite bands with an average inclination around and wedge-like bands on tensile regions in bending, while compression remains more homogeneous. The approach yields qualitative agreement with experimental observations, highlights tension-compression asymmetry due to the interaction energy, and provides a framework for exploring localization under arbitrary loading, with potential extensions to texture-induced anisotropy and non-spherical inclusions.

Abstract

The work presents a thermomechanical model for polycrystalline NiTi-based shape memory alloys developed within the framework of generalized standard solids, which is able to cover loading-mode dependent localization of the martensitic transformation. The key point is the introduction of a novel austenite-martensite interaction term responsible for strain-softening of the material. Mathematical properties of the model are analyzed and a suitable regularization and a time-discrete approximation for numerical implementation to the finite-element method are proposed. Model performance is illustrated on two numerical simulations: tension of a superelastic NiTi ribbon and bending of a superelastic NiTi tube.

Paper Structure

This paper contains 11 sections, 1 theorem, 30 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Local model
  3. Non-local model
  4. Global formulation and mathematical properties
  5. Numerical simulations
  6. Implementation into the finite element method
  7. Tension of a NiTi ribbon
  8. Bending of a NiTi tube
  9. Discussion and Conclusions
  10. Mathematical analysis
  11. Material parameters

Key Result

Proposition 4.2

Let $(\xi_{k-1}\varepsilon^{\rm in}_{k-1}) \in L^\infty(\Omega) \times L^\infty(\Omega; \mathbb{R}^{3 \times 3})$ and that $F_\mathrm{vol} \in C^0(\Omega \times [0,T])$ as well as $F_\mathrm{surf} \in C^0(\Gamma_\mathrm{N} \times [0,T])$. Then there exists a triple $(u_k,\varepsilon^{\rm in}_k, \xi_

Figures (3)

  • Figure 1: (a) Experimental data on strain-softening in NiTi alloys by ALA-SIT-17 -- denoted E1 -- and HAL-KYR -- denoted E2 -- plotted together with their best numerical fit reached by the non-local constitutive model introduced in the Section \ref{['sec-nonloc']}. (b) Character of the evolution of the internal elastic energy $E^{\rm int}$ with variation of the volume fraction of martensite $\xi$. For $C_{\rm MA}^{\rm int} = C_{\rm AM}^{\rm int}$ the classical Mori-Tanaka symmetric case is recovered (marked in green). The symmetry of $E^{\rm int}$ is lost in the best fit of E1 (red dashed line) since $C_{\rm MA}^{\rm int} = 2.8\,C_{\rm AM}^{\rm int}$, see the respective contributions of corresponding energy terms (cyan and magenta dashed lines).
  • Figure 2: Snapshots from a three-dimensional simulation of the NiTi ribbon in tension with distribution of volume fraction of martensite in color (for legend see the VFM colorbar in Figure \ref{['fig:tube']}). The average angle of inclination of the planar martensite band front with respect to the axis of the ribbon is marked in ⑦.
  • Figure 3: Snapshots from a three-dimensional simulation of the NiTi tube in bending. The distribution of the axial component of strain on the left, the distribution of the volume fraction of martensite on the right.

Theorems & Definitions (3)

  • Remark 4.1
  • Proposition 4.2
  • Remark 4.3: Convexity