An energy-based material model for the simulation of shape memory alloys under complex boundary value problems

TL;DR

This work addresses the challenge of realistically simulating shape memory alloys under complex thermo-mechanical boundary value problems by presenting a Hamilton's principle–based energy framework. It formulates a macroscopic constitutive model with internal variables for phase fractions $\boldsymbol{\lambda}$, rotation via Euler-Rodrigues parameters $\boldsymbol{\alpha}$, irreversible plastic strain $\boldsymbol{\varepsilon}_{\mathrm{pl}}$, and hardening $\kappa$, incorporating Reuss homogenization, a sigmoid constraint for volume fractions, and rate-independent dissipation. The authors provide a robust numerical strategy, including Newton-based updates with Lagrange-multiplier corrections and Euler-Rodrigues recentering, implemented in Abaqus through UMAT with a consistent tangent. Demonstrations on tensile tests and two medical-device boundary-value problems (a stent and a compression staple) show mesh-independent results and physically consistent phase transformations, underscoring the model's practical relevance for medical technology, automotive, aerospace, and robotics.

Abstract

Shape memory alloys are remarkable 'smart' materials used in a broad spectrum of applications, ranging from aerospace to robotics, thanks to their unique thermomechanical coupling capabilities. Given the complex properties of shape memory alloys, which are largely influenced by thermal and mechanical loads, as well as their loading history, predicting their behavior can be challenging. Consequently, there exists a pronounced demand for an efficient material model to simulate the behavior of these alloys. This paper introduces a material model rooted in Hamilton's principle. The key advantages of the presented material model encompass a more accurate depiction of the internal variable evolution and heightened robustness. As such, the proposed material model signifies an advancement in the realistic and efficient simulation of shape memory alloys.

Figures (19)

• Figure 1: A graphical representation of the artificial energy-volume fraction $\Psi_{\mathrm{pen}, i}$ correlation with different values for the parameter $\Lambda$
• Figure 2: Principle structure of the user subroutine UMAT
• Figure 3: Functionality of the algorithms for determining the internal variable in the current time step $(\mathbin{\text{\large$\bullet$}})^{n+1}$
• Figure 4: Algorithm to calculate the Euler parameter $\pmb{\alpha}^{n+1}$
• Figure 5: Algorithm to calculate the volume fraction of the crystallographic phases $\pmb{\lambda}^{n+1}$