# A Finite Strain Constitutive Model for Martensitic Transformation in Shape Memory Alloys Based on Logarithmic Strain

## TL;DR

This paper introduces a finite strain constitutive model for Shape Memory Alloys using logarithmic strain, enabling accurate predictions under large deformations and rotations, especially relevant for complex geometries and cyclic loading.

## Contribution

It develops a three-dimensional phenomenological model based on logarithmic strain for SMAs considering finite strains and rotations, addressing limitations of previous infinitesimal strain models.

## Key findings

- Model accurately predicts SMA behavior under large strains and rotations.
- Numerical simulations validate the model with various SMA component geometries.
- Framework sets the stage for future inclusion of transformation-induced plasticity.

## Abstract

Shape Memory Alloys (SMAs) are materials with the ability to recover apparently permanent deformation under specific thermomechanical loading. The majority of constitutive models for SMAs are developed based on the infinitesimal strain theory. However, such assumption may not be proper in the presence of geometric discontinuities, such as cracks, and repeated cycling loading that has been reported to induce irrecoverable strains up to 20% due to transformation induced plasticity. In addition to finite strains, SMA-based devices may also undergo large rotations. Thus, it is indispensable to develop a constitutive model based on the finite strain to provide accurate predictions of these actuators response. A three-dimensional phenomenological constitutive model for SMAs considering finite strains and finite rotations is proposed in this work. This model utilizes the logarithmic strain as the strain measure that is the strain measure whose logarithmic rate in a corotating material frame is equal to the rate of deformation tensor. In the proposed model, the martensitic volume fraction and the second-order logarithmic transformation strain tensor are chosen as the internal state variables associated with the inelastic transformation process. Numerical simulations considering basic SMAs component geometries such as a bar, a beam, and a torque tube are performed to test the capabilities of the proposed model under both mechanically and thermally induced phase transformation. The presented model formulation will be extended in future work for the incorporation of transformation-induced plasticity.