Numerical algorithms for cyclic phase transformation hysteresis in a shape memory plate subject to axisymmetric plane stress

• Published in: International journal for numerical methods in engineering Vol. 69; no. 9; pp. 1819 - 1850
• Main Authors: Chi, Yuwei, Pence, Thomas J., Tsai, Hungyu
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 26.02.2007; Wiley
• Subjects: Algorithms; Computational techniques; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Hysteresis; iterated mappings; Martensite; Martensitic transformations; Mathematical analysis; Mathematical methods in physics; Numerical analysis; Physics; plane stress; shape memory; shooting method; Solid mechanics; Strain; Structural and continuum mechanics; Transformations; Stress analysis; Volume fraction; Biaxial stress; Quasi static theory; Thermal load; Modeling; Axial symmetry; Plate; hysteresis; Phase transformation; Static conditions; Shape memory alloy; iterated mappings; Martensitic transformation; plane stress; shape memory; Microstructure; Shooting method; Numerical stability; Finite difference method
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.1829

We present numerical algorithms for calculating stress fields in an annulus composed of a shape memory material under conditions of quasi‐static edge loading at constant temperature. The algorithms track the material microstructure in terms of the volume fraction of austenite (A) and martensite (M), the latter of which provides a transformation strain. The dependence on load path imparts significant hysteresis in the stress induced transformation between A and M. A previous study that was restricted to proportional loading in the direction of forward transformation (J. Appl. Mech. 2005; 72:44–53) is here generalized to consider arbitrary loadings. The shooting algorithm that was robust for the previously considered proportional loadings is found to be subject to numerical instability for the most general transformation possibilities considered here. This motivates the development of an alternative iterated mapping algorithm that is found to generate a robust semi‐analytical finite difference procedure. The algorithm efficiently determines the operative transformation type, as is illustrated in cases where forward and reverse loading are occurring simultaneously at different plate locations. At those locations where phase transformation is inactive, the algorithm continues to account for martensite reorientation that alters the local transformation strain. Copyright © 2006 John Wiley & Sons, Ltd.