The constitutive relations of 1-D shape memory alloys developed by Brinson (1993, 1996), based on previous work by Liang and Tanaka is derived on a thermomechanical basis and has a relatively simple mathematical expression:
where x is martensite fraction and xo the fraction oriented by stress to provide a transformation strain. Coupled with the constitutive model, a kinetic law is used that robustly handles even complex thermomechanical loading paths involving cycling or partial loading. The kinetic law is described in Bekker and Brinson (1998) and uses a phase diagram as shown in Figure 1. Starting from this constitutive law, kinetic law framework, we have built a robust element subroutine (UEL) for ABAQUS that then allows SMA wire or ribbon elements to be considered as part of structures. By using this code, shape memory alloys or ribbons in which the SMA material behavior is primarily one-dimensional can be simulated. Most importantly, the UEL allows use of a commercial software package for analysis of more complex structures incorporating SMA wires and enables prediction and design of structural response to actuation conditions. Examples implemented in our group include both self-healing composites and adaptive shape control composites.
Figure 1. A typical one-dimensional phase diagram of shape memory alloys. The shaded regions are the transformation strips while other regions are “dead zones” where no change in martensite fractions occurs. G is an arbitrary loading path with points B-G denoted on it, where C and G are the entrance and exit points respectively; F is the “current point”, D is a switching point where transformation stops and and E is the most recent switching point, where transformation restarts and which carries the material history for the current point.
Gao, Qiao, Brinson, Implementation of Reorientation Kinetics in Uniaxial SMA Modeling for ABAQUS, manuscript in preparation, (2007).
Brinson, L. C. and M. S. Huang, Simplifications and Comparisons of Shape Memory Alloy Constitutive Models, Journal of Intelligent Material Systems & Structures , vol. 7, pp. 108-114, (1996).
Brinson, L. C., One Dimensional Constitutive Behavior of Shape Memory Alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable, Journal of Intelligent Material Systems & Structures, vol. 4, pp. 229-242, (1993).
Brinson, L. C. and R. Lammering, Finite Element Analysis of the Behavior of Shape Memory Alloys and their Applications, International Journal of Solids & Structures, vol. 30, pp. 3261-3280, (1993).