Dan Reynolds  Research  Teaching  Publications  Talks  CV, etc.  Links

Shape Memory Alloy Modeling

(Multi-scale Mathematical Modeling and Simulation)

In my thesis research, I developed a new mathematical model for simulating multi-scale thermodynamic phase transformation processes in shape memory alloys (SMA), based on continuum-level thermodynamic PDE modeling of their behavior. Such alloys are at the forefront of research in materials science and engineering applications, as they may be used in the production of microscopic devices with the ability to perform work without the use of moving parts. Such behaviors are made possible by the understanding and control of first-order phase transformations that occur at the atomistic scale of these materials, in which a large number of small-scale crystalline transformations coordinate to allow large-scale nonlinear thermodynamic actions, a diagram of which is presented in Figure 1.

Figure 1:  Shape Memory Effect and Hysteresis: when cooled, austenite transforms into twinned martensite, which is easily deformable under stress. Upon heating, the deformed martensite returns to the austenitic configuration. Hysteresis in these materials provides a measure of the overall energy absorbed by the material during a full loop of the forward and reverse phase transformations.

At the macroscopic (continuum) level, these phase transformations may be modeled through the use of non-convex free energy potentials, which may be tuned to model temperature-dependent hysteresis, super-elasticity, and other nonlinear material behaviors. Mathematically, the use of non-convex Helmholtz free energy potentials in the macroscale model results in a PDE system of the form

where is the macroscopic material deformation, is the material temperature, , is the material-dependent Helmholtz free energy density, , , and are various material-dependent parameters, and and provide external body forces and heat source interactions with the surrounding environment. Of particular interest in such models of shape memory materials is that unlike standard models in linear or even nonlinear elasticity, the potential is non-quasiconvex (see the strain energy component in Figure 2), giving rise to variational PDE formulations that do not satisfy weak lower semicontinuity, thus precluding the use of standard PDE existence theory and computational solution techniques toward their solution. For simplified one-dimensional model systems of these materials we have developed stochastic computational techniques for computing steady-state averaged crystalline properties [1, 2, 3, 4]. In addition, we have derived qualitatively and quantitatively-accurate deterministic PDE models for SMA wires [9].

Figure 2:  Strain Energy Density: The Helmholtz free energy density, , is primarily composed of the deformation- and temperature-dependent strain energy density W, that encodes many of the nonlinear properties of these materials.

In addition to deriving physically-accurate continuum-level models for SMA wires, we then developed an efficient and robust computational solution approach for approximating solutions to the system (1), based on natural parameter continuation approaches employing the material viscosity to regularize the modeling system, thereby allowing the computational solution of the non-convex PDE model to successfully reproduce material phase transitions and time-dependent computations of thermodynamic SMA processes [5]. A graphic showing computed results for the stress-induced martensitic phase transformation is shown at the left of Figure 3. We then used this computational "laboratory" for SMA wires to design techniques for damping vibrational energy through thermal actuation [6, 7, 8] -- a kinetic energy profile showing this vibration damping is shown in Figure 3.

Figure 3:  Stress-Induced Martensitic Phase Transformation (left) and Kinetic Energy Damping Results (right). Through application of tensile and compressive stresses on the model, we achieve the so-called stress-induced martensitic phase transformation, even though the material remains within the austenitic temperature region. Moreover, through appropriate thermal controls, we were able to dampen vibrational energy in the model through conversion of mechanical to thermal energy in the SMA phase transformation.


While these advances have proven fruitful in the modeling and design of one-dimensional SMA wires, there remains significant work on extending such continuum-level descriptions to realistic models for slabs and solids, as these 2D and 3D models require construction of very complex multi-dimensional free-energy potentials. Therefore in addition to investigations of free energy potentials for higher-dimensional models of SMA materials, current research in this area attempts to span the gap between macroscopic-scale systems and atomistic material dynamics through the use of multi-scale modeling approaches. In our upcoming investigations in this area, we look to follow this second path of realistic material thermodynamics through the development of multi-scale and multi-model approaches for SMA thermodynamics, along with the accompanying numerical techniques required for multi-scale computational simulations.


[1] D.D. Cox, P. Kloucek, and D.R. Reynolds. "The computational modeling of crystalline materials using a stochastic variational principle." Lecture Notes in Computer Science, 2330:461--469, 2002.
[2] D.D. Cox, P. Kloucek, and D.R. Reynolds. "A subgrid projection method for relaxation of non-attainable differential inclusions." In Proceedings: ENUMATH 2001 European Conference on Numerical Mathematics, Berlin, 2002. Springer-Verlag.
[3] D.D. Cox, P. Kloucek, and D.R. Reynolds. "On the asymptotically stochastic computational modeling of microstructures." Future Generation Computer Systems, 20:409--424, 2004.
[4] D.D. Cox, P. Kloucek, D.R. Reynolds, and P. Solin. "Stochastic relaxation of variational integrals with non-attainable infima." In Proceedings: ENUMATH 2003 European Conference on Numerical Mathematics, Berlin, 2004. Springer-Verlag.
[5] P. Kloucek and D.R. Reynolds. "On the modeling of nonlinear thermodynamics in SMA wires." Computer Methods in Applied Mechanics and Engineering, 196:180--191, 2006.
[6] P. Kloucek, D.R. Reynolds, and T.I. Seidman. "On thermodynamic active control of shape memory alloy wires." Systems & Control Letters, 48, 2003.
[7] P. Kloucek, D.R. Reynolds, and T.I. Seidman. "Thermal stabilization of shape memory alloy wires." R.C. Smith, editor, In Smart Structures and Materials 2003: Modeling, Signal Processing, and Control, volume 5049 of Proc. of SPIE, 2003.
[8] P. Kloucek, D.R. Reynolds, and T.I. Seidman. "Computational modeling of vibration damping in SMA wires." Continuum Mechanics and Thermodynamics, 16:495--514, 2004.
[9] D.R. Reynolds. A Nonlinear Thermodynamic Model for Phase Transitions in Shape Memory Alloy Wires. PhD thesis, Rice University Dept. of Computational and Applied Mathematics, 2003.
[10] P. Kloucek and D.R. Reynolds. Vibration damping and heat transfer using material phase changes. U.S. Patent Number 7,506,735, issued March 24, 2009.


Maintained by reynolds [at] smu [dot] edu

Copyright ©2014 Southern Methodist University. All rights reserved.
Right to Know, Nondiscrimination, and other legal statements.