Shape Memory Alloy Modeling
(Multiscale Mathematical Modeling and Simulation)
In my thesis research, I developed a new mathematical model for
simulating multiscale thermodynamic phase transformation
processes in shape memory alloys (SMA), based on continuumlevel
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 firstorder phase transformations
that occur at the atomistic scale of these materials, in which a
large number of smallscale crystalline transformations
coordinate to allow largescale 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 nonconvex
free energy potentials, which may be tuned to model
temperaturedependent hysteresis, superelasticity, and other
nonlinear material behaviors. Mathematically, the use of
nonconvex 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 materialdependent Helmholtz free energy density,
,
,
and
are various materialdependent 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 nonquasiconvex (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 onedimensional model systems of these materials
we have developed stochastic computational techniques for
computing steadystate averaged crystalline properties [1, 2, 3,
4]. In addition, we have derived qualitatively and
quantitativelyaccurate deterministic PDE models for SMA wires [9].
Figure 2:
Strain Energy Density: The Helmholtz free energy
density,
,
is primarily composed of the deformation and
temperaturedependent strain energy density W, that encodes
many of the nonlinear properties of these materials.

In addition to deriving physicallyaccurate continuumlevel
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 nonconvex PDE model to
successfully reproduce material phase transitions and
timedependent computations of thermodynamic SMA processes [5].
A graphic showing computed results for the stressinduced
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:
StressInduced Martensitic Phase Transformation
(left) and Kinetic Energy Damping Results (right). Through
application of tensile and compressive stresses on the model,
we achieve the socalled stressinduced 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 onedimensional SMA wires, there remains significant
work on extending such continuumlevel descriptions to realistic
models for slabs and solids, as these 2D and 3D models require
construction of very complex multidimensional freeenergy
potentials. Therefore in addition to investigations of free
energy potentials for higherdimensional models of SMA
materials, current research in this area attempts to span the
gap between macroscopicscale systems and atomistic material
dynamics through the use of multiscale modeling approaches.
In our upcoming investigations in this area, we look to follow
this second path of realistic material thermodynamics through
the development of multiscale and multimodel approaches for
SMA thermodynamics, along with the accompanying numerical
techniques required for multiscale computational simulations.
References
[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,
2330:461469, 2002. 
[2] 
D.D. Cox, P. Kloucek, and D.R. Reynolds. "A subgrid
projection method for relaxation of nonattainable differential
inclusions." In Proceedings: ENUMATH 2001 European Conference
on Numerical Mathematics, Berlin,
2002. SpringerVerlag. 
[3] 
D.D. Cox, P. Kloucek, and D.R. Reynolds. "On the asymptotically
stochastic computational modeling of microstructures." Future
Generation Computer Systems, 20:409424, 2004. 
[4] 
D.D. Cox, P. Kloucek, D.R. Reynolds, and P. Solin.
"Stochastic relaxation of variational integrals with nonattainable
infima." In Proceedings: ENUMATH 2003 European Conference on
Numerical Mathematics, Berlin, 2004. SpringerVerlag. 
[5] 
P. Kloucek and D.R. Reynolds. "On the modeling of nonlinear
thermodynamics in SMA wires." Computer Methods in
Applied Mechanics and Engineering, 196:180191, 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, 16:495514, 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. 
