# ON THE MECHANCAL BEHAVIOR OF NITINOL UNDER MULTIAXIAL LOADING CONDITIONS AND *IN SITU* SYNCHROTRON X-RAY DIFFRACTION

# 

# *V. Imbeni\*,\*\*, A. Mehta\*\*\*, S.W. Robertson\*, T.W. Duerig\*\*\*\*, A. Pelton\*\*\*\* and R.O. Ritchie\**

*\* Materials Sciences Division, Lawrence Berkeley National Laboratory,
and Department of Materials Science and Engineering, University of
California, Berkeley*

*\*\* SRI International and The International Organization on Shape
Memory and Superelastic Technologies*

*\*\*\* Stanford Synchrothron Radiation Laboratory, University of
Stanford, Palo Alto*

*\*\*\*\* Nitinol Devices and Components, Fremont, California*

# ABSTRACT

Although the use of Nitinol in the biomedical industry has been very
successful, there have been distinct problems due to the difficulty of
being able to accurately compute stresses, particularly in the presence
of complex in vivo loading patterns. Constitutive laws for shape-memory
alloys subjected to multiaxial loading, which are based on direct
experimental observations, are generally not available in the
literature. Accordingly, in the present work, tension--torsion tests are
conducted on Nitinol thin-walled tubes (thickness/radius ratio of 1:10)
using various loading/unloading paths under isothermal conditions. The
experimental results showed significant variations in the mechanical
response along the two loading axes. To fully understand the macroscopic
and microscopic aspects of the transformation in situ multiaxial
(tension-torsion) loading experiments were also performed at the SSRL
(Stanford Synchrothron Radiation Laboratory) in order to monitor the
material's state of transformation and identifying the nature, volume
fraction of phases present and the elastic strain in them as a function
of stress and stress state.

# 

Keywords: Nitinol, multiaxial, torsion, X-ray, synchrotron, Lüders
bands, R-phase, *in situ*

# INTRODUCTION

Most of the mechanical testing on polycrystalline Nitinol found in the
literature has been performed on wires and is thus one-dimensional. As a
result, most phenomenological constitutive models are based on uniaxial
data, oftentimes extended to three-dimensions in an *ad hoc* fashion.
Although there are several one-dimensional models, the lack of such
constitutive relationships for these materials presents a real
limitation in the design and *in vivo* reliability of the many medical
devices manufactured from Nitinol. Design challenges have arisen from
the difficulty of measuring and computing stresses and strains in the
presence of complex *in vivo* loading during service life, as well as
the incomplete understanding of Nitinol's constitutive behavior (i.e.,
the relationship between stress and strain) under these conditions.

Very little information exists on the multiaxial loading/unloading of
Nitinol and even less on tubes; to our knowledge, only three previous
studies, all on thick-walled tubing, are available. \[1,2,3\] The
present work focuses on the biaxial testing of thin-walled tubes, the
starting material for the manufacture of endovascular stents and chosen
to minimize the gradient in the torsional strain along the radial
direction. Three distinct multiaxial stress-loading paths are explored
under isothermal conditions.

The resulting data are also reproduced in an *in situ* testing rig at
the Stanford Synchrotron Radiation Laboratory (SSRL), while
simultaneously using synchrotron x-ray diffraction to discern the
differences in phases present between tension and torsion observed
during multiaxial loading. The novel *in situ* synchrotron x-ray
experiments provided insight on the martensitic phase transformation
associated with superelastic transformations under complex strain
states. The long-term goal of this work is to incorporate the
'macroscopic' deformation data measured under tension/torsion loading
with the 'microscopic' *in situ* x-ray diffraction tension/torsion
results in order to develop a three-dimensional constitutive model for
superelasticity in Nitinol.

# EXPERIMENTAL PROCEDURES

# MATERIALS

Nitinol tubing (Ti 49.2 at.%, Ni 50.8 at.%) was received from NDC
(Fremont, CA), with a 4.64 mm outer diameter and 0.37 mm thickness. The
tubing was cut into 75 mm long specimens and ground down along the 25 mm
long center test section to an outer diameter of 4.3 mm in order to
obtain a waisted shape and minimize end effects during testing. The wall
thickness in the gauge section was reduced to 0.2 mm, resulting in a
thickness-to-radius ratio of approximately 1:10. The specimens were
subsequently heat-treated at 485^°^C in air for 5 min and then ice
water-quenched to bring the austenite finish temperature (*A*~f~)
slightly below room temperature. This treatment produces a complex
microstructure consisting of equiaxed low-angle subgrains, \~1 μm in
size, within irregularly shaped parent grains, \~25 μm in size.

The transformation temperatures of the heat-treated samples were
characterized by differential scanning calorimetry (DSC) and were found
to be *A*~s~ = -6.4^°^C, *A*~f~ = 18.1^°^C, and *M*~s~ = -51.6^°^C,
*M*~f~ = -87.4^°^C, respectively. X-ray diffractometry conducted at SSRL
was used to observe the phases present before and after the testing.

# MULTIAXIAL MECHANICAL TESTING

As the shape change during the austenite-to-martensite transformation
can create slippage problems in the grips, reliable tests could not be
performed with straight tubes. Therefore tubes with a gauge section of
reduced diameter were used in order to minimize end effects at the grips
and to lower the stress in the gripped section. Also, a split-grip
design was employed; a hardened steel pin was inserted into the tube
throughout the grip region which allowed application of sufficient
lateral gripping force to inhibit slippage during testing. Strain rates
in the range 10^-5^--10^-4^ s^-1^ were used for both the tensile and
torsional loading in order to avoid strain rate effects. The loading
rates used in the experiments were also shown to be sufficiently low to
suppress significant temperature variations during testing \[4\] such
that all experiments took place under essentially isothermal conditions.
More details on the mechanical test setup can be found in ref. 6.

## Loading Programs

In order to investigate thoroughly the complex nature of the forward and
reverse transformation process and its effect on the constitutive
properties of polycrystalline Nitinol, three specific loading/unloading
programs (referred to as Types I--III) were investigated, as outlined
below:

*Type I:* (Load in tension followed by torsion, unload in reverse order)

1\. Tensile loading to a particular strain level; hold during segments 2
and 3; 2. Torsional loading to 2% shear strain; 3. Unload shear axis to
zero strain; 4. Unload tensile axis to zero strain.

*Type II:* (Load in torsion followed by tension, unload in reverse
order)

1\. Torsional loading to 2% shear strain; hold during segments 2 and 3;
2. Tensile loading to a particular strain level; 3. Unload tensile axis
to zero strain; 4. Unload shear axis to zero strain.

*Type III*: (Load and unload in simultaneous tension--torsion)

1\. Loaded simultaneously in tensile and torsional axes. Rates for each
axis were independently controlled so that the end strain was reached
simultaneously; 2. Unloaded tensile and torsional axes simultaneously to
zero strain.

A series of specific loading paths were considered for Types I--III,
involving tensile strains ranging from 0% to 6.0% and the maximum
attainable shear strain limited to 2% (due to buckling of the
thin-walled structure). As the forward transformation is initiated at
\~1% shear strain, this limit did not substantially restrict the mapping
of the loading in the shear strain dimension. A minimum of two
repetitions were carried out for each loading path considered.

## *IN SITU* X-RAY SYNCHROTRON DIFFRACTION

For the *in situ* investigation of phases and the elastic strain of
Nitinol tubes, the high-resolution x-ray, synchrotron based instrument,
such as the one that was used at SSRL (BL2-1), offers many advantages
over a conventional parafocusing x-ray instrument based on a sealed-tube
as the source of radiation. Such an instrument utilizes the high
brightness of the x-ray beam and a perfect crystal analyzer (float zone
Si 111) to easily achieve angular resolution sufficiently high to detect
changes in lattice parameters smaller than 10 picometers. Not only was
the ultra-high resolution very useful in detecting the R-phase in
Nitinol (Figure 1), but it also enabled us to very accurately measure
the elastic strain at a local level in our *in situ* loading tests. The
high brightness of the synchrotron beams provided two addition
advantages for our measurements. These bright x-ray beams can be focused
to a spatial spot size of 0.5x2 mm (FWHM), which was further reduced by
using slits. This enabled us to probe the phenomenon of inhomogeneous
transformation of the martensite during loading of the tubes in tension.
Furthermore, the high degree of collimation of the synchrotron beam
facilitates using a parallel-beam diffraction geometry. In a
parallel-beam geometry the location of the Bragg peak is not dependent
on the deviation of the diffracting surface of the sample from the
center of the diffractometer, and therefore it is ideal for
investigating a curved sample (a thin-walled tube) in an *in situ*
deformation rig.

Tension and torsion tests were performed *in situ* on the same samples
employed for the multiaxial loading described above. Due to the extreme
complexity of the non-proportional loading tests, where tension and
torsion were combined, only the pure tension and torsion experiments
where reproduced *in situ* using a custom designed rig. The tubes were
placed in the *in situ* rig and gradually loaded up to 6% strain in
tension and 2.5% in torsion (until buckling occurred). X-ray diffraction
patterns were collected at fixed strains along the gauge length of the
tube, and patterns were collected at 1 mm intervals along the tube. This
procedure was chosen in order to study the nucleation and growth of
martensite spiral bands (similar in appearance to Lüders bands), which
is observed during loading of tubes in tension (Figure 2).

![](media/image1.wmf){width="5.992361111111111in"
height="4.627083333333333in"}

*Figure 1: Nitinol diffraction patterns: the characteristic patterns of
austenite, R-phase and martensite are shown. This pattern was collected
on a fully annealed sample to highlight the effect of the R-phase.*

![](media/image2.wmf){width="2.638888888888889in" height="4.0in"}

*Figure 2: Spiral martensite bands appear during tensile loading of an
electropolished tube, similar in appearance to Lüders bands observed in
mild steel*

# RESULTS AND DISCUSSION

# MULTIAXIAL MECHANICAL TESTING

Results of the experiments conducted in this study are presented in
Figs. 3-8 as plots of equivalent true (Cauchy) stress σ~eq~ versus
equivalent referential (Lagrangian) strain ε~eq~, i.e., σ~eq~ =
√(σ~t~^2^ + 3 σ~s~^2^) and ε~eq~ = √(ε~t~^2^ + 4/3ε~s~^2^), where σ~t~
and ε~t~ (respectively, σ~s~ and ε~s~) denote the tensile (respectively,
shear) stress and strain. Within acceptable experimental error margins,
the unloading of all stress--strain curves passed back through the
origin, indicating that the transformation was reversible and displays
the superelastic effect independently of the loading path.

# Uniaxial tension and torsion

Pure tension results are shown in Figure 3 and reveal behavior typical
of shape-memory alloys in the superelastic regime. The repeatability of
the tests appears to be adequate with the onset of the transformation
varying between 1.1% and 1.2% strain and the transformation plateau
stresses lying between 370 and 380 MPa. Under pure torsional loading
(Figure 4), the transition from the elastic loading region to the
transformation plateau region is less sharp. There is a lack of closure
in the loading/unloading curve due to the slippage of the axial
extensometer during torsion; this is deduced by simple comparison with
the corresponding shear (as opposed to equivalent) stress--strain
diagram where the closure is accurately reproduced.

![](media/image4.wmf){width="4.006944444444445in"
height="3.0006944444444446in"}

Figure 3: Equivalent stress--strain plot showing the repeatability of
three separate uniaxial tension tests loaded up to 6% strain.

![](media/image5.wmf){width="3.998611111111111in"
height="3.0006944444444446in"}

Figure 4: Equivalent stress-strain plot showing pure torsion loaded up
to 2% shear

**Tension followed by torsion (Type I)**

Results of Type I loading experiments involving tension up to a maximum
of 6% strain, followed by torsional loading of 2% strain (tension is
held constant) and reverse-order unloading are presented in Figure 6.
Even with this relatively simple loading path, the material response is
quite complex. The stress-induced austenite-to-martensite transformation
is effected by the formation of martensitic structures which correspond
to system energy minimizers known as variants. Although 24 variants of
the less symmetric martensitic phase can be formed by the same crystal
of parent austenite, it has been experimentally observed that only a few
variants are typically active and fully resolve the stress/shape
change.\[1,5\]At low tensile strains (\<1%), the overall behavior is
similar to that of pure torsion; at higher tensile strains, a sharp
increase is observed in the stress upon superposition of torsion, which
is indicative of the generation of a new set of martensitic variants
which are energetically preferable for torsional (rather than the
tensile) deformation. In addition, the hysteresis in the torsional
loading/unloading loop appears to decrease progressively with increasing
applied tensile strain, which is consistent with the decreasing volume
fraction of variants associated with torsion. At all levels of applied
tensile strain, the end of the torsional unloading segment is marked by
an increasing equivalent stress with decreasing applied equivalent
strain. The micromechanical origin of this mechanical response is not
presently understood. However, it does qualitatively indicate that the
material has retained a microstructural ''memory'' of its tensile
transformation state.

![](media/image6.wmf){width="4.006944444444445in"
height="3.0006944444444446in"}

*Figure 5: Equivalent stress--strain plots in tension (0%, 0.7%, 1.10%,
1.5%, 2%, 3%, 6%), followed by torsion (2%) and reverse unloading (Type
I).*

**Torsion followed by tension (Type II)**

In this case, torsion is applied corresponding to 2% strain, followed by
tension up to 5.8% strain. Unloading is subsequently effected in the
reverse order. Type II loading paths differ substantially from Type I
paths, as seen in Figure 6: here the transformation plateau that follows
the torsional loading occurs at monotonically decreasing stress levels.
In addition, upon unloading, the ''elastic'' region is followed by
reverse transformation along a plateau at monotonically increasing
stress levels up to the point that corresponds to the end of the tensile
unloading. Again, the material appears to exhibit 'microstructural'
memory of the transition from proportional to non-proportional loading.

![](media/image7.wmf){width="4.006944444444445in"
height="3.0069444444444446in"}

Figure 6: Equivalent stress--strain plots in torsion (2%), followed by
tension (0%, 0.7%, 1.05%, 3%, 5.8%) and reverse unloading (Type II).

**Simultaneous tension--torsion (Type III)**

The results of simultaneous proportional tension--torsion at up to 6.0%
tensile strains and at 2.0% shear are shown in Figure 7. All loading
paths produce curves that are qualitatively similar to those obtained
under pure tension or torsion. In particular, smooth loading and
unloading transformation plateaus are obtained at approximately constant
stresses. This observation appears to support the conjecture that sharp
changes in the stresses during non-proportional loading are effected by
the generation of new sets of martensitic variants, while existing
variants lead to higher stresses due to their energetic incompatibility
with the current loading state.

![](media/image8.wmf){width="4.006944444444445in"
height="3.0006944444444446in"}

Figure 7: Equivalent stress--strain plots in simultaneous tension (0%,
0.7%, 1.5%, 3%, 6%) and torsion (2%) with simultaneous unloading (Type
III).

![](media/image9.wmf){width="4.006944444444445in"
height="3.0006944444444446in"}

*Figure 8: Comparison of equivalent stress and strain behavior between
loading paths at peak strains of 3.0% in tension and 2.0% in torsion.*

## Comparison of results from various loading path types

Figure 8 compares Types I--III loading paths for 3.0% tensile strains
and shearing strain of 2%. At low tensile strain levels, the loading
paths are quite similar. At moderate tensile strains, Type I and Type II
loading generates widely differing "non-smooth" stress--strain curves,
owing to the sharp transitions in variant formation at the
transformation plateau. At a high tensile strain level, the Types II and
III paths yield remarkably similar curves, which can be explained by the
predominantly proportional nature of both paths during the
transformation process. More detail can be found in ref. 6.

## Analysis of results: qualitative features

The experiments demonstrate that the development of a load-dependent
microstructure could be responsible for the non-smooth stress--strain
curves under non-proportional loading of Types I and II. In addition,
the alloy appears to exhibit memory of its forward transformation
history throughout the loading process, as illustrated by the reverse
transformation paths followed in the non-proportional loading cases.
Another important feature observed in the experiments is that the
forward transformations in tension and torsion follow markedly different
paths. In particular, and contrary to the case of initial yielding in
metals, the transformation stress in torsion is consistently higher than
in tension.

In order to investigate further the different deformation mechanisms in
tension and torsion, electropolished specimens were loaded according to
the Types I and II protocols. During Type I loading, a spiral martensite
band similar in appearance to a Lüders band \[7\] nucleated and grew.
Its formation was observed as soon as the tensile stress reached the
plateau value of \~400 MPa at 1% tensile strain, as shown in Figure 3;
the spiral band remained visible during the torsion cycle, where the
tensile strain is held constant. The martensite band only disappeared at
the end of the tension plateau where the sample was fully martensitic as
confirmed by the *in situ* x-ray experiments. In contrast, no Lüders
bands were observed during Type II loading (even during the tension
cycle applied at a fixed shear stress). The absence of Lüders band in
torsion was also verified by the *in situ* x-ray diffraction
experiments. It is believed that Lüders band formation during the
tensile cycle facilitates the deformation, consistent with the higher
transformation stresses required in torsion where such bands do not
appear to form (at least up to 2% shear). This is also supported by the
simultaneous loading tests (Type III), where the plateau stress
decreases with increasing tensile strain (Figure 7). Another interesting
characteristic of the stress--strain curves is that in tension (Figure
3), there is a significant reduction in the elastic modulus at very low
strains (\~0.3%), which may be attributed to the presence of the
R-phase; interestingly, the torsion experiments show no such reduction
in modulus (Figure 4), as again verified in the *in situ* torsion
experiments.

## Constitutive modeling

The tension--torsion experiments described above were simulated by a
finite element analysis using a three-dimensional superelasticity model
in finite deformations. The model predicts phase transformation
potentially involving all 24 habit plane martensitic variants \[8.9\].
The computational method for solving the constitutive equations
associated with the above model is discussed in a companion paper (Jung
*et al*., SMST 2003 proceedings). The model predicts reasonably well the
stress response in both tension and torsion, due to the inclusion of
different martensitic variants that can nucleate under non-proportional,
non-simultaneous tensile and torsional loading. This feature, which is
clearly exhibited in the experiments and replicated in the numerical
simulation, has been difficult to capture using previous
phenomenological constitutive models \[9\].

## MICROSCOPIC *IN SITU* X-RAY SYNCHROTRON DIFFRACTION

Through *in situ* multiaxial loading of Nitinol in tension and torsion,
quantitative estimates of the nature and volume fraction of phases was
assessed as a function of stress/strain state and of position along the
tube. Also, the perpendicular elastic strain field (Poisson strain) was
measured from the shifts of the characteristic peaks upon loading as a
function of position along the tube, stress and strain. The
perpendicular strain varies with applied strain and with phase
(austenite/martensite). The volume fraction of martensite present is
calculated from the area under the peaks.

### Uniaxial tension and torsion

In uniaxial tensile loading, the stress-induced martensitic
transformation is inhomogeneous and occurs by initiation and growth of a
spiral martensite band (Lüders bands) with a sharp austenite-martensite
interface (Figures 9,10). The Lüders bands observed during the tensile
loading cycle were fully characterized *in situ*. Their presence was
confirmed by x-ray data; outside the Lüders bands the material is fully
austenitic whereas within the bands it is fully martensitic (Fig 9,10).
The bands are actually a spiral that wraps around the tube.

![](media/image10.wmf){width="5.167361111111111in"
height="3.998611111111111in"}

*Fig 9: Poisson elastic strain vs. position along the tube at various
equivalent tensile strains*

![](media/image11.wmf){width="5.167361111111111in"
height="3.998611111111111in"}

*Fig 10: Volume fraction of martensite vs. position along the tube at
various equivalent tensile strains*

![](media/image12.wmf){width="5.03125in" height="4.010416666666667in"}

*Fig 11: Poisson strain vs. position along the tube in torsion*

![](media/image13.wmf){width="5.179166666666666in"
height="3.9895833333333335in"}

*Fig 12: Volume fraction of martensite vs. position along tube in
torsion*

![](media/image14.wmf){width="5.141666666666667in"
height="4.002083333333333in"}

*Fig 13: Volume fraction of martensite vs. equivalent strain in tension
and torsion*

![](media/image15.wmf){width="5.179166666666666in"
height="3.9930555555555554in"}

*Fig 14: Volume fraction of martensite vs. Poisson strain (perpendicular
elastic strain).*

The tubes behaved very differently under torsion. Both *in situ* x-ray
diffraction and surface morphology analysis indicated that the
deformation is homogeneous across the whole tube (Figures 11-12). In
torsion, the volume fraction of martensite and the Poisson strain (as a
function of equivalent strain) varied uniformly (Figures13-14), with no
Lüders band formation. The experiments revealed a surprising path
dependance of Nitinol loading that is bound to affect the in vivo
behavior where multiaxial stresses are present.

# SUMMARY AND CONCLUDING REMARKS

Tension--torsion experiments on thin-walled superelastic Nitinol tubes
have demonstrated a rich spectrum of mechanical behavior involving
significantly different characteristics in each of the two loading axes.
Indeed, the behavior of Nitinol under multiaxial loading is extremely
complex yet displays remarkable microstructural memory. This behavior
provides important insights into the stress-induced phase transformation
under multiaxial loading commonly encountered in a wide array of
engineering applications, in particular in the biomedical industry. The
experiments suggest that the characterization and modeling of the
microstructure is of paramount importance in understanding the
phenomenology of shape-memory alloys. This characterization and modeling
must include both mechanics- and microstructural-based understanding.

Under tensile loading, the phase transformation is highly heterogeneous
and involves Lüders band formation; corresponding behavior in torsion is
essentially homogeneous without the formation of Lüders bands. In
addition, the transformation stress is 50-70 MPa higher in torsion
compared to tensile loading. We surmise that load-dependent
microstructures (i.e., variants of martensite dictated by the loading
conditions) may be responsible for the distinctly differing
stress-strain behavior in tension and torsion.

# ACKNOWLEDGEMENTS

This work was supported by the National Science Foundation Grant no.
CMS-9800006 with the University of California at Berkeley. Special
thanks are due to NDC for additional financial support and for supplying
the Nitinol tubing, to Dr. J. H. McNaney for his initial work on the
tension/torsion testing, and to Prof. Panos Papadopoulos and Youngjean
Jung for many helpful discussions. *In situ* x-ray diffraction
measurements were carried out at the Stanford Synchrotron Radiation
Laboratory (SSRL), a national user facility operated by Stanford
University on behalf of the U.S. Department of Energy

#  

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