An Investigation of the Effect of Anisotropy on the Thermomechanical Behavior of Textured Nickel/Titanium Shape Memory
Alloys
Anthony Wheeler
Advisor: Dr. Atef Saleeb
Honors research Project
Abstract
The commercially utilized form of SMA is typically obtained by casting, followed by
hot-working using rolling or drawing, and followed by heat treatment. This process induces
highly texture microstructure in the polycrystals that result into texture-induced anisotropy in the
material. The work presented here aims at the enhancement of our understanding of the effects of
textured-induced anisotropy on several important SMA response characteristics under different
thermomechanical loading conditions. To this end, a multimechanism based, viscoelastoplastic,
generalized, three dimensional SMA model is utilized in conjunction with the commercial finite
element package, ABAQUS. In this study, it is shown that the textured SMA response (such as
recoverable strains under uniaxial loading, traces of transformation surfaces under combined
state of loading, evolution of transformation strain rate vector at different stages of
transformation) is remarkably different from that of untextured material. Furthermore, the effect
of texture alignment direction and/or the extent of transformation on the relative orientation of
transformation strain rate vector with respect to the transformation surface (defined in terms of
global stress components) are investigated. It is demonstrated that the apparent deviation-fromnormality is linked with intrinsic character of SMA transformation, and cannot be described by
means developed for conventional elastic or plastic materials.
Table of Contents
Introduction……………………………………………………….………………………………4
Texture Effect on SMAs…………………………………………………………………………..5
Numerical Models…………………………………………………………………………………7
Numerical Simulation………………..……………………………………………………………8
Tension-Torsion Strain Controlled Test…………………………………………………………..9
Biaxial Stress Controlled………………………………………………………………………...12
Triaxial Strain Controlled………………………………………………………………………..15
Conclusion……………………………………………………………………………………….20
References………………………………………………………………………………………..21
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List of Figures
Figure 1. Stress-Temperature Diagram……………………………………………………………5
Figure 2. Transformation Curve for SMAs………………………………………………………..6
Figure 3. Experimental Data for Ti-Ni-Cu………………………………………………………..7
Figure 4. Unit Block with Single Mesh used for Modeling……………………………………….8
Figure 5. Loading Path for Case I…………………………………………………………………9
Figure 6. Selected Results from Case I (Low Temperature)…………………………………….11
Figure 7. Selected Results from Case I (High Temperature)…………………………………….12
Figure 8. Loading paths for Case II...……………………………………………………………13
Figure 9. Selected Results from Case II (Low Temperature)…………………...……………….14
Figure 10. Selected Results from Case II (High Temperature)……..………......……………….15
Figure 11. Loading path for Case III…………………………………………………………….16
Figure 12. Selected Results from Case III (Low Temperature)…………………....…………….18
Figure 13. Selected Results from Case III (High Temperature)……………...…....…………….19
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1. Introduction:
Shape Memory Alloys (SMAs) are metallic alloys that are capable of recovering large
inelastic strains as a result of transformation between two phases when heated above a certain
temperature (Patoor et. al., 2006). Most conventional metal recover less than 1% strain before
plastic deformation, whereas SMAs undergo a diffusionless, thermo-elastic, solid-state
martensitic phase transformation that allows complete recovery of strains as large as 8 -12%
(Grabe and Bruhns, 2009). The key characteristic of all SMAs is the occurrence of martensitic
phase transformation between the high temperature, high symmetry, austenitic (A) phase, and
different variants of the low temperature, low symmetry martensitic (M) phase.
The theoretical phase diagram shown in Figure 1 summarizes the characteristics of SMAs
response under uniaxial thermomechanical loading. Depending on the thermomechanical state,
SMAs exist in an A (at high temperature) or M phase (at low temperature), or both phases can
coexist. The M can further be separated in twinned (self-accommodated/random oriented)
martensite and detwinned martensite. Ideally, there are three regions where the material can be in
a pure phase along with the transformation lines (surfaces in three dimensions) in stresstemperature subspace that separate them. When the twinned M is subjected to mechanical
loading at low temperature, it transforms into detwinned M accompanied by large inelastic
strain/deformations. After the removal of the load, a very small amount of strain is recovered,
giving an appearance of plastic/permanent deformation. This character of SMAs is known as
pseudoplasticity. Upon heating above certain temperature (austenite start temperature, As), the
detwinned M starts transforming into the stable A, and completes the transformation with
complete recovery of the inelastic strain when austenite finish temperature (Af) is reached.
Furthermore, during cooling, the material transforms back from A to M between martensite start
temperature (Ms) and martensite finish temperature (Mf). This is known as the characteristic
Shape Memory Effect (SME) (Grabe and Bruhns, 2009). These characteristic temperatures shift
with the current stress state, as shown in Figure 1. Note that the phenomenon of pseudoplasticity
is observed at temperatures below Mf.
An SMA at a higher temperature behaves differently than the same material at a lower
temperature. At high temperature (higher than Af), the material in A phase transforms directly
into detwinned M upon loading, and a large amount of inelastic strain/deformation is observed.
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However, unlike the low temperature response, inelastic strains are completely recovered upon
unloading at the same temperature, giving the characteristic pseudoelastic response.
The existence of two phases, and transformation between them results in high actuation
energy densities) making them attractive for many engineering applications.
Figure 1. Stress–temperature diagram showing the relationship of stress and temperature
and the austenitic and martensitic domains (Grabe and Bruhns, 2009).
2. Texture Effect on SMAs
As metals are forged or subjected to cyclic loading conditions, their crystal lattice grains
get aligned in the certain direction. This alignment of grains is known as texture effect. The
textured material response varies with relative orientation between the direction of load and the
alignment of grains. In Figure 2, the pseudoelastic and pseudoplastic response of textured SMA
(with different texture alignment directions) is shown. Here, an SMA material (Mf = 65 ºC, and
Af = 120 ºC) was modeled with texture angles 0, 15, 25, 35, 45, and 90 degrees with respect to
the loading direction. The strongest uniaxial response was observed for the texture alignment
direction of 0 degrees (parallel to the loading direction). When grain alignment direction was 90
degrees with respect to the loading direction, the weakest material response is observed. The
texture effect was observed at both temperatures. Furthermore, the total amount of recoverable
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strain in pseudoelastic/pseudoplastic response is also dependent upon the texture alignment
direction. Figure 3 shows model prediction for the recoverable strain at high temperature
(pseudoelastic response) as a function of loading direction. Note that the loading directions 0 and
90 correspond to the texture alignment direction of 90 degrees and 0 degrees, respectively. The
recoverable strain was calculated from stress controlled, tension load-unload tests for a fixed
amount of normal stress (600 MPa). In this case recoverable strain is defined as the difference of
normal strain reading 600 MPa, and 0 MPa. The observed response is similar to the experimental
observations presented in the literature (Bhattacharya, 2004); i.e., with increasing load angle
direction, the recoverable strain is decreasing. Note that the model’s material parameters do not
represent either of the SMA systems (Ti-Ni-Cu and Ti-Ni).
Figure 2. Transformation curve for SMAs showing the effect of texture. (a) High
temperature, and (b) Low temperature.
As mentioned earlier, texture effects are unavoidable in any SMA material. Therefore, in
order describe SMA response accurately, a numerical constitutive SMA model must account for
texture effects.
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Figure 3. Experimental data for Ti-Ni-Cu and Ti-Ni taken from (Bhattacharya, 2004)
3. Numerical Models
Most of the existing constitutive models of SMAs are formulated to fit a particular set of
uniaxial test data for a specific SMA system, and therefore do not fully capture the true
material’s response. With merely uniaxial test data, it is impossible to know the relative
orientation of texture alignment with respect to loading direction. For a complete understanding
of textured SMA response, it is not possible to use just one dimensional model to reveal the
effect of texture on these unique material. The true nature of SMA system can only be obtained
by using three dimensional models and experiments, since one dimensional experiments/models
do not reveal the existance of texture.
The three dimensional, multimechanism based, viscoelastoplastic SMA constitutive
model, developed by A. F. Saleeb and his research team, is fully capabale of capturing the
behavior of SMAs (Saleeb et al., 2011). Within their model, it is possible to account for the
texture alignment with respect to the global coordinate system. Thus, it will be utilized in this
research work to study the effect of texture on the compex SMA response. Currently, the effect
of texture on the response of these unique materials have not been experimentally studied in
details. Therefore, with the help of the generalized SMA model (Saleeb et al., 2011), a better
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understanding of the effect of texture on SMAs can be achieved, and thus can provide useful
insights in designing more exploratory experimental programs.
The mathematical model is utilized as the user defined material (UMAT) in the
commercial, large-scale, finite element (FE) package ABAQUS®. Abaqus allows the user to
input material parameters, loads, and boundary conditions to any model for any material. For this
research a unit block with a single element mesh (Figure 4), a UMAT for the SMA constitutive
behavior, and different sets of loads and boundary conditions were used. In the next section, the
test cases and results for these cases will be discussed in details.
Figure 4. Unit block with a single mesh used for modeling in Abaqus.
4. Numerical Simulation
In the present work, three different loading tests were conducted on the unit block. The
following test cases were considered:
I.
Non-proportional tesion-torsion strain controlled (2-D) test.
II.
Proportioanl stress controlled, biaxial (2-D) tests.
III.
Proportioanl, triaxial strain controlled (3-D) test.
Each case was ran at two temperatures (low temperature, T = 30 °C,
and high
temperature, T=150 °C, and six texture angles (TA = 0, 15, 25, 35, 45, and 90 degrees). The
same material parameters, as listed in Table 2 in (Saleeb et al., 2011), were utilized. The
additional parameters, pertinent to the texture effect are listed in Table 1 below. The three
loading cases will be discussed throughly below.
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Table 1. Additional parameters for texture effects, used with the material model created by
Saleeb et al. (2011)
I. Tension-torsion strain controlled test
In this case, the utilized loading control path in normal – shear strain subspace is shown
in Figure 5. Firstly, the block is loaded in strain control, uniaxially, following the path AB. Then
the block is strained with periodic amplitudes along the path BCDEB (to achieve circular path),
and finally it is unloaded to initial position along path BA. The applied strain magnitudes (radius
of the circle, ρ, shown in Figure 5) were varied from 0.1% to 0.5% by increments of 0.1%, 1% to
8% by increments of 1%, and 10% to 12% by increments of 0.5%. This was done in order to
capture the material’s response at different stages of transformations; i.e., how the material
behaved as it transformed from twinned to detwinned M at a low temperature, and from A to
detwinned M at a high temperature.
Figure 5. Loading path for case I
Shown below are six of the output plots from the strain controlled circular path test. As it
can be seen in Figures 6 (T = 30 °C) and 7 (T = 150 °C), with low applied strains (in the nearly
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elastic region) of Figure 2 (Left of the first blue line), the effect of texture is non-existent. When
applying strains in the range of the transformation region of Figure 2 (in between the two blue
lines) the output values of stress vary greatly from each other especially at a low temperature.
When the applied strain is in region of detwinned martensite reorientation of Figure 2 (to the
right of the second blue line), the values of the stresses do not vary greatly from each other for
each texture angle. Within the transformation region, the block with a texture angle of 45 degrees
has the most residual stress at the end of complete unloading for each of the texture angles. Note
that there is a characteristic difference between the response at the low temperature and the
response at the high temperature. Compared to the counterpart (pseudoplastic) response at low
temperature, the material develops much higher stresses at the same strain at high temperature,
and has a very low amount of residual strain after the complete removal of load at high
temperature due to pseudoelasticity.
With the change in texture orientation, and the amount of applied strain, different degrees
of distortion in the obtained response are observed. For example, if the SMA behaved linear
elastically, all the plots in Figure 6 and 7 would be circular. The applied strain magnitude, ρ, of
0.5% corresponds to the nearly linear elastic region in the uniaxial tension test, therefore the
obtained response paths are circular and with not much variation in the stresses. During
transformation, the shape of the response in normal-shear stress subspace changes to distorted
ellipse with different orientation of major axis, depending upon the texture direction.
Another distinct feature of the obtained response is the apparent deviation from normality
of the internal transformation strain rate vectors (shown as arrows in Figures 6 and 7). The
theory of plasticity states that the internal transformation strain rate vectors are perpendicular
with respect to the global transformation stress surface. At low strains the arrows are
perpendicular to the stress response curve, but as the strain increases the arrows become more
and more tangential, as noticeable in the 12% strain plot. This indicates that, during evolution,
the transformation strain rate vectors continuously change the direction, moving from the
extreme condition of nearly normal to the other extreme condition of nearly tangential relative to
the trace of the transformation surface in the concerned stress subspace. Note that, the apparent
deviation from the normality condition has been interpreted by some researchers as requiring non
– associative flow rules. A detailed discussion on the apparent deviation from normality can be
found in (Lim & Mcdowell, 1999; Saleeb et al., 2011).
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Figure 6. Selected results from case I (low temperature).
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Figure 7. Selected results from case I (high temperature).
II. Biaxial Stress Controlled
This biaxial test involves application of stresses along radial direction in σ11 and σ22
subspace. The radial stress is the resultant of two normal stresses: one along the x-axis, σ11 , and
the other along the y-axis, σ 22. The applied stress magnitude, ρσ , was selected to be 1200 MPa
(sufficient enough to initiate material reorientation after complete phase transformation), and the
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proportionality angle φ was varied from 0º to 360º in increments of 15º. Figure 8 shows the
proportional biaxial paths in σ11 and σ22 subspace.
Figure 8. Loading paths for case II.
Here, from the obtained response, the stress values (σ 11 and σ22 ) corresponding to a fixed
amount of transformation strain was extracted. These were normalized with respect to the normal
tensile stress (σ 11 for φ = 0°). The resultant plots, along with the transformation strain rate
vectors, are shown in Figures 9 and 10.
The texture angle of 45 degrees is in between 0 and 90 degrees therefore it is not biased
to have its response surface to be oriented one way or another. It can easily be observed that the
response surface of the 45 degree texture angle is nearly symmetric about an imaginary line
drawn at 45 degrees from the origin. If this line were to be drawn on any of the other plots it can
be seen that they are unsymmetrical, and favor a particular direction. For TA = 0 and 90 degrees
the symmetry is completely broken, due to complete bias towards a particular direction.
Furthermore, the shape of the transformation surface continuously changes during
transformation. This further indicates that the definition of transformation surfaces and
associated phenomena based on the classical plasticity theories cannot directly be applied to
these complex systems.
From the biaxial loading case, the apparent deviation from normality is also evident.
Similar to Case I, the arrows shown in the plots should be perpendicular to the curve. Again, at
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the low transformation strain magnitude, the arrows are perpendicular, and at higher strain they
are no longer perpendicular. This clearly indicates that the apparent deviation from normality
will always remain in the global response space. However, in the internal stress space, the
condition of normality will always be satisfied for the present model.
Figure 9. Selected results from case II (low temperature).
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Figure 10. Selected results from case II (high temperature).
III. Triaxial Strain Controlled
This test involves application of proportional strains along the three axes. The magnitude
of the applied strain was set to a constant value of 12%. The applied strains are different along
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each axis, and are defined by Equations 1-3 (by varying B from 0 to 1.0 by increments of 0.125).
The test procedure and its important implications are explained in details in (Saleeb et al., 2011).
Figure 11. Loading path for case III.
Since phase transformation is essentially governed by shear, it would be more appropriate
to plot the trace of the transformation surface in a deviatoric plane that passes through the origin
(this plane is called π – plane), where every stress point represents a state of pure shear with no
hydrostatic stress (or volumetric strain) component. Let σ´i be the projection of the σi axis (i = 1,
2, 3 for three principal stress/strain axes as in Fig. 13) on the deviatoric plane. Then, the
projection of the deviatoric stress (strain) vector on the σ1 and σ2 axes can be written as,
1
2
where, ρ is the length of the deviatoric stress vector and is defined as
, and si (for i =
1, 2, 3) are the principal deviatoric stresses. The θ is the Lode’s angle. Readers are referred to the
section 2.11 in Chen & Saleeb (1994) for further details. Note that the values of B equaling 0,
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0.5, and 1 correspond to tension path (θ = 0º), shear path (θ = 30º), and compression path (θ =
60º), respectively.
The obtained traces of transformation surfaces are shown in Figures 12 and 13 for
different texture alignment directions. For every texture alignment, the magnitude of the
transformation stress vector monotonically increases from a minimum in for B = 0 (tension path
shown in red) to a maximum for B = 1 (compression path shown in green) with an intermediate
value being obtained in for B = 0.5 (shear path shown in blue). Also, at a particular temperature,
the fanning (or angular separation between the tension and compression path) increased with the
increase in the texture alignment direction. Furthermore, at 90 degrees of texture angle, the
magnitude of the response is least, while the maximum magnitude is observed for the strongest
alignment direction (i.e. TA = 0 degrees). Note that the trend of obtained trace is independent of
temperature; i.e. except the magnitude, the pattern of the trace of transformation surface
remained similar at both 30 ºC and 160 ºC.
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Figure 12. Selected results from case III (low temperature).
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Figure 13. Selected results from case III (high temperature).
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6. Conclusion
SMAs are unique materials that are currently being studied extensively for advanced
engineering applications. Prior to their application as smart engineering material, all aspects of
SMA response must be understood. This is particularly true with regard to the common
commercially utilized forms of these material systems (typically obtained by casting, followed
by hot-working using rolling or drawing, and followed by heat treatment) as polycrystal with
many grains, where strong anisotropy results from the highly textured microstructure of the
polycrystals. The work presented here aims at furthering our understanding of the effects of
textured-induced anisotropy of SMA materials on several important characteristics of their
response under different thermomechanical loading conditions; i.e. (i) the amount of recoverable
strains under uniaxial loading; (ii) the traces of the stress and strain transformation surfaces
under combined tension-torsion as well as plane-stress biaxial loading conditions; and (iii) the
significant qualitative and quantitative differences in the evolution of transformation-strain rate
vectors as straining continues from the early stage of transformation onset, to the well-developed
stages and completion of phase-transformation and/or martensite variants detwinning and
reorientation.
To this end, we have presented, in conjunction with item (i) and (ii) above detailed results
showing how the SMA material responds differently under the same loading when texture angles
change. Furthermore, the test cases investigated here in connection with item (iii) have clearly
shown that, when endowed with texture, the SMA polycrystals exhibit behavior that cannot be
analyzed as elastic or plastic materials, as commonly done for traditional materials such as
“ordinary” metals. In contrast to conventional elasticity, the transformation-induced total strain
vector is not normal to the surface of constant values of Gibb’s complementary energy function
(when plotted in stress space). Also, and in contradiction to the conventional plastic materials,
the transformation-induced strain rate vector is not normal to the “yield” surfaces (in stress
space). This feature of significant deviation-from-normality appears to be intimately connected to
the well known superthermal and supermechanical characteristics of textured SMA materials,
and must therefore be accounted for in any theoretical modeling effort.
7. Acknowledgements
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The author would like to acknowledge the guidance provided by Dr. Atef Saleeb, Dr.
Abhimanyu Kumar and Mr. Binod Dhakal. As well as the Honors College at the University of
Akron.
References
Bhattacharya, K., 2003. Microstructure of martensite : why it forms and how it gives rise to the
shapememory effect. Oxford University Press, Oxford.
Grabe, C., Bruhns, O.T., 2009. Path dependence and multiaxial behavior of a polycrystalline
NiTi alloy within the pseudoelastic and pseudoplastic temperature regimes. Int. J. Plast 25, 513545.
Lim, T.J., McDowell, D.L., 1999. Mechanical Behavior of an Ni-Ti Shape Memory Alloy Under
Axial-Torsional Proportional and Nonproportional Loading. J. Eng. Mater. Technol. 121, 9-18.
Patoor, E., Lagoudas, D.C., Entchev, P.B., Brinson, L.C., Gao, X., 2006. Shape memory alloys,
Part I: General properties and modeling of single crystals. Mech. Mater. 38, 391-429.
Saleeb, A.F., Padula Ii, S.A., Kumar, A., 2011. A multi-axial, multimechanism based
constitutive model for the comprehensive representation of the evolutionary response of SMAs
under general thermo-mechanical loading conditions. Int. J. Plast 27, 655-687.
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