Constitutive Modeling of Shape Memory Alloy Using Artificial
Neural Network
Konštutívne modelovanie zliatiny s tvarovou pamäťou s využitím
umelej neurónovej siete
K. P. Arulshri, K. P. Padmanaban, V. Selladurai
Abstract
This paper makes an attempt to develop constitutive
model for Shape Memory Alloy (SMA) material behavior
using Artificial Neural Network. The thermo mechanical
behavior of SMA depends on many variables such as Prestrain, applied stress and temperature, these variables are
also inter-dependent and characteristics of the SMA wire
are studied by controlling one variable while allowing the
other two variable to vary, determine the material coefficients. One dimensional behaviour of SMA is modeled
using Brinson model which is based on energy balance
equation with quantifiable engineering quantities and material coefficients. In this proposed model, Artificial Neural
Network is used to learn complex non linear relationship in
the modeling of constitutive behaviour of Shape Memory
Alloy. This proposed model learn constitutive behaviour
and store the knowledge in their connection weights.
Conventional model is expressed only in terms of observable variable, but the proposed model has not only used
observable variable but also the variable representing the
internal behaviour. Due to the introduction of additional
variable this type of model has larger degree of freedom in
description and thus being superior to the conventional
model. The Predicted results using Artificial Neural Network (ANN) is found to correlate well with experimental
results when compared with the correlation between conventional models with experiments.
Keywords: Constitutive modeling, SMA, ANN, Shape
recovery stress
Abstrakt
Príspevok sa snaží rozvinúť konštutívny model správania sa materiálu pre zliatiny s tvarovou pamäťou (SMA)
s využitím umelej neurónovej siete. Termo-mechanické vlastnosti SMA závisia na mnohých premenných ako napríklad
predpätie, použitý tlak a teplota, tieto premenné sú taktiež
závislé medzi sebou a charakteristika SMA je zoširoka študovaná ovládaním jednej z premenných, pokiaľ zvyšné dve
sú premenlivé, určujúc materiálové koeficienty. Jednorozmerné chovanie sa SMA je namodelované s využitím Brinsonovho modelu, ktorý je postavený na rovnici energetickej
rovnováhy s kvantifikovateľnými inžinierskymi množstvami a materiálovými koeficientmi. V navrhovanom modely
je použitá umelá neurónová sieť pre pozorovanie nelineárneho vzťahu v modelovaní konštutívneho správania sa zliatiny s tvarovou pamäťou. Navrhnutý model pozoruje konštutívne správanie sa a uchováva tieto znalosti vo svojich
vzťažných hodnotách. Konvenčný model je vyjadrený iba
v zmysle sledovateľnej premennej, ale navrhnutý model nepoužíva iba sledovateľné premenné ale taktiež prezentujúce
vnútorné vlastnosti. Kvôli zavedeniu ďalších premenných
tento typ modelu má väčší počet stupňov voľnosti a preto je
nadradený konvenčnému modelu. Očakávaný výsledok vyu-
žívajúci umelú neurónovú sieť (ANN) pri porovnávaní korelácie medzi konvenčným modelom a experimentmi sa
zisťuje ako dobre korelujúci s experimentálnymi výsledkami
Kľúčové slová: konštutívne modelovanie, SMA, ANN,
napätie pre obnovenie tvaru
1 Introduction
Shape memory alloys (SMAs) are a class of materials
processing very attractive mechanical properties. Under certain conditions they exhibit two unique effects, which are
known in the literature as the shape memory effect and super elasticity. An initially deformed SMA can recover its
predetermined low temperature shape during heating, demonstrating the shape memory effect, which can also be
used to generate large internal recovery stresses. At higher
temperatures super elastic behavior is observed and this effect is associated with large son-linear recovery strain during loading and unloading of the SMA. Azadi et al [1] developed a multi-dimensional continuum-level constitutive
model of shape memory alloys exhibiting pseudo-elastic behavior. In their paper, the constitutive relation was constructed based on the gradient of a transformation potential
function (effective stress), and the kinetics of transformation
are expressed based on a set of transformation surfaces in
stress-temperature space. Yuping Zhu et al [2] developed a
constitutive model for magnetic shape memory alloys
(MSMAs) through a combined consideration of micromechanical and thermodynamic theories. The kinetic equation
was established in terms of the thermodynamic driving force derived from the reduction of Gibbs free energy of
MSMA. The first discoveries in this field were reported in
the literature a few decades ago. Chang and Read found the
shape memory effect in an Au-Cd alloy. This direction of
research has been successfully continued and many new
alloys and materials, particularly polymers and ceramics,
are still intensively investigated due to their shape memory
properties. It can be appreciated from the summary above
that many researchers have used various models of the
SMA behaviour in order to investigate the static or dynamic
performances of different composite material structures
with SMA components. For this reason it is essential to
address and emphasize the major differences and the
similarities between the models available in the literature, in
order to ensure the correctness of any numerical simulation
based on such models. The main objective of the work
reported in this paper is to provide results from a comparative study on the SMA behaviour. This study comprises
results obtained from the use of three of the most popular
one-dimensional models, as developed by Brinson [3] model with by neural network. Major the accuracy of the models is carefully examined against results obtained by the
61
authors from recent experimental measurements of the performance of a Ni-Ti SMA wire. Bodner et al [4] derived and
applied the Constitutive equations for elastic-viscoplastic
strain hardening materials.
2 Thermomechanical behavior of SMA
The martensitic transformation is the basis characteristics of an SMA involved in all the unique characteristics of
the SMAs, The martensitic transformation may be simply
illustrated by the change in the martensitic volume fraction
with respect to temperature and stress shown in figure.1.
These thermomechanical behaviors have been described by
a variety of constitutive models, and the differences between the actual behavior and the model have been presented previously. Therefore, this paper suggests the neural
network model based on the experimental results. In this
work, the suggested neural network model predicts the
shape recovery force, which is the important factor to be
well predicted in terms of control problem of SMA wires.
ady, and it is fixed between two ends of the fixtures after
measuring the length of the wire. Then the oven is set to a
constant temperature. The load is gradually increased from
zero state stress. Change in length of the wire can be taken
from the LVDT; strain value can be calculated from that.
After reaching a certain stress value (within elastic limit),
unloading is done. Temperature vs stress plot at two differrent constant temperatures for three different samples. The
modules increase with increase in temperature, due to phase
transformation. The variation of stress with respect to temperature at lower strain is higher than at higher strain values. Linear segment of the stress-strain curve was used to
determine the Young’s modulus of the wire. The procedure
is repeated for different temperatures and for different samples. When the material has been transformed to the martensitic phase, it can be subjected to a significant plastic
deformation. Such transformation requires a minimum load
or stress value to deform it from its original shape when it is
in Martensite phase, which varies as there is, a variation in
temperature. The values of σ Scr , σ crf could be found by plotting transformation stress start and finish at constant temperature i.e. critical stress values are plotted with respect to the
temperature which are obtained from the stress –strain plot
at different constant temperatures.
Fig. 1 Critical stresses for transformation of Martensite
twin conversion as functions of temperature
Obr. 1 Kritické napätia pre transformáciu martenzitického
zrastenca ako funkcie teploty
3 Experimental details
The tensometer setup, shown in figure 2, has two fixed
end plated where two cylindrical bars that are fixed to the
end plates. Two sliding bars with fixtures to hold the SMA
wire are mounted on the cylindrical bars, where one end is
fixed and other end is allowed to slide over the cylindrical
rod. SMA wire is fixed between fixtures, between the fixed
end and movable end. The movable rod is connected to a
lead screw, which has mechanical rotary actuator controls
loading rate. Lead screw arrangement is used to transfer the
force from rotary actuator from movable end. As a tensile
force acts on the wire when the movable end is pulled as the
actuator is subjected to a clockwise rotation.
Electrical resistive heating controls temperature of the
SMA wire and load is measured using a load cell. The
displacement of the wire is measured by LVDT. The core of
the LVDT is mounted on the movable end, which indicates
the displacement of the wire, and the body of the LVDT is
fixed to the base of the setup. To avoid the influence of the
surroundings, the wire should be kept in an enclosed chamber. Current supplied is measured and it is calibrated to the
temperature values.
Thermo-mechanical characteristics of SMAs are studied
by keeping temperature as constant, and strain value is measured for the corresponding change in stress during loading
and unloading. The temperature kept as constant during
loading and unloading. Wire, which is to be tested, is heated
above austenite finish temperature and cooled to obtain
100% Martensite and to remove the stresses developed alre-
62
Fig. 2 Tensometer Setup
Obr. 2 Nastavenie tenzometra
There is a linear relationship between phase transition
temperature, and stress. The constants CM and C A the material properties, which describe the relationship of temperature and critical stress to induce transformation and it can be
obtained by the rate of change of critical stress value with
respect to change in temperature. The critical stress value
below M S is taken to be constant.
The setup has an oven where the SMA wire that is to be
tested is fixed between two fixtures. The temperature
control unit sets temperatures of the oven. The wire is fixed
between fixtures, where the top end is fixed and the movable bottom end is attached to actuator loading system.
Change in length of the wire can be obtained from the
LVDT (Linear Variable Differential Transformer) Core of
the LVDT is mounted on the wire, and the body is placed
on the top of the oven. A load-cell and a thermocouple are
used to measure the applied load, and to measure the temperature of the wire. Gradual temperature increase can be
measured by the thermocouple, which is wound on the wire.
Tightening of the wire increases the load, and correspondding change in length can be measured using LVDT as a
sensor. The material properties of Ni-Ti alloy is given in
Table 1.
This setup is used to determine stress-strain curve at different temperatures. The SMA wire can also be loaded by
adding the load on the base plate, at equal distance from the
center of the wire. Oven temperature controls the temperature of the wire. The wire is kept in an enclosed chamber so
as to avoid the environmental disturbances on the wire and
heat transfer by, convection and radiation. The temperature
of the chamber can be kept as constant while doing the experiment.
The relations for the Young’s modulus and the phase
transformation coefficient are the same as equations (2) and
(3), respectively. In this model, a modified cosine model for
the Martensite volume fraction is used and is divided into
two parts.
where ζS is the portion of the Martensite that is transformed due to applied stress and ζT is the portion of Martensite that is temperature induced. The equations also vary
during transformation. For the (MÅA) transformation:
For T>Ms and
(3)
σ scr + CM (T − M s ) < σ < σ crf + CM (T − M s )
)
ξT 0
(ξ S − ξ S 0 )
1 − ξS 0
For T<Ms and σ Scr < σ < σ crf
ξT = ξT 0 −
ξS =
⎫⎪ 1 + ξ S 0
π
1− ξS0
⎪⎧
cos ⎨ cr
(σ − σ crf )⎬ +
cr
2
2
⎪⎭
⎪⎩σ s − σ f
ξT = ξT 0 −
ξT 0
(ξ S − ξ S 0 ) + ΔTξ
1− ξS 0
For Mf<T<Ms and T<To
1 − ξT 0
[cos(aM (T − M f )) + 1]
Δ Tξ =
2
end of transformation. These values are approximated from
a stress-strain curve where the initial phase was 100%
Martensite. On the curve, it is clear where transformation
begins and ends so that these values are easily determined.
For MÆA conversion the Martensite volume fraction is
determined from the following relations:
(9)
For T>As and
C A (T − A f ) < σ < C A (T − AS )
C M (T − M s ) ≤ σ eq ≤ C M (T − M f )
(10)
(11)
(12)
4 Evaluation of SMA characteristics
4.1 Isothermal force–displacement relation
(2)
(
start of transformation and σ crf is the critical stress at the
Where all terms with a subscript ‘o’ denote initial condition and are equivalent to those defined in the Liang and
Rogers model.
Brinson [5] developed the third model. Brinson uses the
same constitutive equation with some modifications. The
modified relation is as follows:
(1)
σ − σ 0 = E (ξ )ε − E (ξ 0 )ε 0 + Ω(ξ )ξ 0 − Ω(ξ 0 )ξ S 0 + θ (T − T0 )
⎧⎪ π
⎫⎪ 1 + ξ S 0
1− ξS 0
x cos⎨ cr
σ − σ crf − CM (T − Ms) ⎬ +
cr
2
2
⎪⎩σ s − σ f
⎪⎭
In the above equations, σ Scr is the critical stress for the
ξS = ξS 0 −
3.1 Brinson Model
ξs =
Δ Tξ = 0
ξS0
(ξ 0 − ξ )
ξ0
ξ
ξT = ξT 0 − T 0 (ξ 0 − ξ )
ξS0
Table 1 Material properties of the Ni-Ti alloy
Tab 1. Materiálové vlastnosti zliatiny Ni-Ti
Property
Value
Martensite finish temperature Mf (˚C)
25
Martensite start temperature MS (˚C)
34
Austenite start temperature AS (˚C)
55
Austenite finish temperature AS (˚C)
80
Stress influence coefficient CM MPa ˚C-1 5
Stress influence coefficient CA MPa ˚C-1 11.5
Maximum residual strain εL
0.067
Young’s modulus EM (GPa)
15
Young’s modulus EA (GPa)
30
Coefficient of thermal expansion αM (˚C-1) 6.6 x 10-6
Coefficient of thermal expansion αA (˚C-1) 1.1 x 10-5
100
Critical stress σS (MPa)
Critical stress σS (MPa)
170
ξ = ξ S + ξT
Else
Figure 3 shows the complete loading and unloading process of the SMA wire under various isothermal conditions.
Brinson model the stress range in which the Martensite to
austenite transformation may occur, can be expressed in
equation (9) and the corresponding stress range for the reverse transformation is given by equation (1) and CA and CM
are material constants related to the influence of the stress
on the phase transformation, an increase of temperature increases the transformation stresses as in equations (9) and
(10). Since no residual strain remains above 300C in figure
3, we can determine that Ms will be near 300C. A residual
strain remains after the isothermal loading and unloading
process decreases below about 300C in figure 3 and then
heating above Af can restore the SMA wire with residual
strain to the original shape. We can obtain CA, CM, Af , and
Ms of equations (9) and (10) from the results of an isothermal strain-strain experiment as shown in figure 3, but
the other two constants cannot be obtained because the stroke of the SMA wire is limited. Thus, when the shape recovery force is predicted in the following section, two constants will be properly assumed.
(4)
(5)
(6)
(7)
(8)
Fig. 3 Isothermal Stress-strain curve of an SMA wire
at different temperature
Obr. 3 Izotermická krivka napätie-posunutie drôtu SMA
pri rozdielnych teplotách
63
4.2 Shape recovery force
Figure 4 shows the result of the restrained recovery of
an SMA wire at various initial deflections. Restraining a deformed SMA wire, while heating the wire above its transition temperature, will generate a large recovery force. This
unique behavior can offer active movement to conventional
catheters, which require motorless motion. The important
feature, as an actuator, is that the greater the initial deformation of the SMA wire the larger the recovery force will
be. These characteristics are also shown in figure 4. Based
on the multidimensional constitutive relation, the onedimensional shear stress and strain relation of an SMA can
be expressed as (1) where G is the elastic shear modulus, is
the phase transformation tensor (which can be calculated
from –DεL where D is the elastic modulus), εL is the maximum recoverable strain of an SMA, and is the thermoelastic tensor, which is related to the thermal expansion of
SMA. T is the temperature and ξ is the internal variable,
which describes the degree of the martensitic transformation. This parameter, referred to as the martensitic fraction,
is defined as the ratio of martensite to the total volume of
the SMA. In order to predict the experimental results using
equation (4), the 12 following material constants are required: where DA and DM are Young’s moduli in austenite phase and martensite phase, respectively; ν is Poisson’s ratio, εL
is the maximum recoverable strain, and ξ0 is the initial martensitic fraction.
Fig. 4 Recovery stress against temperature at various initial
stresses
Obr. 4 Obnovovacie napätie verzus teplota pri rozdielnych
počiatočných napätiach
Calculated results using equation (4), with the assumptions of some material properties, do not show good agreement with the experimental data, as shown in figure 5. This
conventional constitutive modeling has the following disadvantages. It is not practical because the constitutive modeling requires many experimental data to determine the 12
material constants. To obtain the shape recovery force, equation (4) should be solved by iteration because of nonlinearity of equation (4). Thus, it is a time consuming method.
Such disadvantages can be improved by a new method of
constitutive modeling using a neural network, which was
originally proposed by Ghaboussi et al [6]. This methodology was then applied to constitutive modeling of concrete,
composite material and current temperature of the SMA wire. One output neuron, which produces the shape recovery
force, constituted the output layer of the neural network.
Two hidden layers were made up of four neurons each.
64
Among the nine experimental data sets with different initial
deflections from each other, three experimental data sets
were used in training the neural network and the remainder.
Fig. 5 Comparison of experimental data and calculated
results of the SMA wire by conventional constitutive
modeling
Obr. 5 Porovnanie experimentálnych údajov a vypočítaných
výsledkov drôtu SMA pri konvenčnom konštutívnom
modelovaní
Three experimental data sets, were compared with the
predicted curves that resulted from the trained neural network. The training data are 24 points, which are randomly
selected, in five experimental data sets. Figure 7 shows that
the trained neural network predicts the shape recovery force
very well. The developed neural network has two advantages compared with the conventional constitutive equations.
Fig. 6 The architecture of an MLP for the prediction
of the shape recovery stress
Obr. 6 Architektúra MLP pre predvídanie napätia
pre obnovu tvaru
Firstly, the prediction of the shape recovery force is possible immediately through the real-time monitoring of the
temperature of the SMA because the neural network method
does not require a history of materials. However, if the conventional constitutive equations are used, it takes time to
obtain the result because nonlinear materials such as SMA
are path dependent. Secondly, the developed neural network
method is so simple, since it requires only two input data.
However, the conventional constitutive equation based on
Brinson theory requires several input data, which may be
impossible to obtain by conventional experiments. The application of the neural network is able to predict the almost
exact shape recovery force with only two input variables.
These two input parameters are the initial deflection and
temperature of the SMA wire. The initial deflection is the
most important design factor in the design of an active
catheter and the temperature may be a feedback variable in
the control of the SMA wire. Thus, the developed neural
network method as shown in figure 6 can be applied to
designing an active catheter and explaining the mathematical model of the SMA wire in a controller design.
that the force feedback control is better than the temperature.
K.P.Arulshri,
Kongu Engg College, Perundurai, Erode,
Tamil Nadu, India.
S_p_arul@redifmail.com
K.P.Padmanaban,
PSNA College of Engineering & Technology,
Dindigul, Tamilnadu, India.
Land Line: +914512431050
padmarubhan@yahoo.co.in
V.Selladurai
Coimbatore Institute of Technology,
Coimbatore, Tamil Nadu, India.
References
Fig. 7 Comparison of the experimental data and the
predicted results of the SMA wire by the neural network
Obr. 7 Porovnanie experimentálnych údajov
a predvídaných výsledkov drôtu SMA neurónovou sieťou
5 Conclusion
This paper was mainly aimed at describing the concept
of the proposed model as the first step, thereby indicating its
applicability only from simple experimental data, so various
further studies are still left open. Depending on the type of
experiment, one may try to describe material behaviors
using various characteristic points in the experiment for the
representation of internal variables. Strategies for selecting
proper internal variables are thus important issues. An adequate technique for extracting training data from experimental data has to be proposed for each internal variable accordingly.
The evaluation of the characteristics of an SMA wire
actuator was performed in this work. The fabricated SMA
wire shows a difference between the experimental results
and the predicted results from the constitutive model by
Brinson. Therefore, in this work, the combination of SMA
modeling and neural network technology was first proposed. The predicted results by the trained neural network
based on the experiments showed good agreement with the
experimental results. In terms of the shape recovery stress,
this technique requires only two input parameters (initial
deflection and temperature), which can be important design
parameters of the active catheter. The load- and temperature-following experiments were conducted in order to evaluate the feedback control characteristic when the feedback
variables were assumed to be temperature and force in the
active catheter. The experimental results showed that cooling and control methods were able to enhance the following characteristic of the SMA wire. It also informed us
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