Study on Dynamic Constitutive Model and Parameter
Analysis of Ti/Ni Shape Memory Alloy Based on Irreversible
Thermodynamics
Y S Yang
1
, X G Zeng
1, *
, J Chen
2
, Y Guo
1
and Y Sheng
3
1
College of Architecture and Environment, Sichuan University, Chengdu, Sichuan,
610065, P.R. China
2
Institute of Applied Physics snd Computational Mathematics, Beijing100094, P.R.
China
3
Southwest University of Science and Technology, Mianyang, Sichuan621000, P.R.
China
Corresponding author and E-mail: X G Zeng, xiangguozeng@scu.edu.cn
Abstract. The dynamic constitutive model of Ti/Ni shape memory alloy based on irreversible
thermodynamics theory was established by introducing two internal variables to characterize
phase transformation and plastic deformation process. The evolution laws of phase
transformation and plastic deformation were deduced respectively by assuming two
generalized force associated with the corresponding internal variables.Latin Hypercube
Sampling method is applied to sample among the whole parameter space for constitutive
parameter of Ti/Ni alloy dynamic constitutive model based on irreversible thermodynamics
theory, and Spearman rank correlation analysis of non-parameter statistics method is
employed to conduct correlation analysis for random-input sample set of constitutive
parameter and the output set of its corresponding objective function. A model is built to solve
parameter sensitivity based on Spearman rank correlation coefficient, in order to achieve
overall analysis of parameter sensitivity. According to results, numerical results, obtained
with this method, are well consistent with experimental data. Furthermore, the optimal
solution to the dynamic constitutive parameters can be realized with high efficiency, accuracy
and reliability via the research method on parameter sensitivity analysis and identification put
forward in this paper.
1. Introduction
Ti/Ni alloy is not only widely used in Aviation & Aerospace, Mechanics, Electronics, Energy, and
Medical field [1-2], but also the Ti/Ni alloy high strain dynamic responding behavior is closely
related to a great many of applications [3]. Meanwhile, the unique Shape Memory Effect and
excellent Super-elasticity of the Ti/Ni alloy draws wide attention to researchers at home and abroad,
and it is currently one of the Shape Memory Alloys that studied and applied most [4]. The Super
elasticity of Ti/Ni alloy refers to the phenomenon that the strain, which produced under effect of
external force, is far greater than its elastic limit, but self recovers when unloaded. The Shape
Memory Effect refers to the phenomenon when material going through plastic deform, and being
100
Yang, Y., Zeng, X., Chen, J., Guo, Y. and Sheng, Y.
Study on Dynamic Constitutive Model and Parameter Analysis of Ti/Ni Shape Memory Alloy Based on Irreversible Thermodynamics.
In Proceedings of the International Workshop on Materials, Chemistry and Engineering (IWMCE 2018), pages 100-106
ISBN: 978-989-758-346-9
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
heated until certain temperature level is reached, it could restore the shape before deform [5]. zone,
such that the fundamental frame of Super elasticity constitutive relations of Memory Alloy is built.
In order to study and apply the super elasticity and Shape Memory Effect of Ti/Ni alloy, a
reasonable description of Ti/Ni alloy constitutive relation model is indeed necessary. Earlier
phenomenological theory model proposed by Tanaka [6] is limited to one-dimension only. Liang C [7]
expand its one-dimension constitutive model to three dimension. But neither of the above two model
take the effect of temperature onMartensitic transformation into consideration, so the Martensitic
reverse transformation cannot be described. Later in the model raised by Brinson[8], Martensitic
transformation was divided into two parts, temperature induced transformation and stress induced
transformation, therefore, Martensitic transformation dynamics equation has two evolution equation,
positive and negative, under different temperature zone, such that the fundamental frame of Super
elasticity constitutive relations of Memory Alloy is built.
Based on irreversible thermodynamics, a macro-phenomenological constitutive model is
developed to describe the dynamic response of Ti/Ni alloy by introducing two internal variables to
characterize phase transformation and plastic deformation evolution. Then this paper adopts Latin
Hyper-cube Sampling method to sample among the whole parameter space of material parameter in
Ti/Ni alloy dynamic constitutive model based on irreversible thermodynamics theory, adopts
Spearman correlation analysis in non-parameter statistic method to conduct correlation analysis on
randomly input sample set of constitutive parameter and the output set of corresponding objective
function, to build an equation which adopts Spearman rank correlation coefficient to get equivalent
solution of parameter sensitivity, in order to achieve overall analysis of parameter sensitivity. Based
on the result obtained through Spearman rank correlation analysis, generic algorithm is applied to
identify material parameter of Ti/Ni alloy dynamic constitutive model.
2. Ti/Ni alloy constitutive parameter identification model and constitutive parameter sensitivity
analysis and generic algorithm procedure
Based on irreversible thermodynamics theory, Ti/Ni alloy dynamic constitutive model is defined[9],
which contains eighteen parameters waiting to be identified. Based on its property, the model can be
divided into four phases and optimize: Austenite elastic phase, transformation phase, Martensite
elastic phase and plastic flow phase. Use the four vector below to represent the to-be-identified
parameter of dynamic constitutive model in the four phases:
Austenite elastic phase:
[]
A
T
A
EX =
(1)
Transformation phase:
1 1 1 2 0 1 2 0
[ , , , , , , , ]
tr tr T
TR s f
X z n k k c css=
(2)
Martensite elastic phase:
[]
M
T
M
EX =
(3)
Plastic flow phase:
2 2 3 4 0 3 1
[ , , , , , , , ]
pT
Py
X z n k k c m Rs=
(4)
The purpose of Ti/Ni alloy constitutive parameter identification is to find out the optimal value of
*
X
in four phases, to achieve the minimum result of the equation of (5) of each phase.
*2
1
( ) min ( ( ) )
n
ii
i
WX

(5)
Study on Dynamic Constitutive Model and Parameter Analysis of Ti/Ni Shape Memory Alloy Based on Irreversible Thermodynamics
101
Since Ti/Ni alloy constitutive model is non-linear, and subject to the mutual effect of numerous
parameters, the correlation analysis among random variables is conducted using non-parameter
statistic method, using Spearman rank correlation coefficient to measure the correlation among
random variables. Before conducting overall analysis on parameter sensitivity, it is necessary to
apply Latin Hypercube Sampling method (LHS) [10] to sample in the whole range of Ti/Ni alloy
constitutive parameters. And sensitivity degree
i
r
of parameter
i
x
, which can be expressed as
below:
(6)
Generic algorithm is an algorithm which mimics natural biological evolution mechanism.
Through encoding initial group, operate based on their adaptability to environment, and achieve
survival of the fittest evolutionary process. Generic algorithm is composed of three basic generic
operators: Selection, Crossover, and Mutation.
3. Ti/Ni alloy constitutive parameter identification
This paper uses Split Hopkinson Pressure Bar (SHPB) device in dynamic mechanics laboratory of
SiChuan University, to test dynamic compressive performance of Ti/Ni alloy.
3.1. Solution of Ti/Ni alloy dynamic constitutive parameter sensitivity
Ti/Ni alloy dynamic constitutive parameter and the LHS range of each parameter are shown as Table
1. 20 times of unit sampling is conducted within the range of each parameter. Based on the Spearman
Rank definition method from previous section, the rank value of Ti/Ni alloy dynamic constitutive
parameter can be obtained, as shown in Table 2. According to equation (6), Spearman correlation
coefficient of Ti/Ni alloy dynamic constitutive parameter can be calculated, as shown in Table 3.
From the calculated result, it can be confirmed that the sensitivity degree value of
1
z
2
z
1
n
2
n
1
c
2
c
3
c
0
tr
s
0
tr
f
0
p
y
1
R
M
E
A
E
is fairly large, which means the effect of parameter on
output value is fairly large, and whose value should be focused on while identifying Ti/Ni alloy
dynamic constitutive parameter. However, sensitivity value of
1
k
,
2
k
,
3
k
,
4
k
is rather small,
which means these parameter has less effect on output value during calculation, and be considered as
sub-influencing factor of Ti/Ni alloy dynamic constitutive.
Based on Spearman rank correlation analysis theory, to make sure the accuracy of generic
algorithm, it is necessary to adjust the range and sampling times of each parameter, in order to make
sensitivity degree value of each parameter close to each other and all less than 0.2.
Table 1. Dynamic constitutive parameters of Ti/Ni alloy.
/a
A
E MP
/a
M
E MP
1
/aZ MP
1
n
2
k
1
k
0
tr
s
/MPa
1
c
2
c
10
5
~4x10
5
10
5
~3x10
5
0-15
0~10
500~900
200~500
200~500
0~1
0~1
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
0
/a
tr
f
MP
2
/aZ MP
2
n
4
k
3
k
0
/
p
y
MPa
3
c
m
1
R
200~500
0~200
0~10
600~900
200~500
500~2000
0~0.1
100~300
0~100
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
102
Table 2. Rank of Latin hypercube sampling data.
1
z
1
n
2
k
1
k
0
tr
s
1
c
2
c
0
tr
f
2
z
2
n
4
k
3
k
0
p
y
3
c
m
1
R
A
E
M
E
16
11
7
17
16
15
4
9
16
11
9
7
4
2
17
16
16
11
19
3
14
4
15
6
14
12
19
3
10
14
8
12
4
15
19
3
6
9
15
9
20
9
8
17
6
9
19
15
15
8
9
20
6
9
11
8
10
3
14
17
16
18
11
8
4
10
6
13
3
14
11
8
15
13
5
20
4
16
15
4
15
13
2
5
12
15
20
4
15
13
5
10
1
19
6
18
13
11
5
10
13
1
14
1
19
6
5
10
8
5
20
6
19
14
18
16
8
5
6
20
20
19
6
19
8
5
14
1
3
12
8
13
3
15
14
1
17
3
7
14
12
8
14
1
17
15
6
10
18
7
20
14
17
15
18
6
16
5
10
18
17
15
1
2
17
7
3
3
11
3
1
2
8
17
10
20
7
3
1
2
3
14
8
5
10
1
2
6
3
14
1
8
19
3
5
10
3
14
7
18
2
18
9
2
9
13
7
18
5
2
18
17
18
9
7
18
20
6
13
13
2
10
12
1
20
6
16
13
17
4
13
2
20
6
18
12
16
14
11
19
5
5
18
12
11
16
3
9
14
11
18
12
4
16
9
8
13
11
19
20
4
16
12
9
2
10
8
13
4
16
9
17
18
11
5
8
1
19
9
17
15
18
11
6
11
5
9
17
13
7
19
16
12
12
17
7
13
7
3
19
1
18
16
12
13
7
2
20
12
2
7
4
7
10
2
20
14
12
13
7
2
7
2
20
12
19
4
15
1
20
10
8
12
19
20
4
5
11
15
1
12
19
10
4
11
1
17
5
6
2
10
4
7
11
9
16
1
17
10
4
Table 3. Dynamic constitutive parameter sensitivity of Ti/ Ni alloy.
/a
A
E MP
/a
M
E MP
1
/aZ MP
1
n
2
k
1
k
0
tr
s
/MPa
1
c
2
c
0.225
0.27
0.242
0.26
0.085
0.018
0.269
0.21
0.2375
0
/a
tr
f
MP
2
/aZ MP
2
n
4
k
3
k
0
/
p
y
MPa
3
c
m
1
R
0.355
0.24
0.261
0.005
0.013
0.72
0.269
0.02
0.269
3.2. Generic algorithm optimization of Ti/Ni Alloy dynamic constitutive parameter
Based on irreversible thermodynamics theory, Ti/Ni alloy dynamics constitutive model can be
created. During identification of constitutive parameter using generic algorithm, this constitutive
model can be divided into four phases: parent elastic phase, transformation phase, Marstensite elastic
phase and plastic flow phase, build objective function on 4 phases respectively. Table 4. is the range
of parameters in generic algorithm. Next, based on the range of all above mentioned parameter, use
generic algorithm to conduct identification and solving for each parameters. During solving, set
crossover rate as 0.5, mutation rate as 0.09, number of generation as 500, binary digit for each
Study on Dynamic Constitutive Model and Parameter Analysis of Ti/Ni Shape Memory Alloy Based on Irreversible Thermodynamics
103
variable is 9.Finally Ti/Ni alloy dynamic constitutive parameter can be obtained, as shown in Table 5.
Substitute the optimal solution of above mentioned parameters into Ti/Ni alloy dynamic
constitutive correlation, based on the characteristics of master equation of Ti/Ni alloy constitutive
correlation, semi-implicit stress integration method is applied, which is to solve non-elastic strain
increment amount using semi-implicit iteration method, and update explicit equation at next
incremental step to get point (ε, σ). On this basis, the incremental format describing phase
deformation plasticity process of the Ti/Ni alloy is derived. In addition, nonlinear iterative algorithms
is raised and the calculation program composed in Fortran language to constructively demonstrate the
impact process under different strain rates. Refer to Figure 1 for details.
Table 4. Parameter range.
/a
A
E MP
/a
M
E MP
1
/aZ MP
1
n
2
k
1
k
0
tr
s
/MPa
1
c
2
c
10
5
~4
×
10
5
10
5
~2
×
10
5
0~15
0~10
500~900
200~500
400~500
0~1
0~1
0
/a
tr
f
MP
2
/aZ MP
2
n
4
k
3
k
0
/
p
y
MPa
3
c
m
1
R
350~600
0~200
0~10
700~900
300~500
1000~2000
0~0.0001
200~300
0~100
Table 5. Optimal solution of dynamic constitutive parameters of Ti/Ni alloy.
/a
A
E MP
/a
M
E MP
1
/aZ MP
1
n
2
k
1
k
0
tr
s
/MPa
1
c
2
c
37807
19288
10.87
3.0098
797.16
410.14
400
0.0378
0
0
/a
tr
f
MP
2
/aZ MP
2
n
4
k
3
k
0
/
p
y
MPa
3
c
m
1
R
556.36
172.3
4
806
401.9
1464.1
0
256.7
49
.2
Based on optimal solution of Ti/Ni alloy dynamic constitutive parameter in Table 5. stress-strain
numerical simulation result of Ti/Ni alloy under strain rate of 500s
-1
, 1300s
-1
, 1500s
-1
, 2100s
-1
,
3000s
-1
can be obtained, and comparison with experimental result is shown as Figure 1. From the
calculated result, numerical result of Ti/Ni alloy dynamic constitutive model matches nicely with
experimental value. Under these five strain rate, Ti /Ni alloy dynamic constitutive curve during the
whole simulation process shows 4 phases: Austenite elastic phase, transformation phase, Martensitic
elastic phase, plastic flow phase, indicates the sensitivity of Ti /Ni alloy dynamic constitutive
coefficient can be achieved using Spearman rank correlation analysis, and Ti /Ni alloy dynamic
constitutive parameter identification can be achieved using generic algorithm. The simulation results
are higher than experimental data after dislocation plastic deformation occurs and the deviations
increase obviously with the increasing strain rate. This phenomenon might result in the following
reasons proposed by Zeng et al[9]. The plastic deformation of materials under a high strain rates is
regarded as an adiabatic process so that the heat produced by plastic work cannot dissipate in time.
And its dynamic stress-strain response is an essentially coupled result of strain-rate strengthening,
strain hardening and adiabatic softening effects. The temperature in material rises will cause the
decrease in the initial yield stress, which is the so-called adiabatic softening effect.
At the same time, overall analysis of parameter sensitivity and generic algorithm in this paper has
important reference meaning to the identification of other engineering materials.
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
104
0 2 4 6 8 10
0
200
400
600
800
1000
1200
1400
500s
-1
Experiment
500s
-1
Calculation
stress/MPa
strain/%
0 2 4 6 8 10 12 14 16 18
0
200
400
600
800
1000
1200
1400
1600
1800
1300s
-1
Experiment
1300s
-1
Calculation
stress/MPa
strain/%
(a)The loading strain rate is 500s
-1
(b) The loading strain rate is 1300s
-1
0 2 4 6 8 10 12 14 16 18
0
250
500
750
1000
1250
1500
1750
2000
1500s
-1
Experiment
1500s
-1
Calculation
stress/MPa
strain/%
0 2 4 6 8 10 12 14 16 18
0
250
500
750
1000
1250
1500
1750
2000
2100s
-1
Experiment
2100s
-1
Calculation
stress/MPa
strain/%
(c)The loading strain rate is 1500s
-1
(d) The loading strain rate is 2100s
-1
0 2 4 6 8 10 12 14 16 18 20
0
250
500
750
1000
1250
1500
1750
2000
3000s
-1
Experiment
3000s
-1
Calculation
stress/MPa
strain/%
(e) The loading strain rate is3000s
-1
Figure 1. Stress and strain curves for Ti/Ni alloy at different strain rate.
4. Conclusions
Applied Super-Latin Cube Sampling method to sample among the whole parameter space, and used
Spearman rank correlation analysis in non-parameter statistic method to conduct correlation analysis
on random input sample set of constitutive parameter and its corresponding objective function output,
hence achieved overall analysis of parameter sensitivity. Based on Spearman rank correlation
coefficient, it reflected the impact extent of Spearman correlation coefficient on objective function of
Ti/Ni alloy dynamic constitutive parameter, and then confirmed the primary-secondary relation
among each parameter, and provided an effective method for multi-factor systematic analysis. This
method overcomed shortcomings of single-factor analysis, so it is a more practical
Study on Dynamic Constitutive Model and Parameter Analysis of Ti/Ni Shape Memory Alloy Based on Irreversible Thermodynamics
105
method.Sensitivity value of Ti/Ni alloy dynamic constitutive parameter, obtained through Spearman
rank correlation analysis, defined the range of constitutive parameter. Generic algorithm was used to
identify Ti/Ni alloy dynamic constitutive parameter, making the workload of identifying constitutive
model parameter much less heavy, and locating the optimal solution in a fast, accurate and reliable
manner.
Acknowledgments
The authors greatly acknowledge the jointly funded by the National Natural Science Foundation of China
and China Institute of Engineering Physics No. U1430119.
References
[1] Elahinia M H, Hashemi M, Tabesh M and Bhaduri S B 2012 Functionally graded Ni-Ti
microstructures synthesised in process by direct laser metal deposition J. Prog. Mater.Sci.
57 911
[2] Es-Souni M, Es-Souni M and Fischer-Brandies H 2005 Biointerface: protein enhanced stem
cells binding to implant surface J. Anal. Bioanal.Chem. 381 557
[3] Millett J C F, Bourne N K and GrayIII G T 2002 High strain-rate, small strain response of a
NiTi shape-memory-alloy J. Appl. Phys. 92 3107
[4] Wang X M, Yue Z F, Wang Y F and et al 2009 Superelastic mechanical properties of NiTi
alloy and its application M. Beijing: Science Press
[5] Yang J and Wu Y H 1993 Shape memory alloy and its application M. Hefei University of
Science and Technology China press
[6] Tanaka K A 1986 Thermomechanical sketch of shape memoryeffect: One-dimension tensile
behavior J.Res.Mechanica (18) 251-263
[7] Liang C and Rogers C A 1990 One-dimension thermomechanical constitutive relation for
shape memory alloy J. J.of Intell.Mater.Syst. and Struct (1) 207-234
[8] Brinson L C and Lammering R 1993 Finite-element analysis of thebehavior of shape-memory
alloys and their applications J. International Journal of Solids and Structures 30(23)
3261-3280
[9] Chen H Y, Chen C, Zeng X G, Chen J and Guo Y 2016 Nified onstitutive odel for hase
ransformation and lastic ehavior of Shape Memory Alloy Based on the irreversible
thermodynamics J.journal of sichuan university (engineering science edition) 48(2)
219-224
[10] Simpson T W, Lin D K J, Chen W 2001 Sampling strategiesb for computer experiments:
Design and analysis J.International Journal of Reliability and Applications 2(3) 209-240
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
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