Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model

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Consistent derivation of non-constant material function for onedimensional shape memory alloy phenomenological model Ravishankar N. Chikkangoudar a, Chetan S. Jarali b,⇑, S. Raja b, Subhas F. Patil a,c a

Department of Mechanical Engineering, K.L.E. Dr. M. S. Sheshgiri College of Engineering and Technology, Belagavi 590008, Karnataka, India Dynamics and Adaptive Structures Group, Structural Technologies Division, CSIR National Aerospace Laboratories, Bengaluru 560017, Karnataka, India c Ph. D Research Centre, Visvesvaraya Technological University, Belagavi 590008, Karnataka, India b

a r t i c l e

i n f o

Article history: Received 22 August 2019 Received in revised form 29 November 2019 Accepted 2 December 2019 Available online xxxx Keywords: Shape memory alloys Constitutive model Material functions Compatibility conditions Phase Transformation

a b s t r a c t Constitutive modelling of shape memory alloys with new material functions is presented within the frame work of thermodynamic consistency. A one- dimensional constitutive model based on the previous work of constant and non-constant material functions is redefined from first principles. In the first step Clausius-Duhem inequality condition for stress is rewritten and an alternate form of differential equation is proposed. The initial and final condition of evolution are applied to obtain a new form of non-constant transformation tensor, which is independent of residual strain the SMA material. As a result in the present work the residual strain is purely defined as a function of transformation stress and not as a function of transformation modulus. The proposed form of new transformation tensor is compared with previously proposed material function and validation results for the consistency are presented. It is observed that newly derived non-constant material function is compatible in both differential and integrated form of the one-dimensional shape memory alloy constitutive relation and satisfies the evolution conditions of phase transformations. Ó 2019 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the First International Conference on Recent Advances in Materials and Manufacturing 2019.

1. Introduction Shape Memory Alloys (SMAs) are smart materials which have a wide range of force and deformation capabilities due making them a potential material to be applied as multifunctional structures. SMA behaviours are characterized by the phase transformation temperatures with respect to material phases; namely austenite phase (A), multiple variant martensite phase (M) and single variant martensite phase (S). The material properties such as Young’s modulus (E), residual strain ðel Þ and transformation stress ðrÞ depend on the phase transformation temperatures. These temperatures are determined experimentally by heating and cooling the SMA material. Basically SMAs behaviours are characterized by austenite start temperature ðAs Þ, austenite finish temperature ðAf Þ, martensite start temperature ðM s Þ and martensite finish temperature ðMf Þ. As a result SMAs are able to show isothermal behavior when temperature T P Af referred to as pseudoelastic effect (PE), and non-isothermal behavior when T < Af referred to as the shape ⇑ Corresponding author. E-mail address: [email protected] (C.S. Jarali).

memory effect (SME). Constitutive modelling is important for the design of SMA based structures. One-dimensional (1-D) Phenomenological modelling is most widely followed to derive the differential and integrated SMA constitutive models in 1-D applications. A thermodynamic 1-D model based on thermodynamic principles is first proposed by Tanaka [1] using Helmholtz free energy ðUÞ and Clausius-Duhem inequality. The martensite fraction ðnÞ used to define the phase transformation kinetics is defined by an exponential function. Further Tanaka model is refined by Laing and Rogers [2] using a cosine function for transformation kinetics. The material functions and constitutive relation are further generalized to model 2-D and 3-D SMA behaviours [3]. Further the martensite fraction is separated into stress induced martensite fraction ðns Þ and temperature induced martensite fraction ðnT Þ by Brinson [4]. A micromechanics based derivation of 1-D model is also presented with observation that the constitutive relation is identical in terms of stress strain relations, and the differences between models is due to the development in evolution equations [5,6]. The internal state variable or the martensite fraction is varied in austenite and martensite phases to propose different evolution

https://doi.org/10.1016/j.matpr.2019.12.006 2214-7853/Ó 2019 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the First International Conference on Recent Advances in Materials and Manufacturing 2019.

Please cite this article as: R. N. Chikkangoudar, C. S. Jarali, S. Raja et al., Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.12.006

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equations in the same constitutive model [7]. 1-D modelling of SMAs is also proposed based on plasticity theory and evolution equations are derived for austenite fraction ðnM Þ, multiple variant and single variant ðnS Þ martensite fractions [8]. Evolution equations are also derived to model two way shape memory effects using the same constitutive relations and different material functions [9]. Experimental studies are also performed on 1-D SMA wires to analyze the stress strain and recovery stress behaviours [10]. It is observed that both residual strain/prestrain and transformation temperatures influence the recovery stress and constrained behavior of SMAs. Thermodynamics frame work is also extended to develop 3-D models for SMAs with additive decomposition of martensite twins between martensite variants [11]. Finite element modelling procedures are also applied to propose constitutive models for SMAs [12], where in c thermodynamic laws and the principle of micro-force balance are applied to derive finite deformation crystal mechanics model to simulate twinning in SMAs. One-Dimensional SMA models based on generalized theory of plasticity is extended to predict 1-D SMA behaviours under different temperature ranges [13]. It is observed that the non-constant form of Young’s modulus is important for constitutive models to simulate the experimental behaviours. Temperature dependent 1-D model is derived addressing the effects of martensite phase transformation and transformation induced plasticity [14]. The transformation modulus is defined to be the equivalent Young’s modulus and a new logarithmic form of expression is derived. Moreover micromechanical model based on crystal plasticity is also derived to describe anisotropic deformation behaviours of SMAs under cyclic loading [15]. In this work it is clearly observed that residual strain is occurs due to transformation induced plasticity within SMAs. Analytical and experimental studies on cyclic deformation of SMA wires shows that the residual strain accumulates in corresponding variants of transformed martensite in addition to the habit plane variants [16]. Finite element implementation of 1-D SMA model using commercial software is also performed with alternate expression for elastic modulus [17]. It is observed that in the final constitutive relation the transformation tensor ðXtr Þ is equivalent to the transformation modulus ðEðnÞÞ and independent of residual strain ðel Þ. This means that the residual strain is used to define the transformation strain ðetr Þ, which is now defined as ðnel Þ. In the original phenomenological model by Tanaka [1] it is observed that the constant material function used to define the transformation tensor is obtained as function of residual strain and elastic modulus. The same expression is adopted in the model by Liang and Rogers [2]. Although Brinson [3] separated the total martensite fractions into stress and temperature induced phase fractions, the similar form of transformation tensor as proposed previously by Tanaka [1] and Liang and Rogers [2] is obtained in the SMA model by Brinson [3]. Further it is observed that compatability conditions are satisfied in case of constant material function, that is Xtr  Eel , and E is constant during phase transformations. [1–3]. However, incompatibility is observed for the conditions on stress when the transformation tensor is a non-constant material function, that is Xtr  EðnÞel , and is non-constant E  EðnÞ [18– 20]. As a result new expression for the non-constant material function, which in present case is defined as the transformation tensor is then proposed [18,19]. Further, a new expression for thermal transformation tensor is separately defined by Brinson and Panico [19] as compared to the previous SMA models by Tanaka [1], Liang and Rogers [2], Brinson [3], Brinson and Panico [4], Brinson and Huang [5], Bekker and Brinson [6]. Further alternate forms of non-constant material functions for transformation and thermal tensors are defined by simplifying the models in [18–19] by Khandal et al. [20]. Oslen et al. [21] also defined alternate expression for

the non-constant material transformation tensor. Recently Islam et al., [22] studied the uncertainty in the elastic modulus proposed by Tanaka [1] and Laing and Rogers [2] on the SMA behaviours. In the present work exact simplifications are provided regarding the inconsistencies in the development of non-constant material functions in the previous SMA models [1–4,18–21]. Next it is clearly presented that redefined non-constant material functions proposed recently in SMA models [18–21], are not derived from first principles. Consequently a new non-constant material transformation tensor for 1-D SMA model is derived in the present work based on first principles. Subsequently new form of differential equation to implement in finite difference form is also proposed for the non-constant transformation tensor. Finally validations are provided to clearly present the consistency of the proposed derivations. 2. SMA constitutive models The phenomenological 1-D SMA constitutive equations and the phase transformation kinetics are derived following ClaussiusDuhem inequality. The 1-D models proposed are presented with constant and non-constant material functions. The inconsistencies are also highlighted. 2.1. One-dimensional SMA model of Tanaka [1] The 1-D SMA constitutive relation is proposed using the internal energy balance and the Clausius-Duhem inequality (entropy production) principles. These thermodynamic equations are expressed, respectively, as

q U_  r^ L þ q T

q S_  q þ

@qsur  q q ¼ 0; @X

ð1Þ

@ qsur  P 0; @X T

ð2Þ

where q is the density of the SMA material, U is internal energy ^ Cauchy stress, qsur heat flux, q is heat production term, density, r S is entropy, T is temperature and X is the material coordinate in the current material configuration, respectively. L is length of 1-D SMA wire. The Helmholtz free energy ðUÞ is function of state variable ðKÞ of the material. Helmholtz free energy UðKÞ may be applied to describe the kinetics of SMA phase transformation as follows

UðKÞ ¼ U  T S:

ð3Þ

In Eq. (3) the internal energy U will try to minimize the function by pulling all material particles into the depths of the potential well while the entropy S causes to maximize by distributing the particles uniformly over the available range of shear lengths. Next the state variable for the SMA material ðKÞ is defined in terms of the independent state variables as

K  ðe; T; nÞ;

ð4Þ

where e is the strain corresponding to large deformations in SMA and n is the total martensite fraction. By virtue of Eqs. (3) and (4) the Clausius-Duhem inequality for the SMA material may written in terms of independent state variables as









r @U _ @U _ @U _ 1 @T T P 0;  e S þ n q cof F T @X q0 @ e @T @n q0 T sur ð5Þ

where r is the second Piola–Kirchoff stress, F is the deformation gradient, q0 is the SMA material density in the reference configuration, respectively. In case of continuum mechanics, for the inequality condition in Eq. (5) to hold good for all the material process the

Please cite this article as: R. N. Chikkangoudar, C. S. Jarali, S. Raja et al., Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.12.006

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equality condition for UðKÞ and entropy S needs to be imposed. As a result the coefficients of state variables e and T needs to vanish resulting in the mechanical constitutive equation as

r ¼ q0

@ Uðe; n; TÞ ¼ rðe; n; TÞ; @e

ð6Þ

The 1-D integrated form of SMA constitutive relation as derived by Tanaka [1] is

r ¼ Eel e þ X;

ð7Þ

where Eel is the elastic modulus and X is the defined in Tanaka model [1] as a material constant. The constant X is proposed to be the function of residual strain ðel Þ such that

X ¼ el E

ð8Þ

2.2. One-dimensional SMA model of Liang and Rogers [2] In the constitutive model by Liang and Rogers [2], the differential Eq. (6) is defined as

dr ¼ Eel de þ Xtr dn þ H dT;

ð9Þ

where H relates the thermal expansion coefficient for SMA material. Therefore, the integrated form of Eq. (9) is obtained as

r  r0 ¼ Eel ðe  e0 Þ þ Xtr ðn  n0 Þ þ Hth ½ðT  T 0 Þ;

ð10Þ

where Xtr is the material constant, which is defined as transformation tensor. r0 ; e0 ; n0 and T 0 are material values corresponding to the initial conditions of phase transformations. The material constant is now defined using the initial and final condition of martensite transformation as [2]

Xtr ¼ el Eel :

ð11Þ

2.3. One-dimensional SMA model of Brinson [3,4] In the 1-D model by Brinson [3] the martensite fraction is redefined into stress induced martensite fraction ðns Þ and temperature induced martensite fraction ðnT Þ as follows

n ¼ ns þ nT :

ð12Þ

As a result, the integrated form of Eq. (10) is rewritten as [3]

r  r0 ¼ Eel ðe  e0 Þ þ Xtr ðns  ns0 Þ þ XT ðnT  nT 0 Þ þ Hth ½ðT  T 0 Þ;

ð13Þ

ð14Þ

Using Eqs. (13) and (14), the transformation tensors are obtained as follows

Xtr ¼ el Eel ; XT ¼ 0:

ð15Þ

Further in the 1-D SMA model by Brinson [3], a non-constant form of material functions in Eq. (13) are proposed in terms of the modulus of austenite phase ðEa Þ and martensite phase ðEs Þ as

Eel ðnÞ ¼ Ea þ ns ðEm  Ea Þ;

Xtr ðnÞ ¼ el Eðns Þ:

In consistency in compatibility conditions is observed in the 1-D model by Brinson [3] due to the non-constant material function Xtr ðns Þ[18]. Subsequently a new form of transformation tensor is proposed in the model by Buravalla and Khandelwal [18]. The transformation tensors are re-defined as follows

Xs ðe; nÞ ¼ el Eel ðnÞ þ ð e  el ns Þ ðEm  Ea Þ; XT ðe; nÞ ¼ 0:

ð16Þ

ð17Þ

Further in the 1-D SMA model by Brinson [3], a non-constant form of material functions in Eq. (13) is also proposed in terms of the modulus of austenite phase ðEa Þ and martensite phase ðEs Þ. 2.5. One-dimensional SMA model of Brinson and Huang [19] and Khandan et al. [20] The non-constant material function model [3] is used to rederive the forms of non-constant material functions Xtr ðe; ns ; nT Þ and XT ðe; ns ; nT Þ in the model by Brinson and Huang [19]. The material functions are proposed in the following form

Xs ðe; nÞ ¼ el Eel ðnÞ þ ð e  el ns Þ ðEm  Ea Þ; XT ðe; nÞ ¼ ð e  el ns Þ ðEm  Ea Þ:

ð18Þ

Further, the modifications in the model [18,19] are simplified in the work of Khandan et al., [20] to obtain the following form of new expressions for the material functions

Xs ðe; nÞ ¼ eðEm  Ea Þ  2el ns ðEm  Ea Þ  el nT ðEm  Ea Þ  el Ea ; XT ðe; nÞ ¼ e ðEm  Ea Þ  el ns ðEm  Ea Þ: ð19Þ The 1-D SMA constitutive relation is redefined as

r ¼ e½Ea þ nðEm  Ea Þ  el ðEm  Ea Þn2s  el ðEm  Ea Þns nT  el Ea ns ðEm  Ea Þ þ hT:

ð20Þ

2.6. One-dimensional SMA model of Olsen et al. [21] New improvements have been further observed in the derivation of transformation tensors representing the non-constant material functions in 1-D SMA models. An alternate form of the material function Xtr ðe; ns ; nT Þ is proposed as [21]

XðnÞ ¼

where XT is defined as temperature induced phase transformation tensor and ns0 corresponds to the initial conditions of phase transformations. The initial and final conditions for martensite phase transformations are defined as

r0 ¼ 0; e0 ¼ 0; ns0 ¼ 0; nT 0 ¼ 0; T 0 ¼ T 0 r ¼ 0; e ¼ el ; ns ¼ 1; nT ¼ 0; T ¼ T 0

2.4. One-Dimensional SMA model of Buravalla and Khandelwal [18]

½rmax  rmin  Eel ðnÞðEm  Ea Þ ; nmax

ð21Þ

where the derivative or the tangent modulus of Eq. (21) is further presented as

X0 ðnÞ ¼

½EMA ðemax  emin Þ : nmax

ð22Þ

2.7. One-dimensional SMA model of Islam and Karadogan [22] In a very recent study [22], sensitivity and uncertainty analysis is presented on the two original SMA constitutive models of Tanaka [1] and Liang and Rogers [2]. In this study it is observed that 1-D model of Tanaka [1] as well as 1-D model of Liang and Rogers [2] the variability or inconsistency is primarily caused due to the effect of working temperature and loading conditions on the elastic modulus ðEel Þ of SMA material. As a result Sobol and extended Fourier Amplitude Sensitivity Testing (eFAST) method is applied to understand the parameters influencing the stability of these 1-D models. It is proposed that these models may need further refinements in terms of the material functions, especially

Please cite this article as: R. N. Chikkangoudar, C. S. Jarali, S. Raja et al., Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.12.006

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the elastic modulus ðEel Þ, phase transformation coefficient/tensor for stress ðXtr Þ and temperature ðXT Þ, temperature coefficient ðHÞ and transformation temperatures ðAf ; As ; M f ; M s Þ. It is observed that in the constitutive equation of Tanaka [1]

r  r0 ¼ Eel ðnÞðe  e0 Þ þ Xtr ðn  n0 Þ þ Hth ½ðT  T 0 Þ;

ð23Þ

The elastic modulus ðEel Þ is the single material function influencing the sensitivity of 1-D SMA model. In the transformation region, the 1-D SMA model is sensitive to the transformation coefficient ðXtr Þ. However, in the 1-D model of Liang and Rogers [2] high variability is observed in the phase transformation regions, which is influenced by stress, temperature, slope of loading and martensite fraction. Further in the elastic unloading both Eel and Xtr define the model sensitivity [22]. The following influential material parameters for original 1-D SMA models are listed in Table 1. 3. Definition of inconsistency in the previously proposed 1-D SMA models

4. Derivation of non-constant material function for 1-D SMA phenomenological model Consider the martensitic phase transformations when the SMA material is unloaded after the formation of single variant martensite. At the end of complete unloading it is very clear that the model parameter n ¼ 1. Now this condition exists only when the stress r ¼ 0. This means that substituting the evolution conditions defined in Eq. (14) in to the relation defined by Eq. (13), the constitutive relation defined in Eq. (13) reduces to

0  0 ¼ ½ Ea þ 1ðEm  Ea Þðel  0Þ þ Xtr ðns Þ ð1  0Þ þ XT ð0  0Þ þ Hth ½ðT  TÞ;

Xtr ðns Þ ¼ Em el

ð24Þ

However, in the previous models it is observed that the evolution conditions defined in Eq. (14) are not substituted as defined. This is understood from Fig. 1.That is, the condition for strain e ¼ el exists only when n ¼ 1 as shown in region (a–d) in Fig. 1. The condition e ¼ el n exists only when n–1 meaning the stress

Based on the previous developments of the 1-D SMA constitutive models [1–6,18–22] the following consistency problems have been observed in the material functions and the constitutive models. These sensitivities have be analyzed in the present work and present in Table 2. It is observed that the above problems in the material functions and parameters may be lead to incorrect definition of 1-D SMA constitutive equations. In order to overcome these inconsistencies, a new definition for the non-constant material function representing the transformation coefficient Xtr ðnÞ is derived in the present work.

Table 1 Parameters influencing 1-D SMA models [1,2]. Tanaka Model [1]

Liang and Rogers Model [2]

SMA 1-D Behaviour

ðT CÞ T > Af

Eel ; As ; Ms

Eel ; M f

As < T < Af

Eel ; Xtr ; M s ; T

Eel ; Xtr ; Ms ; T

Pseudoelastic Effect (PE) Shape Memory Effect (SME)

Temperature 

Fig. 1. Stress–strain diagram for SMA in the martensitic state, when the volume fracture of martensite is 100% under T < M f :

Table 2 Inconsistencies in the previously reported one-dimensional constitutive models. Model

Material Function with Inconsistency

Potential Problem

Tanaka [1] Liang and Rogers [2]

Eel ; As ; M s ; M f X ¼ Eel – Derived only for SME ðT < Af Þ where el is present upon unloading Xs ¼ Eel , Xs ¼ Eðns Þel , XT ¼ 0 – Incorrect since this relation is derived only for SME ðT 6 Af Þ where el exists upon unloading Xðe; nÞ ¼ eðEm  Ea Þ  el ½Ea þ ð2ns þ nT ÞðEm  Ea Þ – In this relation martensite fraction ns is defined as 2ns . As a result for ns ¼ 1 and nT ¼ 0, n ¼ ns þ nT ¼ 2, which violates kinetic of martensite phase evolution. Further in the above relation when ns ¼ 0 and nT ¼ 0 the term Xðe; nÞ ¼ eðEm  Ea Þ, which is incorrect XT ¼ ðe  el ns ÞðEm  Ea Þ – in this relation XT should be function of martensite fraction nT due to temperature and not function of ns , which further violates the evolution conditions for phase transformations. In the above relation when ns ¼ 0 the term XT ðe; nÞ ¼ eðEm  Ea Þ is inconsistent definition Xs ¼ ðe  2el ns  el nT ÞðEm  Ea Þ  el Ea – Incorrect since the correct definition of residual strain is eres ¼ el ns

Material model is sensitive for PE and SME X ¼ Eel – This relation cannot be derived for PE ðT P Af Þ

Brinson [3], Brinson and Huang [4] Buravalla and Khandelwal [18], Brinson and Panico [19] Brinson and Panico [19], Khandan et al. [20]

Khandan et al. [20] Oslen et al. [21]

XðnÞ ¼ ½Em ðE1m Ea Þ ; when n ¼ 1 and rmax ¼ rmin X0 ðnÞ ¼ 0; when emax ¼ emin

Xs ¼ Eðns Þel – This relation cannot be derived for PE ðT P Af Þ and XT ¼ 0 may not be not be true. n ¼ ns þ nT is of the form n ¼ 2ns þ nT : Since Xðe; nÞ ¼ eðEm  Ea Þ when n ¼ 0, there is no physical definition for ðEm  Ea Þ in the elastic loading and unloading regions. That is ðEm  Ea Þ may not represent any SMA material phases for the either austenite or martensite state when n ¼ 0: n ¼ ns þ nT is of the form n ¼ 2ns þ nT and XT cannot be function of ns . In the term Xðe; nÞ ¼ eðEm  Ea Þ when n ¼ 0, ðEm  Ea Þ has no physical meaning since ðEm  Ea Þ does not represent the initial state of modulus before loading

el is obtained as 2el , which is undefined for SMA material property XðnÞ needs to be generalized for the conditions where the recovery process starts from the condition of complete unloading after martensite phase transformations

Please cite this article as: R. N. Chikkangoudar, C. S. Jarali, S. Raja et al., Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.12.006

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r–0. Further in the elastic unloading for T < Af the elastic modulus

is Eel ¼ Em in region (c-d) since the phase transformation is complete with n ¼ 1 at point c. Based on the sensitivity and uncertainty analysis in the work of Islam and Karadogan [22] as well as the inconsistencies for the previous models presented in in Table 1, the constitutive relation in Eq. (6) proposed by Tanaka [1], Liang and Rogers [2] and Brinson [3] may be redefined as

r ¼ q0

@ Uðe; etr ; eth Þ ¼ rðe; etr ðn; eÞ; eth ða; TÞÞ; @e

ð25Þ

Therefore, the constitutive Eq. (9) is rewritten as

dr ¼ Eel de þ Xtr detr þ H deth ;

Further, in Fig. 3, the variation in transformation coefficient for complete loading and unloading is presented. The new definition for Xtr ðnÞ ¼ Eel ðnÞ is able to simulate the pseudoelastic effect with appropriate slopes in the elastic region and transformation regions. However, the previous definitions for Xtr ðnÞ violate the prediction of the loading and unloading paths. Moreover it may observed that when the martensite fraction n ¼ 0 the transformation modulus must define the modulus of austenite phase ðEa Þ. Similarly in the elastic unloading transformation modulus must define the modulus of martensite phase ðEm Þ. Consider the shape memory effect with SMA material in T < Af as shown in Fig. 4. A similar variability exists in the definition of

ð26Þ

where de is the change in total strain, detr is the change in transformation strain and deth is the thermal strain in the SMA material. The integrated form of the Eq. (26) is written as

r  r0 ¼ Eel ðe  e0 Þ þ Xtr ðetr  etr0 Þ þ H ðeth  eth0 Þ;

ð27Þ

Substituting the initial and final conditions of evolution defined in Eq. (14) into Eq. (27), the transformation coefficient may be obtained as

Xtr ¼ Eel :

ð28Þ

Further the non-constant elastic modulus is defined by Tanaka [1] and Laing and Rogers [2] as

Eel ðnÞ ¼ Ea þ ns ðEm  Ea Þ

ð29Þ

Following the procedure [3], expanding Xtr using Taylor series with respect to n0 and neglecting higher order terms, results in

Xtr ðnÞ ¼ Xtr ðn0 Þ þ ðn  n0 ÞX0tr ðn0 Þ

ð30Þ

Applying the Xtr  Eel relationship, the initial value of Xtr ðnÞ is

Xtr ðn0 Þ ¼ Eel ðn0 Þ

ð31Þ

Fig. 2. Variation in transformation coefficient for pseudoelastic effect T ¼ 60:

By substituting Eqs. (29) and (31) into Eq. (30) the new form of non-constant material function representing the transformation coefficient is obtained as

Xtr ðnÞ ¼ Eel ðnÞ

ð32Þ

In this work the form of thermal coefficient is assumed constant as proposed in the previous models. Eq. (32) now defined the new non-constant transformation coefficient for the 1-D SMA models. 5. Results and discussions The variation in transformation coefficient for different models is presented. The material properties used to compute the results have been presented in Table 3. In Fig. 2 the variation in constant and non-constant transformation coefficient is presented for isothermal loading with T > Af : In this result it may be observed that the constant transformation coefficient, which when defined in terms of residual strain, that is, Xtr ¼ Eel el will result in higher elastic modulus Xtr > Ea . However in the final constitutive the transformation coefficient is precisely defined to be Xtr ¼ Eel . Therefore, sensitivity in the definition of transformation coefficient is observed.

Fig. 3. Variation in transformation modulus with respect to transformation coefficient for pseudoelastic effect T ¼ 60:

Table 3 Material properties for the 1-D SMA model [2,14,23]. Moduli

Transformation temperatures

Transformation parameters

Maximum residual strain

Thermal expansion coefficient

Ea ¼ 67  103 MPa Em ¼ 26:3  103 MPa

M s ¼ 18:4 C M f ¼ 9 C As ¼ 34:5 C Af ¼ 49 C

C M ¼ 8MPa= C C A ¼ 13:8MPa= C rcrs ¼ 100MPa rcrf ¼ 170MPa

el ¼ 0:067

aa ¼ 8:208e6 = c am ¼ 8:208e6 = c

Please cite this article as: R. N. Chikkangoudar, C. S. Jarali, S. Raja et al., Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.12.006

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rial coefficients for XT ðnÞ and HðnÞ. The sensitivity and uncertainty analysis [22] must be extended to observe the improvements in material functions. 6. Conclusions

Fig. 4. Variation in transformation modulus with respect to transformation coefficient for SME effect T ¼ 30:

transformation coefficients in the model proposed by Buravalla and Khandelwal [18], Brinson and Panico [19] with respect to the new definition of transformation coefficient. In Fig. 4. The SME is presented when the SMA material is completely in multiple variant martensite phase. In this condition the elastic modulus does not change and remains constant under the application of isothermal loading. it may also be important to analyse the variation in thermal coefficient XT ðnÞ. The thermal coefficient should be invariance to the stress induced martensite. However, in the previous models [18,19] XT ðnÞ is derived to be function of ns and not nT . Moreover it may observed that ðEm  Ea Þ is used to define the modulus function instead of the change in modulus ½Ea þ nðEm  Ea Þ. As a result the definition of XT ðnÞ needs to be revisited in the context of compatibility conditions. In Fig. 5 it may also be noticed that the transformation coefficient corresponding to ns shows variance with respect to initial modulus of martensite fraction. Subsequently the proposed derivation for Xtr ðnÞ reduces to Xtr ðnÞ  Em , which correctly defines the SMA phase and the modulus value. It may be understood that further modifications may be required to correctly derive the new and consistent forms of mate-

Fig. 5. Variation in transformation modulus with respect to transformation coefficient for SME effect T < M f :

In the present work the definitions for constant and nonconstant transformation coefficients used to define stress induced phase transformations are revisited. Further the definition of thermal coefficient is also analysed. Appropriate simplifications and justifications are provided to highlight the sensitivity and uncertainty in the definition of material functions. Alternatively strain based development of constitutive equation is proposed based on Helmholtz free energy equation by defining strains within the material and results have been presented. It is concluded that the new definition of transformation coefficient is able to clearly define the modulus of SMA material in respective phases as well as under forward and reverse phase transformations. It is also brought out that the definition for thermal coefficient obtained in the one-dimensional SMA model by Tanaka [1], Liang and Rogers [2], Brinson [3], Brinson and Panico [19] may be rederived under consistency. The new definition of transformation coefficient presented needs to applied into constitutive relations defined in Buravalla and Khandelwal [18], Brinson and Panico [19] and in the model of Khandan et al. [20]. The simplifications, observations, results and discussions suggested in the present work may be further extended to improve the definitions for material functions and differential equations. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] K. Tanaka, A thermomechanical sketch of shape memory effect: onedimensional tensile behaviour, Res. Mech. 18 (1986) 251–263. [2] C. Liang, C.A. Rogers, One-dimensional thermomechanical constitutive relations for shape memory materials, J. Intell. Mater. Syst. Struct. 1 (2) (1990) 207–234. [3] C. Liang, C.A. Rogers, A multi-dimensional constitutive model for shape memory alloys, J. Eng. Math. 26 (1992) 429–443. [4] L.C. Brinson, One-dimensional constitutive behaviour of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variables, J. Intell. Mater. Syst. Struct. 4 (1993) 229–242. [5] L.C. Brinson, M.S. Huang, Simplifications and comparisons of shape memory alloy constitutive models, J. Intell. Mater. Syst. Struct. 7 (1996) 108–114. [6] A. Bekker, L.C. Brinson, Phase diagram based description of the hysteresis behavior of shape memory alloys, Acta Mater. 46 (1998) 3649–3665. [7] J.G. Boyd, D. Lagoudas, A thermodynamic constitutive model for the shape memory materials. Part I. The monolithic shape memory alloy, Int. J. Plast. 12 (1996) 805–841. [8] F. Auricchio, R.L. Taylor, J. Lubliner, Shape memory alloys: macromodelling and numerical simulations of the superelastic behavior, Comput. Methods Appl. Mech. 146 (3–4) (1997) 281–312. [9] F. Auricchio, S. Marfia, E. Sacco, Modelling of SMA materials: training and two way memory effects, Comput. Struct. 81 (2003) 2301–2317. [10] Y. Li, L.S. Cui, H.B. Xu, D.Z. Yang, Constrained phase-transformation of a TiNi shape-memory alloy, Metall Mater. Trans. A 34A (2003) 219–223. [11] Y. Chemisky, A. Duval, E. Patoor, T. Ben, Zineb, Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation, Mech. Mater. 43 (2011) 361–376. [12] P. Thamburaja, H. Pan, F.S. Chau, The evolution of microstructure during twinning: constitutive equations, finite-element simulations and experimental verification, Int. J. Plasticity 25 (2009) 2141–2168. [13] T. Videnic, M. Brojan, J. Kunavar, F. Kosel, A simple one-dimensional model of constrained recovery in shape memory alloys, Mech. Adv. Mater. Struct. 21 (2014) 376–383. [14] C. Yu, G. Kang, Q. Kan, A physical mechanism based constitutive model for Temperature-dependent transformation ratchetting of NiTi shape memory alloy: one-dimensional model, Mech. Mater. 78 (2014) 1–10.

Please cite this article as: R. N. Chikkangoudar, C. S. Jarali, S. Raja et al., Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.12.006

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Please cite this article as: R. N. Chikkangoudar, C. S. Jarali, S. Raja et al., Consistent derivation of non-constant material function for one-dimensional shape memory alloy phenomenological model, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.12.006