Determination of superplastic properties from the results of technological experiments

Advances in Engineering Software 112 (2017) 54–65

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Determination of superplastic properties from the results of technological experiments A.A. Kruglov a, V.R. Ganieva b,∗, F.U. Enikeev b a b

Institute for Metals Superplasticity Problems, Russian Academy of Sciences, Khalturina 39, Ufa 450001, Russia Ufa state petroleum technological university, Department of Computer Sciences, Kosmonavtov 1, Ufa 450062, Russia

a r t i c l e

i n f o

Article history: Received 18 November 2016 Revised 15 May 2017 Accepted 28 June 2017

Keywords: Superplasticity Strain rate sensitivity Technological experiments Superplastic forming Finite element modeling

a b s t r a c t The problem to determine experimentally the values of material parameters for two material models of  superplastic flow, σ =Kξ m and σ = K ξ m ε n , from the results of technological trials is considered. With this in view, a special computational procedure is developed to minimize the deviation of the theoretically predicted forming times from experimental data recorded during constant pressure forming trials of a sheet into a circular die. As compared with similar procedures known in the literature the methods suggested enable one to obtain a unique set of material parameters by using the whole set of available experimental data. The validity of the procedures suggested is confirmed by means of comparing the results obtained with corresponding finite element solutions. The accuracy of modeling of the experimentally measured values of the forming time is found to be better than 5% for all cases considered. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Structural superpalsticity is known to be the ability of finegrained materials to exhibit unique tensile elongations under relatively low strain rates and high homologous temperatures [1]. Therefore, tensile tests are often used to establish the boundaries of superplastic flow. The most important characteristic of superplastic flow is the high value of the strain rate sensitivity index, m, which is defined as

σ = Kξ m

(1.1)

where σ is flow stress,ξ is strain rate, K is a material parameter which depends on the average grain size and other structural characteristics. Material model (1.1) has been suggested in the pioneering work [2]. Mathematically, Eq. (1.1) represents a straight line when plotted in the logarithmic coordinate’s logσ – logξ . However, the experimental dependencies logσ – logξ have conventionally a specific sigmoidal shape with the point of inflection corresponding to the optimum values of the strain rate, ξ opt , and flow stress, σ opt . The slope of the sigmoidal curve, M = ∂ logσ /∂ logξ , depends on the strain rate so that M(logξ ) curve has a specific dome like shape, the maximum slope, Mmax , being corresponding to the optimum

∗

Corresponding author. E-mail addresses: [email protected] (A.A. Kruglov), [email protected] (V.R. Ganieva). http://dx.doi.org/10.1016/j.advengsoft.2017.06.014 0965-9978/© 2017 Elsevier Ltd. All rights reserved.

strain rate ξ opt . The boundaries of the optimum strain rate interval are conventionally determined from the condition M > 0.3 [1,3]. In spite of the strain rate dependency of M is recognized for a limited number of materials and for specific temperature conditions…, the material model (1.1) is often used in practical calculations when considering superplastic metal working techniques [4–13]. Most superplastic crystalline materials have this unique property because they are fine-grained (grain size less than about 10 μm). Starting from 1990th the ultra-fine grained materials of grain size less than about 1 μm have attracted the attention of due to their unique microstructures and exceptional properties [14–18]. Analysis shows [19] that one of the most serious problems in developing new technologies in metal working of nanostructured materials (grain size less than 0.1 μm) is concerned with the necessity to take into account the influence of the grain growth in describing the mechanical behavior of such kind materials. Therefore, the strain hardening index, n, is introduced into the phenomenological model of superplastic flow σ = Kξ m as follows 

σ = K  ξ m εn

(1.2)

where ε is the strain, K , m and n are the material constants to be determined experimentally. The material model (1.2) is also used in practical calculations of superplastic metal working processes [20–24].  Thus, the material models σ = Kξ m and σ = K ξ m ε n are widely used in practical calculations when finite element modeling the superplastic metal working processes. The values of material param-

A.A. Kruglov et al. / Advances in Engineering Software 112 (2017) 54–65

Nomenclature D Fm H Im Jm K m N p R R0 R0 s, s0 ti

Depth of the die Common notation for the functions 2Im and Jm Current dome height Definite integral determined by Eq. (4.4) Definite integral determined by Eq. (4.4J) Material constant Strain rate sensitivity index Total number of available experimental data Gas pressure Radius of the dome Radius of the die Radius of the die Current and initial sheet thickness respectively Forming time α Angle between the axis of symmetry and the radius of the dome corresponding to the fastened boundary of the circular membrane σ Flow stress σ m , σ t Meridian and tangential stress respectively σe Effective von Mises stress ξ Strain rate ε Strain  (K,m) Goal function eters needs to be known in advance to fulfill the finite element calculations, it explains the interest to the methods of experimental determination of the values of material parameters K, m and K , m , n has revived within last decade. The aim of the present paper is to develop robust computational procedures enabling one to determine reliably the values of K, m in Eq. (1.1), as well as the values of K , m , n in Eq. (1.2) from the results of technological experiments. 2. Superplastic properties Presently, the commercial finite element software available on the market, such as ABAQUS, LS-DYNA, ANSYS, DEFORM, MARC, etc are now wide-spread and constantly in use by most of employers. Therefore, now it is possible to fulfill the finite element modeling of superplastic metal working processes without appealing to the developers of the specific software, e.g., SPLEN [25] or similar products (see, e.g, the corresponding references in the review [21]). However, the availability of the software is not enough to solve the problem of constructing the effective finite element models of the technological processes of interest. The point is that one has to solve at least three problems before starting the modeling: (i) to select the material model for describing the superplastic flow (ii) to determine the values of material constants for the material model chosen (iii) to state the boundary value problem in the mechanics of solids Finite element software used in modeling the superplastic metal working processes is intended to solve the boundary value problem in the mechanics of continuum. This problem can be stated in different ways [3,21]. For example, the boundary value problem can be stated in terms of mechanics of fluids [25], or theory of creep [3], or in terms of some another approach [26]. Independently of the method chosen to state the boundary value problem, the same values of material parameters (e.g., K and m) should be introduced into the FEM-software when fulfilling the calculations.

55

Thus, the material constants to be used in practical calculations are to be determined from the results of mechanical experimentation independently of the finite element software used and/or method to state the boundary value problem chosen, based on that, the methods to determine the material constants from the results of experiments needs to be developed and then tested in accordance with the requirements that are well known in the mechanics of solids [3]. It is to be emphasized that the material parameters are to be considered to be constant in value from the very beginning and up to the end of calculations. In the case when the strain rate dependency of m-value is to be taken into account, the material model σ = Kξ m is to be rejected and replaced by another material model containing different material constants. Correspondingly, other methods to determine experimentally the specific values of these new material constants are to be developed and tested in accordance with the requirements of the general theory of constitutive equations. The term ‘superplastic properties’ means, from the mechanical point of view, the specific values of the material constants to be used in calculations. For elastic body the value of Young’s modulus as well as that of Poisson coefficient do not depend upon the strain by definition. Otherwise, the Hook’s law is to be rejected and then replaced by some another material model which is more appropriate for the case of, say, non-linear elastic body. Similarly, from the mechanical point of view, the values of material parameters, K and m, do not depend on the strain rate by definition. Otherwise, the standard power law σ = Kξ m is to be rejected and replaced by another material model. In doing so, it is necessary to remember, that to change the material model means to redefine the term ‘superplastic properties’. In this case, the term ‘superplastic properties’ will be accounted for the specific values of the material constants for the new material model chosen. For example, when introducing the well-known concept of threshold stress, the material model   σ = Kξ m can be modified, e.g., as follows: σ = σ 0 +K ξ m . Correspondingly, the term ‘superplastic properties’ will be accounted for the values of material constants σ 0 , K and m , the value of σ 0 being accounted for the mechanical threshold. Considering the above considerations the values of material parameters K, m in Eq. (1.1), as well as those of K , m , n in Eq. (1.2) are assumed to be material constants by definition in the present work. As stated by Barnes [27], current and likely future trends in further developing superplastic metals forming techniques include applications in the aerospace and automotive markets, faster-forming techniques to improve productivity, the increasing importance of computer modeling and simulation in tool design and process optimization and new alloy developments including superplastic magnesium alloys. The following task among above mentioned ones is placed at the center of attention in present work: the increasing importance of computer modeling and simulation in tool design and process optimization. It is noted that the inaccuracies in determining the material parameters from the results of mechanical experimentation cannot be reduced for the sake of usage the advanced numerical methods and/or powerful computers. Such kind inaccuracies could occur both as due to inappropriate experimental procedures used as well as due to ambiguous procedures of treating the primary experimental data. Classic example is the report of Hedworth and Stowell [28] where 5 independent procedures to treat the same set of experimental data have been suggested. The analysis of the inaccuracies arising from the inappropriate experimental procedures used lies beyond the framework of the present study. The main attention is paid to developing robust computational methods enabling one to determine the values of material constants of interest from the experimental data recorded.

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3. Technological experiments The values of material constants K, m in Eq. (1.1) can be found from the tensile experiments in different ways, e.g., as described by Hedworth and Stowell [28]. However, these procedures are conventionally used with the aim to discuss various mechanisms of superplastic deformation rather than to be used in practical calculations. In practice, much more simple procedures are conventionally used, e.g., as described by Davies et al. [29]. Recently, Luckey et al. [30] have suggested an elegant procedure enabling one to relate the material constants K , m , n established from tensile test data to typical superplastic forming conditions. As the structural superplasticity is observed under relatively high homologous temperatures the fulfillment of standard tensile tests is sometimes concerned with a number of difficulties. For example, when studying the mechanical response of the commercial titanium sheet alloys such as Ti-6Al-4V the mechanical experimentation needs to be conducted in a vacuum to avoid possible oxidation. This problem is accomplished by the necessity to provide sufficiently high values of elongation, δ , under isothermal conditions of straining. Therefore, the values of the material constants, K and m, in the material model (1.1) are sometimes determined from the results of technological experiments, e.g., [31–36]. The general scheme of such kind an approach has been stated by Padmanabhan et al. [3]. In accordance with this scheme, the following order to determine the material constants from the technological experiments should be observed: 1. Development of a simplified model for the technological process. 2. Development of methods to determine the material constants directly from the technological experiment, these methods being based on the above simplified model. 3. Calculation of material properties and technological parameters. The following problem remains to be obscure. It is not yet clear are the values of material constants found by means of the above procedure suitable to be used in further finite element modeling of the technological process under consideration? Therefore, the following additional step is suggested in the present paper with the aim to validate the appropriateness of the procedure for the further finite element considerations. It appears reasonable to input the values of material constants found from the technological experiments into an appropriate finite element software with the aim to obtain corresponding finite element solutions. After that, it would be possible to validate the approach involved by means of comparison the finite element solutions with the experimental data used to calculate the material constants in accordance with Steps 1, 2, 3. The well-known and most investigated technological scheme of superplastic forming is a circular die sheet metal forming, whose is chosen for analysis in the present report. This scheme has been firstly reported by Backofen et al. in 1964 [2] and since then being placed at the center of attention of many researchers. As a result, different analytical approaches have been developed to a day [3]. In this paper, the simplified model, suggested by Enikeev and Kruglov [32], is chosen for analysis. Besides, the classic model of Jovane [37] is also used for comparison purposes.

Fig. 4.1. Schematic of the superplastic forming a sheet into a circular die; R0 is the die radius, D is the die depth, H is the dome height.

in the material model (1.2). However, in this case one has to calculate in advance the pressure-time cycle providing such regime of loading. As the corresponding analysis is conventionally based on the usage of approximate principal equations of the thin shell theory, this circumstance could introduce some additional inaccuracies in the resulting values of the material constants to be determined. It is noted that constant pressure regime of loading is much more often used in practice, e.g. [31–34,36–40]. For example, Cheng [33] has used the experimentally measured time dependency of the dome height, H, corresponding to the constant pressure regime of forming to determine the values of K and m for Ti-6Al-4V sheet alloy. Similar dependencies have been used also by Giuliano and Franchitti [40] with the aim to determine the values of material constants K , m , n for Pb-Sn eutectic alloy. It is important to analyze the theoretical time dependencies of the dome height, H, with the aim to find out the typical features of these curves that can be used later for developing the robust numerical procedures intending to determine the values of material constants for the constitutive model under consideration. As the material model (1.1) can be considered as a particular case of application for the material model (1.2) (with n = 0), the theoretical analysis is given below for the material model (1.2) only. From the principal equations of the thin shell theory it follows that the following relations take place at the dome apex: 

n σea = σma = σta = pR/2sa = K  ξeam εea

where σ ea is equivalent von Mises stress, σ ma , σ ta are meridian and tangential principal stresses respectively, p is gas pressure, R is the radius of the dome, sa is the thickness at the pole, ξ ea is strain rate intensity and ε ea is equivalent strain. Let α be the angle between the axis of symmetry and the radius of the dome passing through the boundary of the dome (Fig. 4.1). As shown in [32], one can derive the following expressions for the effective strain rate and strain

ξea = 2α˙ ·

1

α

− cot anα

In principle, various records from the technological experiments can be used to develop the procedures to determine the values of material constants of interest. For example, Aoura et al. [35] have chosen the regime of constant stress loading to develop the procedures for estimating the values of material constants K , m , n



εea = 2 ln

α

(4.2)

sin α

where the point over symbol denotes the time derivative. Combining Eqs. (4.1) and (4.2) one obtains



4. Theoretical analysis

(4.1)



1 pR0 α2 1 · = K  · 2α˙ − 2s0 sin3 α α tan α

m 

· 2 ln

α n sin α

(4.3)

For the case of constant pressure regime of forming, one can obtain by integrating Eq. (4.3) the following relation

 pR 1/m 0 2K  s

0

t = 2Im n (α ) = 2

 α  3 1/m sin x 0

x2

·

1 x



− cot anx

A.A. Kruglov et al. / Advances in Engineering Software 112 (2017) 54–65



· 2 ln

x sin x

n/m dx

(4.4)

where Im n (α ) is the notation used for the definite integral at the right hand side of Eq. (4.4). For the case when n = 0, Eq. (4.4) coincides with corresponding equation in [32]. It is noted, that similar considerations can be developed within framework of Jovane’s model [37]

ξea =

2H H˙ α = α˙ tan 2 R20 + H 2



εea = ln 1 + H 2 /R20 = ln

1 cos2 (α /2 ) (4.2J)

Substituting Eq. (4.2J) into Eq. (4.1) one obtains

1 pR0 · = K · 2s0 sin α cos2 (α /2 )



α˙ tan

α m 2





·

ln

1

n

cos2 (α /2 ) (4.3J)

Integrating Eq. (4.3J) one has

 pR 1/m 0 2K  s0



· ln

t = Jm n (α ) =

1 cos2 (x/2 )

n/m tan

 α

sin xcos2

0

x dx 2

x 2

1/m

(4.4J)

where Jm n (α ) is the notation used for the definite integral at the right hand side of Eq. (4.4J). It is noted that Eq. (4.4J) coincides with equation reported by Belk [31] (for the case when n = 0). One can rewrite Eqs. (4.4) and (4.4J) in the following normalized form: τ = Im n (α ) and τ = Jm n (α ) respectively. Here  τ = t(pR0 /2K s0 )1/m is normalized time. Besides, the relative dome height of the dome, h, can also be introduced through the simple geometrical relation h = (H/R0 ) = tan(α /2). Thus, now one can investigate the main peculiarities of time dependencies of the relative dome height, h(τ ), by using the following way. As τ = Imn (α ), h = tan(α /2) one can derive that time dependency of h(τ ) from the conditionτ = Im n [2atan(h)] as follows





−1 h = tan (α /2 ) = h tan Im  n (τ )/2 ,

(4.5)

−1 where Imn (τ ) is the function reciprocal to Imn (α ). Similarly, one can derive the following theoretical expression within framework of Jovane’s model:





−1 h = tan (α /2 ) = h tan Jm  n (τ )/2 ,

(4.5J)

−1 where Jm  n (τ ) is the function reciprocal to Jm n (α ). Time dependencies of the relative dome height, h(τ ), calculated according to Eq. (4.5) are shown in Fig. 4.2. As seen in Fig. 4.2(a), as the value of m is larger the time forming to the maximum dome height, τ f = τ (π /2), is also greater. On the other hand, as seen in Fig. 4.2(b), as the value of the strain hardening index, n, is greater, the forming time is diminishing. These conclusions can be made for both simplified models under consideration. As one can see in Fig. 4.2, the time dependencies have the typical shape which is often observed experimentally (e.g., Jovane, [37]; Cheng, [33]; Song and Zhao, [38]; Guo and Ridley, [39]). In order to investigate more thoroughly the shape of these curves, one can introduce the relative time, τ /τ f = t/tf , where t is  current time, tf is the total forming time and τ f = tf (pR0 /2K s0 )1/m . One can find the corresponding theoretical dependencies of the normalized dome height, h, on the relative time τ /τ f =t/tf , in Figs. 4.3 and 4.4. As seen in Fig. 4.3, the typical feature of the curves shown in Fig. 4.3(a) and (b) is that they have the common point of inflection τ ∗ , h∗ , the co-ordinates of which depends on the simplified model chosen for analysis. If the geometrical model by Enikeev

57

and Kruglov [32] is chosen for analysis, then the co-ordinates of the point of inflection are as follows: τ ∗ ∼ =0.3τ f , h∗ ∼ =0.5 (see Fig. 4.3(a)). In the opposite case, when Jovane’s model is chosen for analysis, the coordinates of the point of inflection are as follows: τ ∗∼ =0.45τ f , h∗ ∼ =0.62 (see Fig. 4.3(b)). As seen in Fig. 4.4, the influence of the value of n on the behavior of the curves h(t/tf ) reduces to the displacement of the curves towards lower values of forming time, the point of inflection being gradually disappeared. As one can conclude from Fig. 4.3(a), the value of the dome height, h∗ = 0.5, can be considered as specific characteristic of h(t) curves recorded experimentally. The corresponding value of the relative time, τ ∗ , can be easily calculated from the experimental curves by means of dividing t/tf . That is why it is of interest to investigate the dependencies of the coordinates of this specific point on the values of material constants, m and n. The corresponding theoretical dependencies are presented in Figs. 4.5 and 4.6. As one can see in Figs. 4.5 and 4.6, the characteristic time, τ ∗ , is restricted in value for all cases considered. The following important conclusion can be derived from Figs. 4.5 and 4.6: the point is that the value of τ ∗ in all cases considered turned out to be less than some upper limit, τ ∗ max , the value of which is different depending on the choice of the simplified model used.

0 ≤ τ ∗ ≤ 0.29

(4.6)

0 ≤ τ ∗ ≤ 0.32

(4.6J)

τ∗

As a result, one can expect that the prediction ≤ 0.32 corresponding to h∗ = 0.5 can be used in practice to estimate the ap plicability of the constitutive model σ = K ξ m ε n in describing the superplastic flow when blowing a sheet material into a circular die. 5. Determination of material constants K and m 5.1. Determination of the material constants K, m from the minimum set of data Minimum set of data includes one pair of the experimentally measured values of the forming time t1 , t2 under constant pressure p1 , p2 respectively. The following simple formula has been suggested in [32] to estimate m-value:

m=

ln ( p2 / p1 ) ln (t1 /t2 )

(5.1)

where t1 , t2 are the forming times under constant pressures p1 , p2 respectively. It is noted that the same expression (5.1) can also be derived within framework of the approaches developed by Jovane [32], Belk [31] and Song and Zhao [38]. However, this equation has not been reported by them. Later this formula has been used in [36,40]. Safiullin et al. [34] has shown the validity of this formula for the case of superplastic forming a sheet material into a rectangular die. It is possible to show that the same formula is valid also for the case of superplastic forming a sheet into an elliptical die [41]. As far as the value of the second material constant is concerned, the following expressions have been suggested in [32]:

Ki =

pi R0 · 2s0



ti

2Im (π /2 )



i = 1, 2

(5.2)

where ti is the duration of forming under the constant pressure pi . It is noted that in practice the condition K1 = K2 is to be observed when fulfilling the calculations. It is noted that the same equation is valid when determining the values of K, m by using the simplified models of Jovane

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Fig. 4.2. Dependencies of the relative dome height, h = H/R0 , on the normalized time, τ = t(pR0 /2K s0 )1/m , calculated in accordance with Eq. (4.5) (solid lines) and (4.5J) (dashed lines):(a) for n = 0 and different values of m ; (b) for m = 0.5 and different values of n (indicated by the numbers near the curves).

Fig. 4.3. Dependencies of the relative dome height, h = H/R0 , on the relative time, τ /τ f = t/tf , calculated for n = 0 and m = 0.3;0.4;0.5;0.6;0.7;0.7;0.8;0.9;1 in accordance with Eq. (4.5) (a) and (4.5J) (b).

[37] and Song and Zhao [38], the corresponding meaning of the definite integrals being used at the right hand side of Eq. (5.2). As the model of Belk [31] gives the same results as the model of Jovane [37] there is no need to consider this model separately. 5.2. Determination of the material constants K, m from the full set of data The main disadvantage of the procedure described above is concerned with the ambiguity of the results obtained when more than one pair of experimental data is available. Actually, the Eqs. (5.1) and (5.2) are applied for two meanings of p and t from the experimental set {pi ,ti }, i = 1,2,…,N. Therefore, it is of interest to

develop the modification of the procedure involved enabling one to use the full set of data {pi ,ti } with the aim to obtain the unique solution. With this as a target, the equation was rewritten (4.4) for the case when n = 0 as follows:

 pR 1/m 0 2K  s

0

t = 2Im (α ) = 2

 α  3 1/m sin x 0

x2

·

1 x



− cot anx dx (5.3)

 pR 1/m 0 2K  s

0

t = Jm (α ) =

 α   x 1/m x sin xcos2 tan dx 2 2 0

(5.3J)

A.A. Kruglov et al. / Advances in Engineering Software 112 (2017) 54–65

59

Fig. 4.4. Dependencies of the relative dome height, h = H/R0 , on the relative time, τ /τ f = t/tf , calculated in accordance with Eq. (4.5) (a) and (4.5J) (b) for m = 0.5 and different n (indicated by the numbers near the curves).

Fig. 4.5. Dependencies of the characteristic time, τ ∗ , on the values of material constants m (a), and n (b), calculated according to Eq. (4.5).

Thus, Eq. (5.3) can be rewritten as (pR0 /2Ks0 )1/m t = Fm where Fm = Jm for the case of the model of Jovane [37] or Belk [31] is used, while Fm = 2Im for the case when the model of Enikeev and Kruglov [32] is used. One of possible ways to use the full set of available data is to rewrite Eq. (5.3) in the following equivalent form: ptm = constant. Then one can use one pair, pr ,tr , from the available set of data {pi ,ti } to normalize this relation as follows: p/pr = (tr /t)m , 1 ≤ r ≤ N. The pair pr ,tr can be considered further as the so-called reference point. It is noted that such kind reference point has been introduced when determining the value of the threshold stress by Enikeev

[42]. In general, the introducing of the reference point diminishes the dimensionality of the problem under consideration however, it has two major disadvantages. Firstly, the calculated values of K and m depend on the reference point chosen. Secondly, only conditional minimum for the goal function used can be found when introducing the reference point. This second disadvantage is of principal importance and cannot be overwhelmed. Actually, the introducing of the reference point means that the values of materials constants become dependable. Therefore, the solution of the problem to minimize the goal function will be found by using more narrow class of functions. In other words, the theoretical curve corresponding to the global minimum of the goal function

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Fig. 4.6. Dependencies of the characteristic time, τ ∗ , on the values of material constants m (a), and n (b), calculated according to Eq. (4.5J).

can be placed apart from all points from the available set {pi ,ti }, i = 1,2,…,N. Taking into account the above considerations a special method to minimize the average deviation of the theoretically predicted values of time forming from corresponding experimental data has been used in present study. In this case, the reference point is not introduced. The following goal function is considered:

(K, m ) =

2 N   2K s0 tim − · [Fm (π /2 )]m → min pi R0

(5.4)

i=1

Then, the values of m and K(m) can be substituted into the expression (5.4) with the aim to calculate the value of the goal function  (K(m),m), let notate it through  m . As a result, the dependency  m (m) can be obtained. The last step to be done is to investigate the behavior of the function  m (m). If this function has a minimum, one can introduce an appropriate subroutine of numerical minimization with respect to the function  m . As a result, a unique solution for the problem can be obtained in terms of the goal function  (K,m). 6. Determination of material constants K , m , n

5.3. Procedure of ‘external cycle’ In general, the goal function  (K,m) can be minimized by means of any known numerical procedure. However, the direct application of standard procedures of numerical minimization is rather complicated due to the non-linearity in the dependency of  (K,m) on the material constant m. That is why the following nonstandard procedure to minimize the goal function  (K,m) is suggested which is termed further for the sake of convenience as the ‘procedure of external cycle’. Let us suppose that the value of the material constant, m, is known in advance. Then, the goal function  (K,m) transforms into the function of the only variable, K. Let us denote it as θ (K). This function is described by the same mathematical expression which is written at the right-hand side of Eq. (5.4), the value of m being assumed to be known, say, e.g., m = 0.5. In this case, one can obtain the following equation from the standard condition dθ /dK = 0: N

R0 K (m ) = · N 2s0 i=1

i=1

tim / pi (5.5)

m [Fm (π /2 )] /p2i

where notation K(m) is used to emphasize the fact that this value of K is found for known in advance value of m. The following step to be done is the organizing of the external cycle with respect to m. Let as assume m be equal to, say, 0.01, 0.02, 0.03, etc and calculate the corresponding values of K(m) in accordance with Eq. (5.5).

As one can conclude from the results of theoretical analysis presented in Figs. 4.2–4.6 the values of the material constants, m and n, influences considerably time dependencies of the dome height to be recorded. That is why it appears to be reasonable to develop the procedures to determine these values from the set of experimentally measured values of the forming time under different gas pressures. Let t1 , t2 be two meanings of the forming time corresponding to the same dome height, H1 = H2 , achieved under constant pressure p1 , p2 respectively. Then one can see from Eq. (4.4) that the same Eq. (5.1) can be used in practice to estimate the value of the material constant m . As far as the value of the strain hardening index, n, is concerned, one more measurement is required, the dome height being different from H1 = H2 , that is, it is necessary to select H3 = H1 = H2 . It is more convenient to use the same pressure from above two mentioned values, p1 and p2 . In this case, one can derive from Eq. (4.4) the following condition:

t3 Im n (α3 ) = t1 Im n (α1 )

(6.1)

where t3 is time forming up to the dome height H3 under constant pressure p1 , α 1 = 2atan(H1 /R0 ), α 3 = 2atan(H3 /R0 ). Eq. (6.1) can be solved numerically with respect to the unknown value of n if the value of the material constant m is already found from Eq. (5.1). In practice, it is convenient to select the value of H3 = H1 /2. In this case, according to the estimates (4.6) the value of the ratio t3 /t1 could not be greater than ∼0.3. If this is the case (that is,

A.A. Kruglov et al. / Advances in Engineering Software 112 (2017) 54–65

• m = 0.430, K = 411.22 MPas–m by using the model of Enikeev and Kruglov [32]

Table 7.1 The values of K,m calculated according to Eqs. (5.1) and (5.2). Pair

M

K∗ , MPas–m

K∗∗ , MPas–m

1–2 1–3 2–3

0.429 0.431 0.432

409.72 414.16 417.54

407.97 412.02 415.91

Rem. ∗ Calculated within the framework of the model suggested by Jovane [37]. ∗∗ Calculated within the framework of the model suggested by Enikeev and Kruglov [32].

t3 /t1 > 0.3) then the Eq. (6.1) cannot be resolved. This means, that constitutive model (1.2) is not an adequate material model and so one has to reject it and consider some another material model to describe the experimental data with t3 /t1 > 0.3 (for H3 = H1 /2). Thus, it is assumed below that the condition t3 /t1 < 0.3 is satisfied. The value of the third material constant, K , can be then estimated by using known values of m and n found as described above in accordance with the following expressions derived from (4.4)

Ki =

pi R0 · 2s0

ti

Im n (αi )

m 

,

i = 1, 2, 3

(6.2)

It is noted, that the condition K1 = K2 = K3 is to be satisfied in this case within the accuracy of calculations. The reasons of such coincidence are related with that the unique solution of the problem under study is to be found when using the minimal set of input data since only three equations are used to calculate three unknowns values of K , m and n. Thus, the following minimum set of data is required to estimate all three material constants, K , m and n for the material model (1.2): (i) t1 , p1 , H1 (ii) t2 , p2 =p1 , H1 = H2 (iii) t3 , p3 = p1 , H3 = H1 Eq. (4.4) can be used to develop the procedure to determine the values of material constants, K , m , n from the whole set of available data. However, corresponding analysis is omitted here for the sake of brevity. 7. Examination of the methods proposed 7.1. Determination of the values of K, m The following set of experimental data on constant pressure forming of Ti-6Al-4V sheets of initial thickness s0 = 1 mm into a circular die of radius R0 = 35 mm has been reported in [32]. The duration of forming under constant pressure 0.5, 0.7 and 1.0 MPa under 900 °C is equal to 1500, 685 and 300 s respectively. This set of data can be treated according to the procedures described above in Section 5. The results of calculations of the values of the material constants K, m by using three various pairs of data from available set of data are incorporated into the Table 7.1. As seen in Table 1, there is no notable difference in the values of material constants obtained by using different simplified models of the superplastic forming a sheet into a circular die. At the same time, there is no the unique solution of the problem in this case. The calculations fulfilled in accordance with the procedure of external cycle in accordance with Eqs. (5.4) and (5.5) give the following results • m = 0.430, K = 413.10 Jovane [37]

MPas–m

by

using

the

61

model

of

Thus, the unique solution can be obtained for each simplified model of the process under consideration. It is interesting to note that the values of K,m determined by using two different simplified models of the technological process under consideration turned out to be practically coinciding within the accuracy of calculations. This result appears to be rather unexpected. From intuitive considerations, one can suppose that as simpler the model of the technological process is chosen for the development of the procedure to determine the values of material constants as coarser will be the results obtained. Or, at least, the values of material constants determined by using two different simplified models are to be different depending on the level of accuracy of the simplified model used. This prediction appears to be rather evident at first glance. However, the results obtained in the present report, contradict to these instinctive propositions. It is pertinent to note in this respect that similar results have been reported by Cheng [33]. In his study, the numerical procedures to estimate the material constants for two different material models from the experimentally measured time dependency H(t) have been developed within framework of the model of Jovane [32]. Besides, another simplified model suggested by Chandra and Kannan [43] has been used by Cheng [33] to develop an alternative procedure to determine the values of material constants for the same two material models. Experiments have been effected on the titanium sheet alloy Ti-6Al-4V of average grain size less than 10 μm and initial thickness s0 = 1 mm under temperature T = 925 °C into the die of radius R0 = 25 mm. The following values of the material constants, K and m, determined by using two above mentioned different simplified models have been reported by Cheng [33]: • m = 0.613 and K = 1030.78 MPas–m by using the model of Jovane [32]; • m = 0.613 and K = 1053.021 MPas–m by using the model of Chandra and Kannan [43]. As one can see from the results obtained, no difference have been found by Cheng [33]in the values of material constants, determined by using two different simplified models. This result also seems to be rather unexpected. Is it an accidental coincidence or alternatively this fact reflects some more general tendency? The analysis of the procedures used by Cheng [33] enable us to suppose that such kind coincidence can be, in principle, attributed to the mathematical background of these procedures. The point is that both procedures used by Cheng [33] are based on differentiating the experimentally measured curve H(t). In doing so, the primary experimental measurements of the first time derivative, H˙ (t ), have not been done, the procedures of numerical differentiation of the primary experimental curve H(t) being used instead. At the same time, the results obtained by Vasin et al. [44] when determining the values of material constants for Bingham-type material model show that considerable instabilities when effecting the numerical procedures to determine the material constants could be observed in a similar situation. Thus, it remains unclear, whether or not the necessity to differentiate the primary experimental curve H(t) influence considerably the results obtained by Cheng [33]. In other words, it is of interest to exclude the influence of the mathematical procedures of numerical differentiation of the experimentally measured curve on the values of the material constants to be determined. As far as the results obtained on Ti-6Al-4V sheets reported by Enikeev and Kruglov [32] the following alternative explanation can be suggested. May be, this set of experimental data is somewhat favorable one with respect to the determination of the material

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constants, K and m? Therefore, it appears to be reasonable to use some another set from independent researchers for the practical approbation of the procedures suggested in the present report. One can find two such kind sets in the Table 1 of the report of Belk [31]. Though the blank’s dimensions are not clearly indicated by Belk one can use for calculations the dimensions indicated in Table on page 507: s0 = 1.27 mm and R0 = 51 mm for the Sheet A and s0 = 1.07 mm and R0 = 51 mm for the Sheet B. Then, the following results can be obtained. The duration of forming under constant pressure 0.364, 0.21 and 0.14 MPa is equal to 8, 22 and 50 s respectively (for Sheet 4). The results of calculations with s0 = 1.27 mm and R0 = 51 mm are as follows: • m = 0.513, K= 43.80 MPas–m by using the model of Jovane [37] • m = 0.513, K= 42.67 MPas–m by using the model of Enikeev and Kruglov [32] • m = 0.522, K= 85.24 MPas–m reported by Belk [31] The duration of forming under constant pressure 0.399, 0.21 and 0.14 MPa is equal to 35, 121 and 365 s respectively (for Sheet 014). The results of calculations with s0 = 1.07 mm and R0 = 51 mm are as follows: • m = 0.410, K = 75.09 MPas–m by using the model of Jovane [37] • m = 0.410, K = 75.13 MPas–m by using the model of Enikeev and Kruglov [32] • m = 0.448, K = 121.79 MPas–m reported by Belk [31] Thus, as one conclude from the results obtained, in both cases considered (Sheet 4 and Sheet 014) the values of K, m determined by using two different simplified models of the process turned out to be again practically coinciding. The difference in the results obtained in the present report from those reported by Belk can be explained as follows. The approximate semi-empirical approach has been used by Belk [31] to estimate m-value (see Eq. (4.5) in his report) and so the value of m reported by Belk is different from those obtained in the present study by using the rigorous computational procedures. As far as the value of K is concerned, the deviation can be explained by the difference in the input data used (the values s0 and R0 are not clearly indicated by Belk [31]). At the same time, it is not the aim of the present study to reconsider the results obtained by Belk [31]. Attention is concentrated on the significance of the choice of the simplified model. And the following conclusion can be made now: the values of material constants do not depend on such choice when the experimental data from independent researcher (Belk [31]) are using as input data into the procedures developed. It is pertinent to note that the results of calculation of m-value in accordance with the procedure suggested above do not depend upon the choice of blank’s geometry (the values of s0 and R0 ). This confirmation are justifying by means of additional calculations made with different meanings of s0 and R0 . Besides, the same conclusion can be made from the Eq. (5.1) which does not contain the values of s0 and R0 . It is noted that the computer programs developed to be used in determining the values of K and m from the experimental data have been tested by means of standard procedures. As a result, the possible program’s errors have been excluded from the consideration. Besides, the stability of the procedures used has also been verified. As a result, the conclusion was made that the procedures developed enable one to obtain acceptable results obtained up to 20% level of noise.

Table 7.2 Forming times, at different pressures, obtained for Pb-Sn eutectic sheet alloy by experimental activities [40]. Gas pressure, MPa

0.10 0.18

Forming time, s H = 0.5R0

H=R0

39.936 13.130

151.534 48.770

7.2. Determination of the values of K , m , n Experimental data obtained on Pb-Sn eutectic alloy of the initial thickness s0 = 0.3 mm into the die of R0 = 30 mm reported by Giuliano and Franchitti [40] are listed in Table 7.2. The experimental data listed in Table 7.2 have been used to determine the values of material constants K , m , n. In doing so, the values of forming time corresponding to H=R0 indicated in the last column of Table 7.2, have been used to estimate m-value. The value of material constants K and n have been estimated by using one additional point (1 or 2) corresponding to H = 0.5R0 . The results obtained are incorporated into the Table 7.3. The corresponding values reported by Giuliano and Franchitti [40] are also indicated in the last line of Table 7.3 for the sake of comparison. As seen from Table 7.3, the results obtained are rather adjacent, but not coinciding. In particular, nonzero strain hardening exponent, n, is found for the case when the simplified model of Enikeev and Kruglov [32] is used. On the other hand, the results obtained by using the simplified models of Jovane [37] and Giuliano and Franchitti [40] are very close. That is why it is necessary to check thoroughly the validity of the procedures suggested. The most appropriate way to do it appears to be concerned with the comparison of the results obtained with corresponding finite element solutions. 8. Finite element modeling 8.1. Material model σ =Kξ m Finite element modeling has been fulfilled by using the educational version of the commercial ANSYS code. The formulation of the boundary value problem in the mechanics of solids is given in details by Vasin et al. [34]. For the sake of comparison the following calculations have been done. Firstly, the set of material constants m = 0.430, K = 413.10 MPas–m determined by using the model of Jovane [37] has been introduced into ANSYS program and corresponding time dependencies of the dome height H(t) have been plotted in Fig. 8.1. Similar calculations with the set of material constants m = 0.430, K = 411.22 MPas–m determined by using the model of Enikeev and Kruglov [32], have also been fulfilled. The corresponding time dependencies of the dome height H(t) are shown in Fig. 8.2. In both figures the corresponding theoretical predictions of the simplified models of Jovane [37] (in Fig. 8.1) and the model of Enikeev and Kruglov [32] (in Fig. 8.2) are also shown by dashed lines for the sake of comparison. As one can see in Figs. 8.1 and 8.2, a satisfactory agreement is found for both cases studied. The deviation in curves observed in Figs. 8.1 and 8.2 can be attributed to the following two reasons: (1) inadequacy of the simplified model of the process used; (2) inadequacy of the material model chosen. In order to estimate quantitatively the discrepancy found the calculated values of the forming time for the hemispheres obtained are incorporated into the Table 8.1. As one can see from the Table 8.1, the maximum deviation of the results of finite element modeling from the corresponding experimental data does not exceed 5% in value for all cases considered in spite of sim-

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63

Table 7.3  The values of material constants for the material model σ = K ξ m ε n found by using the experimental data from Table 7.2. The values of material constants K , MPas–m 1 2 1 2 –



150.1 148.4 142.4 142.1 144

Accuracy of modeling

m

n



ζ

0.5185 0.5185 0.5185 0.5185 0.518

0.0859 0.0732 0.0015 0.0 0 0 0 ≈0

1.2310–29 1.2310–29 3.2910–25 3.2910–25 –

1.2410–16 1.1310–16 4.9410–14 4.9310–14 –

Simplified model used

[32] [31] [40]

Table 8.1 The comparison of the results of finite element modeling with experimental data on Ti-6Al-4 V alloy and corresponding predictions of the simplified approaches. Experimental values for Ti-6Al-4 V sheet alloy

Calculated by using the model of Jovane (1968)

Calculated by using the model of Enikeev and Kruglov (1995)

pi , MPa 0.5 0.7 1.0

tanalyt , s 1499.4 686.1 299.5

tanalyt , s 1499.4 686.1 299.5

ti , s 1500 685 300

tFEM , s 1553 715 312

Fig. 8.1. Time dependencies of the dome height, H, calculated by means of finite element method with m = 0.430, K = 413.10 MPas–m (solid lines) and within framework of the model of Jovane, [37] (dashed lines) for different values of the forming pressure, p, MPa (indicated by the numbers near the curves).

Fig. 8.2. Time dependencies of the dome height, H, calculated by means of finite element method m = 0.430, K = 411.22 MPas–m (solid lines) and within framework of the model of Enikeev and Kruglov, [32] (dashed lines) for different values of the forming pressure, p, MPa (indicated by the numbers near the curves).

plified models have been used to determine the material constants, K and m. These values have been used as input data into the ANSYS program and corresponding finite element solutions have been obtained.

tFEM , s 1544 714 310

As mentioned by Vasin et al [34], for the case of the technological experiments the strain paths in the space of Iljushin (1948) are close to the class of trajectories inherent to the technological process under consideration. Therefore, there is no need to investigate separately the applicability of a material model chosen for this class of trajectories. On the other hand, the assumptions made in developing the simplified models concerning the boundary conditions, loads applied and kinematics of the process under study can be partially compensated by the corresponding adjusting the values of material constants. As a result, approximate solutions obtained within framework of a simplified model can be brought closer to the exact solution of corresponding boundary value problem. As far as the comparing of the values of material constants obtained from the technological experiments with those found from standard uni-axial experiments, more comprehensive study is yet to be fulfilled and will be reported elsewhere. It is to be mentioned that this interesting and useful for practical applications problem is discussed by Belk [31]. Actually, it is of interest to compare the values of material constants obtained by means of blowforming trials with those found from the results of conventional tensile tests. However, this problem cannot be reduced to the simple hypothesis adopted by Belk, [31], without discussion. This hypothesis is sometimes called in the literature as the hypothesis on the existence of the so-called ‘unique curves’ for superplastic materials (see, e.g., Padmanabhan et al., [3]). This is a fundamental problem which is worth to be discussed in details elsewhere. The point is that the deformation trajectories in the space of Iljushin (1948) inherent to the processes of tensile tests and blowforming trials can differ considerably and so the existence of this ‘unique curve’ is yet to be testified by means of special experimental programs to be fulfilled in future. The consideration of this problem is beyond the framework of the present study. Analysis of the results obtained enables one to conclude that the deviation observed between the results of finite element modeling and corresponding analytical predictions can be attributed mostly to the inadequacy of the simplified models involved rather than those of the material model used. The following argument can also be taken into account when analyzing the results obtained. The dashed curves in Fig. 8 are disposed closer to the FEM solutions as compared with those in Fig. 7, which can be explained by more adequacy of the model suggested by Enikeev and Kruglov [32]. Actually, the boundary conditions is not observed within the framework of the model of Jovane [37]. This circumstance is not conventionally mentioned in the literature. Actually, according to

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Fig. 8.3. Time dependencies of the dome height, H, calculated by means of finite element method with m = 0.43, K = 410 MPas–m (solid lines) and within framework of the model of Enikeev and Kruglov, [32] (short-dashes lines) and the model of Jovane, [37] (long-dashes lines) for different values of the forming pressure, p, MPa (indicated by the numbers near the curves).

the model of Jovane [37] the meridian and tangential strains and strain rates are equal in value throughout the whole dome, including the zone in the vicinity of the boundary. However, the die is evidently not deformed, so that the tangential strain as well as tangential strain rate are to be equal to zero along the boundary. At the same time, one can find by integrating the expression for the tangential strain rate ξt = H H˙ /(R20 + H 2 ) the following relation for the tangential strain, ε t :

εt =

 0

t1

ξt dt =



R0 0

HdH 1 = ln 2 2 R20 + H 2

Taking into account that ε t = ln(2π Rf /2π R0 )=ln(Rf /R0 ), one has to conclude that, in accordance with the model of Jovane [37] the sheet of the initial radius R0 is to be transformed into the dome of √ radius Rf =R0 2, that is approximately 1.4 times greater than the die radius R0 . Additional finite-element calculations have been fulfilled by using the set of material constants, reported earlier by Enikeev and Kruglov [32]: m = 0.43, K = 410 MPas–m (see Fig. 8.3). The theoretical predictions from both simplified models considered have also been plotted for the sake of comparison. As one can see in Fig. 8.3, in this case the theoretical predictions made from the model of Enikeev and Kruglov [32] fell closer to the FEM-solutions as compared with those predicted within framework of the model of Jovane [37]. This result also confirms the possible explanation of the discrepancies found by the inaccuracy of the simplified models used rather than the material model chosen. One can show that the results obtained enable us to conclude that the principal assumption of the geometrical model suggested by Enikeev and Kruglov [32] concerning the kinematics of the process under study is valid for the case of Newtonian viscous flow (when m = 1). In this case, the thickness distribution predicted analytically is rather close to that reported in classic work of Cornfield and Johnson (1970). As the value m is lower the thickness non-uniformity becomes more notable. Therefore, the difference in the results of finite element modeling and analytical solutions obtained within simplified model appears to be as less as the value of m is closer to 1. Second remark is concerned with corresponding comparing of the results obtained when analyzing the time dependencies of the dome height for the case of constant strain rate tests. It was found that similar findings can be derived from the comparison of the finite-element solutions with corresponding predictions of both analytical models considered for all 5 target strain rates used experimentally by Enikeev and Kruglov, [32]. The following useful remark can be now cited from the report of

Fig. 8.4. Time dependencies of the dome height, calculated by means of finite element modeling techniques (solid lines) and within framework of the simplified model of Enikeev and Kruglov, [32] (long dash lines) for constant pressure forming of Pb-Sn eutectic alloy. Experimental values from Table 2 are shown by short dash lines for the sake of comparison.

Belk [31]. ‘The real benefit of determining m and K from blowing tests is that they can be used to predict the results of other blow forming trials.’ 

8.2. Material model σ =Kξ m ε n In this case, the following model from the ANSYS library of creep strain equations has been used in the present study:

εcr = C1 σ C2 εC3 e−C4 /T t

(8.1)

where ε is equivalent strain (based on modified total strain), σ is equivalent stress, T is absolute temperature, t is time at end of substep, e is natural logarithm base. It is easy to show that the following relations take place:



C1 = 1/K 

1/m

C2 = 1/m

C3 = −n/m  = K ξ m ε n

(8.2)

Thus, the material model σ can be used in finite element modeling by means of using the model (8.1) with the values of material constants, C1 , C2 , C3 , calculated according to Eq. (8.2). For example, for the set of material constants K = 150.1 MPas–m , m = 0.5185 and n = 0.086 one can obtain from Eq. (8.2) the following values: C1 = 1.69710–16 MPa–n s–1 , C2 = 1.929, C3 = –0.166. The results of calculations are presented in Fig. 8.4, where the finite element solutions are shown by solid lines. The corresponding analytical solutions calculated in accordance with Eq. (4.4) with K = 150.1 MPas–m , m = 0.5185, n = 0.086, s0 = 0.3 mm, R0 = 30 mm are also shown in Fig. 8.4 for the sake of comparison. As seen in Fig. 8.4, the analytical solutions are practically coinciding with the finite element solutions found. As a result, it is rather difficult to distinguish the long-dashed lines corresponding to the analytical solutions obtained in accordance with Eq. (4.4) with corresponding finite element solutions found by using the educational version of the commercial ANSYS-code. At the same time, the corresponding accuracy of modeling the forming time is also sufficiently good (see Table 8.2).

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65

Table 8.2 Comparison of the theoretical values of the forming time with corresponding experimental data on Pb-Sn eutectic sheet alloy [9]. Gas pressure, MPa

0.10 0.18

Forming time, t, s, to H = 0.5R0

Forming time, t, s, to H=R0

Exp

Analytical

FEM

Exp

Analytical

FEM

39.936 13.130

39.93 12.85

39.8 13.2

151.534 48.770

151.48 48.75

151.2 49.15

9. Conclusions The results obtained in the present study confirm the applicability of the common scheme suggested by Padmanabhan et al. [3] for determining the values of material constants for the ma terial models σ = Kξ m and σ = K ξ m ε n from the results of technological experiments. It is shown that the values of material constants K, m determined by means of using two different simplified models of the superplastic forming suggested by Jovane [31] and Enikeev and Kruglov [32] turned out to be coinciding within the accuracy of calculations. At the same time, the values of material constants K , m , n determined by using the same two simplified models turned out to be somewhat different as indicated in Table 7.3. Finite element consideration of the results obtained confirms the validity of the procedures developed. Thus, the methods to determine the values of material constants K, mfor the material model σ =Kξ m and those for deter mining the values of K , m , n in the material model σ = K ξ m ε n from the results of constant pressure forming trials can be used in practice as an independent procedures along with other methods known in the literature. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.advengsoft.2017.06. 014. References [1] Padmanabhan KA, Davies JJ. Superplasticity. Berlin: Springer–Verlag; 1980. [2] Backofen WA, Turner IR, Avery DH. Superplasticity in an Al–Zn Alloy. Trans ASM 1964;57:980–90. [3] Padmanabhan KA, Vasin RA, Enikeev FU. Superplastic flow: phenomenology and mechanics. Berlin: Springer-Verlag; 2001. [4] Carrino L, Giuliano G, Palmieri C. On the optimisation of superplastic forming processes by the finite-element method. J Mater Process Technol 2003;143–144:373–7. [5] Hwang YM, Lay HS. Study on superplastic blow-forming in a rectangular closed-die.2003. J Mater Process Technol 2003;140:426–31. [6] Li GY, Tan MJ, Liew KM. Three-dimensional modeling and simulation of superplastic forming. J Mater Process Technol 2004;150:76–83. [7] Chan KC, Wang GF, Wang CL, Zhang KF. Low temperature superplastic gas pressure forming of electrodeposited Ni/SiCp nanocomposites. Mater Sci Eng 2005;A 404:108–16. [8] Kumar VSS, Viswanathan D, Natarajan S. Theoretical prediction and FEM analysis of superplastic forming of AA7475 aluminium alloy in a hemispherical die. J Mater Process Technol 2006;173:247–51. [9] El-Morsy A, Akkus N, Manabe K, Nishimura H. Evaluation of superplastic characteristics of tubular materials by multi-tube bulge test. Mater Lett 2006;60:559–64. [10] Hojjati MH, Zoorabadi M, Hosseinipour SJ. Optimization of superplastic hydro forming process of aluminium alloy 5083. J Mater Process Technol 2008;205:482–8. [11] Yoon JH, Lee HS, Yi YM. Finite element simulation on superplastic blow forming of diffusion bonded 4 sheets. J Mater Process Technol 2008;201:68–72. [12] Wang GC, Fub MW, Cao CX, Dong HB. Study on the maximum m superplasticity deformation of Ti–6.5Al–3.5Mo–1.5Zr–0.3Si alloy. Mater Sci Eng 2009;A513–514:32–41. [13] Faraji Gh, Mashhadi MM, Norouzifard V. Evaluation of effective parameters in metal bellows forming process. J Mater Process Technol 2009;209:3431–7. [14] Gleiter H. Nanocrystalline materials. Progr Mater Sci 1989;33:224–302. [15] Valiev RZ, Islamgaliev RK, Semenova IP. Superplasticity in nanostructured materials: new challenges. Mater Sci Eng 2007;463:2–7.

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