A constitutive equation for hot deformation range of 304 stainless steel considering grain sizes

A constitutive equation for hot deformation range of 304 stainless steel considering grain sizes

Materials and Design 52 (2013) 412–421 Contents lists available at SciVerse ScienceDirect Materials and Design journal homepage: www.elsevier.com/lo...

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Materials and Design 52 (2013) 412–421

Contents lists available at SciVerse ScienceDirect

Materials and Design journal homepage: www.elsevier.com/locate/matdes

A constitutive equation for hot deformation range of 304 stainless steel considering grain sizes M.H. Parsa a,b,c,⇑, D. Ohadi a a

School of Metallurgy and Materials Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran Center of Excellence for High Performance Materials, School of Metallurgy and Materials Engineering, University of Tehran, Tehran, Iran c Advanced Metalforming and Thermomechanical Processing Laboratory, School of Metallurgy and Materials Engineering, University of Tehran, Tehran, Iran b

a r t i c l e

i n f o

Article history: Received 22 July 2012 Accepted 3 May 2013 Available online 18 May 2013 Keywords: Constitutive equation 304 Stainless steel Invariant theory Grain size

a b s t r a c t A general constitutive equation based on the framework of invariant theory by consideration of hot deformation key variables and also the properties of the material such as initial grain size is presented in the current work. Soundness of the considered parameters to be used in the developed formula was initially verified based on the important axioms such as objectivity, entropy principle, and thermodynamics stability. To access the prediction ability of the method, the formula was simplified for the simple hot compression test. To evaluate the simplified formula, single-hit hot compression tests were carried out at the temperature range of 900–1100 °C under true strain rate of 0.01–1 s1 on a AISI 304 stainless steel. The capability of proposed formula for reproducing the variation of flow stress with strain and the strain hardening rate with stress for the resultant flow stress data was examined. The good agreement between model predictions and actual results signified the applicability of this method as a general constitutive equation in hot deformation studies. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The major purpose of thermomechanical treatment during hot deformation processes such as hot rolling, forging, and extruding, are the achievement of required mechanical properties with desirable microstructure. Many factors such as temperature, stress, strain, strain rate and many other internal variables affect the behavior of material during and after hot working. Establishment of logical relation between these factors for mathematical prediction of response of hot deformed material to the deformation process is essential. Vast amount of information regarding the relation between various factors encountered in hot forming of materials can be expressed in the form of constitutive equations. Constitutive equations can be established based on the empirical or mathematical procedures. These equations can be derived from macroscopic view or microscopic view. Phenomenological equations are derived based on the macroscopic view. This type of modeling establishes the correlation between the measurable parameters such as flow stress, strain, strain rate, and temperature, in the framework of mathematical functions [1,2]. Although phenomenological constitutive equations are well-matched with the experimental observations, but they have no physical background ⇑ Corresponding author at: School of Metallurgy and Materials Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran. Tel.: +98 21 61114069; fax: +98 21 88006076. E-mail address: [email protected] (M.H. Parsa). 0261-3069/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.matdes.2013.05.021

and they are consistent in limited range of deformation conditions [3]. However, according to physical theories of plasticity, at the microscopic level, the plastic flow is confined to certain crystallographic planes in certain directions and it is not uniform throughout a body [4]. Slip and twining are crystallographic phenomena that cause plastic flow of materials [5]. Among physical plasticity theories, the dislocation theory [6–10] and plasticity theories for single crystals [11–13] and polycrystals [14–17] are the most important ones which are considered in constitutive modeling. Other considered theories in physical based constitutive equations are theory of thermodynamics [18–22], theory of kinetics [23,24], etc. In comparison with phenomenological models, physical based models can be employed in wider range of deformation condition; however, they have larger number of materials constants [3]. From other point of view, the mathematical form of formulation is another important aspect of constitutive equations. Various mathematical forms of formulations have been suggested and applied for high temperature behavior of metals; such as polynomial [23–25], power law [26–28], and hyperbolic sine [29–31]. The applied mathematical form of formulation depends on the experience and researcher’s view on theories of materials. Most of the procedures that were employed to develop constitutive equations were purely experimental, except some physically-based models [3,31]. These procedures incorporate experimental data along with curve fitting to the collected data using one of the mentioned mathematical forms of formulations.

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413

Nomenclature

W Ia

Ua U0a a, b u, x Uk eijk

r r⁄ Q QT F R U V V⁄ B1 e e_ L D D⁄

matrix polynomial polynomials in invariants belonging to the integrity basis for the symmetric and skew-symmetric argument tensors Matrix products transpose of Ua symmetric tensors skew-symmetric tensors axial vectors third-order alternating tensor cauchy stress tensor cauchy stress tensor in non-inertial frame rotation (proper orthogonal tensor) transpose of Q deformation gradient orthogonal rotation tensor right stretch tensor left stretch tensor left stretch tensor in non-inertial frame cauchy deformation tensor Hencky logarithmic strain tensor rate of Hencky strain tensor velocity gradient tensor rate of deformation tensor rate of deformation tensor in non-inertial frame

A number of theoretical foundations have been also established for construction of constitutive equations [31–35]. Among these theoretical foundations, invariant theory and its principles have had many applications in solving continuum problems. The theory of invariants states that the material symmetries impose certain invariance requirements on the form of the constitutive equations. This theory plays a significant role in mathematics, physics, and quantum mechanics, and has been extensively used in continuum mechanics and constitutive equations of non-Newtonian fluids and viscoelastic materials [36]. In the current work, invariant theory is applied to establish a constitute equation for 304 stainless steel considering grain size effect. Hot compression tests for determination of constitutive equation constants have been carried out. In the following sections, details of deriving such a constitutive equation will be explained. 2. Mathematical model 2.1. Foundation of model It is assumed that constitutive equation can be expressed as a tensor polynomial function of vectors and tensors. This polynomial subjects to certain invariance, which is defined by invariant theory. A form invariant symmetric tensor polynomial function of symmetric and skew-symmetric tensors can be expressed by Eq. (1) [36]:

X W¼ Ia ðUa þ U0a Þ

ð1Þ

a

where W is the matrix polynomial. Ia are polynomials in invariants belonging to the integrity basis for the symmetric and skew-symmetric argument tensors, which can be read from Table 1. Ua are matrix products composed from tensors that are listed in Table 2 by making appropriate substitutions. U0a is the transpose of Ua. ‘‘a’’ and ‘‘b’’ are symmetric tensors and ‘‘u’’ and ‘‘x’’ are skew-symmetric tensors.

W g G X(i) J

spin tensor or the vorticity grain size vector skew-symmetric grain size tensor force vector flux vector ai equation constants ai equation constants l length at deformed configuration r radius at deformed configuration l0 length at undeformed state radius at undeformed state r0 x, y, z coordinates at current configuration X, Y, Z coordinates at reference configuration k1 and k2 principal stretch ratios p axial load A0 initial area c1, c2, c3 constants I01 ; I02 ; I03 equation constants which are functions of different variables bI 1 ; bI 2 ; bI 3 equation constants which are functions of different variables c hot deformation activation energy j gas constant T absolute temperature of deformation m constant friction factor

In this formulation, form-invariance under the proper orthogonal group is considered and there is no distinction between axial vectors and absolute vectors. Axial vectors are associated with skew-symmetric tensors according to the Eq. (2) [36]:

uij ¼ eijk U k

ð2Þ

where eijk is the third-order alternating tensor, defined by:

eijk

8 i; j; k is an even permutation of 1; 2; 3 > <1 ¼ 1 i; j; k is an odd permutation of 1; 2; 3 > : 0 otherwise

ð3Þ

2.2. Selected parameters and deformation space The parameters of the proposed tensor polynomial function are considered to be representative of the strain, strain rate, initial grain size and temperature. It has been tried to express the derived constitutive equation in the strain space, since according to the Drucker’s stability postulate, the softening part of stress–strain curve cannot be described in the stress space. Material behavior and related constitutive equation can be expressed in the material coordinates (Langrangian formulation) or spatial coordinate (Eulerian formulation) [4]. In current work, all the variables are described in Eulerian space. 2.3. Evaluation of selected parameters Selected parameters representing by scalars, vectors, and tenors, should satisfy important criterions, such as objectivity, entropy principle, thermodynamics stability and equipresence axiom in order to be entered into the constitute formulation. 2.3.1. Objectivity One of the main tools for restricting the appearance of any considered parameters in the constitutive equations is the principle of material frame indifference or objectivity [37].

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Table 1 Matrix products whose traces form the integrity basis for the proper orthogonal group [36]. Matrices

Matrix products

a a, b u u, a u, a, b

a; a2; a3 ab; ab2; ba2; a2b2 u2 u2a; u2a2; u2aua2 uab; ua2b, ub2a; ua2b2; ua2ba; ub2ab; ua2b2a; ub2a2b; u2ab; u2a2b; u2b2a; u2aub; u2aub2; u2bua2

Table 2 Form invariant symmetric tensor polynomial functions of symmetric and skewsymmetric tensors [36]. Matrices

Ua

a a, b x x, a x, a, b

I a; a2 ab; a2b; b2a; a2b2 x2 xa; xa2; axa2; x2a; x2a2; x2ax; x2a2x xab; bxa; xba; xa2b; xb2a; bxa2; axb2; xba2; xab2; axa2b; bxb2a; xa2b2; xb2a2; x2ab; ax2b; x2a2b; x2b2a; x2axb

One of primitive parameters that normally appear in the constitutive equations is the Cauchy Stress Tensor, r, which is a symmetric objective tensor and the objectivity of this tensor was proved [4]. Another important parameter which effects on high temperature behavior of metals have been proved is strain [23,25]. In order to demonstrate the effect of strain, various parameters have been used. In the present research, strain tensor based on the Left Stretch Tensor, V, has been selected to show the effect of strain. V results from multiplicative decomposition of deformation gradient (F = R  U = V  R). Both U the Right Stretch Tensor and V the Left Stretch Tensor are symmetric [38] and objective [39]. Derived formulation is represented in the Eulerian coordinates, therefore Cauchy Deformation Tensor B1 or any single-valued, monotonic, and tensor valued function of B1 can be described as the Eulerian-type strain tensor. This tensor is related to the left stretch tensor as shown by Eq. (4) [38]:

B1 ¼ ðF 1 ÞT ðF 1 Þ ¼ V 2

ð4Þ

Considering the logarithmic strain tensor e or Hencky strain tensor as the criterion of strain, it can be shown that there is a relation between e; B1 and V as shown by the following equation:

e ¼ 

1 ln B1 ¼ lnðB1 Þ1=2 ¼ ln V 2

ð5Þ

The Hencky strain tensor is also objective and symmetric. Strain rate is a parameter, which has important effects on the hot deformation behavior of materials [2]. Rate of deformation tensor D is selected to represent the strain rate effects. By adding the rate of Deformation tensor D to the spin tensor W, the velocity gradient tensor, L, is obtained as shown by Eq. (6) [38]. Neither L nor W is objective:

( L¼DþW

D ¼ 12 ðL þ LT Þ W ¼ 12 ðL  LT Þ

ð6Þ

It can be proved that there is a relation between velocity gradient tensor and left stretch tensor as shown in Eqs. (7) and (8):

If it is assumed that the velocity field is irrotational, the spin-tensor components will be all zero (W = 0). Therefore, only the rate of deformation tensor remains and the second term in left hand side of the Eq. (8) will be also zero. As a result, the rate of deformation tensor is equal to rate of Hencky strain (Eq. (9)). The rate of deformation tensor is an objective tensor (Eq. (10)) [39]. This proves that rate of Hencky strain is also an objective and symmetric tensor. It has been proved that the rate of deformation tensor, D, is the corotational rate of the Hencky strain tensor associated with the logarithmic spin tensor [40]:

_ 1 e_ ¼ D ¼ VV

ð9Þ

D ¼ Q  D  Q T

ð10Þ

Temperature is always an important parameter in the hot deformation processes [41]. The temperature T is a scalar objective quantity and since only the symmetric and skew-symmetric tensors are directly put in the formulation, the effect of temperature is considered to be inherent in the polynomial coefficients. One of the objectives of the present research is to consider the grain size effects in the constitutive equation [42]. The grain shapes are assumed to be a geometrically closed space. The initial grain size is represented by a vector ‘‘g’’ which components gi (i = 1, 2, 3) are the distances between the assumed center of grain and the surface of close surface in three perpendicular directions (Eq. (11)). This vector can be transformed to skew-symmetric tensors Gij according to the relation (12). This tensor is also objective:

2

g1

3

6 7 g ¼ 4 g2 5

ð11Þ

g3 Gij ¼ eijk gk

ð12Þ

2.3.2. Entropy principle and thermodynamic stability Constitutive equations must be consistent with the fundamental theorems of thermodynamics. The first fundamental theorem of thermodynamics (principle of conservation of energy) does not impose any restriction on constitutive equations and the direction of process, but it shows which parameters are conjugate to each other [38]. The second most important principle, which any constitutive equation should fulfill, is the Entropy Principle. This theorem excludes processes with negative entropy production and puts limits on the direction of the processes [43]. The entropy principle also shows which thermodynamic processes are dissipative and irreversible. The second law of thermodynamics can be rephrased in the notation of continuum mechanics, namely Clausius–Duhem inequality [38]. It can be proved that the Cauchy stress tensor, r, deformation rate tensor, D and the left stretch tensor, V fulfill the fundamental thermodynamics principles and entropy principle. Based on different deformation mechanisms, there are different relations between flow stress and grain size, as a result of grain boundary diffusion and lattice diffusion [26,33]. These equations clearly show the effects of grain size on the high temperature flow stress. Therefore, it is also possible to show that the grain size also fulfills the entropy principle. 2.4. Formulating the general equation

_ 1 ¼ ðV_  R þ V  RÞ _  ðR1  V 1 Þ L ¼ FF ¼ V_  R  R1  V 1 þ V  R_  R1  V 1

ð7Þ

L ¼ D þ W ¼ V_  V 1 þ V  R_  R1  V 1

ð8Þ

It has been shown (in Section 2.3) that all selected parameters satisfy the important axioms of mechanics. Now it is possible to formulate the general equation using procedure and parameters mentioned in Sections 2.1 and 2.2. The coefficients of general equa-

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tion, namely Ia, can be expressed as polynomials composed of traces of symmetric tensors and skew-symmetric tensors using Table 1. Example of Ia is shown in Appendix A. Using calculated coefficients, the final form of invariant polynomial tensor function can be expressed by function of selected parameters in the strain space as presented in Appendix B. Derived general Eq. (B1) is very complicated and it has many constants (I1, . . . , I42). Therefore, determination of coefficients using experimental examinations would be a tedious task. This complication can be reduced by limiting the order of matrix products. For this reason, the matrix products are limited to three which result to the simplified equation presented in Appendix B as Eq. (B2). By rearranging Eq. (B2) considering transformation of transpose of symmetric and skew-symmetric tensors, Eq. (13) is resulted:

þ Dr Þ þ I8 ðD

2

2

ð13Þ

Derived invariant polynomial Eq. (13) relates strain tensor to the selected independent parameters. As mentioned before the coefficients Ia encompass of traces of all tensors and scalars which consist of temperature. This equation can be more simplified, if experimental procedures for identifying the coefficients are clarified. The most important point of derived formulation is that it can be easily used for multi-axial loading, since six component of strain tensor related to components of stress tensor, six components of deformation rate and components of tensor representing the grain size. It depends on the decision of researcher for using of formula for a special loading case. For multiaxial test it is necessary to develop especial instruments or special test for recording multi loadings and multi displacements. In the first stage of introducing this method, simple compression tests are used for showing the validity of procedure. 2.5. Simplification of formulation In the research work, hot compression tests were used to investigate the behavior of 304 stainless steel specimens. Cylindrical specimens were subjected to the uniaxial compression as shown in Fig. 1. The length and radius of cylindrical specimen at current configuration or deformed state is defined by l and r respectively. While the corresponding length and radius in the undeformed state is defined by l0 and r0. Using the coordinate system shown in Fig. 1, deformed configuration can be related to the undeformed configuration using Eq. (14):

x ¼ k1 X;

y ¼ k2 Y; z ¼ k2 Z

3 2 lnk1 0 0 lnk1 e ¼ lnV ¼ 4 0 lnk2 0 5¼4 0 0 0 0 lnk2 2 l 3 ln l 0 0 6 0 0 7 ¼ 4 0  12 ln ll0 5 1 0 0  2 ln ll0

3 0 0 5  12 lnk1

0  12 lnk1 0

ð17Þ

2

_ e_ ¼ D ¼ L ¼ F:F

¼4

2

r þ rD Þ þ I10 ðG Þ þ I11 ðGr  rGÞ 2 2 þ I12 ðGr  r GÞ þ I14 ðG2 r þ rG2 Þ þ I18 ðGD  DGÞ þ I19 ðGD2  D2 GÞ þ I21 ðG2 D þ DG2 Þ þ I25 ðGrD  DrGÞ þ I26 ðDGr  rGDÞ þ I27 ðGDr  rDGÞ

2

2

e ¼ I1 I þ I2 r þ I3 r2 þ I4 D þ I5 D2 þ I6 ðrD þ DrÞ þ I7 ðr2 D 2

After calculating left stretch tensor, V, it is possible to calculate Hencky strain tensor and strain rate tensor as shown by Eqs. (17) and (18) for cylindrical specimen:

k_1 =k1 0 0

1

32 1 3 0 0 k1 0 0 1 0 5 k_2 0 54 0 k2 0 k_2 0 0 k1 2 3 3 2_ l=l 0 0 0 6 7 0 5 0 5 ¼ 4 0  12 ð_l=lÞ _ 1 _ k2 =k2 0 0  2 ðl=lÞ

k_1 ¼40 0

0 k_2 =k2 0

If the axial load, p, is known, then the Cauchy stress tensor can be calculated using Eq. (19) for cylindrical specimen [4]:

2

p=pr 2

r¼6 4 0 0

0 0

3

2

7 6 0 05 ¼ 4

p=k22 pr 20

0 0

0 0

3

2

0

7 6 0 05 ¼ 4

0

0 0

p=k22 A0

k2 ¼ r=r 0

k1

0

0

0 0 ð19Þ

In the cylindrical specimen under uniform one-dimensional deformation, the stress, and strain rate tensors are defined using only two values of r11 and D11. The components of grain size vector are considered to be equal in all the three perpendicular directions (g1 = g2 = g3). This vector can be transformed into skew-symmetric tensor according to the relation (12) as shown by the following equation:

2

0

6 G ¼ 4 g 3 g2

g3 0 g 1

g 2

3

2

0

7 6 g 1 5 ¼ 4 g 1 0 g1

g1 0 g 1

g 1

3

7 g1 5 0

6 F¼U¼V ¼40

k2

e11 ¼

2 2 1 4 I2 r11 þ I3 r211 þ I4 e_ 11 þ I5 e_ 211 þ I6 r11 e_ 11 3 3 2 3 4 4 2  2 þ I7 r11 e_ 11 þ I8 e_ 11 r11 3 3

ð14Þ

0

0

3

7 05 k2

ð20Þ

Eq. (21) is obtained by considering mentioned assumptions for cylindrical specimen and substituting of calculated of stress, strain, strain rate and grain size tensors into Eq. (13). Eq. (21) relates the strain and stress components in the deformed condition for the simple compression test:

ð15Þ

0

3

7 0 05

Considering ideal condition for compression test and ignoring the effect of friction leads to disregarding the rotation during deformation. Therefore, rotation tensor, R, equal to unity and deformation gradient, F, right stretch tensor, U, and left stretch tensor, V, can be calculated using the following equation:

2

0 0

0

where coefficients k1 and k2 are the principal stretch ratios and they can be calculated using initial and final configurations using Eq. (15) [4]:

k1 ¼ l=l0 ;

ð18Þ

ð16Þ Fig. 1. Schematic of uniaxial compression test.

ð21Þ

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All the coefficients Ia relating to grain size tensor G are eliminated in Eq. (21). The effect of initial grain size and temperature can be considered in the multipliers of remaining Ia coefficients. The details of applied procedures for deriving Eq. (21) are presented in Appendix C. The Eq. (13) is the general tensor form while Eq. (21) is the simplified component form for simple compression test. It is very difficult to express the whole behavior of metals in the hot deformation range by one equation. Therefore, in the present research, for formulating of constitutive equation of 304 stainless steel at high temperatures, the stress–strain curves are divided into two parts: ascending part (hardening behavior), and descending part (softening behavior) as shown in Fig. 2. For two mentioned parts, two series of coefficients, Ia, for the developed invariant polynomial equations were evaluated using hot compression tests data. At the introduction level of procedure, there was no intention to incorporate the interesting hot deformation behavior of 304 stainless steel during static, dynamic and metha-dynamic recrystallizations [44]. Presenting the capability of developed constitutive equation based on the invariant theory for showing hardening and softening behavior is significant which is not possible by using many available models. Incorporation of mentioned phenomena were not among the main goals of present research. Introduction of them into the derived formula can be subject of future researches.

The hot compression tests were performed on the annealed samples using Instron-8502 universal testing machine at the temperature range of 900–1100 °C and strain rate of 0.01–1 s1. Mica sheet lubricants were employed to reduce the frictional effects between specimen and the anvils. The samples were loaded onto the bottom die in the furnace and for attaining uniformity of temperature, they held for 10 min at temperatures of 900, 1000, and 1100 °C. Heating of test samples during hot compression was measured via embedded thermocouples located at bottom die, near the specimen-die interface. The compression testing machine operates at constant true strain rate by incremental calculation of the current true strain. The velocity of moving die is varied via computer control in order to apply deformation at a constant strain rate of 0.01, 0.1, and 1 s1. The tests were stopped at a specific true strain   e ¼ ln l0l ffi 0:5 and then the samples were immediately quenched in 25 °C water in order to prevent further change in microstructure. The strain was assumed uniform throughout the sample. The tests results showed the variation of stress with strain at definite temperatures and strain rates. Correction of stress–strain curves were carried out based on the method suggested by Ebrahimi and Najafizadeh [45]. Although friction effect can cause non-uniform deformation, but since the tests were stopped at low strain (ffi0.5), this corrections was negligible.

3. Experimental procedure

4. Results and discussion

An austenite stainless steel AISI 304L was selected in this work. The chemical composition (wt.%) of this steel is given in Table 3. One of the main softening mechanisms in hot deformation of austenite is dynamic recrystallization [41,44]. Cylindrical compression test samples with 12 mm height and 8 mm diameter were machined from the rolled bars, with specimen axis parallel to bar axis. In order to attain different grain sizes (for studying the effect of initial grain size) selected samples were annealed at 1100 °C for 10 and 20 min and at 1200 °C for 60 min. They were immediately quenched in 25 °C water after annealing. Optical examinations revealed three different initial average grain sizes of 30, 60, and 140 lm according to ASTM: E112.

4.1. Invariant polynomial formula

Stress (MPa)

Fig. 3 shows typical results of compression tests which were carried out at temperatures of 900, 1000, and 1100 °C and strain rate of 0.01, 0.1, and 1 s1 on a specimen with initial grain size of 30 lm. Stress–strain curves representing dynamic recrystallization can be observed in this figure. Coefficients of Eq. (21) for hardening and softening cases have been extracted by using results of compression tests at temperatures of 900–1100 °C and strain rate of 0.01–1 s1. Extracted coefficients of the derived invariant polynomial formula are presented in Table 4. Data presented in Table 4 showed sensitivity to the small data variations. Employing the polynomial curve fitting on the gathered test data proved the possibility of expressing data using Eq. (22) for both the ascending and the descending parts of curves:

e11 ¼ I01 r211 þ I02 r11 þ I03

ð22Þ

in which, I01 ; I02 , and I03 are constants. Therefore, the Eq. (21) was rearranged as Eq. (23):

Descending part

e11 ¼ Ascending part

Strain Fig. 2. Division of the stress–strain curve into two parts.

Table 3 Chemical composition of 304L stainless steel. Elements

Fe

C

Mn

Cr

Ni

Mo

Cu

Avg.wt.%

base

0.02

1.76

18.38

8.37

0.40

0.54

    2 4 2 4 4 I3 þ I7 e_ 11 r211 þ I2 þ I6 e_ 11 þ I8 e_ 211 r11 3 3 3 3 3   1 þ I4 e_ 11 þ I5 e_ 211 2

ð23Þ

Therefore, I01 ; I02 , and I03 are functions of temperature, rate of deformation, and grain size. Based on the experimental results, strain ranges of 0.05–0.23 for ascending part and strain ranges of 0.30– 0.44 for descending part were considered due to the assumed division of hardening and softening according to Fig. 2. The material constants of this type of equation for ascending part are calculated for different deformation conditions as they represented in Table 5. It seems that material constants are affected by different hot deformation parameters like, temperature, rate of deformation, and grain size and it is possible to find a relation between these variables and material constants. Therefore, Eq. (22) can be converted into Eq. (24):

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where c is hot deformation activation energy (J mol1), j is the gas constant, and T is the absolute temperature of deformation (K). The value of c is estimated to be about 400 kJ mol1 for 304 stainless steel [46]. According to Dehghan et al. the grain size might have an effect on hot deformation activation energy [47]. As declared in the same reference, the estimated activation energy for initial grain sizes of 8 and 35 lm are 354 and 457 kJ mol1 respectively. Using results of experiments, value of activation energy for grain sizes of 30, 60 and 140 lm were found to be equal to 404, 433, and 471 kJ mol1 respectively. Using reported values and estimated values of activation energies for 304L stainless steel with different grain sizes, Eq. (26) is suggested for showing the effect of grain size on the activation energy:

c ¼ 287:5  g 0:1 1

ð26Þ

The final form of Eq. (24) for ascending part can be written as Eq. (27). The approximated values of material constants are listed in Table 6:

e11ascending

part

b 2 ¼  logðe_ 11 Þ  expð287:5  g 0:1 1 =jTÞ  ð I1 r11 þ Ib2 r11 þ Ib3 Þ

ð27Þ

Considering the same general Eq. (22) for the descending part leaded to the different materials constants and formula. As it can be seen from Table 7, there is a linear correlation between stress and strain and I01 tends to zero. This type of behavior has also been reported by other researches [23]. For the descending part, general equation such as Eq. (28) can be suggested:

e11descending

part

¼ f ðT; D; GÞ  ð Ib2 r11 þ Ib3 Þ

ð28Þ

Using the experimental data, for the coefficient f and its relation with the grain size, rate of deformation and temperature, Eq. (29) can be suggested:

f ðT; D; GÞdescending

part

_ 0:1 ¼ g 1:5 1  e11 =T

ð29Þ

Using this coefficient, the final form of Eq. (28) can be written as Eq. (30):

e11descending

Fig. 3. Stress–strain curves obtained from hot deformation tests.

e11ascending

part

  ¼ f ðT; D; GÞ  Ib1 r211 þ Ib2 r11 þ Ib3

ð24Þ

where Ib1 ; Ib2 , and Ib3 are functions of stress. By examining of the experimental data, it was found that the f function can be expressed by Eq. (25). This factor is very similar to the Zener–Hollomon parameter [41]:

f ðT; D; GÞascending

part

¼  logðe_ 11 Þ  expðc=jTÞ

ð25Þ

part

  b b _ 0:1 ¼ g 1:5 1 :e11 =T ð I 2 r11 þ I3 Þ

ð30Þ

The estimated values of material constants for descending part are listed in Table 8. By using Eqs. (27) and (30), it has been tried to show the variation of stress and strain during hot compression, considering grain size effect. These equations show the capability of applied procedure for developing phenomenological constitutive equation for plastic region of hot deformation considering time and temperature. In Fig. 4, some results of hot compression tests and derived equation have been superimposed. Fig. 4a shows the effect of grain size on ascending part of stress–strain curves. Although the model underestimates strains at lower stresses, it seems that the proposed model is capable of showing the grain size effect. Important

Table 4 Estimated for different strain rates. Ascending part

I2 I3 I4 I5 I6 I7 I8

Descending part

e_ ¼ 0:01

e_ ¼ 0:1

e_ ¼ 0:01

e_ ¼ 0:1

511020775168 3280719261 13490354205493700 1349035420549340000 19366969402344 164035963050 618406935605635

173589842974 4852894404 2115599350111990 21155993501120000 4743017488016 24264472018 56109667028886

233004046862 20273246 5407511083447330 540751108344874000 7777477151664 1013662318 387272519145670

229919308238 138025510 440573842358 4405738426524 853549550818 690127550 2960469903696

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Table 5 Materials constants of ascending part obtained for modified equation. T (°C)

e_ ðs1 Þ

g (lm)

I01

I02

I03

1000 1000 1000 900 1000 1100 1000 1000 1000 900 1000

0.1 0.1 0.1 0.1 0.1 0.1 0.01 0.01 0.01 0.01 0.01

30 60 140 60 60 60 30 60 140 60 60

0.0003 0.0001 0.0003 0.0001 0.0001 0.0001 0.0010 0.0003 0.0000 0.0001 0.0003

0.0557 0.0157 0.0469 0.0112 0.0157 0.0167 0.1670 0.0404 0.0019 0.0090 0.0404

2.9057 0.5665 1.8145 0.6430 0.5665 0.4927 7.0251 1.4405 0.2644 0.3923 1.4405

Table 6 Finalized materials constants obtained for equation for ascending part. Ib1

Ib2

Ib3

0.0002

0.0231

1.0371

Table 7 Materials constants of descending part obtained for modified equation. T (°C)

e (s1)

g (lm)

I01

I02

I03

900 1000 900 1000 1000

0.1 0.1 0.01 0.01 0.01

30 30 30 30 60

0 0 0 0 0

0.0305 0.0373 0.027 0.031 0.0871

6.3184 5.3862 4.6131 3.3199 8.7489

Table 8 Finalized materials constants obtained for equation for descending part. Ib2

Ib3

0.2653

37.3474

point in this figure is the clear representing of grain size effect on the flow stress variation for 304L stainless steel. Fig. 4b illustrates the temperature effect on the stress–strain curves. Again, it can be seen that the model and test data are more comparable at higher stresses. The effect of rate of deformation and temperature at descending part are shown in Fig. 4c. The model and test data are in good correlation. As it can be seen in Fig. 4c, the results from invariant model are almost well matched with experimental tests, especially for the descending part. In Fig. 4d, the ascending and descending part are presented together. It seems that there is a gap between predicted strains in transition from ascending part to descending part. In other words, for ascending part results are more compatible at higher stresses, while descending part seems to be more reliable at lower stresses. 4.2. Strain hardening behavior and temperature effect Mechanical behaviors of metals change under different hot deformation conditions. Effects of strain, rate of deformation, and temperature on the hot deformation behavior are well known and have been investigated by many researchers [2]. Any proposed formula should clearly show the effects of mentioned parameters. In the current model, the effects of temperature, strain and grain size are considered to appear in the Ia coefficients (I1, I2, . . . , I8). The hardening can be calculated using Eq. (31) for the most general formula:

Fig. 4. Comparing stress–strain curves obtained from experimental data with curves obtained from modified invariant polynomial model: (a) effect of grain size in the ascending part, (b) effect of temperature in ascending part, (c) effect of temperature and strain rate in descending part, and (d) illustrating ascending and descending parts together.

  I4 þ I5 e_ 11 þ 43 I6 r11 þ 43 I7 r211 þ 83 I8 e_ 11 r11 dr 1  dD ¼ 2 de 4  de I þ 3 I3 r11 þ 43 I6 e_ 11 þ 83 I7 r11 e_ 11 þ 43 I8 e_ 211 3 2

ð31Þ

As the relation (31) shows, the rate of hardening is a function of stress, rate of deformation and their variations with strain. However, according to this equation, the exclusive effect of any of these variables cannot be easily explained. This problem requires more investigation in order to find more precise equation. By modifying the general equation and simplifying of it to Eqs. (27) and (30), the hardening rate can be recalculated using this new approach, which leads to an equation such as Eq. (32):

h i   b b dr=de ¼ 1=  log e_ 11 Þ  expð287:5  g 0:1 1 =jT ð2 I1 r11 þ I 2 Þ

ð32Þ

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M.H. Parsa, D. Ohadi / Materials and Design 52 (2013) 412–421

used to form polynomial tensor function, which represents hot deformation constitutive equation in the framework of Invariant Theory. The proposed general equation which is incorporated the grain size effects, can be formulated as Eq. (13). This general equation can be simplified for simple compression test as demonstrated by Eq. (21). For defining the unknown and material relating coefficients of the proposed constitutive equation, hot compression tests have been carried out on the AISI 304L Stainless steel in the temperature range of 900–1100 °C and the strain rate of 0.01–1 s1. The experimental results show that besides the well known parameters, grain size has effects on the hot deformation behavior of used material. At initial level for formulating of constitutive equation of 304 stainless steel at high temperatures based on the invariant, the stress–strain curves are divided into two parts: ascending part (hardening behavior), and descending part (softening behavior). Final form of derived Eq. (27) for the hardening behavior during compression test show that stress, strain, strain rate, temperature and grain size play important rules. For the descending part, general equation can be simplified by using the experimental data for the coefficients which show relations with the grain size, rate of deformation, strain, stress and temperature. Descending part can be formulated by Eq. (30). Constants of Eqs. (27) and (30) for the considered ranges of temperatures, strain and strain rates have been determined as represented in Tables 7 and 8. It is interesting to note that reported and estimated values of activation energies show grain size dependence as illustrated by Eq. (26) for 304L stainless steel. Comparison of derived hardening-stress curves from experimental evaluations and proposed formula are in good agreement. These confirm the effects of grain size on the hardening behavior during hot deformation as expressed by Eq. (32). Appendix A Any of coefficients Ia can be formulated using general formulation shown in Eq. (A1): I1 a0 þ a1 ðtrrÞa1 þ a2 ðtrr2 Þa2 þ a3 ðtrr3 Þa3 þ a4 ðtrDÞa4 þ a5 ðtrD2 Þa5 



6 3 a



a 7



8 2 a



9 2 a

a 10

þ a6 ðtrD Þ þ a7 ðtrrDÞ þ a8 ðtrrD Þ þ a9 ðtrDr Þ þ a10 ðtrr2 D2 Þ

Fig. 5. Modified hardening rate-stress curves for experimental data and model at: (a) 900 °C and 0.1 s1, (b) 1000 °C, and 0.1 s1, and (c) 1000 °C, and 0.01 s1.

Using modified equations, the hardening rate-stress curves were calculated and plotted as shown in Fig. 5. The hardening effect decreases and reaches to zero at the peak stress. Comparison of Fig. 5a and b illustrates that in the same conditions of hot deformation, hardening rate decreases by increasing of temperature. This behavior has been reported for stainless steel [47]. Comparison of Fig. 5b and c illustrate that increasing of the strain rate increases the rate of hardening. The effect of strain rate on hardening behavior has been also reported before [47]. The hardening rate-stress curves have also been obtained for the proposed model (dashed lines). The predicted behaviors of material by changing the testing conditions show accordance with the experimental results.

 11 2 a

þ a11 ðtrG Þ

2

þ a12 ðtrG

a 12



2

þ a13 ðtrG

 13 2 a



2

þ a14 ðtrG

 14 2 a

rGr Þ

þ a15 ðtrG2 DÞa15 þ a16 ðtrG2 D2 Þa16 þ a17 ðtrG2 DGD2 Þa17 þ a18 ðtrGrDÞ 





ðA1Þ

þ a19 ðtrGr2 DÞþ a20 ðtrGD2 rÞþ a21 ðtrGr2 D2 Þþ a22 ðtrGr2 DrÞþ a23 ðtrGD2 rDÞ þ a24 ðtrGr2 D2 rÞþ a25 ðtrGD2 r2 DÞþ a26 ðtrG2 rDÞþ a27 ðtrG2 r2 DÞþ a29 ðtrG2 D2 rÞ þ a30 ðtrG2 rGDÞþ a31 ðtrG2 rGD2 Þþ a32 ðtrG2 DGr2 Þ

The ai and a0i are constants that should be defined for specified material considering experimental results. Appendix B Using calculated Ia, considered independent tenor valued parameters and Eq. (1), final form of invariant polynomial tensor function can be expressed as Eq. (B1). e ¼ I1 I þ I2 r þ I3 r2 þ I4 D þ I5 D2 I6 ½rD þ ðrDÞT  þ I7 ½r2 D þ ðr2 DÞT þ I8 ½D2 r þ ðD2 rÞT  þ I9 ½r2 D2 þ ðr2 D2 ÞT  þ I10 ðG2 Þ þ I11 ½Gr þ ðGrÞT  þ I12 ½Gr2 þ ðGr2 ÞT  þ I13 ½rGr2 þ ðrGr2 ÞT 

5. Conclusions In an attempt to propose a phenomenological based constitutive equation, for hot deformation of metals considering strain, rate of deformation, temperature, and initial grain size effects, first soundness of considered representing values for parameters have been evaluated. Selected parameters are the Cauchy stress tensor, (r), the left stretch tensor, (V), the rate of deformation tensor, (D), the temperature, (T), and grain size (g). These parameters were

þ I14 ½G2 r þ ðG2 rÞT  þ I15 ½G2 r2 þ ðG2 r2 ÞT  þ I16 ½G2 rG þ ðG2 rGÞT  þ I17 ½G2 r2 G þ ðG2 r2 GÞT  þ I18 ½GD þ ðGDÞT  þ I19 ½GD2 þ ðGD2 ÞT  þ I20 ½DGD2 þ ðDGD2 ÞT  þ I21 ½G2 D þ ðG2 DÞT þ þ I22 ½G2 D2 þ ðG2 D2 ÞT  þ I23 ½G2 DG þ ðG2 DGÞT  þ I24 ½G2 D2 G þ ðG2 D2 GÞT  þ I25 ½GrD þ ðGrDÞT  þ I26 ½DGr þ ðDGrÞT  þ I27 ½GDr þ ðGDrÞT  þ I28 ½Gr2 D þ ðGr2 DÞT  þ I29 ½GD2 r þ ðGD2 rÞT  þ I30 ½DGr2 þ ðDGr2 ÞT  þ I31 ½rGD2 þ ðrGD2 ÞT  þ I32 ½GDr2 þ ðGDr2 ÞT  þ I33 ½GrD2 þ ðGrD2 ÞT  þ I34 ½rGr2 þ ðrGr2 ÞT  þ I35 ½DGD2 r þ ðDGD2 rÞT  þ I36 ½Gr2 D2 þ ðGr2 D2 ÞT  þ I37 ½GD2 r2 þ ðGD2 r2 ÞT  þ I38 ½G2 rD þ ðG2 rDÞT  þ I39 ½rG2 D þ ðrG2 DÞT  þ I40 ½G2 r2 D þ ðG2 r2 DÞT  þ I41 ½G2 D2 r þ ðG2 D2 rÞT  þ I42 ½G2 rGD þ ðG2 rGDÞT 

ðB1Þ

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Limiting the matrix products to the order of three in Eq. (B1) leads to Eq. (B2): e ¼ I1 I þ I2 r þ I3 r2 þ I4 D þ I5 D2 þ I6 ½rD þ ðrDÞT  þ I7 ½r2 D þ ðr2 DÞT  þ I8 ½D2 r þ ðD2 rÞT  þ I10 ðG2 Þ þ I11 ½Gr þ ðGrÞT  þ I12 ½Gr2 þ ðGr2 ÞT  þ I14 ½G2 r þ ðG2 rÞT  þ I18 ½GD þ ðGDÞT  þ I19 ½GD2 þ ðGD2 ÞT  þ I21 ½G2 D þ ðG2 DÞT  þ I25 ½GrD þ ðGrDÞT  þ I26 ½DGr þ ðDGrÞT  þ I27 ½GDr þ ðGDrÞT  ðB2Þ

Appendix C Substituting Eqs. (17), (19) and (20) in Eq. (B2) for simple compression test leads to Eq. (C1): 3 3 2 2 2 e11 0 0 1 0 0 r 0 03 7 7 6 0  1 e 6 6 0 7 5 ¼ I1 4 0 1 0 5 þ I 2 4 0 0 0 5 4 2 11

 12 e11 0 0 1 2 32 e_ 11 0 r 0 0 6 6 7 1 6 þ I3 4 0 0 0 5 þ I4 4 0  2 e_ 11 0

0

2

0 0

0 0 0 3 7 7 5

 12 e_ 11 02 3 2 e_ 11 0 0 0 0 r 0 0 3 e_ 11 B6 7 7 6 76 1 1 _ _ B 6 0 7 0 7 þ I5 6 5 þ I6 @4 0 0 0 5:4 0  2 e11 5 4 0  2 e11 0 0 0 0 0  12 e_ 11 0 0  12 e_ 11 1 32 2 e_ 11 0 0 r 0 03 76 6 7C 1 _ C 0 7 þ6 5:4 0 0 0 5A 4 0  2 e11 0 0 0 0 0  12 e_ 11 02 3 32 2 _ e11 0 0 r 0 0 B6 7 7 6 1 _ 6 0 7 þ I7 B @4 0 0 0 5 :4 0  2 e11 5 0 0 0 0 0  12 e_ 11 32 2 1 e_ 11 0 0 r 0 0 32 7 6 6 7 C 1 _ C 0 7 þ6 5:4 0 0 0 5 A 4 0  2 e11 1 _ 0 0 0 0 0  2 e11 02 32 2 e_ 0 0 r 0 03 B6 11 7 B 0  1 e_ 7 6 0 7 þ I8 B6 2 11 5 :4 0 0 0 5 @4 0 0 0 0 0  12 e_ 11 32 1 2 2 3 2 0 0 r 0 0 3 e_ 11 0 g 1 g 1 2 C 7 6 C 6 7 6 7 0  1 e_ 0 7 þ4 0 0 0 5:6 þ I10 4 g 1 0 g1 5 2 11 5 C 4 A g 1 g 1 0 0 0 0 0 0  12 e_ 11 32 02 3 0 g 1 g 1 r 0 0 76 B6 7 þ I11 @4 g 1 0 g 1 5:4 0 0 0 5 0 0 0

0

2

0

32

2

g1

g 1

r 0 03 2 0

6 76 4 0 0 0 5:4 g 1

0 g1 0

0 0 0 31

g 1

7C g 1 5A

g 1 g 1 0 0 0 0 02 32 0 g 1 g 1 r 0 0 32 B6 6 7 7 þ I12 B g 1 5:4 0 0 0 5 @4 g 1 0 0 0 0 g 1 g 1 0 2 32 2 31 r 0 0 0 g 1 g 1 6 7 6 7C 4 0 0 0 5 :4 g 1 0 g 1 5C A 0 0 0

g1

g 1

0

02 0 B6 B þ I14 @4 g 1

g1 0

g 1 g 1 2 32 r 0 0 0 6 76 þ4 0 0 0 5:4 g 1

32 2

3 r 0 0 7 6 7 g 1 5 :4 0 0 0 5

g 1 0

0 0 0 32 1 g 1 g 1 7 C 0 g1 5 C A g 1 g 1 0 0 0 0 02 3 32_ e11 0 0 0 g 1 g 1 B6 7 7 6 0  1 e_ 0 7 þ I18 B g 1 5:6 2 11 @4 g 1 0 5 4 g 1 g 1 0 0 0  12 e_ 11 32 2 31 e_ 11 0 0 0 g 1 g 1 7 6 7C 6 1 _ 0 7 6 g 1 5C A 5:4 g 1 0 4 0  2 e11 g 1 g 1 0 0 0  12 e_ 11 32 02 32_ e11 0 0 0 g 1 g 1 7 B6 7 6 0  1 e_ 0 7 þ I19 B g 1 5:6 2 11 5 @4 g 1 0 4 g 1 g 1 0 0 0  12 e_ 11 32 2 2 31 e_ 11 0 0 0 g 1 g 1 7 6 6 7C 1 _ 0 7 6 g 1 5C A 5 :4 g 1 0 4 0  2 e11 g 1 g 1 0 0 0  12 e_ 11 02 3 3 2 0 0 0 g 1 g 1 2 e_ 11 B6 7 7 6 0  1 e_ 0 7 þ I21 B g 1 5 :6 2 11 @4 g 1 0 4 5 g 1 g 1 0 0 0  12 e_ 11 2 32 3 1 e_ 11 0 0 0 g 1 g 1 2 6 76 7 C 1 _ 0 7 þ6 g1 5 C 4 0  2 e11 A 5:4 g 1 0 g 1 g 1 0 0 0  12 e_ 11 02 3 32 32 0 0 0 g 1 g 1 r 0 0 e_ 11 B6 7 76 7 6 0  1 e_ 0 7 þ I25 B g 1 5:4 0 0 0 5:6 2 11 @4 g 1 0 5 4 g 1 g 1 0 0 0 0 0 0  12 e_ 11 32 2 31 32 e_ 11 0 0 r 0 0 0 g 1 g 1 76 6 7C 76 1 _ 0 7 6 g 1 5C 5:4 0 0 0 5:4 g 1 0 A 4 0  2 e11 g 1 g 1 0 0 0 0 0 0  12 e_ 11 32 02 32 3 e_ 11 0 0 0 g 1 g 1 r 0 0 7 B6 6 76 7 1 _ 6 0 7 þ I26 B g 1 5:4 0 0 0 5 5:4 g 1 0 @4 0  2 e11 g 1 g 1 0 0 0 0 0 0  12 e_ 11 31 32_ 2 32  e11 0 0 r 0 0 0 g 1 g 1 7C 6 7 0  1 e_ 6 76 C 0 7 4 0 0 0 5:4 g 1 0 g 1 5:6 2 11 5A 4 1 _ g 1 g 1 0 0 0 0 0 0  2 e11 32 02 32_ 3  0 0 e 0 g 1 g 1 r 0 0 11 7 6 B6 6 7 0  1 e_ 7 0 7 þ I27 B g 1 5:6 2 11 5:4 0 0 0 5 4 @4 g 1 0 g 1 g 1 0 0 0 0 0 0  12 e_ 11 32 31 2 32_ 0 0 0 g 1 g 1 r 0 0 e11 76 7C 6 7 6 0  1 e_ 0 7 ðC1Þ 4 0 0 0 5:6 g 1 5C 2 11 5:4 g 1 0 4 A g 1 g 1 0 0 0 0 0 0  12 e_ 11 Expanding the left hand side of Eq. (C1) and equating calculated components of derived left hand side matrix to the components of strain tensor matrix, following equalities will be resulted. Equating  e12 to the corresponding resulted left hand side matrix component leads to Eq. (C2):

I10 ¼ I21 e_ 11

ðC2Þ

M.H. Parsa, D. Ohadi / Materials and Design 52 (2013) 412–421

Equating e13 to the corresponding resulted left hand side matrix component leads to Eq. (C3):

3 3 I10 g 21  I11 g 1 r11  I12 g 1 r211 þ I14 g 21 r11  I18 g 1 e_ 11  I19 g 1 e_ 211 2 4 1 1 2 þ I21 g 1 e_ 11  I25 g 1 r11 e_ 11 þ I26 g 1 r11 e_ 11  I27 g 1 r11 e_ 11 ¼ 0 2 2 ðC3Þ Equating e23 to the corresponding resulted left hand side matrix component leads to Eq. (C4):

3 3 I10 g 21 þ I11 g 1 r11 þ I12 g 1 r211 þ I14 g 21 r11 þ I18 g 1 e_ 11 þ I19 g 1 e_ 211 2 4 1 1 2 þ I21 g 1 e_ 11 þ I25 g 1 r11 e_ 11  I26 g 1 r11 e_ 11 þ I27 g 1 r11 e_ 11 ¼ 0 2 2 ðC4Þ Adding Eqs. (C3), (C4) and considering Eq. (C2) leads to Eq. (C5):

3  I10 ¼ I14 r11 2

ðC5Þ

Equating e22 to the corresponding resulted left hand side matrix component and using Eq. (C2) leads to Eq. (C6):

I1 ¼

1 _ 1 1 I4 e11  I5 e_ 211  e11 2 4 2

ðC6Þ

Equating e11 to the corresponding resulted left hand side matrix component leads to Eq. (C7):

e11 ¼ I1 þ I2 r11 þ I3 r211 þ I4 e_ 11 þ I5 e_ 211 þ 2I6 r11 e_ 11 þ 2I7 r211 e_ 11 þ 2I8 e_ 211 r11  2I10 g 21  4I14 g 21 r11  4I21 g 21 e_ 11

ðC7Þ

Using Eqs. (C2), (C5), (C6), and substituting them in Eq. (C7) simplifying it leads to Eq. (C8):

e11 ¼

2 2 1 4 I2 r11 þ I3 r211 þ I4 e_ 11 þ I5 e_ 211 þ I6 r11 e_ 11 3 3 2 3 4 4 2  2 þ I7 r11 e_ 11 þ I8 e_ 11 r11 3 3

ðC8Þ

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